Cover......Page 1
Title Page......Page 2
Contents......Page 4
Preface......Page 10
Acknowledgments......Page 13
Credits......Page 15
1.1. Functions and Their Graphs......Page 16
1.2. Combining Functions; Shifting and Scaling Graphs......Page 29
1.3. Trigonometric Functions......Page 36
1.4. Graphing with Software......Page 44
1.5. Exponential Functions......Page 48
1.6. Inverse Functions and Logarithms......Page 53
2.1. Rates of Change and Tangents to Curves......Page 66
2.2. Limit of a Function and Limit Laws......Page 73
2.3. The Precise Definition of a Limit......Page 84
2.4. One-Sided Limits......Page 93
2.5. Continuity......Page 100
2.6. Limits Involving Infinity; Asymptotes of Graphs......Page 111
Practice Exercises......Page 125
3.1. Tangents and the Derivative at a Point......Page 130
3.2. The Derivative as a Function......Page 134
3.3. Differentiation Rules......Page 143
3.4. The Derivative as a Rate of Change......Page 153
3.5. Derivatives of Trigonometric Functions......Page 162
3.6. The Chain Rule......Page 168
3.7. Implicit Differentiation......Page 176
3.8. Derivatives of Inverse Functions and Logarithms......Page 181
3.9. Inverse Trigonometric Functions......Page 191
3.10. Related Rates......Page 197
3.11. Linearization and Differentials......Page 205
Questions to Guide Your Review......Page 216
Practice Exercises......Page 217
4.1. Extreme Values of Functions......Page 224
4.2. The Mean Value Theorem......Page 232
4.3. Monotonic Functions and the First Derivative Test......Page 240
4.4. Concavity and Curve Sketching......Page 245
4.5. Indeterminate Forms and L’Hôpital’s Rule......Page 256
4.6. Applied Optimization......Page 265
4.7. Newton’s Method......Page 276
4.8. Antiderivatives......Page 280
Questions to Guide Your Review......Page 290
Practice Exercises......Page 291
5.1. Area and Estimating with Finite Sums......Page 298
5.2. Sigma Notation and Limits of Finite Sums......Page 308
5.3. The Definite Integral......Page 315
5.4. The Fundamental Theorem of Calculus......Page 327
5.5. Indefinite Integrals and the Substitution Method......Page 338
5.6. Definite Integral Substitutions and the Area Between Curves......Page 346
Practice Exercises......Page 356
6.1. Volumes Using Cross-Sections......Page 362
6.2. Volumes Using Cylindrical Shells......Page 373
6.3. Arc Length......Page 381
6.4. Areas of Surfaces of Revolution......Page 387
6.5. Work......Page 391
6.6. Moments and Centers of Mass......Page 397
Practice Exercises......Page 405
7.1. The Logarithm Defined as an Integral......Page 408
7.2. Exponential Change and Separable Differential Equations......Page 418
7.3. Hyperbolic Functions......Page 427
Practice Exercises......Page 435
8. Techniques of Integration......Page 437
8.1. Integration by Parts......Page 438
8.2. Trigonometric Integrals......Page 444
8.3. Trigonometric Substitutions......Page 450
8.4. Integration of Rational Functions by Partial Fractions......Page 455
8.5. Integral Tables and Computer Algebra Systems......Page 462
8.6. Numerical Integration......Page 467
8.7. Improper Integrals......Page 477
Practice Exercises......Page 488
9.1. Sequences......Page 493
9.2. Infinite Series......Page 505
9.3. The Integral Test......Page 514
9.4. Comparison Tests......Page 521
9.5. Absolute Convergence; The Ratio and Root Tests......Page 525
9.6. Alternating Series and Conditional Convergence......Page 531
9.7. Power Series......Page 537
9.8. Taylor and Maclaurin Series......Page 547
9.9. Convergence of Taylor Series......Page 552
9.10. The Binomial Series and Applications of Taylor Series......Page 558
Questions to Guide Your Review......Page 566
Practice Exercises......Page 567
10.1. Parametrizations of Plane Curves......Page 572
10.2. Calculus with Parametric Curves......Page 579
10.3. Polar Coordinates......Page 589
10.4. Graphing Polar Coordinate Equations......Page 593
10.5. Areas and Lengths in Polar Coordinates......Page 596
10.6. Conics in Polar Coordinates......Page 601
Questions to Guide Your Review......Page 607
Practice Exercises......Page 608
11.1. Three-Dimensional Coordinate Systems......Page 611
11.2. Vectors......Page 616
11.3. The Dot Product......Page 625
11.4. The Cross Product......Page 633
11.5. Lines and Planes in Space......Page 639
11.6. Cylinders and Quadric Surfaces......Page 647
Questions to Guide Your Review......Page 652
Practice Exercises......Page 653
12.1. Curves in Space and Their Tangents......Page 657
12.2. Integrals of Vector Functions; Projectile Motion......Page 665
12.3. Arc Length in Space......Page 671
12.4. Curvature and Normal Vectors of a Curve......Page 676
12.5. Tangential and Normal Components of Acceleration......Page 682
12.6. Velocity and Acceleration in Polar Coordinates......Page 684
Practice Exercises......Page 688
13.1. Functions of Several Variables......Page 691
13.2. Limits and Continuity in Higher Dimensions......Page 699
13.3. Partial Derivatives......Page 708
13.4. The Chain Rule......Page 719
13.5. Directional Derivatives and Gradient Vectors......Page 728
13.6. Tangent Planes and Differentials......Page 736
13.7. Extreme Values and Saddle Points......Page 745
13.8. Lagrange Multipliers......Page 754
Questions to Guide Your Review......Page 763
Practice Exercises......Page 764
14.1. Double and Iterated Integrals over Rectangles......Page 770
14.2. Double Integrals over General Regions......Page 775
14.3. Area by Double Integration......Page 784
14.4. Double Integrals in Polar Form......Page 788
14.5. Triple Integrals in Rectangular Coordinates......Page 794
14.6. Moments and Centers of Mass......Page 803
14.7. Triple Integrals in Cylindrical and Spherical Coordinates......Page 810
14.8. Substitutions in Multiple Integrals......Page 821
Practice Exercises......Page 831
15.1. Line Integrals......Page 836
15.2. Vector Fields and Line Integrals: Work, Circulation, and Flux......Page 843
15.3. Path Independence, Conservative Fields, and Potential Functions......Page 855
15.4. Green’s Theorem in the Plane......Page 866
15.5. Surfaces and Area......Page 878
15.6. Surface Integrals......Page 889
15.7. Stokes’ Theorem......Page 900
15.8. The Divergence Theorem and a Unified Theory......Page 912
Practice Exercises......Page 923
A.1. Real Numbers and the Real Line......Page 928
A.2. Mathematical Induction......Page 933
A.3. Lines and Circles......Page 937
A.4. Conic Sections......Page 943
A.5. Proofs of Limit Theorems......Page 951
A.6. Commonly Occurring Limits......Page 954
A.7. Theory of the Real Numbers......Page 955
A.8. Complex Numbers......Page 958
A.9. The Distributive Law for Vector Cross Products......Page 966
A.10. The Mixed Derivative Theorem and the Increment Theorem......Page 968
Index......Page 1036
A Brief Table of Integrals......Page 1052
Formulas and Theorems......Page 1058
Back Cover......Page 1068

##### Citation preview

University Calculus

Early Transcendentals

For these Global Editions, the editorial team at Pearson has collaborated with educators across the world to address a wide range of subjects and requirements, equipping students with the best possible learning tools. This Global Edition preserves the cutting-edge approach and pedagogy of the original, but also features alterations, customization, and adaptation from the North American version.

Global edition

Global edition

Global edition

Third edition

Hass • Weir • Thomas

This is a special edition of an established title widely used by colleges and universities throughout the world. Pearson published this exclusive edition for the benefit of students outside the United States and Canada. If you purchased this book within the United States or Canada, you should be aware that it has been imported without the approval of the Publisher or Author.

University Calculus Early Transcendentals Third edition

Joel Hass • Maurice D. Weir • George B. Thomas, Jr.

Pearson Global Edition

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University

Calculus Early Transcendentals

Third Edition Global Edition

Joel Hass University of California, Davis Maurice D. Weir Naval Postgraduate School George B. Thomas, Jr. Massachusetts Institute of Technology with the assistance of

Christopher Heil Georgia Institute of Technology

Boston Columbus Indianapolis New York San Francisco Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montréal Toronto Delhi Mexico City São Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo

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Contents

Preface  9

Credits  14

1

Functions  15 1.1 1.2 1.3 1.4 1.5 1.6

2

Functions and Their Graphs  15 Combining Functions; Shifting and Scaling Graphs  28 Trigonometric Functions  35 Graphing with Software  43 Exponential Functions  47 Inverse Functions and Logarithms  52

Limits and Continuity

65

2.1 Rates of Change and Tangents to Curves  65 2.2 Limit of a Function and Limit Laws  72 2.3 The Precise Definition of a Limit  83 2.4 One-Sided Limits  92 2.5 Continuity 99 2.6 Limits Involving Infinity; Asymptotes of Graphs  110 Questions to Guide Your Review  124 Practice Exercises  124 Additional and Advanced Exercises  126

3 Derivatives

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11

129

Tangents and the Derivative at a Point  129 The Derivative as a Function  133 Differentiation Rules  142 The Derivative as a Rate of Change  152 Derivatives of Trigonometric Functions  161 The Chain Rule  167 Implicit Differentiation  175 Derivatives of Inverse Functions and Logarithms  180 Inverse Trigonometric Functions  190 Related Rates  196 Linearization and Differentials  204 Questions to Guide Your Review  215 Practice Exercises  216 Additional and Advanced Exercises  220

3

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4

Contents

4

Applications of Derivatives

223

4.1 Extreme Values of Functions  223 4.2 The Mean Value Theorem  231 4.3 Monotonic Functions and the First Derivative Test  239 4.4 Concavity and Curve Sketching  244 4.5 Indeterminate Forms and L’Hôpital’s Rule  255 4.6 Applied Optimization  264 4.7 Newton’s Method  275 4.8 Antiderivatives 279 Questions to Guide Your Review  289 Practice Exercises  290 Additional and Advanced Exercises  294

5 Integrals  297

5.1 5.2 5.3 5.4 5.5 5.6

6

Area and Estimating with Finite Sums  297 Sigma Notation and Limits of Finite Sums  307 The Definite Integral  314 The Fundamental Theorem of Calculus  326 Indefinite Integrals and the Substitution Method  337 Definite Integral Substitutions and the Area Between Curves  345 Questions to Guide Your Review  355 Practice Exercises  355 Additional and Advanced Exercises  358

Applications of Definite Integrals

361

6.1 Volumes Using Cross-Sections  361 6.2 Volumes Using Cylindrical Shells  372 6.3 Arc Length  380 6.4 Areas of Surfaces of Revolution  386 6.5 Work 390 6.6 Moments and Centers of Mass  396 Questions to Guide Your Review  404 Practice Exercises  404 Additional and Advanced Exercises  406

7

7.1 7.2 7.3

Integrals and Transcendental Functions

407

The Logarithm Defined as an Integral  407 Exponential Change and Separable Differential Equations  417 Hyperbolic Functions  426 Questions to Guide Your Review  434 Practice Exercises  434 Additional and Advanced Exercises  435

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Contents

8

Techniques of Integration

8.1 8.2 8.3 8.4 8.5 8.6 8.7

9

5

436

Integration by Parts  437 Trigonometric Integrals  443 Trigonometric Substitutions  449 Integration of Rational Functions by Partial Fractions  454 Integral Tables and Computer Algebra Systems  461 Numerical Integration  466 Improper Integrals  476 Questions to Guide Your Review  487 Practice Exercises  487 Additional and Advanced Exercises  490

Infinite Sequences and Series

492

9.1 Sequences 492 9.2 Infinite Series  504 9.3 The Integral Test  513 9.4 Comparison Tests  520 9.5 Absolute Convergence; The Ratio and Root Tests  524 9.6 Alternating Series and Conditional Convergence  530 9.7 Power Series  536 9.8 Taylor and Maclaurin Series  546 9.9 Convergence of Taylor Series  551 9.10 The Binomial Series and Applications of Taylor Series  557 Questions to Guide Your Review  565 Practice Exercises  566 Additional and Advanced Exercises  568

10

10.1 10.2 10.3 10.4 10.5 10.6

11

Parametric Equations and Polar Coordinates

571

Parametrizations of Plane Curves  571 Calculus with Parametric Curves  578 Polar Coordinates  588 Graphing Polar Coordinate Equations  592 Areas and Lengths in Polar Coordinates  595 Conics in Polar Coordinates  600 Questions to Guide Your Review  606 Practice Exercises  607 Additional and Advanced Exercises  608

Vectors and the Geometry of Space

610

11.1 Three-Dimensional Coordinate Systems  610 11.2 Vectors  615 11.3 The Dot Product  624 11.4 The Cross Product  632

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6

Contents

11.5 11.6

12

12.1 12.2 12.3 12.4 12.5 12.6

13

13.1 13.2 13.3 13.4 13.5 13.6 13.7 13.8

14

14.1 14.2 14.3 14.4 14.5 14.6 14.7 14.8

Lines and Planes in Space  638 Cylinders and Quadric Surfaces  646 Questions to Guide Your Review  651 Practice Exercises  652 Additional and Advanced Exercises  654

Vector-Valued Functions and Motion in Space

656

Curves in Space and Their Tangents  656 Integrals of Vector Functions; Projectile Motion  664 Arc Length in Space  670 Curvature and Normal Vectors of a Curve  675 Tangential and Normal Components of Acceleration  681 Velocity and Acceleration in Polar Coordinates  683 Questions to Guide Your Review  687 Practice Exercises  687 Additional and Advanced Exercises  689

Partial Derivatives

690

Functions of Several Variables  690 Limits and Continuity in Higher Dimensions  698 Partial Derivatives  707 The Chain Rule  718 Directional Derivatives and Gradient Vectors  727 Tangent Planes and Differentials  735 Extreme Values and Saddle Points  744 Lagrange Multipliers  753 Questions to Guide Your Review  762 Practice Exercises  763 Additional and Advanced Exercises  766

Multiple Integrals

769

Double and Iterated Integrals over Rectangles  769 Double Integrals over General Regions  774 Area by Double Integration  783 Double Integrals in Polar Form  787 Triple Integrals in Rectangular Coordinates  793 Moments and Centers of Mass  802 Triple Integrals in Cylindrical and Spherical Coordinates  809 Substitutions in Multiple Integrals  820 Questions to Guide Your Review  830 Practice Exercises  830 Additional and Advanced Exercises  833

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Contents

15

15.1 15.2 15.3 15.4 15.5 15.6 15.7 15.8

16

Integrals and Vector Fields

7

835

Line Integrals  835 Vector Fields and Line Integrals: Work, Circulation, and Flux  842 Path Independence, Conservative Fields, and Potential Functions  854 Green’s Theorem in the Plane  865 Surfaces and Area  877 Surface Integrals  888 Stokes’ Theorem  899 The Divergence Theorem and a Unified Theory  911 Questions to Guide Your Review  922 Practice Exercises  922 Additional and Advanced Exercises  925

First-Order Differential Equations  online

16.1 Solutions, Slope Fields, and Euler’s Method 16.2 First-Order Linear Equations 16.3 Applications 16.4 Graphical Solutions of Autonomous Equations 16.5 Systems of Equations and Phase Planes

17

Second-Order Differential Equations  online

17.1 Second-Order Linear Equations 17.2 Nonhomogeneous Linear Equations 17.3 Applications 17.4 Euler Equations 17.5 Power Series Solutions

Appendices  927 A.1 A.2 A.3 A.4 A.5 A.6 A.7 A.8 A.9 A.10

Real Numbers and the Real Line  927 Mathematical Induction  932 Lines and Circles  936 Conic Sections  942 Proofs of Limit Theorems  950 Commonly Occurring Limits  953 Theory of the Real Numbers  954 Complex Numbers    957 The Distributive Law for Vector Cross Products  965 The Mixed Derivative Theorem and the Increment Theorem  967

Index  1035

A Brief Table of Integrals  1051

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Preface This third edition of University Calculus provides a streamlined treatment of the material in a standard three-semester or four-quarter course taught at the university level. As the title suggests, the book aims to go beyond what many students may have seen at the high school level. By emphasizing rigor and mathematical precision, supported with examples and exercises, this book encourages students to think more clearly than if they were using rote procedures. Generalization drives the development of calculus and is pervasive in this book. Slopes of lines generalize to slopes of curves, lengths of line segments to lengths of curves, areas and volumes of regular geometric figures to areas and volumes of shapes with curved boundaries, rational exponents to irrational ones, and finite sums to series. Plane analytic geometry generalizes to the geometry of space, and single variable calculus to the calculus of many variables. Generalization weaves together the many threads of calculus into an elegant tapestry that is rich in ideas and their applications. Mastering this beautiful subject is its own reward, but the real gift of mastery is the ability to think through problems clearly—distinguishing between what is known and what is assumed, and using a logical sequence of steps to reach a solution. We intend this book to capture the richness and powerful applicability of calculus, and to support student thinking and understanding for mastery of the material.

New to this Edition In this new edition, we have followed the basic structure of earlier editions. Taking into account helpful suggestions from readers and users of previous editions, we continued to improve clarity and readability. We also made the following improvements:

• Updated and added numerous exercises throughout, with emphasis on the mid-level and more in the life science areas

• Reworked many figures and added new ones • Moved the discussion of conditional convergence to follow the Alternating Series Test • Enhanced the discussion defining differentiability for functions of several variables with more emphasis on linearization

• •

Showed that the derivative along a path generalizes the single-variable chain rule

• •

Developed surface integrals of vector fields as generalizations of line integrals

Added more geometric insight into the idea of multiple integrals and the meaning of the Jacobian in substitutions for their evaluations Extended and clarified the discussion of the curl and divergence, and added new figures to help visualize their meanings

9

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10

Preface

Continuing Features RIGOR  The level of rigor is consistent with that of earlier editions. We continue to distinguish between formal and informal discussions and to point out their differences. We think starting with a more intuitive, less formal, approach helps students understand a new or difficult concept so they can then appreciate its full mathematical precision and outcomes. We pay attention to defining ideas carefully and to proving theorems appropriate for calculus students, while mentioning deeper or subtler issues they would study in a more advanced course. Our organization and distinctions between informal and formal discussions give the instructor a degree of flexibility in the amount and depth of coverage of the various topics. For example, while we do not prove the Intermediate Value Theorem or the Extreme Value Theorem for continuous functions on a … x … b, we do state these theorems precisely, illustrate their meanings in numerous examples, and use them to prove other important results. Furthermore, for those instructors who desire greater depth of ­coverage, in Appendix 7 we discuss the reliance of these theorems on the completeness of the real numbers. WRITING EXERCISES  Writing exercises placed throughout the text ask students to explore and explain a variety of calculus concepts and applications. In addition, the end of each chapter contains a list of questions to help students review and summarize what they have learned. Many of these exercises make good writing assignments. END-OF-CHAPTER REVIEWS  In addition to problems appearing after each section, each chapter culminates with review questions, practice exercises covering the entire chapter, and a series of Additional and Advanced Exercises providing more challenging or synthesizing problems. WRITING AND APPLICATIONS  As always, this text continues to be easy to read, conversational, and mathematically rich. Each new topic is motivated by clear, easy-tounderstand examples and is then reinforced by its application to real-world problems of immediate interest to students. A hallmark of this book has been the application of calculus to science and engineering. These applied problems have been updated, improved, and extended continually over the past several editions. TECHNOLOGY  In a course using the text, technology can be incorporated at the discretion of the instructor. Each section contains exercises requiring the use of technology; these are marked with a “T” if suitable for calculator or computer use, or they are labeled “Computer Explorations” if a computer algebra system (CAS, such as Maple or Mathematica) is required.

Additional Resources Instructor’s Solutions Manual (download only) Single Variable Calculus (Chapters 1–10) Multivariable Calculus (Chapters 9–15) The Instructor’s Solutions Manual contains complete worked-out solutions to all of the exercises. These files are available to qualified instructors through the Pearson Instructor Resource Center, www.pearsonglobaleditions.com/Hass, and MyMathLab.

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Preface

11

Video Lectures with Optional Captioning The Video Lectures with Optional Captioning feature an engaging team of mathematics instructors who present comprehensive coverage of topics in the text. The lecturers’ presentations include examples and exercises from the text and support an approach that emphasizes visualization and problem solving. Available only through MyMathLab. TestGen® TestGen® (www.pearsoned.com/testgen) enables instructors to build, edit, print, and administer tests using a computerized bank of questions developed to cover all the objectives of the text. TestGen is algorithmically based, allowing instructors to create multiple but equivalent versions of the same question or test with the click of a button. Instructors can also modify test bank questions or add new questions. The software and test bank are available for download from Pearson Education’s online catalog.

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12

Preface

PowerPoint® Lecture Slides These classroom presentation slides are geared specifically to the sequence and philosophy of University Calculus: Early Transcendentals. Key graphics from the book are included to help bring the concepts alive in the classroom. These files are available to qualified instructors through the Pearson Instructor Resource Center, www.pearsonglobaleditions .com/Hass, and MyMathLab.

Acknowledgments We wish to express our gratitude to the reviewers of this and previous editions, who provided such invaluable insight and comment. Harry Allen, Ohio State University Edoh Amiran, Western Washington  University Anthony Bedenikovic, Bradley University Robert A. Beezer, University of Puget Sound Przemyslaw Bogacki, Old Dominion  University Deborah Brandon, Carnegie Mellon  University Samuel Chamberlin, Park University Leonard Chastofsky, University of  Georgia Meighan Dillon, Southern Polytechnic State University Anne Dougherty, University of Colorado Said Fariabi, San Antonio College Klaus Fischer, George Mason University Tim Flood, Pittsburg State University Rick Ford, California State University— Chico Toni Fountain, Chattanooga State   Community College Robert Gardner, East Tennessee State  University Mark Greer, University of North Alabama Ivan Gotchev, Central Connecticut State  University Christopher Heil, Georgia Institute of  Technology David Hemmer, SUNY—Buffalo Joshua Brandon Holden, Rose-Hulman   Institute of Technology Alexander Hulpke, Colorado State  University Jacqueline Jensen, Sam Houston State  University Jennifer M. Johnson, Princeton  University

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Hideaki Kaneko, Old Dominion  University Przemo Kranz, University of Mississippi John Kroll, Old Dominion University Krystyna Kuperberg, Auburn University Glenn Ledder, University of Nebraska—  Lincoln Matthew Leingang, New York University Xin Li, University of Central Florida Abey Lopez-Garcia, University of South  Alabama Maura Mast, University of  Massachusetts—Boston Val Mohanakumar, Hillsborough   Community College—Dale Mabry  Campus Aaron Montgomery, Central Washington University Yibiao Pan, University of Pittsburgh Christopher M. Pavone, California State   University at Chico Cynthia Piez, University of Idaho Brooke Quinlan, Hillsborough   Community College—Dale Mabry  Campus Paul Sacks, Iowa State University Rebecca A. Segal, Virginia   Commonwealth University Andrew V. Sills, Georgia Southern  University Edward E. Slaminka, Auburn University Alex Smith, University of Wisconsin—   Eau Claire Mark A. Smith, Miami University Donald Solomon, University of  Wisconsin—Milwaukee John Sullivan, Black Hawk College Stephen Summers, University of Florida Maria Terrell, Cornell University

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Preface

Blake Thornton, Washington University in St. Louis Ruth Trubnik, Delaware Valley College Ilie Ugarcovici, Rice University David Walnut, George Mason University Adrian Wilson, University of Montevallo

13

Bobby Winters, Pittsburg State  University Dennis Wortman, University of  Massachusetts—Boston Yilian Zhang, University of South   Carolina, Aiken

The publishers would like to thank the following for their contribution to the Global Edition: Contributor Nalinakshi N., Atria Institute of  Technology Reviewers B. R. Shankar, National Institute of Technology Karnataka Antonio Behn Von Schmieden, University of Chile Sunil Jacob John, National Institute of Technology Calicut

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Credits Chapter 3 Page 159, Exercise 19, PSSC PHYSICS, 2nd ed., Reprinted by permission of Education Development Center, Inc. Chapter 6 Page 397, Figure 6.40, PSSC PHYSICS, 2nd ed., Reprinted by permission of Educational Development Center, Inc. Chapter 9 Page 507, Figure 9.9, PSSC PHYSICS, 2nd ed., Reprinted by permission of Educational Development Center, Inc. Chapter 13 Page 694, Figure 13.7, Reprinted by permission of the Appalachian Mountain Club; Page 727, Figure 13.26, U.S. Geological Survey Chapter 15 Page 842, Figures 15.6 and 15.7, National Committee for Fluid Mechanics, edited by Shapiro, ILLUSTRATED EXPERIMENTS IN FLUID MECHANICS: THE NCFMF BOOK OF FILM NOTES, reprinted by permission of Educational Development Center, Inc.; Page 844, Figure 15.15, InterNetwork Media, Inc., courtesy of NASA/JPL

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chapter

1

Functions

Overview  Functions are fundamental to the study of calculus. In this chapter we review

what functions are and how they are pictured as graphs, how they are combined and transformed, and ways they can be classified. We review the trigonometric functions, and we discuss misrepresentations that can occur when using calculators and computers to obtain a function’s graph. We also discuss inverse, exponential, and logarithmic functions. The real number system, Cartesian coordinates, straight lines, circles, parabolas, ellipses, and hyperbolas are reviewed in the Appendices.

1.1  Functions and Their Graphs

Functions are a tool for describing the real world in mathematical terms. A function can be represented by an equation, a graph, a numerical table, or a verbal description; we will use all four representations throughout this book. This section reviews these function ideas.

Functions; Domain and Range The temperature at which water boils depends on the elevation above sea level (the boiling point drops as you ascend). The interest paid on a cash investment depends on the length of time the investment is held. The area of a circle depends on the radius of the circle. The distance an object travels at constant speed along a straight-line path depends on the elapsed time. In each case, the value of one variable quantity, say y, depends on the value of another variable quantity, which we might call x. We say that “y is a function of x” and write this symbolically as y = ƒ(x)

(“y equals ƒ of x”).

In this notation, the symbol ƒ represents the function, the letter x is the independent variable representing the input value of ƒ, and y is the dependent variable or output value of ƒ at x. Definition A function ƒ from a set D to a set Y is a rule that assigns a unique (single) element ƒ(x)∊Y to each element x∊D. The set D of all possible input values is called the domain of the function. The set of all output values of ƒ(x) as x varies throughout D is called the range of the function. The range may not include every element in the set Y. The domain and range of a function can be any sets of objects, but often in calculus they are sets of real numbers interpreted as points of a coordinate line. (In Chapters 12–15, we will encounter functions for which the elements of the sets are points in the coordinate plane or in space.)

15

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16

x

Chapter 1: Functions

f

Input (domain)

Output (range)

f(x)

Figure 1.1  A diagram showing a function as a kind of machine.

x a D = domain set

f (a)

f(x)

Y = set containing the range

Figure 1.2  A function from a set D to a set Y assigns a unique element of Y to each element in D.

Often a function is given by a formula that describes how to calculate the output value from the input variable. For instance, the equation A = pr 2 is a rule that calculates the area A of a circle from its radius r (so r, interpreted as a length, can only be positive in this formula). When we define a function y = ƒ(x) with a formula and the domain is not stated explicitly or restricted by context, the domain is assumed to be the largest set of real x-values for which the formula gives real y-values, which is called the natural domain. If we want to restrict the domain in some way, we must say so. The domain of y = x2 is the entire set of real numbers. To restrict the domain of the function to, say, positive values of x, we would write “y = x2, x 7 0.” Changing the domain to which we apply a formula usually changes the range as well. The range of y = x2 is [0, q). The range of y = x2, x Ú 2, is the set of all numbers obtained by squaring numbers greater than or equal to 2. In set notation (see Appendix 1), the range is 5x2  x Ú 26 or 5y  y Ú 46 or 3 4, q). When the range of a function is a set of real numbers, the function is said to be realvalued. The domains and ranges of most real-valued functions of a real variable we consider are intervals or combinations of intervals. The intervals may be open, closed, or half open, and may be finite or infinite. Sometimes the range of a function is not easy to find. A function ƒ is like a machine that produces an output value ƒ(x) in its range whenever we feed it an input value x from its domain (Figure 1.1). The function keys on a calculator give an example of a function as a machine. For instance, the 2x key on a calculator gives an output value (the square root) whenever you enter a nonnegative number x and press the 2x key. A function can also be pictured as an arrow diagram (Figure 1.2). Each arrow associates an element of the domain D with a unique or single element in the set Y. In Figure 1.2, the arrows indicate that ƒ(a) is associated with a, ƒ(x) is associated with x, and so on. Notice that a function can have the same value at two different input elements in the domain (as occurs with ƒ(a) in Figure 1.2), but each input element x is assigned a single output value ƒ(x).

EXAMPLE 1   Let’s verify the natural domains and associated ranges of some simple functions. The domains in each case are the values of x for which the formula makes sense. Function y y y y y

= = = = =

2

x 1>x 2x 24 - x 21 - x2

Domain (x)

Range ( y)

(- q, q) (- q, 0) ∪ (0, q) 3 0, q) (- q, 44 3 -1, 14

3 0, q) (- q, 0) ∪ (0, q) 3 0, q) 3 0, q) 3 0, 14

Solution  The formula y = x2 gives a real y-value for any real number x, so the domain is (- q, q). The range of y = x2 is 3 0, q) because the square of any real number is non2 negative and every nonnegative number y is the square of its own square root, y = 1 2y 2 for y Ú 0. The formula y = 1>x gives a real y-value for every x except x = 0. For consistency in the rules of arithmetic, we cannot divide any number by zero. The range of y = 1>x, the set of reciprocals of all nonzero real numbers, is the set of all nonzero real numbers, since y = 1>(1>y). That is, for y ≠ 0 the number x = 1>y is the input assigned to the output value y. The formula y = 2x gives a real y-value only if x Ú 0. The range of y = 2x is 3 0, q) because every nonnegative number is some number’s square root (namely, it is the square root of its own square). In y = 24 - x, the quantity 4 - x cannot be negative. That is, 4 - x Ú 0, or x … 4. The formula gives real y-values for all x … 4. The range of 24 - x is 3 0, q), the set of all nonnegative numbers.

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1.1  Functions and Their Graphs

The formula y = 21 - x2 gives a real y-value for every x in the closed interval from -1 to 1. Outside this domain, 1 - x2 is negative and its square root is not a real number. The values of 1 - x2 vary from 0 to 1 on the given domain, and the square roots of these values do the same. The range of 21 - x2 is 3 0, 14 .

Graphs of Functions

If ƒ is a function with domain D, its graph consists of the points in the Cartesian plane whose coordinates are the input-output pairs for ƒ. In set notation, the graph is 5(x, ƒ(x))  x∊D6 .

The graph of the function ƒ(x) = x + 2 is the set of points with coordinates (x, y) for which y = x + 2. Its graph is the straight line sketched in Figure 1.3. The graph of a function ƒ is a useful picture of its behavior. If (x, y) is a point on the graph, then y = ƒ(x) is the height of the graph above (or below) the point x. The height may be positive or negative, depending on the sign of ƒ(x) (Figure 1.4). y

f (1)

y

f (2) x y=x+2

0

1

x

2 f(x)

2 (x, y)

y = x2

x

−2

- 2 4 - 1 1 0 0 1 1 9 3 2 4 2 4

0

Figure 1.3  The graph of ƒ(x) = x + 2 is the set of points (x, y) for which y has the value x + 2.

Figure 1.4  If (x, y) lies on the graph of ƒ, then the value y = ƒ(x) is the height of the graph above the point x (or below x if ƒ(x) is negative).

EXAMPLE 2   Graph the function y = x2 over the interval 3 -2, 24 .

Solution  Make a table of xy-pairs that satisfy the equation y = x2 . Plot the points (x, y) whose coordinates appear in the table, and draw a smooth curve (labeled with its equation) through the plotted points (see Figure 1.5).

y (−2, 4)

x

How do we know that the graph of y = x2 doesn’t look like one of these curves? (2, 4)

4

y

y

y = x2

3 3 9 a2 , 4b

2 (−1, 1)

1

−2

0

−1

1

2

x

Figure 1.5  Graph of the function in Example 2.

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y = x 2?

y = x 2?

(1, 1)

x

x

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18

Chapter 1: Functions

To find out, we could plot more points. But how would we then connect them? The basic question still remains: How do we know for sure what the graph looks like between the points we plot? Calculus answers this question, as we will see in Chapter 4. Meanwhile, we will have to settle for plotting points and connecting them as best we can.

Representing a Function Numerically We have seen how a function may be represented algebraically by a formula (the area function) and visually by a graph (Example 2). Another way to represent a function is numerically, through a table of values. Numerical representations are often used by engineers and experimental scientists. From an appropriate table of values, a graph of the function can be obtained using the method illustrated in Example 2, possibly with the aid of a computer. The graph consisting of only the points in the table is called a scatterplot.

EXAMPLE 3   Musical notes are pressure waves in the air. The data associated with Figure 1.6 give recorded pressure displacement versus time in seconds of a musical note produced by a tuning fork. The table provides a representation of the pressure function over time. If we first make a scatterplot and then connect approximately the data points (t, p) from the table, we obtain the graph shown in the figure. p (pressure)

Time

Pressure

Time

Pressure

0.00091 -0.080 0.00362 0.217 0.00108 0.200 0.00379 0.480 0.00125 0.480 0.00398 0.681 0.00144 0.693 0.00416 0.810 0.00162 0.816 0.00435 0.827 0.00180 0.844 0.00453 0.749 0.00198 0.771 0.00471 0.581 0.00216 0.603 0.00489 0.346 0.00234 0.368 0.00507 0.077 0.00253 0.099 0.00525 -0.164 0.00271 -0.141 0.00543 -0.320 0.00289 -0.309 0.00562 -0.354 0.00307 -0.348 0.00579 -0.248 0.00325 -0.248 0.00598 -0.035 0.00344 -0.041

1.0 0.8 0.6 0.4 0.2 −0.2 −0.4 −0.6

Data

0.001 0.002 0.003 0.004 0.005 0.006

t (sec)

Figure 1.6  A smooth curve through the plotted points gives a graph of the pressure function represented by the accompanying tabled data (Example 3).

The Vertical Line Test for a Function Not every curve in the coordinate plane can be the graph of a function. A function ƒ can have only one value ƒ(x) for each x in its domain, so no vertical line can intersect the graph of a function more than once. If a is in the domain of the function ƒ, then the vertical line x = a will intersect the graph of ƒ at the single point (a, ƒ(a)). A circle cannot be the graph of a function, since some vertical lines intersect the circle twice. The circle graphed in Figure 1.7a, however, does contain the graphs of functions of x, such as the upper semicircle defined by the function ƒ(x) = 21 - x2 and the lower semicircle defined by the function g (x) = - 21 - x2 (Figures 1.7b and 1.7c).

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1.1  Functions and Their Graphs

y

−1

0

y

1

x

(a) x 2 + y 2 = 1

y = 0x0

1 −3 −2 −1 0

1

2

x

3

Figure 1.8  The absolute value function has domain (- q, q) and range 30, q). y

y = −x

−1

0

x

(c) y = −"1 − x 2

(b) y = "1 − x 2

y = f (x)

Sometimes a function is described in pieces by using different formulas on different parts of its domain. One example is the absolute value function

0x0 = e

x, -x,

x Ú 0 First formula        x 6 0, Second formula

whose graph is given in Figure 1.8. The right-hand side of the equation means that the function equals x if x Ú 0, and equals -x if x 6 0. Piecewise-defined functions often arise when real-world data are modeled. Here are some other examples.

2

EXAMPLE 4   The function

y=1

1 −2

1

−1

Piecewise-Defined Functions

y=x

2

1

x

Figure 1.7  (a) The circle is not the graph of a function; it fails the vertical line test. (b) The upper semicircle is the graph of a function ƒ(x) = 21 - x2. (c) The lower semicircle is the graph of a function g (x) = - 21 - x2.

y y = −x 3

0

−1

y

0

y=x

2

1

2

x

-x, ƒ(x) = c x2, 1,

Figure 1.9  To graph the function y = ƒ(x) shown here, we apply different formulas to different parts of its domain (Example 4).

x 6 0 0 … x … 1    x 7 1

First formula Second formula Third formula

is defined on the entire real line but has values given by different formulas, depending on the position of x. The values of ƒ are given by y = -x when x 6 0, y = x2 when 0 … x … 1, and y = 1 when x 7 1. The function, however, is just one function whose domain is the entire set of real numbers (Figure 1.9).

y y=x

3 2

y = :x;

1 1

−2 −1

2

3

x

−2

Figure 1.10  The graph of the greatest integer function y = : x ; lies on or below the line y = x, so it provides an integer floor for x (Example 5).

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EXAMPLE 5   The function whose value at any number x is the greatest integer less than or equal to x is called the greatest integer function or the integer floor function. It is denoted : x ; . Figure 1.10 shows the graph. Observe that

: 2.4 ; = 2, : 2 ; = 2,

: 1.9 ; = 1, : 0.2 ; = 0,

: 0 ; = 0, : -0.3 ; = -1,

: -1.2 ; = -2,  : -2 ; = -2.

EXAMPLE 6   The function whose value at any number x is the smallest integer greater than or equal to x is called the least integer function or the integer ceiling function. It is denoted < x = . Figure 1.11 shows the graph. For positive values of x, this function might represent, for example, the cost of parking x hours in a parking lot that charges \$1 for each hour or part of an hour.

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Chapter 1: Functions

Increasing and Decreasing Functions

y

2

y = x, then sometimes it is said that y is inversely proportional to x (because 1>x is the multiplicative inverse of x). Power Functions  A function ƒ(x) = xa, where a is a constant, is called a power function. There are several important cases to consider.

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22

Chapter 1: Functions

(a)  a = n, a positive integer. The graphs of ƒ(x) = xn, for n = 1, 2, 3, 4, 5, are displayed in Figure 1.15. These functions are defined for all real values of x. Notice that as the power n gets larger, the curves tend to flatten toward the x-axis on the interval (-1, 1), and to rise more steeply for 0 x 0 7 1. Each curve passes through the point (1, 1) and through the origin. The graphs of functions with even powers are symmetric about the y-axis; those with odd powers are symmetric about the origin. The even-powered functions are decreasing on the interval (- q, 04 and increasing on 3 0, q); the odd-powered functions are increasing over the entire real line (- q, q). y

y

y=x

1

1 0

−1 −1

y

y = x2

1

x

−1

y

y = x3

1

0

1

x

−1

−1

0

y y = x5

y = x4

1 x

1

−1

1

0

−1

1

x

−1

−1

0

1

x

−1

Figure 1.15  Graphs of ƒ(x) = xn, n = 1, 2, 3, 4, 5, defined for - q 6 x 6 q.

(b)  a = -1 or a = -2. The graphs of the functions ƒ(x) = x-1 = 1>x and g(x) = x-2 = 1>x2 are shown in Figure 1.16. Both functions are defined for all x ≠ 0 (you can never divide by zero). The graph of y = 1>x is the hyperbola xy = 1, which approaches the coordinate axes far from the origin. The graph of y = 1>x2 also approaches the coordinate axes. The graph of the function ƒ is symmetric about the origin; ƒ is decreasing on the intervals (- q, 0) and (0, q). The graph of the function g is symmetric about the y-axis; g is increasing on (- q, 0) and decreasing on (0, q). y y y = 1x

y = 12 x

1 0

1

x

Domain: x ≠ 0 Range: y ≠ 0

(a)

1 0

x 1 Domain: x ≠ 0 Range: y > 0

(b)

Figure 1.16  Graphs of the power functions ƒ(x) = xa for part (a) a = -1 and for part (b) a = -2.

(c)  a =

1 1 3 2 , , , and . 3 2 3 2

3 The functions ƒ(x) = x1>2 = 2x and g(x) = x1>3 = 2 x are the square root and cube root functions, respectively. The domain of the square root function is 3 0, q), but the cube root function is defined for all real x. Their graphs are displayed in Figure 1.17, along with the graphs of y = x3>2 and y = x2>3. (Recall that x3>2 = (x1>2)3 and x2>3 = (x1>3)2.)

Polynomials  A function p is a polynomial if p(x) = an xn + an - 1xn - 1 + g + a1 x + a0 where n is a nonnegative integer and the numbers a0, a1, a2, c, an are real constants (called the coefficients of the polynomial). All polynomials have domain (- q, q). If the

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23

1.1  Functions and Their Graphs

y

y

y

y

y = x 32

y = !x

y = x 23

3

y = !x

1 1 0

1 Domain: 0 ≤ x < ∞ Range: 0 ≤ y < ∞

x

0

1

1 x

1 Domain: −∞ < x < ∞ Range: −∞ < y < ∞

0

Figure 1.17  Graphs of the power functions ƒ(x) = xa for a =

x

x

0 1 Domain: −∞ < x < ∞ Range: 0 ≤ y < ∞

1 Domain: 0 ≤ x < ∞ Range: 0 ≤ y < ∞

2 1 1 3 , , , and . 2 3 2 3

leading coefficient an ≠ 0 and n 7 0, then n is called the degree of the polynomial. Linear functions with m ≠ 0 are polynomials of degree 1. Polynomials of degree 2, usually written as p(x) = ax2 + bx + c, are called quadratic functions. Likewise, cubic functions are polynomials p(x) = ax3 + bx2 + cx + d of degree 3. Figure 1.18 shows the graphs of three polynomials. Techniques to graph polynomials are studied in Chapter 4. 3 2 y = x − x − 2x + 1 3 2 3 y

4 y 2

−4

−2

y y=

8x 4

14x 3

9x 2

y = (x − 2)4(x + 1)3(x − 1)

+ 11x − 1

16

2

0

2

−1

x

4

1

−2 −4 −6 −8 −10 −12

−2 −4

x

2

−1

0

1

x

2

−16

(a)

(c)

(b)

Figure 1.18  Graphs of three polynomial functions.

Rational Functions A rational function is a quotient or ratio ƒ(x) = p(x)>q(x), where p and q are polynomials. The domain of a rational function is the set of all real x for which q(x) ≠ 0. The graphs of several rational functions are shown in Figure 1.19. y y

8 2 y = 5x +2 8x − 3 3x + 2

y

4 2 2 y = 2x − 3 2 7x + 4

−4

2

−2

4

x

−5

4

Line y = 5 3

1

5

0

2 10

−1

−2

−2 −4

6

y = 11x3 + 2 2x − 1

x

−4 −2 0 −2

2

4

6

x

−4 NOT TO SCALE

−6 −8 (a)

(b)

(c)

Figure 1.19  Graphs of three rational functions. The straight red lines approached by the graphs are called asymptotes and are not part of the graphs. We discuss asymptotes in Section 2.6.

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Chapter 1: Functions

Algebraic Functions  Any function constructed from polynomials using algebraic operations (addition, subtraction, multiplication, division, and taking roots) lies within the class of algebraic functions. All rational functions are algebraic, but also included are more complicated functions (such as those satisfying an equation like y3 - 9xy + x3 = 0, studied in Section 3.7). Figure 1.20 displays the graphs of three algebraic functions. y = x 13(x − 4)

y 4 3 2 1 −1 −1 −2 −3

x

4

−1

1 x

1

0

y = x(1 − x)25

y

y = 3 (x 2 − 1) 23 4 y

0

5 7

x

1

−1

(b)

(a)

(c)

Figure 1.20  Graphs of three algebraic functions.

Trigonometric Functions  The six basic trigonometric functions are reviewed in Section 1.3. The graphs of the sine and cosine functions are shown in Figure 1.21. y

y

1

3p 0

−p

p

2p

1

− p2

x

0 −1

−1 (a) f (x) = sin x

3p 2

5p 2

x

p 2

(b) f (x) = cos x

Figure 1.21  Graphs of the sine and cosine functions.

Exponential Functions  Functions of the form ƒ(x) = ax, where the base a 7 0 is a positive constant and a ≠ 1, are called exponential functions. All exponential functions have domain (- q, q) and range (0, q), so an exponential function never assumes the value 0. We discuss exponential functions in Section 1.5. The graphs of some exponential functions are shown in Figure 1.22. y

y y=

10 x

y=

10 –x

12

12

10

10

8

8

6 4 2 −1 −0.5

0 (a)

y=

y = 3x

1

6 4

y = 2x 0.5

3 –x

y = 2 –x x

−1 −0.5

2 0 (b)

0.5

1

x

Figure 1.22  Graphs of exponential functions.

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25

1.1  Functions and Their Graphs

Logarithmic Functions  These are the functions ƒ(x) = loga x, where the base a ≠ 1 is a positive constant. They are the inverse functions of the exponential functions, and we discuss these functions in Section 1.6. Figure 1.23 shows the graphs of four logarithmic functions with various bases. In each case the domain is (0, q) and the range is (- q, q). y

y

y = log 2 x y = log 3 x

1 0

x

1

y = log5 x

−1

1

y = log10 x −1

Figure 1.23  Graphs of four logarithmic functions.

0

1

x

Figure 1.24  Graph of a catenary or hanging cable. (The Latin word catena means “chain.”)

Transcendental Functions  These are functions that are not algebraic. They include the trigonometric, inverse trigonometric, exponential, and logarithmic functions, and many other functions as well. A particular example of a transcendental function is a catenary. Its graph has the shape of a cable, like a telephone line or electric cable, strung from one support to another and hanging freely under its own weight (Figure 1.24). The function defining the graph is discussed in Section 7.3.

Exercises  1.1 y

8. a.

Functions In Exercises 1–6, find the domain and range of each function.

b.  y

ƒ(x) = 1 - 2x 1. ƒ(x) = 18 + x2 2.

3. F(x) = 25x + 10 4. g(x) = 2x2 - 3x 5. ƒ(t) =

5 4 6. G(t) = 2 3 - t t - 4

In Exercises 7 and 8, which of the graphs are graphs of functions of x, and which are not? Give reasons for your answers. 7. a.

b.  y

y

0

x

0

x

Finding Formulas for Functions 9. Express the area and perimeter of an equilateral triangle as a function of the triangle’s side length x.

10. Express the side length of a square as a function of the length d of the square’s diagonal. Then express the area as a function of the diagonal length.

0

M01_THOM9580_03_SE_C01_015-064.indd 25

x

0

x

11. Express the edge length of a cube as a function of the cube’s diagonal length d. Then express the surface area and volume of the cube as a function of the diagonal length.

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26

Chapter 1: Functions

12. A point P in the first quadrant lies on the graph of the function ƒ(x) = 2x. Express the coordinates of P as functions of the slope of the line joining P to the origin.

31. a.

b.

y (−1, 1) (1, 1) 1

2

13. Consider the point (x, y) lying on the graph of the line 2x + 4y = 5. Let L be the distance from the point (x, y) to the origin (0, 0). Write L as a function of x. 14. Consider the point (x, y) lying on the graph of y = 2x - 3. Let L be the distance between the points (x, y) and (4, 0). Write L as a function of y. Functions and Graphs Find the natural domain and graph the functions in Exercises 15–20.

x

3

b.

y

A 0 0

19. F(t) = t> 0 t 0 20. G(t) = 1> 0 t 0

x + 7 . 12 - 2x2 - 25 x2 2 2. Find the range of y = 2 + 2 . x + 4 21. Find the domain of y =

a. 0 y 0 = x b. y2 = x2

0x + y0 = 1 a. 0 x 0 + 0 y 0 = 1 b.

26. g(x) = e

27. F(x) = e

1 - x, 2 - x,

0 … x … 1 1 6 x … 2

4 - x 2, x2 + 2x,

28. G(x) = e

1>x, x,

x 6 0 0 … x b.

0

30. a.

x

0

1

2

2

3

5

x

−2 −3

M01_THOM9580_03_SE_C01_015-064.indd 26

: x ;, 8 44. y = - 4 2x

45. y = - x3>2 46. y = (- x)2>3

t

ƒ(x) = x-5 47. ƒ(x) = - 8 48. 49. g(x) = 9x3 - 3 50. ƒ(x) = x2 + x g(x) = x4 + 3x2 - 1 51. g(x) = x3 + x 52. x 1 54. g(x) = 2 x2 - 4 x - 1 1 55. h(t) = 3 56. h(t) =  t 3  t + 1 5 7. h(t) = 5t - 1 58. h(t) = 2  t  + 1 53. g(x) =

2 1 −1 −1

4

y 3

(2, 1)

ƒ(x) = e

Even and Odd Functions In Exercises 47–58, say whether the function is even, odd, or neither. Give reasons for your answer.

b.

y 2

2

2

34. What real numbers x satisfy the equation : x ; = < x = ?

35. Does < - x = = - : x ; for all real x? Give reasons for your answer.

41. y = 2 0 x 0 42. y = 2- x

y

(1, 1)

1

< x = = 0? a.  : x ; = - 1? b.

1 37. y = - x3 38. y = - 2 x 1 1 39. y = - x 40. y = 0x0

x … 1 x 7 1

y

−A

Increasing and Decreasing Functions Graph the functions in Exercises 37–46. What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing.

Find a formula for each function graphed in Exercises 29–32. 29. a.

x

t

Why is ƒ(x) called the integer part of x?

Piecewise-Defined Functions Graph the functions in Exercises 25–28.

0 … x … 1 1 6 x … 2

T

T 3T 2T 2

36. Graph the function

24. Graph the following equations and explain why they are not graphs of functions of x.

x, 2 - x,

T 2

T 2

The Greatest and Least Integer Functions 33. For what values of x is

23. Graph the following equations and explain why they are not graphs of functions of x.

25. ƒ(x) = e

y

(T, 1)

1

ƒ(x) = 1 - 2x - x2 15. ƒ(x) = 5 - 2x 16. 17. g(x) = 2 0 x 0 18. g(x) = 2- x

x 1 (1, −1) (3, −1)

(−2, −1)

32. a.

y

1

x 2 (2, −1)

Theory and Examples 59. The variable s is proportional to t, and s = 15 when t = 105. Determine t when s = 40.

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27

1.1  Functions and Their Graphs

60. Kinetic energy  The kinetic energy K of a mass is proportional to the square of its velocity y. If K = 12,960 joules when y = 18 m>sec, what is K when y = 10 m>sec?

66. a.  y = 5x b.  y = 5x c.  y = x5 y

61. The variables r and s are inversely proportional, and r = 7 when s = 4. Determine s when r = 10.

g

62. Boyle’s Law  Boyle’s Law says that the volume V of a gas at constant temperature increases whenever the pressure P decreases, so that V and P are inversely proportional. If P = 14.7 lb>in2 when V = 1000 in3, then what is V when P = 23.4 lb>in2?

h

14

x

x

x

x

x x

x 4 7 1 + x. 2 . Confirm your findings in part (a) algebraically. b

x

64. The accompanying figure shows a rectangle inscribed in an isosceles right triangle whose hypotenuse is 2 units long. a. Express the y-coordinate of P in terms of x. (You might start by writing an equation for the line AB.) b. Express the area of the rectangle in terms of x.

70. Three hundred books sell for \$40 each, resulting in a revenue of (300)(\$40) = \$12,000. For each \$5 increase in the price, 25 fewer books are sold. Write the revenue R as a function of the number x of \$5 increases.

B

P(x, ?) A 0

x

T 68.  a. Graph the functions ƒ(x) = 3>(x - 1) and g(x) = 2>(x + 1) together to identify the values of x for which 3 2 6 . x - 1 x + 1 b. Confirm your findings in part (a) algebraically. 69. For a curve to be symmetric about the x-axis, the point (x, y) must lie on the curve if and only if the point (x, - y) lies on the curve. Explain why a curve that is symmetric about the x-axis is not the graph of a function, unless the function is y = 0.

y

−1

f

T 67.  a. Graph the functions ƒ(x) = x>2 and g(x) = 1 + (4>x) to­gether to identify the values of x for which

22 x

x

0

63. A box with an open top is to be constructed from a rectangular piece of cardboard with dimensions 14 in. by 22 in. by cutting out equal squares of side x at each corner and then folding up the sides as in the figure. Express the volume V of the box as a function of x.

x

1

In Exercises 65 and 66, match each equation with its graph. Do not use a graphing device, and give reasons for your answer. y = x7 c.  y = x10 65. a.  y = x4 b.

71. A pen in the shape of an isosceles right triangle with legs of length x ft and hypotenuse of length h ft is to be built. If fencing costs \$2/ft for the legs and \$8/ft for the hypotenuse, write the total cost C of construction as a function of h. 72. Industrial costs  A power plant sits next to a river where the river is 800 ft wide. To lay a new cable from the plant to a location in the city 2 mi downstream on the opposite side costs \$180 per foot across the river and \$100 per foot along the land. 2 mi P

y

x

Q

City

g h

800 ft

Power plant 0 f

x

NOT TO SCALE

a. Suppose that the cable goes from the plant to a point Q on the opposite side that is x ft from the point P directly opposite the plant. Write a function C(x) that gives the cost of laying the cable in terms of the distance x. b. Generate a table of values to determine if the least expensive location for point Q is less than 2000 ft or greater than 2000 ft from point P.

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Chapter 1: Functions

1.2  Combining Functions; Shifting and Scaling Graphs In this section we look at the main ways functions are combined or transformed to form new functions.

Sums, Differences, Products, and Quotients Like numbers, functions can be added, subtracted, multiplied, and divided (except where the denominator is zero) to produce new functions. If ƒ and g are functions, then for every x that belongs to the domains of both ƒ and g (that is, for x∊D(ƒ) ¨ D(g)), we define functions ƒ + g, ƒ - g, and ƒg by the formulas (ƒ + g)(x) = ƒ(x) + g(x) (ƒ - g)(x) = ƒ(x) - g(x) (ƒg)(x) = ƒ(x)g(x). Notice that the + sign on the left-hand side of the first equation represents the operation of addition of functions, whereas the + on the right-hand side of the equation means addition of the real numbers ƒ(x) and g(x). At any point of D(ƒ) ¨ D(g) at which g(x) ≠ 0, we can also define the function ƒ>g by the formula ƒ ƒ(x) a g b (x) = g(x)

(where g(x) ≠ 0).

Functions can also be multiplied by constants: If c is a real number, then the function cƒ is defined for all x in the domain of ƒ by (cƒ)(x) = cƒ(x).

EXAMPLE 1   The functions defined by the formulas ƒ(x) = 2x

and

g(x) = 21 - x

have domains D(ƒ) = 3 0, q) and D(g) = (- q, 14 . The points common to these domains are the points 3 0, q) ¨ (- q, 14 = 3 0, 14 .

The following table summarizes the formulas and domains for the various algebraic combinations of the two functions. We also write ƒ # g for the product function ƒg. Function ƒ + g ƒ - g g - ƒ

ƒ # g

ƒ>g g>ƒ 

Formula (ƒ + g)(x) = 2x + 21 - x (ƒ - g)(x) = 2x - 21 - x

(g - ƒ)(x) = 21 - x - 2x

(ƒ # g)(x)

= ƒ(x)g(x) = 2x(1 - x)

ƒ ƒ(x) g (x) = g(x) = A 1 g g(x) 1 (x) = = ƒ ƒ(x) A

x - x

- x x

Domain 3 0, 14 = D(ƒ) ¨ D(g) 3 0, 14

3 0, 14 3 0, 14

3 0, 1) (x = 1 excluded)

(0, 14 (x = 0 excluded)

The graph of the function ƒ + g is obtained from the graphs of ƒ and g by adding the corresponding y-coordinates ƒ(x) and g(x) at each point x∊D(ƒ) ¨ D(g), as in Figure 1.25. The graphs of ƒ + g and ƒ # g from Example 1 are shown in Figure 1.26.

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1.2  Combining Functions; Shifting and Scaling Graphs

y

y

y=f+g g(x) = "1 − x

8 6

y = ( f + g)(x) y = g(x)

4 2

y = f (x)

f (a)

g(a)

1 2

f (a) + g(a)

x

a

0

f(x) = "x

1

Figure 1.25  Graphical addition of two functions.

0

y=f•g

1 5

2 5

3 5

4 5

x

1

Figure 1.26  The domain of the function ƒ + g is the intersection of the domains of ƒ and g, the interval 30, 14 on the x-axis where these domains overlap. This interval is also the domain of the function ƒ # g (Example 1).

Composite Functions Composition is another method for combining functions.

Definition  If ƒ and g are functions, the composite function ƒ ∘ g (“ƒ composed with g”) is defined by (ƒ ∘ g)(x) = ƒ(g(x)). The domain of ƒ ∘ g consists of the numbers x in the domain of g for which g(x) lies in the domain of ƒ.

The definition implies that ƒ ∘ g can be formed when the range of g lies in the domain of ƒ. To find (ƒ ∘ g)(x), first find g(x) and second find ƒ(g(x)). Figure 1.27 pictures ƒ ∘ g as a machine diagram, and Figure 1.28 shows the composite as an arrow diagram. f ∘g f(g(x)) x g

x

g

g(x)

f

f (g(x))

Figure 1.27  A composite function ƒ ∘ g uses the output g(x) of the first function g as the input for the second function ƒ.

f

g(x)

Figure 1.28  Arrow diagram for ƒ ∘ g. If x lies in the domain of g and g(x) lies in the domain of ƒ, then the functions ƒ and g can be composed to form (ƒ ∘ g)(x).

To evaluate the composite function g ∘ ƒ (when defined), we find ƒ(x) first and then g(ƒ(x)). The domain of g ∘ ƒ is the set of numbers x in the domain of ƒ such that ƒ(x) lies in the domain of g. The functions ƒ ∘ g and g ∘ ƒ are usually quite different.

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30

Chapter 1: Functions

EXAMPLE 2   If ƒ(x) = 2x and g(x) = x + 1, find

(a) (ƒ ∘ g)(x)  (b)  (g ∘ ƒ)(x)  (c)  (ƒ ∘ ƒ)(x)  (d)  (g ∘ g)(x).

Solution Composite (a) (b) (c) (d)

(ƒ ∘ g)(x) (g ∘ ƒ)(x) (ƒ ∘ ƒ)(x) (g ∘ g)(x)

= = = =

ƒ(g(x)) g(ƒ(x)) ƒ(ƒ(x)) g(g(x))

Domain = = = =

2g(x) = 2x + 1

ƒ(x) + 1 = 2x + 1 2ƒ(x) = 2 1x = x1>4 g(x) + 1 = (x + 1) + 1 = x + 2

3 -1, q) 3 0, q) 3 0, q) (- q, q)

To see why the domain of ƒ ∘ g is 3 -1, q), notice that g(x) = x + 1 is defined for all real x but belongs to the domain of ƒ only if x + 1 Ú 0, that is to say, when x Ú -1. Notice that if ƒ(x) = x2 and g(x) = 2x, then (ƒ ∘ g)(x) = 1 2x 2 = x. However, the domain of ƒ ∘ g is 3 0, q), not (- q, q), since 2x requires x Ú 0. 2

Shifting a Graph of a Function

A common way to obtain a new function from an existing one is by adding a constant to each output of the existing function, or to its input variable. The graph of the new function is the graph of the original function shifted vertically or horizontally, as follows.

Shift Formulas Vertical Shifts y

y=

x2

+2

y=

x2

+1

y=x

2

y = ƒ(x) + k Horizontal Shifts y = ƒ(x + h)

y = x2 − 2 1 unit

−2

Shifts the graph of ƒ up k units if k 7 0 Shifts it down 0 k 0 units if k 6 0

Shifts the graph of ƒ left h units if h 7 0 Shifts it right 0 h 0 units if h 6 0

2

EXAMPLE 3

1 0 −1

2

x

2 units

−2

Figure 1.29  To shift the graph of ƒ(x) = x2 up (or down), we add positive (or negative) constants to the formula for ƒ (Examples 3a and b).

(a) Adding 1 to the right-hand side of the formula y = x2 to get y = x2 + 1 shifts the graph up 1 unit (Figure 1.29). (b) Adding -2 to the right-hand side of the formula y = x2 to get y = x2 - 2 shifts the graph down 2 units (Figure 1.29). (c) Adding 3 to x in y = x2 to get y = (x + 3)2 shifts the graph 3 units to the left, while adding -2 shifts the graph 2 units to the right (Figure 1.30). (d) Adding -2 to x in y = 0 x 0 , and then adding -1 to the result, gives y = 0 x - 2 0 - 1 and shifts the graph 2 units to the right and 1 unit down (Figure 1.31).

Scaling and Reflecting a Graph of a Function To scale the graph of a function y = ƒ(x) is to stretch or compress it, vertically or horizontally. This is accomplished by multiplying the function ƒ, or the independent variable x, by an appropriate constant c. Reflections across the coordinate axes are special cases where c = -1.

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31

1.2  Combining Functions; Shifting and Scaling Graphs

Add a positive constant to x. y = (x +

Add a negative constant to x.

y

3) 2

y=

x2

y

y = (x −

y = 0 x − 20 − 1

4

2) 2

1 1

−4 1

0

−3

−2

x

2

Figure 1.30  To shift the graph of y = x2 to the left, we add a positive constant to x (Example 3c). To shift the graph to the right, we add a negative constant to x.

2

−1

4

x

6

Figure 1.31  The graph of y = 0 x 0 shifted 2 units to the right and 1 unit down (Example 3d).

Vertical and Horizontal Scaling and Reflecting Formulas For c + 1, the graph is scaled: y = cƒ(x) Stretches the graph of ƒ vertically by a factor of c. 1 y = c ƒ(x) Compresses the graph of ƒ vertically by a factor of c. y = ƒ(cx) y = ƒ(x>c)

Compresses the graph of ƒ horizontally by a factor of c. Stretches the graph of ƒ horizontally by a factor of c.

For c = −1, the graph is reflected: y = -ƒ(x) Reflects the graph of ƒ across the x-axis. y = ƒ(-x) Reflects the graph of ƒ across the y-axis.

EXAMPLE 4   Here we scale and reflect the graph of y = 2x.

(a) Vertical: Multiplying the right-hand side of y = 2x by 3 to get y = 3 2x stretches the graph vertically by a factor of 3, whereas multiplying by 1>3 compresses the graph by a factor of 3 (Figure 1.32). (b) Horizontal: The graph of y = 23x is a horizontal compression of the graph of y = 2x by a factor of 3, and y = 2x>3 is a horizontal stretching by a factor of 3 (Figure 1.33). Note that y = 23x = 23 2x so a horizontal compression may correspond to a vertical stretching by a different scaling factor. Likewise, a horizontal stretching may correspond to a vertical compression by a different scaling factor. (c) Reflection: The graph of y = - 2x is a reflection of y = 2x across the x-axis, and y = 2-x is a reflection across the y-axis (Figure 1.34). y

y y = 3"x

5

4

4 stretch

y = "x

2 compress

1 −1

0

y = "3x

3

3

1

2

3

y= 4

x

Figure 1.32  Vertically stretching and compressing the graph y = 1x by a factor of 3 (Example 4a).

M01_THOM9580_03_SE_C01_015-064.indd 31

compress

2

1 x 3"

stretch

1 −1

0

y

y = " −x

1

2

3

4

y = "x y = "x3 x

Figure 1.33  Horizontally stretching and compressing the graph y = 1x by a factor of 3 (Example 4b).

y = "x

1 1

−3 −2 −1 −1

2

3

x

y = −" x

Figure 1.34  Reflections of the graph y = 1x across the coordinate axes (Example 4c).

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32

Chapter 1: Functions

EXAMPLE 5   Given the function ƒ(x) = x4 - 4x3 + 10 (Figure 1.35a), find formulas to (a) compress the graph horizontally by a factor of 2 followed by a reflection across the y-axis (Figure 1.35b). (b) compress the graph vertically by a factor of 2 followed by a reflection across the x-axis (Figure 1.35c). y 20

f (x) =

4x 3

+ 10

0 −10

1

20

10 −1

y

y = 16x 4 + 32x 3 + 10 y x4

10

10 1

2

3

4

x

−2

−20

−1

0 −10 −20

1

x

−1

0

y = − 2 x 4 + 2x 3 − 5

1

3

4

x

−10

(b)

(a)

2

(c)

Figure 1.35  (a) The original graph of f. (b) The horizontal compression of y = ƒ(x) in part (a) by a factor of 2, followed by a reflection across the y-axis. (c) The vertical compression of y = ƒ(x) in part (a) by a factor of 2, followed by a reflection across the x-axis (Example 5).

Solution (a) We multiply x by 2 to get the horizontal compression, and by -1 to give reflection across the y-axis. The formula is obtained by substituting -2x for x in the right-hand side of the equation for ƒ: y = ƒ(-2x) = (-2x)4 - 4(-2x)3 + 10 = 16x4 + 32x3 + 10. (b) The formula is

1 1 y = - ƒ(x) = - x4 + 2x3 - 5. 2 2

Exercises  1.2 Algebraic Combinations In Exercises 1 and 2, find the domains and ranges of ƒ, g, ƒ + g, and ƒ # g.

1. ƒ(x) = x, g(x) = 2x - 1

2. ƒ(x) = 2x + 1, g(x) = 2x - 1

In Exercises 3 and 4, find the domains and ranges of ƒ, g, ƒ>g, and g>ƒ. 3. ƒ(x) = 2, g(x) = x2 + 1 4. ƒ(x) = 1, g(x) = 1 + 2x

Composites of Functions 5. If ƒ(x) = x + 5 and g(x) = x2 - 3, find the following.

g(ƒ(0)) a. ƒ(g(0)) b. c. ƒ(g(x)) d. g(ƒ(x))

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e. ƒ(ƒ(-5)) f. g(g(2)) g. ƒ(ƒ(x)) h. g(g(x)) 6. If ƒ(x) = x - 1 and g(x) = 1>(x + 1), find the following. g(ƒ(1>2)) a. ƒ(g(1>2)) b. c. ƒ(g(x)) d. g(ƒ(x)) e. ƒ(ƒ(2)) f. g(g(2)) g. ƒ(ƒ(x)) h. g(g(x)) In Exercises 7–10, write a formula for ƒ ∘ g ∘ h. 7. ƒ(x) = x + 1,  g(x) = 3x,  h(x) = 4 - x 8. ƒ(x) = 3x + 4,  g(x) = 2x - 1,  h(x) = x2 3 1 9. ƒ(x) = 2x + 3,  g(x) = ,  h(x) = x x + 4 x + 2 x2 10. ƒ(x) = ,  g(x) = 2 ,  h(x) = 22 - x 3 - x x + 1

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1.2  Combining Functions; Shifting and Scaling Graphs

Let ƒ(x) = x - 3, g(x) = 2x, h(x) = x3, and j(x) = 2x. Express each of the functions in Exercises 11 and 12 as a composite involving one or more of ƒ, g, h, and j. 11. a. y = 2x - 3 b. y = 2 2x

c. y = x1>4 d. y = 4x e. y = 2(x - 3)3 f. y = (2x - 6)3

12. a. y = 2x - 3 b. y = x3>2

x . Find a function y = g(x) so that x - 2 (ƒ ∘ g)(x) = x.

19. Let ƒ(x) =

20. Let ƒ(x) = 2x3 - 4. Find a function y = g(x) so that (ƒ ∘ g)(x) = x + 2. Shifting Graphs 21. The accompanying figure shows the graph of y = - x2 shifted to two new positions. Write equations for the new graphs.

c. y = x9 d. y = x - 6 e. y = 2 2x - 3 f. y = 2x3 - 3

13. Copy and complete the following table.

(ƒ ∘ g) (x)

g(x)

ƒ(x)

a. x - 7

2x ?

b. x + 2 3x ? c. ? d.

x x - 1

e. ?

2x - 5

2x2 - 5

1 1 + x

x

x ? x - 1

y 0

−7

Position (a)

Position (b)

22. The accompanying figure shows the graph of y = x2 shifted to two new positions. Write equations for the new graphs. y Position (a)

14. Copy and complete the following table. ƒ(x)

1 a . x - 1

(ƒ ∘ g) (x)

0 x 0 ?

x - 1 x

b. ? c. ?

2x

d. 2x ?

x

4

y = −x 2

1 f. x ? x

g(x)

33

x x + 1 0x0

y = x2 3

x

0

0x0

Position (b)

15. Evaluate each expression using the given table of values:   x

- 2

- 1

0 1 2

ƒ(x) 1 0 - 2 1 2 g(x) 2 1 0 -1 0 a. ƒ(g(- 1)) b. g(ƒ(0)) c. ƒ(ƒ(-1))

−5

23. Match the equations listed in parts (a)–(d) to the graphs in the accompanying figure. y = (x - 2)2 + 2 a. y = (x - 1)2 - 4 b. y = (x + 3)2 - 2 c. y = (x + 2)2 + 2 d. y

d. g(g(2)) e. g(ƒ(- 2)) f. ƒ(g(1)) 16. Evaluate each expression using the functions ƒ(x) = 2 - x, g(x) = b

- x, x - 1,

-2 … x 6 0 0 … x … 2.

Position 2

Position 1

a. ƒ(g(0)) b. g(ƒ(3)) c. g(g(-1)) 3

d. ƒ(ƒ(2)) e. g(ƒ(0)) f. ƒ(g(1>2)) In Exercises 17 and 18, (a) write formulas for ƒ ∘ g and g ∘ ƒ and find the (b) domain and (c) range of each. 1 17. ƒ(x) = 2x + 1, g(x) = x 18. ƒ(x) = x2, g(x) = 1 - 2x

(−2, 2) Position 3

2 1

−4 −3 −2 −1 0

(2, 2) 1 2 3 Position 4

x

(−3, −2) (1, −4)

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34

Chapter 1: Functions

24. The accompanying figure shows the graph of y = - x2 shifted to four new positions. Write an equation for each new graph. y

55. The accompanying figure shows the graph of a function ƒ(x) with domain 30, 24 and range 30, 14. Find the domains and ranges of the following functions, and sketch their graphs. y

(1, 4) (−2, 3) (b)

(a) (2, 0)

1

y = f(x)

0

2

x

(−4, −1)

x

a. ƒ(x) + 2 b. ƒ(x) - 1 (c)

(d)

c. 2ƒ(x) d. -ƒ(x) e. ƒ(x + 2) f. ƒ(x - 1) g. ƒ(-x) h. -ƒ(x + 1) + 1

Exercises 25–34 tell how many units and in what directions the graphs of the given equations are to be shifted. Give an equation for the shifted graph. Then sketch the original and shifted graphs together, labeling each graph with its equation.

56. The accompanying figure shows the graph of a function g(t) with domain 3-4, 04 and range 3-3, 04. Find the domains and ranges of the following functions, and sketch their graphs. y

25. x2 + y2 = 49 Down 3, left 2 2

−4

2

26. x + y = 25 Up 3, left 4 27. y = x3 Left 1, down 1

0

−2

t

y = g(t)

28. y = x2>3 Right 1, down 1

−3

29. y = 2x Left 0.81

a. g(-t) b. -g(t)

31. y = 2x - 7 Up 7

c. g(t) + 3 d. 1 - g(t)

30. y = - 2x Right 3

32. y =

1 (x + 1) + 5 Down 5, right 1 2

33. y = 1>x Up 1, right 1 34. y = 1>x2 Left 2, down 1 Graph the functions in Exercises 35–54.

e. g(-t + 2) f. g(t - 2) g. g(1 - t) h. -g(t - 4) Vertical and Horizontal Scaling Exercises 57–66 tell by what factor and direction the graphs of the given functions are to be stretched or compressed. Give an equation for the stretched or compressed graph.

y = 29 - x 35. y = 2x + 4 36.

57. y = x2 - 1, stretched vertically by a factor of 3

37. y =  x - 2  38. y = 1 - x - 1

58. y = x2 - 1, compressed horizontally by a factor of 2

y = 1 - 2x 39. y = 1 + 2x - 1 40.

59. y = 1 +

41. y = (x + 1)2>3 42. y = (x - 8)2>3 43. y = 1 - x2>3 44. y + 4 = x2>3 3 45. y = 2 x - 1 - 1 46. y = (x + 2)3>2 + 1

47. y =

1 1 48. y = x - 2 x - 2

1 , compressed vertically by a factor of 2 x2 1 6 0. y = 1 + 2 , stretched horizontally by a factor of 3 x 61. y = 2x + 1, compressed horizontally by a factor of 4 62. y = 2x + 1, stretched vertically by a factor of 3

1 1 y = 49. y = x + 2 50. x + 2

63. y = 24 - x2, stretched horizontally by a factor of 2

1 1 52. y = 2 - 1 (x - 1)2 x 1 1 y = 5 3. y = 2 + 1 54. x (x + 1)2

65. y = 1 - x3, compressed horizontally by a factor of 3

51. y =

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64. y = 24 - x2, compressed vertically by a factor of 3

66. y = 1 - x3, stretched horizontally by a factor of 2

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35

1.3  Trigonometric Functions

Graphing In Exercises 67–74, graph each function, not by plotting points, but by starting with the graph of one of the standard functions presented in Figures 1.14–1.17 and applying an appropriate transformation.

Combining Functions 77. Assume that ƒ is an even function, g is an odd function, and both ƒ and g are defined on the entire real line (- q, q). Which of the following (where defined) are even? odd?

ƒ>g c. g>ƒ a. ƒg b.

x 67. y = - 22x + 1 68. y = 1 2 A

g2 = gg f. ƒ∘g d. ƒ2 = ƒƒ e.

y = (1 - x)3 + 2 69. y = (x - 1)3 + 2 70. 1 2 7 1. y = - 1 72. y = 2 + 1 2x x 3 73. y = - 2 x 74. y = (- 2x)2>3

75. Graph the function y = 0 x2 - 1 0 . 76. Graph the function y = 2 0 x 0 .

g. g ∘ ƒ h. ƒ ∘ ƒ i. g∘g 78. Can a function be both even and odd? Give reasons for your answer. T 79. (Continuation of Example 1.) Graph the functions ƒ(x) = 2x and g(x) = 21 - x together with their (a) sum, (b) product, (c) two differences, (d) two quotients. T 80. Let ƒ(x) = x - 7 and g(x) = x2. Graph ƒ and g together with ƒ ∘ g and g ∘ ƒ.

1.3  Trigonometric Functions This section reviews radian measure and the basic trigonometric functions.

Angles B′

Angles are measured in degrees or radians. The number of radians in the central angle A′CB′ within a circle of radius r is defined as the number of “radius units” contained in the arc s subtended by that central angle. If we denote this central angle by u when measured in radians, this means that u = s>r (Figure 1.36), or

s B u 1 A C r Un l it c irc

e

A′

C ir

cle of ra diu

s = ru

sr

Figure 1.36  The radian measure of the central angle A′CB′ is the number u = s>r. For a unit circle of radius r = 1, u is the length of arc AB that central angle ACB cuts from the unit circle.

If the circle is a unit circle having radius r = 1, then from Figure 1.36 and Equation (1), we see that the central angle u measured in radians is just the length of the arc that the angle cuts from the unit circle. Since one complete revolution of the unit circle is 360° or 2p radians, we have p radians = 180°(2)

and

180 1 radian = p (≈57.3) degrees

or

1 degree =

Table 1.1 shows the equivalence between degree and radian measures for some basic angles. Table 1.1  Angles measured in degrees and radians Degrees

−180

−135

−90

−45 0 30 45 60 90 120 135 150 180 270 360

−P

−3P 4

−P 2

−P P 0 4 6

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P 4

P 3

P 2

2P 3

3P 4

5P 6

P

3P 2

2P

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Chapter 1: Functions

An angle in the xy-plane is said to be in standard position if its vertex lies at the origin and its initial ray lies along the positive x-axis (Figure 1.37). Angles measured counterclockwise from the positive x-axis are assigned positive measures; angles measured clockwise are assigned negative measures. y

y Terminal ray Initial ray x Positive measure

Initial ray

Negative measure

Terminal ray

x

Figure 1.37  Angles in standard position in the xy-plane.

Angles describing counterclockwise rotations can go arbitrarily far beyond 2p radians or 360°. Similarly, angles describing clockwise rotations can have negative measures of all sizes (Figure 1.38). y

y

y

y 5p − 2

3p x

x

9p 4

Angle Convention: Use Radians  From now on, in this book it is assumed that all angles are measured in radians unless degrees or some other unit is stated explicitly. When we talk about the angle p>3, we mean p>3 radians (which is 60°), not p>3 degrees. We use radians because it simplifies many of the operations in calculus, and some results we will obtain involving the trigonometric functions are not true when angles are measured in degrees.

The Six Basic Trigonometric Functions

Figure 1.39  Trigonometric ratios of an acute angle.

y

u

x

O

You are probably familiar with defining the trigonometric functions of an acute angle in terms of the sides of a right triangle (Figure 1.39). We extend this definition to obtuse and negative angles by first placing the angle in standard position in a circle of radius r. We then define the trigonometric functions in terms of the coordinates of the point P(x, y) where the angle’s terminal ray intersects the circle (Figure 1.40).

y

r

x

opposite

u

P(x, y)

x

Figure 1.38  Nonzero radian measures can be positive or negative and can go beyond 2p.

hypotenuse

opp sin u = hyp adj cos u = hyp opp tan u = adj

3p − 4

r

x

y sine:  sin u = r x cosine:  cos u = r y tangent:  tan u = x

r cosecant:  csc u = y r secant:  sec u = x x cotangent:  cot u = y

These extended definitions agree with the right-triangle definitions when the angle is acute. Notice also that whenever the quotients are defined, Figure 1.40  The trigonometric functions of a general angle u are defined in terms of x, y, and r.

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sin u 1   cot u = cos u tan u 1 1 sec u =   csc u = cos u sin u tan u =

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1.3  Trigonometric Functions

p 4

"2 p 4

As you can see, tan u and sec u are not defined if x = cos u = 0. This means they are not defined if u is {p>2, {3p>2, c. Similarly, cot u and csc u are not defined for values of u for which y = 0, namely u = 0, {p, {2p, c. The exact values of these trigonometric ratios for some angles can be read from the triangles in Figure 1.41. For instance,

p 6

2

1

"3

p 3

p 2

1

p 2

p 1 = 4 22 p 1 cos = 4 22 p tan = 1 4

1

Figure 1.41  Radian angles and side lengths of two common triangles.

sin

sin

p 1 = 2 6

sin

23 p = 3 2

cos

23 p = 2 6

cos

p 1 = 3 2

tan

p = 23 3

tan

p 1 = 6 23

The CAST rule (Figure 1.42) is useful for remembering when the basic trigonometric functions are positive or negative. For instance, from the triangle in Figure 1.43, we see that y S sin pos

sin

23 2p = , 3 2

2p 1 = - , 3 2

tan

A all pos

2p = - 23. 3

3 acos 2p , sin 2pb = a− 1 , " b 2 2 3 3 y

x T tan pos

cos

C cos pos

P "3 2

Figure 1.42  The CAST rule, remembered by the statement “Calculus Activates Student Thinking,” tells which trigonometric functions are positive in each quadrant.

1

2p 3 x

1 2

Figure 1.43  The triangle for calculating the sine and cosine of 2p>3 radians. The side lengths come from the geometry of right triangles.

Using a similar method we determined the values of sin u, cos u, and tan u shown in Table 1.2. Table 1.2  Values of sin u, cos u, and tan u for selected values of u Degrees

−180

−135

−90

−45 0 30 45 60 90 120 135 150 180 270 360

−P

−3P 4

−P 2

−P P 0 4 6

P 4

-1

- 22 1 0 2 2

22

- 22 sin U 0 2 cos U tan U

2

P 3

23

2

P 2 1

2P 3

3P 4

23

22

2

- 22 22 23 22 1 1 0 1 0 - 2 2 2 2 2 2 23 0 1   -1 0 1 23   - 23 3

-1

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2

5P 6

P

3P 2P 2

1 0 -1 0 2

- 22 - 23 -1 0 1 2 2 - 23 -1 0   0 3

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Chapter 1: Functions

Periodicity and Graphs of the Trigonometric Functions When an angle of measure u and an angle of measure u + 2p are in standard position, their terminal rays coincide. The two angles therefore have the same trigonometric function values: sin (u + 2p) = sin u, tan (u + 2p) = tan u, and so on. Similarly, cos (u - 2p) = cos u, sin (u - 2p) = sin u, and so on. We describe this repeating behavior by saying that the six basic trigonometric functions are periodic. Periods of Trigonometric Functions Period P: tan (x + p) = tan x cot (x + p) = cot x sin (x + 2p) = sin x cos (x + 2p) = cos x sec (x + 2p) = sec x csc (x + 2p) = csc x

Period 2P:

Definition  A function ƒ(x) is periodic if there is a positive number p such that ƒ(x + p) = ƒ(x) for every value of x. The smallest such value of p is the period of ƒ. When we graph trigonometric functions in the coordinate plane, we usually denote the independent variable by x instead of u. Figure 1.44 shows that the tangent and cotangent functions have period p = p, and the other four functions have period 2p. Also, the symmetries in these graphs reveal that the cosine and secant functions are even and the other four functions are odd (although this does not prove those results). y y

Even

y y = cos x

cos (-x) = cos x sec (-x) = sec x

−p

= = = =

p 2

0

p 2

p

y = sin x

3p 2p 2

x

Domain: −∞ < x < ∞ Range: −1 ≤ y ≤ 1 Period: 2p (a)

Odd sin (-x) tan (-x) csc (-x) cot (-x)

-sin x -tan x -csc x -cot x

y

−p

p 2

0

p p 3p 2 2

p

p 2

3p 2p 2

x

y

x

Domain: x ≠±p , ± 3p , . . . 2 2 Range: y ≤ −1 or y ≥ 1 Period: 2p (d)

3p −p p − 2 2

p 2

p 3p 2p 2

0 p p 3p 2 2

y

y = csc x

1 −p p 0 − 2

x

Domain: x ≠±p , ± 3p , . . . 2 2 Range: −∞ < y < ∞ Period: p (c)

Domain: −∞ < x < ∞ Range: −1 ≤ y ≤ 1 Period: 2p (b)

y = sec x

1 −3p −p−p 0 2 2

y = tan x

x

Domain: x ≠ 0, ±p, ±2p, . . . Range: y ≤ −1 or y ≥ 1 Period: 2p

y = cot x

1 −p p 0 − 2

p 2

p 3p 2p 2

x

Domain: x ≠ 0, ±p, ±2p, . . . Range: −∞ < y < ∞ Period: p (f)

(e)

Figure 1.44  Graphs of the six basic trigonometric functions using radian measure. The shading for each trigonometric function indicates its periodicity. y P(cos u, sin u)

0 sin u 0

Trigonometric Identities

x 2 + y2 = 1

The coordinates of any point P(x, y) in the plane can be expressed in terms of the point’s distance r from the origin and the angle u that ray OP makes with the positive x-axis (Figure 1.40). Since x>r = cos u and y>r = sin u, we have

u 0 cos u 0 O

1

x

x = r cos u,

y = r sin u.

When r = 1 we can apply the Pythagorean theorem to the reference right triangle in Figure 1.45 and obtain the equation

Figure 1.45  The reference triangle for a general angle u.

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cos2 u + sin2 u = 1.(3)

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1.3  Trigonometric Functions

This equation, true for all values of u, is the most frequently used identity in trigonometry. Dividing this identity in turn by cos2 u and sin2 u gives

1 + tan2 u = sec2 u 1 + cot2 u = csc2 u

The following formulas hold for all angles A and B (Exercise 58).

Addition Formulas cos (A + B) = cos A cos B - sin A sin B sin (A + B) = sin A cos B + cos A sin B

(4)

There are similar formulas for cos (A - B) and sin (A - B) (Exercises 35 and 36). All the trigonometric identities needed in this book derive from Equations (3) and (4). For example, substituting u for both A and B in the addition formulas gives

Double-Angle Formulas cos 2u = cos2 u - sin2 u sin 2u = 2 sin u cos u 

(5)

Additional formulas come from combining the equations cos2 u + sin2 u = 1,

cos2 u - sin2 u = cos 2u.

We add the two equations to get 2 cos2 u = 1 + cos 2u and subtract the second from the first to get 2 sin2 u = 1 - cos 2u. This results in the following identities, which are useful in integral calculus.

Half-Angle Formulas

cos2 u =

1 + cos 2u (6) 2

sin2 u =

1 - cos 2u (7) 2

The Law of Cosines If a, b, and c are sides of a triangle ABC and if u is the angle opposite c, then

c2 = a2 + b2 - 2ab cos u.(8)

This equation is called the law of cosines.

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Chapter 1: Functions

We can see why the law holds if we introduce coordinate axes with the origin at C and the positive x-axis along one side of the triangle, as in Figure 1.46. The coordinates of A are (b, 0); the coordinates of B are (a cos u, a sin u). The square of the distance between A and B is therefore

y B(a cos u, a sin u)

c

a u

C

A(b, 0)

b

x

Figure 1.46  The square of the distance between A and B gives the law of cosines.

c2 = (a cos u - b)2 + (a sin u)2 = a2 (cos2 u + sin2 u) + b2 - 2ab cos u (++)++* 1 = a2 + b2 - 2ab cos u.

The law of cosines generalizes the Pythagorean theorem. If u = p>2, then cos u = 0 and c2 = a2 + b2.

Two Special Inequalities For any angle u measured in radians, the sine and cosine functions satisfy - 0 u 0 … sin u … 0 u 0

To establish these inequalities, we picture u as a nonzero angle in standard position (Figure 1.47). The circle in the figure is a unit circle, so 0 u 0 equals the length of the circular arc AP. The length of line segment AP is therefore less than 0 u 0 . Triangle APQ is a right triangle with sides of length

y P

u O cos u

sin u

1

Q

- 0 u 0 … 1 - cos u … 0 u 0 .

and

u

A(1, 0)

1 − cos u

x

QP = 0 sin u 0 ,

AQ = 1 - cos u.

From the Pythagorean theorem and the fact that AP 6 0 u 0 , we get

sin2 u + (1 - cos u)2 = (AP)2 … u 2.(9)

The terms on the left-hand side of Equation (9) are both positive, so each is smaller than their sum and hence is less than or equal to u 2 : Figure 1.47  From the geometry of this figure, drawn for u 7 0, we get the inequality sin2 u + (1 - cos u)2 … u 2.

sin2 u … u 2

and

(1 - cos u)2 … u 2.

By taking square roots, this is equivalent to saying that so

0 sin u 0 … 0 u 0

and

- 0 u 0 … sin u … 0 u 0

and

0 1 - cos u 0 … 0 u 0 ,

- 0 u 0 … 1 - cos u … 0 u 0 .

These inequalities will be useful in the next chapter.

Transformations of Trigonometric Graphs The rules for shifting, stretching, compressing, and reflecting the graph of a function summarized in the following diagram apply to the trigonometric functions we have discussed in this section. Vertical stretch or compression; reflection about y = d if negative

Vertical shift

y = aƒ(b(x + c)) + d Horizontal stretch or compression; reflection about x = - c if negative

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Horizontal shift

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1.3  Trigonometric Functions

The transformation rules applied to the sine function give the general sine function or sinusoid formula ƒ(x) = A sin a

2p (x - C )b + D, B

where 0 A 0 is the amplitude, 0 B 0 is the period, C is the horizontal shift, and D is the vertical shift. A graphical interpretation of the various terms is given below. y y = A sin a2p (x − C)b + D B

D+A Horizontal shift (C)

Amplitude (A)

D

D−A

This axis is the line y = D.

Vertical shift (D) This distance is the period (B).

x

0

Exercises  1.3 Radians and Degrees 1. On a circle of radius 10 m, how long is an arc that subtends a central angle of (a) 4p>5 radians? (b) 110°?

2. A central angle in a circle of radius 8 is subtended by an arc of length 10p. Find the angle’s radian and degree measures. 3. You want to make an 80° angle by marking an arc on the perimeter of a 12-in.-diameter disk and drawing lines from the ends of the arc to the disk’s center. To the nearest tenth of an inch, how long should the arc be? 4. If you roll a 1-m-diameter wheel forward 30 cm over level ground, through what angle will the wheel turn? Answer in radians (to the nearest tenth) and degrees (to the nearest degree). Evaluating Trigonometric Functions 5. Copy and complete the following table of function values. If the function is undefined at a given angle, enter “UND.” Do not use a calculator or tables.

U

-3P , 2

−P

−2P , 3 0 P , 2

3P , 4

sin u cos u tan u cot u sec u csc u 6. Copy and complete the following table of function values. If the function is undefined at a given angle, enter “UND.” Do not use a calculator or tables.

M01_THOM9580_03_SE_C01_015-064.indd 41

−P , 6

P , 4

5P , 6

sin u cos u tan u cot u sec u csc u

In Exercises 7–12, one of sin x, cos x, and tan x is given. Find the other two if x lies in the specified interval. 7. sin x = 9. cos x = 11. tan x =

U

−P , 3

p p 3 , x∊ c , p d 8. tan x = 2, x∊ c 0, d 2 2 5

p 5 p 1 , x∊ c - , 0 d 10. cos x = - , x∊ c , p d 3 2 13 2

3p 3p 1 1 , x∊ c p, d 12. sin x = - , x∊ c p, d 2 2 2 2

Graphing Trigonometric Functions Graph the functions in Exercises 13–22. What is the period of each function?

sin (x>2) 13. sin 2x 14. px 15. cos px 16. cos 2 px 17. -sin 18. -cos 2px 3 19. cos ax -

p p b 20. sin ax + b 2 6

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42

Chapter 1: Functions

21. sin ax -

p 2p b + 1 22. cos ax + b - 2 4 3

Graph the functions in Exercises 23–26 in the ts-plane (t-axis horizontal, s-axis vertical). What is the period of each function? What symmetries do the graphs have? 23. s = cot 2t 24. s = -tan pt 25. s = sec a

pt t b 26. s = csc a b 2 2

T 27.  a. Graph y = cos x and y = sec x together for -3p>2 … x … 3p>2. Comment on the behavior of sec x in relation to the signs and values of cos x. b. Graph y = sin x and y = csc x together for -p … x … 2p. Comment on the behavior of csc x in relation to the signs and values of sin x. T 28. Graph y = tan x and y = cot x together for - 7 … x … 7. Comment on the behavior of cot x in relation to the signs and values of tan x.

Solving Trigonometric Equations For Exercises 51–54, solve for the angle u, where 0 … u … 2p. 3 51. sin2 u = 52. sin2 u = cos2 u 4

53. sin 2u - cos u = 0 54. cos 2u + cos u = 0 Theory and Examples 55. The tangent sum formula  The standard formula for the tangent of the sum of two angles is

tan(A + B) = Derive the formula.

56. (Continuation of Exercise 55.) Derive a formula for tan (A - B). 57. Apply the law of cosines to the triangle in the accompanying figure to derive the formula for cos (A - B). y 1

29. Graph y = sin x and y = : sin x ; together. What are the domain and range of : sin x ; ?

30. Graph y = sin x and y = < sin x = together. What are the domain and range of < sin x = ?

A

33. sin ax +

p p b = cos x 34. sin ax - b = - cos x 2 2

36. sin (A - B) = sin A cos B - cos A sin B 37. What happens if you take B = A in the trigonometric identity cos (A - B) = cos A cos B + sin A sin B? Does the result agree with something you already know? 38. What happens if you take B = 2p in the addition formulas? Do the results agree with something you already know? In Exercises 39–42, express the given quantity in terms of sin x and cos x. sin (2p - x) 39. cos (p + x) 40. 3p 3p - xb 42. + xb cos a 2 2

43. Evaluate sin

44. Evaluate cos 45. Evaluate cos

7p p p as sin a + b. 12 4 3

11p p 2p as cos a + b. 12 4 3

x

1

58. a.  Apply the formula for cos (A - B) to the identity sin u = cos a

5p 3p 48. cos2 4 7. cos2 8 12 3p 5p 50. sin2 12 8

M01_THOM9580_03_SE_C01_015-064.indd 42

p - ub to obtain the addition formula for sin (A + B). 2

b. Derive the formula for cos (A + B) by substituting -B for B in the formula for cos (A - B) from Exercise 35. 59. A triangle has sides a = 2 and b = 3 and angle C = 60°. Find the length of side c. 60. A triangle has sides a = 2 and b = 3 and angle C = 40°. Find the length of side c. 61. The law of sines  The law of sines says that if a, b, and c are the sides opposite the angles A, B, and C in a triangle, then sin C sin A sin B a = b = c . Use the accompanying figures and the identity sin(p - u) = sin u, if required, to derive the law. A

p 5p . 46. Evaluate sin . 12 12

Using the Half-Angle Formulas Find the function values in Exercises 47–50.

49. sin2

0

p p b = sin x 32. cos ax + b = -sin x 2 2

35. cos (A - B) = cos A cos B + sin A sin B (Exercise 57 provides a different derivation.)

41. sin a

B

1

Using the Addition Formulas Use the addition formulas to derive the identities in Exercises 31–36.

31. cos ax -

tan A + tan B . 1 - tan A tan B

c

B

h

a

A

c

b

C

B

a

b

h

C

62. A triangle has sides a = 2 and b = 3 and angle C = 60° (as in Exercise 59). Find the sine of angle B using the law of sines.

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1.4  Graphing with Software

63. A triangle has side c = 2 and angles A = p>4 and B = p>3. Find the length a of the side opposite A. T 64. The approximation sin x ? x  It is often useful to know that, when x is measured in radians, sin x ≈ x for numerically small values of x. In Section 3.11, we will see why the approximation holds. The approximation error is less than 1 in 5000 if 0 x 0 6 0.1.  a. With your grapher in radian mode, graph y = sin x and y = x together in a viewing window about the origin. What do you see happening as x nears the origin?

b. With your grapher in degree mode, graph y = sin x and y = x together about the origin again. How is the picture different from the one obtained with radian mode? General Sine Curves For

ƒ(x) = A sin a

2p (x - C)b + D, B

identify A, B, C, and D for the sine functions in Exercises 65–68 and sketch their graphs. 1 1 65. y = 2 sin (x + p) - 1 66. y = sin (px - p) + 2 2 p 2pt 2 L 1 y = 67. y = - p sin a tb + p 68. sin , L 7 0 L 2 2p

Computer Explorations In Exercises 69–72, you will explore graphically the general sine function

ƒ(x) = A sina

43

69. The period B  Set the constants A = 3, C = D = 0.  a. Plot ƒ(x) for the values B = 1, 3, 2p, 5p over the interval -4p … x … 4p. Describe what happens to the graph of the general sine function as the period increases. b. What happens to the graph for negative values of B? Try it with B = -3 and B = -2p. 70. The horizontal shift C  Set the constants A = 3, B = 6, D = 0.  a. Plot ƒ(x) for the values C = 0, 1, and 2 over the interval -4p … x … 4p. Describe what happens to the graph of the general sine function as C increases through positive values. b. What happens to the graph for negative values of C?  c. What smallest positive value should be assigned to C so the graph exhibits no horizontal shift? Confirm your answer with a plot. 71. The vertical shift D  Set the constants A = 3, B = 6, C = 0.  a. Plot ƒ(x) for the values D = 0, 1, and 3 over the interval -4p … x … 4p. Describe what happens to the graph of the general sine function as D increases through positive values. b. What happens to the graph for negative values of D? 72. The amplitude A  Set the constants B = 6, C = D = 0.  a. Describe what happens to the graph of the general sine function as A increases through positive values. Confirm your answer by plotting ƒ(x) for the values A = 1, 5, and 9. b. What happens to the graph for negative values of A?

2p (x - C)b + D B

as you change the values of the constants A, B, C, and D. Use a CAS or computer grapher to perform the steps in the exercises.

1.4  Graphing with Software Today a number of hardware devices, including computers, calculators, and smartphones, have graphing applications based on software that enables us to graph very complicated functions with high precision. Many of these functions could not otherwise be easily graphed. However, some care must be taken when using such graphing software, and in this section we address some of the issues that may be involved. In Chapter 4 we will see how calculus helps us determine that we are accurately viewing all the important features of a function’s graph.

Graphing Windows When using software for graphing, a portion of the graph is displayed in a display or viewing window. Depending on the software, the default window may give an incomplete or misleading picture of the graph. We use the term square window when the units or scales used on both axes are the same. This term does not mean that the display window itself is square (usually it is rectangular), but instead it means that the x-unit is the same length as the y-unit. When a graph is displayed in the default mode, the x-unit may differ from the y-unit of scaling in order to capture essential features of the graph. This difference in scaling can cause visual distortions that may lead to erroneous interpretations of the function’s behavior.

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Chapter 1: Functions

Some graphing software allows us to set the viewing window by specifying one or both of the intervals, a … x … b and c … y … d, and it may allow for equalizing the scales used for the axes as well. The software selects equally spaced x-values in 3 a, b4 and then plots the points (x, ƒ(x)). A point is plotted if and only if x lies in the domain of the function and ƒ(x) lies within the interval 3 c, d4 . A short line segment is then drawn between each plotted point and its next neighboring point. We now give illustrative examples of some common problems that may occur with this procedure.

EXAMPLE 1   Graph the function ƒ(x) = x3 - 7x2 + 28 in each of the following

display or viewing windows:

(a) 3 -10, 104 by 3 -10, 104   (b)  3 -4, 44 by 3 -50, 104   (c)  3 -4, 104 by 3 -60, 604

Solution (a) We select a = -10, b = 10, c = -10, and d = 10 to specify the interval of x-values and the range of y-values for the window. The resulting graph is shown in Figure 1.48a. It appears that the window is cutting off the bottom part of the graph and that the interval of x-values is too large. Let’s try the next window. 10

10

4

−4 10

−10

60

10

−4

−10

−50

(a)

(b)

−60 (c)

Figure 1.48  The graph of ƒ(x) = x3 - 7x2 + 28 in different viewing windows. Selecting a window that gives a clear picture of a graph is often a trial-and-error process (Example 1). The default window used by the software may automatically display the graph in (c).

(b) We see some new features of the graph (Figure 1.48b), but the top is missing and we need to view more to the right of x = 4 as well. The next window should help. (c) Figure 1.48c shows the graph in this new viewing window. Observe that we get a more complete picture of the graph in this window, and it is a reasonable graph of a third-degree polynomial.

EXAMPLE 2   When a graph is displayed, the x-unit may differ from the y-unit, as in the graphs shown in Figures 1.48b and 1.48c. The result is distortion in the picture, which may be misleading. The display window can be made square by compressing or stretching the units on one axis to match the scale on the other, giving the true graph. Many software systems have built-in options to make the window “square.” If yours does not, you may have to bring to your viewing some foreknowledge of the true picture. Figure 1.49a shows the graphs of the perpendicular lines y = x and y = -x + 3 22, together with the semicircle y = 29 - x2, in a nonsquare 3 -4, 44 by 3 -6, 84 display window. Notice the distortion. The lines do not appear to be perpendicular, and the semicircle appears to be elliptical in shape. Figure 1.49b shows the graphs of the same functions in a square window in which the x-units are scaled to be the same as the y-units. Notice that the scaling on the x-axis for Figure 1.49a has been compressed in Figure 1.49b to make the window square. Figure 1.49c gives an enlarged view of Figure 1.49b with a square 3 -3, 34 by 3 0, 44 window.

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1.4  Graphing with Software

8

4

4

4

−4

45

6

−6

3

−3 −6 (a)

0 (c)

−4 (b)

Figure 1.49  Graphs of the perpendicular lines y = x and y = - x + 3 22 and of the semicircle y = 29 - x2 appear distorted (a) in a nonsquare window, but clear (b) and (c) in square windows (Example 2). Some software may not provide options for the views in (b) or (c).

If the denominator of a rational function is zero at some x-value within the viewing window, graphing software may produce a steep near-vertical line segment from the top to the bottom of the window. Example 3 illustrates steep line segments. Sometimes the graph of a trigonometric function oscillates very rapidly. When graphing software plots the points of the graph and connects them, many of the maximum and minimum points are actually missed. The resulting graph is then very misleading.

EXAMPLE 3   Graph the function ƒ(x) = sin 100x. Solution  Figure 1.50a shows the graph of ƒ in the viewing window 3 -12, 124 by 3 -1, 14 . We see that the graph looks very strange because the sine curve should oscillate periodically between -1 and 1. This behavior is not exhibited in Figure 1.50a. We might experiment with a smaller viewing window, say 3 -6, 64 by 3 -1, 14 , but the graph is not better (Figure 1.50b). The difficulty is that the period of the trigonometric function y = sin 100x is very small (2p>100 ≈ 0.063). If we choose the much smaller viewing window 3 -0.1, 0.14 by 3 -1, 14 we get the graph shown in Figure 1.50c. This graph reveals the expected oscillations of a sine curve. 1

1

12

−12

−1 (a)

1

6

−6

0.1

−0.1

−1 (b)

−1 (c)

Figure 1.50  Graphs of the function y = sin 100x in three viewing windows. Because the period is 2p>100 ≈ 0.063, the smaller window in (c) best displays the true aspects of this rapidly oscillating function (Example 3).

EXAMPLE 4    Graph the function y = cos x + 1 sin 200x. 200

Solution  In the viewing window 3 -6, 64 by 3 -1, 14 the graph appears much like the cosine function with some very small sharp wiggles on it (Figure 1.51a). We get a better look when we significantly reduce the window to 3 -0.2, 0.24 by 3 0.97, 1.014 , obtaining the graph in Figure 1.51b. We now see the small but rapid oscillations of the second term, (1>200) sin 200x, added to the comparatively larger values of the cosine curve.

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46

Chapter 1: Functions 1.01

1

6

−6

0.2

−0.2 0.97 (b)

−1 (a)

Figure 1.51  In (b) we see a close-up view of the function 1 sin 200x graphed in (a). The term cos x clearly dominates y = cos x + 200 1 the second term, sin 200x, which produces the rapid oscillations along the 200 cosine curve. Both views are needed for a clear idea of the graph (Example 4).

Obtaining a Complete Graph Some graphing software will not display the portion of a graph for ƒ(x) when x 6 0. Usually that happens because of the algorithm the software is using to calculate the function values. Sometimes we can obtain the complete graph by defining the formula for the function in a different way, as illustrated in the next example.

EXAMPLE 5   Graph the function y = x1>3. Solution  Some graphing software displays the graph shown in Figure 1.52a. When we 3 compare it with the graph of y = x1>3 = 2 x in Figure 1.17, we see that the left branch for x 6 0 is missing. The reason the graphs differ is that the software algorithm calculates x1>3 as e(1>3)ln x. Since the logarithmic function is not defined for negative values of x, the software can produce only the right branch, where x 7 0. (Logarithmic and exponential functions are introduced in the next two sections.) 2

2

3

−3

3

−3

−2 (a)

−2 (b)

Figure 1.52  The graph of y = x1>3 is missing the left branch in (a). In (b) we x # 1>3 0 x 0 , obtaining both branches. (See Example 5.) graph the function ƒ(x) = 0x0

To obtain the full picture showing both branches, we can graph the function ƒ(x) =

x

0x0

# 0 x 0 1>3.

This function equals x1>3 except at x = 0 (where ƒ is undefined, although 01>3 = 0). A graph of ƒ is displayed in Figure 1.52b.

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1.5  Exponential Functions

47

Exercises  1.4 Choosing a Viewing Window T In Exercises 1–4, use graphing software to determine which of the given viewing windows displays the most appropriate graph of the specified function.

1. ƒ(x) = x4 - 7x2 + 6x a. 3- 1, 14 by 3- 1, 14

b. 3- 2, 24 by 3-5, 54

c. 3- 10, 104 by 3- 10, 104 d. 3- 5, 54 by 3-25, 154

2. ƒ(x) = x3 - 4x2 - 4x + 16

3- 3, 34 by 3-10, 104 a. 3- 1, 14 by 3- 5, 54 b.

3- 20, 204 by 3-100, 1004 c. 3- 5, 54 by 3- 10, 204 d.

3. ƒ(x) = 5 + 12x - x3

3- 5, 54 by 3-10, 104 a. 3- 1, 14 by 3- 1, 14 b. 3- 4, 54 by 3-15, 254 c. 3- 4, 44 by 3- 20, 204 d.

4. ƒ(x) = 25 + 4x - x2

3- 2, 64 by 3-1, 44 a. 3- 2, 24 by 3- 2, 24 b. c. 3- 3, 74 by 30, 104

d. 3- 10, 104 by 3-10, 104

Finding a Viewing Window T In Exercises 5–30, find an appropriate graphing software viewing window for the given function and use it to display its graph. The window should give a picture of the overall behavior of the function. There is more than one choice, but incorrect choices can miss important aspects of the function. x3 x2 5. ƒ(x) = x4 - 4x3 + 15 6. ƒ(x) = - 2x + 1 3 2

11. y = 2x - 3x2>3 12. y = x1>3(x2 - 8) y = x2>3(5 - x) 13. y = 5x2>5 - 2x 14. y = 0 x2 - x 0 15. y = 0 x2 - 1 0 16. x + 3 1 17. y = 18. y = 1 x + 2 x + 3 19. ƒ(x) =

x2 + 2 x2 - 1 20. ƒ(x) = 2 2 x + 1 x + 1

21. ƒ(x) =

x - 1 8 22. ƒ(x) = 2 x2 - x - 6 x - 9

23. ƒ(x) =

6x2 - 15x + 6 x2 - 3 24. ƒ(x) = x - 2 4x2 - 10x

25. y = sin 250x 26. y = 3 cos 60x x x 1 27. y = cos a b 28. y = sin a b 10 10 50 29. y = x +

1 1 sin 30x 30. y = x2 + cos 100x 10 50

Use graphing software to graph the functions specified in Exercises 31–36. Select a viewing window that reveals the key features of the function. 31. Graph the lower half of the circle defined by the equation x2 + 2x = 4 + 4y - y2. 32. Graph the upper branch of the hyperbola y2 - 16x2 = 1. 33. Graph four periods of the function ƒ(x) = - tan 2x. x 34. Graph two periods of the function ƒ(x) = 3 cot + 1. 2

ƒ(x) = 4x3 - x4 7. ƒ(x) = x5 - 5x4 + 10 8.

35. Graph the function ƒ(x) = sin 2x + cos 3x.

9. ƒ(x) = x 29 - x2 10. ƒ(x) = x2(6 - x3)

36. Graph the function ƒ(x) = sin3 x.

1.5 Exponential Functions Exponential functions are among the most important in mathematics and occur in a wide variety of applications, including interest rates, radioactive decay, population growth, the spread of a disease, consumption of natural resources, the earth’s atmospheric pressure, temperature change of a heated object placed in a cooler environment, and the dating of fossils. In this section we introduce these functions informally, using an intuitive approach. We give a rigorous development of them in Chapter 7, based on important calculus ideas and results.

Exponential Behavior

Don’t confuse the exponential 2x with the power function x2 . In the exponential, the variable x is in the exponent, whereas the variable x is the base in the power function.

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When a positive quantity P doubles, it increases by a factor of 2 and the quantity becomes 2P. If it doubles again, it becomes 2(2P) = 22P, and a third doubling gives 2(22P) = 23P. Continuing to double in this fashion leads us to consider the function ƒ(x) = 2x. We call this an exponential function because the variable x appears in the exponent of 2x. Functions such as g(x) = 10 x and h(x) = (1>2)x are other examples of exponential functions. In general, if a ≠ 1 is a positive constant, the function ƒ(x) = ax, a > 0 is the exponential function with base a.

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Chapter 1: Functions

EXAMPLE 1   In 2014, \$100 is invested in a savings account, where it grows by accruing interest that is compounded annually (once a year) at an interest rate of 5.5%. Assuming no additional funds are deposited to the account and no money is withdrawn, give a formula for a function describing the amount A in the account after x years have elapsed. Solution  If P = 100, at the end of the first year the amount in the account is the original amount plus the interest accrued, or P + a

5.5 b P = (1 + 0.055)P = (1.055)P. 100

At the end of the second year the account earns interest again and grows to (1 + 0.055) # (1.055P) = (1.055)2P = 100 # (1.055)2.   P

= 100

Continuing this process, after x years the value of the account is A = 100 # (1.055)x.

This is a multiple of the exponential function with base 1.055. Table 1.3 shows the amounts accrued over the first four years. Notice that the amount in the account each year is always 1.055 times its value in the previous year. Table 1.3  Savings account growth Year

Amount (dollars)

Increase (dollars)

2014 100   2015 100(1.055) = 105.50 5.50 2016 100(1.055)2 = 111.30 5.80 2017 100(1.055)3 = 117.42 6.12 2018 100(1.055)4 = 123.88 6.46 In general, the amount after x years is given by P(1 + r)x, where r is the interest rate (expressed as a decimal). For integer and rational exponents, the value of an exponential function ƒ(x) = ax is obtained arithmetically as follows. If x = n is a positive integer, the number an is given by multiplying a by itself n times:

an = (++)++* a # a # g # a. n factors

If x = 0, then a0 = 1, and if x = -n for some positive integer n, then 1 1 n n = aab . a

a-n =

If x = 1>n for some positive integer n, then

n

a1>n = 2a,

which is the positive number that when multiplied by itself n times gives a. If x = p>q is any rational number, then q

ap>q = 2ap =

q p 12 a2 .

If x is irrational, the meaning of ax is not so clear, but its value can be defined by considering values for rational numbers that get closer and closer to x. This informal approach is based on the graph of the exponential function, as we are about to describe. In Chapter 7 we define the meaning in a rigorous way.

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1.5  Exponential Functions

We displayed the graphs of several exponential functions in Section 1.1, and show y –x them again in Figure 1.53. These graphs indicate the values of the exponential functions y = 10 12 for all real inputs x. The value at an irrational number x is chosen so that the graph of ax has no “holes” 10 or “jumps.” Of course, these words are not mathematical terms, but they do convey the informal idea. We mean that the value of ax , when x is irrational, is chosen so 8 that the function ƒ(x) = ax is continuous, a notion that will be carefully explored in the –x y= 3 chapter. 6This choice ensures the graph retains its increasing behavior when a 7 1, or next decreasing behavior when 0 6 a 6 1 (see Figure 1.53). 4 Arithmetically, the graphical idea can be described in the following way, using the 2 y =exponential 2 –x function ƒ(x) = 2x as an illustration. Any particular irrational number, say x x −1 = 2 3, has 0a decimal expansion −0.5 0.5 1

y y=

10 x

12 10 8 6

y = 3x

4 2

y = 2x x

−1 −0.5 0 0.5 1 (a) y = 2 x, y = 3 x, y = 10 x y

We then consider the list of numbers, given as follows in the order of taking more and more digits in the decimal expansion,

y = 10 –x

y=

y = 3x

10

We know the meaning of each number in list (1) because the successive decimal approximations to 23 given by 1, 1.7, 1.73, 1.732, and so on, are all rational numbers. As these decimal approximations get closer and closer to 23, it seems reasonable that the list of numbers in (1) gets closer and closer to some fixed number, which we specify to be 223 . Table 1.4 illustrates how taking better approximations to 23 gives better approximations to the number 223 ≈ 3.321997086. It is the completeness property of the real numbers (discussed briefly in Appendix 7) which guarantees that this procedure gives a single number we define to be 223 (although it is beyond the scope of this text to give a proof). In a similar way, we can identify the number 2x (or ax, a 7 0) for any irrational x. By identifying the number ax for both rational and irrational x, we eliminate any “holes” or “gaps” in the graph of ax . In practice you can use a calculator to find the number ax for irrational x by taking successive decimal approximations to x and creating a table similar to Table 1.4. Exponential functions obey the familiar rules of exponents listed below. It is easy to check these rules using algebra when the exponents are integers or rational numbers. We prove them for all real exponents in Chapters 4 and 7.

0 0.5 1 = 3 x, y = 10 x

6 4

y = 2x

2

y = 2 –x x

−1 −0.5 0 0.5 1 (b) y = 2 –x, y = 3 –x, y = 10 –x

x

Figure 1.53  Graphs of exponential functions.

Table 1.4  Values of 223 for

rational r closer and closer to 23

21, 21.7, 21.73, 21.732, 21.7320, 21.73205, c.(1)

12

8 3 –x

23 = 1.732050808 c.

(b) y = 2 –x, y = 3 –x, y = 10 –x

y y = 10 x

49

r

2r

1.0 2.000000000 1.7 3.249009585 1.73 3.317278183 1.732 3.321880096 1.7320 3.321880096 1.73205 3.321995226 1.732050 3.321995226 1.7320508 3.321997068 1.73205080 3.321997068 1.732050808 3.321997086

Rules for Exponents If a 7 0 and b 7 0, the following rules hold true for all real numbers x and y. 1. ax # ay = ax + y 3. ( ax ) y ax 5. x = b

ax = ax - y ay 4. ax # bx = (ab)x 2.

= ( ay ) x = axy a x a b b

EXAMPLE 2   We illustrate using the rules for exponents to simplify numerical expressions. 1. 31.1 # 30.7 = 31.1 + 0.7 = 31.8

2.

1 210 23 210

=

1 210 23 - 1

3. 1 522 222 = 522 4. 7p # 8p = (56)p

# 22

4 1>2 41>2 2 5. a b = 1>2 = 9 3 9

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=

1 210 22

= 52 = 25

Rule 1

= 10

Rule 2 Rule 3 Rule 4 Rule 5

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Chapter 1: Functions

The Natural Exponential Function e x The most important exponential function used for modeling natural, physical, and economic phenomena is the natural exponential function, whose base is the special number e. The number e is irrational, and its value is 2.718281828 to nine decimal places. (In Section 3.8 we will see a way to calculate the value of e.) It might seem strange that we would use this number for a base rather than a simple number like 2 or 10. The advantage in using e as a base is that it simplifies many of the calculations in calculus. If you look at Figure 1.53a you can see that the graphs of the exponential functions y = ax get steeper as the base a gets larger. This idea of steepness is conveyed by the slope of the tangent line to the graph at a point. Tangent lines to graphs of functions are defined precisely in the next chapter, but intuitively the tangent line to the graph at a point is a line that just touches the graph at the point, like a tangent to a circle. Figure 1.54 shows the slope of the graph of y = ax as it crosses the y-axis for several values of a. Notice that the slope is exactly equal to 1 when a equals the number e. The slope is smaller than 1 if a 6 e, and larger than 1 if a 7 e. This is the property that makes the number e so useful in calculus: The graph of y = ex has slope 1 when it crosses the y-axis. y

y

y = 2x

y

y = ex

m ≈ 0.7

m ≈ 1.1

m=1

1

1 x

0

1 x

0

x

0 (c)

(b)

(a)

y = 3x

Figure 1.54  Among the exponential functions, the graph of y = ex has the property that the slope m of the tangent line to the graph is exactly 1 when it crosses the y-axis. The slope is smaller for a base less than e, such as 2x , and larger for a base greater than e, such as 3x.

Exponential Growth and Decay The exponential functions y = ekx , where k is a nonzero constant, are frequently used for modeling exponential growth or decay. The function y = y0 ekx is a model for exponential growth if k 7 0 and a model for exponential decay if k 6 0. Here y0 represents a constant. An example of exponential growth occurs when computing interest compounded continuously modeled by y = P # ert, where P is the initial monetary investment, r is the interest rate as a decimal, and t is time in units consistent with r. An example of exponential decay is the -4 model y = A # e-1.2 * 10 t , which represents how the radioactive isotope carbon-14 decays over time. Here A is the original amount of carbon-14 and t is the time in years. Carbon-14 decay is used to date the remains of dead organisms such as shells, seeds, and wooden artifacts. Figure 1.55 shows graphs of exponential growth and exponential decay. y y

20 1.4

15

1

10

y = e1.5x

y = e –1.2 x

0.6

5

0.2 −1 −0.5

0

0.5 (a)

1

1.5

2

x

−0.5

0

0.5

1

1.5

2

2.5

3

x

(b)

Figure 1.55  Graphs of (a) exponential growth, k = 1.5 7 0, and (b) exponential decay, k = -1.2 6 0.

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1.5  Exponential Functions

EXAMPLE 3   Investment companies often use the model y = Pert in calculating the growth of an investment. Use this model to track the growth of \$100 invested in 2014 at an annual interest rate of 5.5%.

Solution  Let t = 0 represent 2014, t = 1 represent 2015, and so on. Then the exponential growth model is y(t) = Pert , where P = 100 (the initial investment), r = 0.055 (the annual interest rate expressed as a decimal), and t is time in years. To predict the amount in the account in 2018, after four years have elapsed, we take t = 4 and calculate

y(4) = 100e0.055(4) = 100e0.22 = 124.61.   Nearest cent using calculator

This compares with \$123.88 in the account when the interest is compounded annually from Example 1.

EXAMPLE 4   Laboratory experiments indicate that some atoms emit a part of their mass as radiation, with the remainder of the atom re-forming to make an atom of some new element. For example, radioactive carbon-14 decays into nitrogen; radium eventually decays into lead. If y0 is the number of radioactive nuclei present at time zero, the number still present at any later time t will be y = y0 e-rt,

r 7 0.

The number r is called the decay rate of the radioactive substance. (We will see how this formula is obtained in Section 7.2.) For carbon-14, the decay rate has been determined experimentally to be about r = 1.2 * 10-4 when t is measured in years. Predict the percent of carbon-14 present after 866 years have elapsed. Solution  If we start with an amount y0 of carbon-14 nuclei, after 866 years we are left with the amount

-4

y(866) = y0 e(-1.2 * 10 ≈ (0.901)y0.

) (866)

Calculator evaluation

That is, after 866 years, we are left with about 90% of the original amount of carbon-14, so about 10% of the original nuclei have decayed. In Example 7 in the next section, you will see how to find the number of years required for half of the radioactive nuclei present in a sample to decay (called the half-life of the substance). You may wonder why we use the family of functions y = ekx for different values of the constant k instead of the general exponential functions y = ax. In the next section, we show that the exponential function ax is equal to ekx for an appropriate value of k. So the formula y = ekx covers the entire range of possibilities, and we will see that it is easier to use.

Exercises  1.5 Sketching Exponential Curves In Exercises 1–6, sketch the given curves together in the appropriate coordinate plane and label each curve with its equation.

In each of Exercises 7–10, sketch the shifted exponential curves.

1. y = 2x, y = 4x, y = 3-x, y = (1>5)x

9. y = 1 - ex and y = 1 - e-x

2. y = 3x, y = 8x, y = 2-x, y = (1>4)x 3. y = 2-t and y = - 2t 4. y = 3-t and y = -3t

7. y = 2x - 1 and y = 2-x - 1 8. y = 3x + 2 and y = 3-x + 2 10. y = -1 - ex and y = -1 - e-x

5. y = ex and y = 1>ex 6. y = - ex and y = -e-x

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Chapter 1: Functions

Applying the Laws of Exponents Use the laws of exponents to simplify the expressions in Exercises 11–20.

91>3 # 91>6 11. 497 # 49-6.5 12. 13. 15. 17.

4.2

5>3

3 4 14. 43.7 32>3

1 641>12 22 16. 1 1322 222>2 223 # 723 18. 1 23 21>2 # 1 212 21>2

19. a

4 26 2 b 20. a b 3 22

2

Composites Involving Exponential Functions Find the domain and range for each of the functions in Exercises 21–24. 1 21. ƒ(x) = 22. g(t) = cos (e-t) 2 + ex

3 23. g(t) = 21 + 3-t 24. ƒ(x) = 1 - e2x Applications T In Exercises 25–28, use graphs to find approximate solutions. ex = 4 25. 2x = 5 26.

27. 3x - 0.5 = 0 28. 3 - 2-x = 0 T In Exercises 29–36, use an exponential model and a graphing calculator to estimate the answer in each problem. 29. Population growth  The population of Knoxville is 500,000 and is increasing at the rate of 3.75% each year. Approximately when will the population reach 1 million?

30. Population growth  The population of Silver Run in the year 1890 was 6250. Assume the population increased at a rate of 2.75% per year. a. Estimate the population in 1915 and 1940. b. Approximately when did the population reach 50,000? 31. Radioactive decay  The half-life of phosphorus-32 is about 14 days. There are 6.6 grams present initially. a. Express the amount of phosphorus-32 remaining as a function of time t. b. When will there be 1 gram remaining? 32. If Jean invests \$2300 in a retirement account with a 6% interest rate compounded annually, how long will it take until Jean’s account has a balance of \$4150? 33. Doubling your money  Determine how much time is required for an investment to double in value if interest is earned at the rate of 6.25% compounded annually. 34. Tripling your money  Determine how much time is required for an investment to triple in value if interest is earned at the rate of 5.75% compounded continuously. 35. Cholera bacteria  Suppose that a colony of bacteria starts with 1 bacterium and doubles in number every half hour. How many bacteria will the colony contain at the end of 24 hr? 36. Eliminating a disease  Suppose that in any given year the number of cases of a disease is reduced by 20%. If there are 10,000 cases today, how many years will it take a. to reduce the number of cases to 1000? b. to eliminate the disease; that is, to reduce the number of cases to less than 1?

1.6 Inverse Functions and Logarithms A function that undoes, or inverts, the effect of a function ƒ is called the inverse of ƒ. Many common functions, though not all, are paired with an inverse. In this section we present the natural logarithmic function y = ln x as the inverse of the exponential function y = ex , and we also give examples of several inverse trigonometric functions.

One-to-One Functions A function is a rule that assigns a value from its range to each element in its domain. Some functions assign the same range value to more than one element in the domain. The function ƒ(x) = x2 assigns the same value, 1, to both of the numbers -1 and +1; the sines of p>3 and 2p>3 are both 23>2. Other functions assume each value in their range no more than once. The square roots and cubes of different numbers are always different. A function that has distinct values at distinct elements in its domain is called one-to-one. These functions take on any one value in their range exactly once. Definition  A function ƒ(x) is one-to-one on a domain D if ƒ(x1) ≠ ƒ(x2) whenever x1 ≠ x2 in D.

EXAMPLE 1   Some functions are one-to-one on their entire natural domain. Other functions are not one-to-one on their entire domain, but by restricting the function to a smaller domain we can create a function that is one-to-one. The original and restricted functions are not the same functions, because they have different domains. However, the

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1.6  Inverse Functions and Logarithms

y

y

x

0

two functions have the same values on the smaller domain, so the original function is an extension of the restricted function from its smaller domain to the larger domain.

y = "x

y = x3

x

0

(a) One-to-one: Graph meets each horizontal line at most once. y

y = x2

Same y-value y Same y-value

1 −1

0

0.5 1

x

p 6

5p 6

x

(a) ƒ(x) = 2x is one-to-one on any domain of nonnegative numbers because 2x1 ≠ 2x2 whenever x1 ≠ x2. (b) g(x) = sin x is not one-to-one on the interval 3 0, p4 because sin (p>6) = sin (5p>6). In fact, for each element x1 in the subinterval 3 0, p>2) there is a corresponding element x2 in the subinterval (p>2, p] satisfying sin x1 = sin x2, so distinct elements in the domain are assigned to the same value in the range. The sine function is one-toone on 3 0, p>24 , however, because it is an increasing function on 3 0, p>24 giving distinct outputs for distinct inputs. The graph of a one-to-one function y = ƒ(x) can intersect a given horizontal line at most once. If the function intersects the line more than once, it assumes the same y-value for at least two different x-values and is therefore not one-to-one (Figure 1.56).

y = sin x (b) Not one-to-one: Graph meets one or more horizontal lines more than once.

Figure 1.56  (a) y = x3 and y = 1x are one-to-one on their domains (- q, q) and 30, q). (b) y = x2 and y = sin x are not one-to-one on their domains (- q, q).

The Horizontal Line Test for One-to-One Functions A function y = ƒ(x) is one-to-one if and only if its graph intersects each horizontal line at most once.

Inverse Functions Since each output of a one-to-one function comes from just one input, the effect of the function can be inverted to send an output back to the input from which it came.

Definition  Suppose that ƒ is a one-to-one function on a domain D with range R. The inverse function ƒ -1 is defined by ƒ -1(b) = a if ƒ(a) = b. The domain of ƒ -1 is R and the range of ƒ -1 is D. Caution  Do not confuse the inverse function ƒ -1 with the reciprocal function 1>ƒ.

The symbol ƒ -1 for the inverse of ƒ is read “ƒ inverse.” The “-1” in ƒ -1 is not an exponent; ƒ -1(x) does not mean 1>ƒ(x). Notice that the domains and ranges of ƒ and ƒ -1 are interchanged.

EXAMPLE 2   Suppose a one-to-one function y = ƒ(x) is given by a table of values   x 1 2 3 4 5 6 7 8 ƒ(x) 3 4.5 7 10.5 15 20.5 27 34.5 A table for the values of x = ƒ -1(y) can then be obtained by simply interchanging the values in the columns (or rows) of the table for ƒ:   y 

ƒ

3 4.5 7 10.5 15 20.5 27 34.5

−1

( y) 1 2 3 4 5 6 7 8

If we apply ƒ to send an input x to the output ƒ(x) and follow by applying ƒ -1 to ƒ(x), we get right back to x, just where we started. Similarly, if we take some number y in the range of ƒ, apply ƒ -1 to it, and then apply ƒ to the resulting value ƒ -1( y), we get back the

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Chapter 1: Functions

value y with which we began. Composing a function and its inverse has the same effect as doing nothing. (ƒ -1 ∘ ƒ) (x) = x, (ƒ ∘ ƒ -1) (y) = y,

for all x in the domain of ƒ for all y in the domain of ƒ -1 (or range of ƒ)

Only a one-to-one function can have an inverse. The reason is that if ƒ(x1) = y and ƒ(x2) = y for two distinct inputs x1 and x2, then there is no way to assign a value to ƒ -1(y) that satisfies both ƒ -1(ƒ(x1)) = x1 and ƒ -1(ƒ(x2)) = x2. A function that is increasing on an interval satisfies the inequality ƒ(x2) 7 ƒ(x1) when x2 7 x1 , so it is one-to-one and has an inverse. Decreasing functions also have an inverse. Functions that are neither increasing nor decreasing may still be one-to-one and have an inverse, as with the function ƒ(x) = 1>x for x ≠ 0 and ƒ(0) = 0, defined on (- q, q) and passing the horizontal line test.

Finding Inverses The graphs of a function and its inverse are closely related. To read the value of a function from its graph, we start at a point x on the x-axis, go vertically to the graph, and then move horizontally to the y-axis to read the value of y. The inverse function can be read from the graph by reversing this process. Start with a point y on the y-axis, go horizontally to the graph of y = ƒ(x), and then move vertically to the x-axis to read the value of x = ƒ -1(y) (Figure 1.57). y

DOMAIN OF

y = f(x)

f RANGE OF

f –1

y

y

0

x

x DOMAIN OF

x = f –1(y) y

0

f

f –1

(b) The graph of f –1 is the graph of f, but with x and y interchanged. To find the x that gave y, we start at y and go over to the curve and down to the x-axis. The domain of f –1 is the range of f. The range of f –1 is the domain of f.

(a) To find the value of f at x, we start at x, go up to the curve, and then over to the y-axis.

y

f –1

x y=x x = f –1(y) (b, a) 0

y DOMAIN OF

(c) To draw the graph of f –1 in the more usual way, we reflect the system across the line y = x.

f

–1

y = f –1(x)

f –1

(a, b)

RANGE OF

RANGE OF

x

x RANGE OF

x

0 DOMAIN OF

f

–1

(d) Then we interchange the letters x and y. We now have a normal-looking graph of f –1 as a function of x.

Figure 1.57  The graph of y = ƒ -1(x) is obtained by reflecting the graph of y = ƒ(x) about the line y = x.

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1.6  Inverse Functions and Logarithms

We want to set up the graph of ƒ -1 so that its input values lie along the x-axis, as is usually done for functions, rather than on the y-axis. To achieve this we interchange the xand y-axes by reflecting across the 45° line y = x. After this reflection we have a new graph that represents ƒ -1. The value of ƒ -1(x) can now be read from the graph in the usual way, by starting with a point x on the x-axis, going vertically to the graph, and then horizontally to the y-axis to get the value of ƒ -1(x). Figure 1.57 indicates the relationship between the graphs of ƒ and ƒ -1. The graphs are interchanged by reflection through the line y = x. The process of passing from ƒ to ƒ -1 can be summarized as a two-step procedure. 1. Solve the equation y = ƒ(x) for x. This gives a formula x = ƒ -1(y) where x is expressed as a function of y. 2. Interchange x and y, obtaining a formula y = ƒ -1(x) where ƒ -1 is expressed in the conventional format with x as the independent variable and y as the dependent variable. y

y = 2x − 2

EXAMPLE 3   Find the inverse of y = 1 x + 1, expressed as a function of x. y=x

2

Solution 1. Solve for x in terms of y:

y= 1x+1 2

x = 2y - 2.

x

1

−2

The graph is a straight line satisfying the 1 x + 1  horizontal line test (Fig. 1.58). 2

2y = x + 2

1

y =

2. Interchange x and y:  y = 2x - 2. The inverse of the function ƒ(x) = (1>2)x + 1 is the function ƒ -1(x) = 2x - 2. (See Figure 1.58.) To check, we verify that both composites give the identity function:

−2

Figure 1.58  Graphing ƒ(x) = (1>2)x + 1 and ƒ -1(x) = 2x - 2 together shows the graphs’ symmetry with respect to the line y = x (Example 3).

1 ƒ -1(ƒ(x)) = 2a x + 1b - 2 = x + 2 - 2 = x 2 1 ƒ(ƒ -1(x)) = (2x - 2) + 1 = x - 1 + 1 = x.  2

EXAMPLE 4   Find the inverse of the function y = x2, x Ú 0, expressed as a function

y

of x.

Solution  For x Ú 0, the graph satisfies the horizontal line test, so the function is one-toone and has an inverse. To find the inverse, we first solve for x in terms of y:

y = x 2, x ≥ 0 y=x

y = "x

y = x2 2y = 2x2 = 0 x 0 = x   0 x 0

= x because x Ú 0

We then interchange x and y, obtaining

0

x

Figure 1.59  The functions y = 1x and y = x2, x Ú 0, are inverses of one another (Example 4).

2

y = 2x.

The inverse of the function y = x , x Ú 0, is the function y = 1x (Figure 1.59). Notice that the function y = x2, x Ú 0, with domain restricted to the nonnegative real numbers, is one-to-one (Figure 1.59) and has an inverse. On the other hand, the function y = x2, with no domain restrictions, is not one-to-one (Figure 1.56b) and therefore has no inverse.

Logarithmic Functions If a is any positive real number other than 1, the base a exponential function ƒ(x) = ax is oneto-one. It therefore has an inverse. Its inverse is called the logarithm function with base a. Definition The logarithm function with base a, y = loga x, is the inverse of the base a exponential function y = ax (a 7 0, a ≠ 1).

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Chapter 1: Functions y

y = 2x y=x

2 1

y = log 2 x

0

x

1 2

The domain of loga x is (0, q), the range of ax. The range of loga x is (- q, q), the domain of ax. Figure 1.23 in Section 1.1 shows the graphs of four logarithmic functions with a 7 1. Figure 1.60a shows the graph of y = log2 x. The graph of y = ax, a 7 1, increases rapidly for x 7 0, so its inverse, y = loga x, increases slowly for x 7 1. Because we have no technique yet for solving the equation y = ax for x in terms of y, we do not have an explicit formula for computing the logarithm at a given value of x. Nevertheless, we can obtain the graph of y = loga x by reflecting the graph of the exponential y = ax across the line y = x. Figure 1.60 shows the graphs for a = 2 and a = e. Logarithms with base 2 are commonly used in computer science. Logarithms with base e and base 10 are so important in applications that many calculators have special keys for them. They also have their own special notation and names:

(a)

loge x is written as ln x. log10 x is written as log x.

y 8

y = ex

The function y = ln x is called the natural logarithm function, and y = log x is often called the common logarithm function. For the natural logarithm,

7 6

ln x = y 3 ey = x.

5 4

In particular, if we set x = e, we obtain

e

(1, e)

2

y = ln x

ln e = 1

1 −2 −1

0

1

2

e

4

x

because e1 = e.

Properties of Logarithms (b)

Figure 1.60  (a) The graph of 2x and its inverse, log2 x. (b) The graph of ex and its inverse, ln x.

Historical Biography* John Napier (1550–1617)

Logarithms, invented by John Napier, were the single most important improvement in arithmetic calculation before the modern electronic computer. What made them so useful is that the properties of logarithms reduce multiplication of positive numbers to addition of their logarithms, division of positive numbers to subtraction of their logarithms, and exponentiation of a number to multiplying its logarithm by the exponent. We summarize these properties for the natural logarithm as a series of rules that we prove in Chapter 3. Although here we state the Power Rule for all real powers r, the case when r is an irrational number cannot be dealt with properly until Chapter 4. We also establish the validity of the rules for logarithmic functions with any base a in Chapter 7. THEOREM 1—Algebraic Properties of the Natural Logarithm  For any numbers b 7 0 and x 7 0, the natural logarithm satisfies the following rules: 1.  Product Rule:

ln bx = ln b + ln x

2.  Quotient Rule:

b ln x = ln b - ln x

3.  Reciprocal Rule:

1 ln x = -ln x

4.  Power Rule:

ln xr = r ln x

Rule 2 with b = 1

*To learn more about the historical figures mentioned in the text and the development of many major elements and topics of calculus, visit www.aw.com/thomas.

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1.6  Inverse Functions and Logarithms

EXAMPLE 5   Here we use the properties in Theorem 1 to simplify three expressions. (a) ln 4 + ln sin x = ln (4 sin x) x + 1 (b) ln = ln (x + 1) - ln (2x - 3) 2x - 3 1 (c) ln = -ln 8 8 = -ln 23 = -3 ln 2

Product Rule Quotient Rule Reciprocal Rule Power Rule

Because ax and loga x are inverses, composing them in either order gives the identity function. Inverse Properties for ax and loga x 1.  Base a: aloga x = x, loga ax = x, ln x 2.  Base e: e = x, ln ex = x,

a 7 0, a ≠ 1, x 7 0 x 7 0

Substituting ax for x in the equation x = eln x enables us to rewrite ax as a power of e: x

ax = eln (a )

Substitute ax for x in x = eln x.

x ln a

= e = e(ln a)x.

Power Rule for logs Exponent rearranged

Thus, the exponential function ax is the same as ekx for k = ln a. Every exponential function is a power of the natural exponential function. ax = ex ln a That is, ax is the same as ex raised to the power ln a: ax = ekx for k = ln a. For example, 2x = e(ln 2)x = ex ln 2,

and

5 - 3x = e(ln 5) ( - 3x) = e - 3x ln 5.

Returning once more to the properties of ax and loga x, we have

ln x = ln (aloga x) = (loga x) (ln a).

Inverse Property for ax and loga x Power Rule for logarithms, with r = loga x

Rewriting this equation as loga x = (ln x)>(ln a) shows that every logarithmic function is a constant multiple of the natural logarithm ln x. This allows us to extend the algebraic properties for ln x to loga x. For instance, loga bx = loga b + loga x. Change of Base Formula Every logarithmic function is a constant multiple of the natural logarithm. loga x =

ln x   (a 7 0, a ≠ 1) ln a

Applications In Section 1.5 we looked at examples of exponential growth and decay problems. Here we use properties of logarithms to answer more questions concerning such problems.

EXAMPLE 6   If \$1000 is invested in an account that earns 5.25% interest compounded annually, how long will it take the account to reach \$2500?

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Chapter 1: Functions

Solution  From Example 1, Section 1.5, with P = 1000 and r = 0.0525, the amount in the account at any time t in years is 1000(1.0525)t, so to find the time t when the account reaches \$2500 we need to solve the equation 1000(1.0525)t = 2500. Thus we have

(1.0525)t = 2.5 ln (1.0525)t = ln 2.5 t ln 1.0525 = ln 2.5

t =

Divide by 1000. Take logarithms of both sides. Power Rule

ln 2.5 ≈ 17.9 ln 1.0525

Values obtained by calculator

The amount in the account will reach \$2500 in 18 years, when the annual interest payment is deposited for that year.

EXAMPLE 7  The half-life of a radioactive element is the time required for half of the radioactive nuclei present in a sample to decay. It is a notable fact that the half-life is a constant that does not depend on the number of radioactive nuclei initially present in the sample, but only on the radioactive substance. To see why, let y0 be the number of radioactive nuclei initially present in the sample. Then the number y present at any later time t will be y = y0 e-kt. We seek the value of t at which the number of radioactive nuclei present equals half the original number: 1 y 2 0 1 e-kt = 2 1 -kt = ln = -ln 2  Reciprocal Rule for logarithms 2 ln 2 t = .(1) k

y0 e-kt =

This value of t is the half-life of the element. It depends only on the value of k; the number y0 does not have any effect. The effective radioactive lifetime of polonium-210 is so short that we measure it in days rather than years. The number of radioactive atoms remaining after t days in a sample that starts with y0 radioactive atoms is

Amount present y0 y = y0 e–5×10

–3t

-3

y = y0 e-5 * 10 t.

1y 2 0 1y 4 0

The element’s half-life is

0

139

278

t (days)

Half-life

Figure 1.61  Amount of polo­ nium-210 present at time t, where y0 represents the number of radioactive atoms initially present (Example 7).

ln 2 k ln 2 = 5 * 10-3 ≈ 139 days.

Half@life =

Eq. (1) The k from polonium’s decay equation

This means that after 139 days, 1>2 of y0 radioactive atoms remain; after another 139 days (or 278 days altogether) half of those remain, or 1>4 of y0 radioactive atoms remain, and so on (see Figure 1.61).

Inverse Trigonometric Functions The six basic trigonometric functions of a general radian angle x were reviewed in Section 1.3. These functions are not one-to-one (their values repeat periodically). However, we can restrict their domains to intervals on which they are one-to-one. The sine function

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1.6  Inverse Functions and Logarithms

y x = sin y y = sin –1x Domain: −1 ≤ x ≤ 1 Range: −p2 ≤ y ≤ p2

p 2

Domain restrictions that make the trigonometric functions one-to-one

sin x −

p 2

p 2

1

1 0

−1

Figure 1.62  The graph of y = sin-1 x.

y

y

y

x

1

−1

increases from -1 at x = -p>2 to +1 at x = p>2. By restricting its domain to the interval 3 -p>2, p>2] we make it one-to-one, so that it has an inverse sin-1 x (Figure 1.62). Similar domain restrictions can be applied to all six trigonometric functions.

p 2

x

0 −1

y = sin x Domain: 3- p>2, p>24 Range: 3-1, 14 y

cos x p 2

p

x

p 2

0

p 2

x

y = tan x Domain: (- p>2, p>2) Range: (- q, q)

y = cos x Domain: 30, p4 Range: 3- 1, 14 y

tan x

y

sec x

csc x

cot x 1

0

p p y y = sin x, − 2 ≤ x ≤ 2 Domain: [−p2, p2] Range: [−1, 1] 1 −p 2

x

p 2

0 −1 (a) y

x = sin y y = sin –1x Domain: [−1, 1] Range: [−p2, p2]

p 2 0

−1 −

1

x

p 2

(b)

Figure 1.63  The graphs of (a) y = sin x, - p>2 … x … p>2, and (b) its inverse, y = sin-1 x. The graph of sin-1 x, obtained by reflection across the line y = x, is a portion of the curve x = sin y.

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p 2

p

y = cot x Domain: (0, p) Range: (- q, q)

x

1 p 2

0 −1

x

p

y = sec x Domain: 30, p>2) ∪ (p>2, p4 Range: (- q, - 14 ∪ 31, q)

p 2

0 −1

p 2

x

y = csc x Domain: 3- p>2, 0) ∪ (0, p>24 Range: (- q, - 14 ∪ 31, q)

Since these restricted functions are now one-to-one, they have inverses, which we denote by y y y y y y

= = = = = =

sin-1 x cos-1 x tan-1 x cot-1 x sec-1 x csc-1 x

or or or or or or

y y y y y y

= = = = = =

arcsin x arccos x arctan x arccot x arcsec x arccsc x

These equations are read “y equals the arcsine of x” or “y equals arcsin x” and so on. Caution The -1 in the expressions for the inverse means “inverse.” It does not mean reciprocal. For example, the reciprocal of sin x is (sin x)-1 = 1>sin x = csc x. The graphs of the six inverse trigonometric functions are obtained by reflecting the graphs of the restricted trigonometric functions through the line y = x. Figure 1.63b shows the graph of y = sin-1 x and Figure 1.64 shows the graphs of all six functions. We now take a closer look at two of these functions.

The Arcsine and Arccosine Functions We define the arcsine and arccosine as functions whose values are angles (measured in radians) that belong to restricted domains of the sine and cosine functions.

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Chapter 1: Functions Domain: −1 ≤ x ≤ 1 Range: −p ≤ y ≤ p 2 2 y p 2

Domain: −∞ < x < ∞ Range: −p < y < p 2 2 y

Domain: −1 ≤ x ≤ 1 0≤y≤p Range: y p

y = sin –1x x

1

−1 −

p 2

y = cos –1x p 2

p 2

−2 −1 −

x

1

−1 (b)

(a)

p 2

p p 2

y=

sec –1x

1

2

Domain: −∞ < x < ∞ 024 for which sin y = x. y = cos−1 x is the number in 3 0, p4 for which cos y = x.

The “Arc” in Arcsine and Arccosine For a unit circle and radian angles, the arc length equation s = ru becomes s = u, so central angles and the arcs they subtend have the same measure. If x = sin y, then, in addition to being the angle whose sine is x, y is also the length of arc on the unit circle that subtends an angle whose sine is x. So we call y “the arc whose sine is x.”

The graph of y = sin-1 x (Figure 1.63b) is symmetric about the origin (it lies along the graph of x = sin y). The arcsine is therefore an odd function:

The graph of y = cos-1 x (Figure 1.65b) has no such symmetry.

EXAMPLE 8  Evaluate (a)  sin-1 a

y x2 + y2 = 1

0

Arc whose cosine is x x Angle whose cosine is x

1

23

2

Solution (a) We see that

Arc whose sine is x

Angle whose sine is x

sin-1(-x) = -sin-1 x.(2)

x

sin-1 a

1 b  and (b)  cos-1 a- b . 2

23

2

b =

p 3

because sin (p>3) = 23>2 and p>3 belongs to the range 3 -p>2, p>24 of the arcsine function. See Figure 1.66a. (b) We have 2p 1 cos-1 a- b = 3 2

because cos (2p>3) = -1>2 and 2p>3 belongs to the range 3 0, p4 of the arccosine function. See Figure 1.66b.

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1.6  Inverse Functions and Logarithms

y

Using the same procedure illustrated in Example 8, we can create the following table of common values for the arcsine and arccosine functions.

y = cos x, 0 ≤ x ≤ p Domain: [0, p] Range: [−1, 1]

1

p 2

0 −1 (a) y

x = cos y p

−1 0

Figure 1.65  The graphs of (a) y = cos x, 0 … x … p, and (b) its inverse, y = cos-1 x. The graph of cos-1 x, obtained by reflection across the line y = x, is a portion of the curve x = cos y.

Chicago 179 180 62 b

St. Louis

12 Plane position

a

c

23>2 22>2 1>2 -1>2

p>3 p>4 p>6 -p>6

p>6 p>4 p>3 2p>3

- 22>2

-p>4

3p>4

-p>3

5p>6 y

sin –1" 3 = p 2 3

2 p 3 0 1

(b)

61

cos-1 x

y

x

1

sin-1 x

- 23>2

y = cos –1 x Domain: [−1, 1] Range: [0, p]

p 2

x

x

p

61

2

"3 x

"3

sin p = " 3 2 3 (a)

cos–1a− 1b = 2 p 3 2 2p 3 x

−1 0

cos a2 pb = – 1 3 2 (b)

Figure 1.66  Values of the arcsine and arccosine functions (Example 8).

EXAMPLE 9   During a 240 mi airplane flight from Chicago to St. Louis, after flying 180 mi the navigator determines that the plane is 12 mi off course, as shown in Figure 1.67. Find the angle a for a course parallel to the original correct course, the angle b, and the drift correction angle c = a + b. Solution  From the Pythagorean theorem and given information, we compute an approximate hypothetical flight distance of 179 mi, had the plane been flying along the original correct course (see Figure 1.67). Knowing the flight distance from Chicago to St. Louis, we next calculate the remaining leg of the original course to be 61 mi. Applying the Pythagorean theorem again then gives an approximate distance of 62 mi from the position of the plane to St. Louis. Finally, from Figure 1.67, we see that 180 sin a = 12 and 62 sin b = 12, so

Figure 1.67  Diagram for drift correction (Example 9), with distances surrounded to the nearest mile (drawing not to scale).

12 ≈ 0.067 radian ≈ 3.8° 180 12 ≈ 0.195 radian ≈ 11.2° b = sin-1 62 c = a + b ≈ 15°. 

a = sin-1

y

Identities Involving Arcsine and Arccosine

cos–1(−x) cos–1x −1 −x

0

x

1

x

As we can see from Figure 1.68, the arccosine of x satisfies the identity or

Figure 1.68  cos-1 x and cos-1(- x) are supplementary angles (so their sum is p).

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cos-1 x + cos-1(-x) = p,(3) cos-1 (-x) = p - cos-1 x.(4)

Also, we can see from the triangle in Figure 1.69 that for x 7 0,

sin-1 x + cos-1 x = p>2.(5)

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Chapter 1: Functions

cos–1x

1

x

sin–1x

Figure 1.69  sin-1 x and cos-1 x are complementary angles (so their sum is p>2).

Equation (5) holds for the other values of x in 3 -1, 1] as well, but we cannot conclude this from the triangle in Figure 1.69. It is, however, a consequence of Equations (2) and (4) (Exercise 76). The arctangent, arccotangent, arcsecant, and arccosecant functions are defined in Section 3.9. There we develop additional properties of the inverse trigonometric functions in a calculus setting using the identities discussed here.

Exercises  1.6 Identifying One-to-One Functions Graphically Which of the functions graphed in Exercises 1–6 are one-to-one, and which are not? y

1.

2. y y=

−3x 3

Graphing Inverse Functions Each of Exercises 11–16 shows the graph of a function y = ƒ(x). Copy the graph and draw in the line y = x. Then use symmetry with respect to the line y = x to add the graph of ƒ -1 to your sketch. (It is not necessary to find a formula for ƒ -1.) Identify the domain and range of ƒ -1.

11. x

0

−1

0 1 y = x4 − x2

12.

y

x

y y = f (x) = 2 1 , x ≥ 0 x +1

1

1

3.

4.

y

0

0 y = 20 x 0

x

14.

y = x13

−

p 2

y y = f (x) = tan x, p p − 0

0

3

−1 0

x

x + 1, −1 ≤ x ≤ 0 −2 + 2 x, 0 < x < 3 3 3

x

−2

17.  a. Graph the function ƒ(x) = 21 - x2, 0 … x … 1. What symmetry does the graph have? b. Show that ƒ is its own inverse. (Remember that 2x2 = x if x Ú 0.) 18.  a. Graph the function ƒ(x) = 1>x. What symmetry does the graph have? b. Show that ƒ is its own inverse.

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1.6  Inverse Functions and Logarithms

Formulas for Inverse Functions Each of Exercises 19–24 gives a formula for a function y = ƒ(x) and shows the graphs of ƒ and ƒ -1. Find a formula for ƒ -1 in each case.

ƒ(x) = x2, x … 0 19. ƒ(x) = x2 + 1, x Ú 0 20. y

b. What can you conclude about the inverse of a function y = ƒ(x) whose graph is a line through the origin with a nonzero slope m?

y y = f (x)

38. Show that the graph of the inverse of ƒ(x) = mx + b, where m and b are constants and m ≠ 0, is a line with slope 1>m and y-intercept -b>m.

y = f (x) 1

y=f

0

1

–1(x)

0 x

1

x

1

21. ƒ(x) = x3 - 1 22. ƒ(x) = x2 - 2x + 1, x Ú 1 y y = f (x)

b. Find the inverse of ƒ(x) = - x + b (b constant). What angle does the line y = -x + b make with the line y = x?

1 y = f (x)

1

x

1

−1 −1

0

x

1

23. ƒ(x) = (x + 1)2, x Ú - 1 24. ƒ(x) = x2>3, x Ú 0 y y = f (x)

c. What can you conclude about the inverses of functions whose graphs are lines perpendicular to the line y = x? Logarithms and Exponentials 41. Express the following logarithms in terms of ln 2 and ln 3.

y

c. What can you conclude about the inverses of functions whose graphs are lines parallel to the line y = x? 40.  a. Find the inverse of ƒ(x) = -x + 1. Graph the line y = -x + 1 together with the line y = x. At what angle do the lines intersect?

y = f –1(x)

y = f –1(x)

39.  a. Find the inverse of ƒ(x) = x + 1. Graph ƒ and its inverse together. Add the line y = x to your sketch, drawing it with dashes or dots for contrast. b. Find the inverse of ƒ(x) = x + b (b constant). How is the graph of ƒ -1 related to the graph of ƒ?

y = f –1(x)

y

Inverses of Lines 37.  a. Find the inverse of the function ƒ(x) = mx, where m is a constant different from zero.

a. ln 0.75

b. ln (4>9)

c. ln (1>2)

3 d. ln 2 9

e. ln 3 22

y = f –1(x)

f. ln 213.5

42. Express the following logarithms in terms of ln 5 and ln 7. ln 9.8  a. ln (1>125) b.

y = f –1(x)

1 −1

0

1

ln 1225  c. ln 7 27 d.

(ln 35 + ln (1>7))>(ln 25)  e. ln 0.056 f.

y = f (x) x

1

−1

0

1

x

Each of Exercises 25–36 gives a formula for a function y = ƒ(x). In each case, find ƒ -1(x) and identify the domain and range of ƒ -1. As a check, show that ƒ(ƒ -1(x)) = ƒ -1(ƒ(x)) = x. 25. ƒ(x) = x5 26. ƒ(x) = x4, x Ú 0 ƒ(x) = (1>2)x - 7>2 27. ƒ(x) = x3 + 1 28. ƒ(x) = 1>x3, x ≠ 0 29. ƒ(x) = 1>x2, x 7 0 30. 2x x + 3 31. ƒ(x) = 32. ƒ(x) = x - 2 2x - 3 33. ƒ(x) = x2 - 2x, x … 1 34. ƒ(x) = (2x3 + 1)1>5 (Hint: Complete the square.)

35. ƒ(x) =

x + b , b 7 - 2 and constant x - 2

36. ƒ(x) = x2 - 2bx, b 7 0 and constant, x … b

Use the properties of logarithms to write the expressions in Exercises 43 and 44 as a single term. sin u 1 43. a. ln sin u - ln a b b. ln (3x2 - 9x) + ln a b 2 3x c.

1 ln (4t 4) - ln b 2

44. a. ln sec u + ln cos u

b. ln (8x + 4) - 2 ln c

3 2

c. 3 ln 2t - 1 - ln (t + 1)

Find simpler expressions for the quantities in Exercises 45–48. 6

b. e-ln 6x

45. a. eln 8.3 ln (x2 + y2)

46. a. e

47. a. 2 ln 2e

48. a. ln (esec u)

-ln 0.3

b. e

b. ln (ln ee) x

b. ln (e(e ))

c. eln 3x - ln 5y c. eln px - ln 2 2

c. ln (e-x

- y2

)

c. ln (e2 ln x)

In Exercises 49–54, solve for y in terms of t or x, as appropriate. ln y = - t + 5 49. ln y = 4t + 5 50. ln (c - 2y) = t 51. ln (y - 30) = 5t 52. 53. ln (y - 4) - ln 5 = x + ln x 54. ln (y2 - 1) - ln (y + 1) = ln (sin x)

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64

Chapter 1: Functions

In Exercises 55 and 56, solve for k. 3k

55. a. e

8k

k>8

= 27 b. 35e = 175 c. e = a

56. a. e5k =

1 b. 80ek = 1 c. e(ln 0.8)k = 0.8 4

1 57. a. e-0.3t = 64 b. ekt = c. e(ln 0.7)t = 0.8 7 1 1 c. e(ln 2)t = 10 2 2

59. e2t = x6 60. e(x )e(2x + 1) = et Simplify the expressions in Exercises 61–64. 8log822 c. 1.3log1.3 75 61. a. 5log5 7 b. 1 d. log4 16 e. log3 23 f. log4 a b 4 62. a. 2log2 3 b. 10log10 (1>2) c. plogp 7

1 d. log11 121 e. log121 11 f. log3 a b 9

63. a. 2log4 x b. 9log3 x c. log2 (e(ln 2)(sin x)) 2

x

loge (ex) c. log4 (2e sin x) 64. a. 25log5 (3x ) b. Express the ratios in Exercises 65 and 66 as ratios of natural logarithms and simplify. log11 x log5 x logx a b. c. 65. a. log12 x log125 x logx5 a log 210 x log9 x loga b b. c. 66. a. log3 x log 22 x logb a Arcsine and Arccosine In Exercises 67–70, find the exact value of each expression.

- 23 1 1 sin-1 a b c. sin-1 a b 67. a. sin-1 a b b. 2 2 22 23 1 -1 68. a. cos-1 a b b. cos-1 a b c. cos-1 a b 2 2 22

69. a. arccos (- 1) b. arccos (0) arcsin a 70. a. arcsin (- 1) b.

1 22

b

Theory and Examples 71. If ƒ(x) is one-to-one, can anything be said about g(x) = - ƒ(x)? Is it also one-to-one? Give reasons for your answer.

72. If ƒ(x) is one-to-one and ƒ(x) is never zero, can anything be said about h(x) = 1>ƒ(x)? Is it also one-to-one? Give reasons for your answer. 73. Suppose that the range of g lies in the domain of ƒ so that the composite ƒ ∘ g is defined. If ƒ and g are one-to-one, can anything be said about ƒ ∘ g? Give reasons for your answer.

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75. Find a formula for the inverse function ƒ -1 and verify that (ƒ ∘ ƒ -1)(x) = (ƒ -1 ∘ ƒ)(x) = x. a. ƒ(x) =

In Exercises 57–60, solve for t.

58. a. e-0.01t = 1000 b. ekt =

74. If a composite ƒ ∘ g is one-to-one, must g be one-to-one? Give reasons for your answer.

100 50 b. ƒ(x) = 1 + 2-x 1 + 1.1-x

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chapter

2

Limits and Continuity

Overview  Mathematicians of the seventeenth century were keenly interested in the

study of motion for objects on or near the earth and the motion of planets and stars. This study involved both the speed of the object and its direction of motion at any instant, and they knew the direction at a given instant was along a line tangent to the path of motion. The concept of a limit is fundamental to finding the velocity of a moving object and the tangent to a curve. In this chapter we develop the limit, first intuitively and then formally. We use limits to describe the way a function varies. Some functions vary continuously; small changes in x produce only small changes in ƒ(x). Other functions can have values that jump, vary erratically, or tend to increase or decrease without bound. The notion of limit gives a precise way to distinguish between these behaviors.

2.1 Rates of Change and Tangents to Curves Calculus is a tool that helps us understand how a change in one quantity is related to a change in another. How does the speed of a falling object change as a function of time? How does the level of water in a barrel change as a function of the amount of liquid poured into it? We see change occurring in nearly everything we observe in the world and universe, and powerful modern instruments help us see more and more. In this section we introduce the ideas of average and instantaneous rates of change, and show that they are closely related to the slope of a curve at a point P on the curve. We give precise developments of these important concepts in the next chapter, but for now we use an informal approach so you will see how they lead naturally to the main idea of this chapter, the limit. The idea of a limit plays a foundational role throughout calculus.

Average and Instantaneous Speed HISTORICAL BIOGRAPHY* Galileo Galilei (1564–1642)

In the late sixteenth century, Galileo discovered that a solid object dropped from rest (not moving) near the surface of the earth and allowed to fall freely will fall a distance proportional to the square of the time it has been falling. This type of motion is called free fall. It assumes negligible air resistance to slow the object down, and that gravity is the only force acting on the falling object. If y denotes the distance fallen in feet after t seconds, then Galileo’s law is y = 16t 2, where 16 is the (approximate) constant of proportionality. (If y is measured in meters, the constant is 4.9.) A moving object’s average speed during an interval of time is found by dividing the distance covered by the time elapsed. The unit of measure is length per unit time: kilometers per hour, feet (or meters) per second, or whatever is appropriate to the problem at hand. *To learn more about the historical figures mentioned in the text and the development of many major elements and topics of calculus, visit www.aw.com/thomas.

65

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Chapter 2: Limits and Continuity

Example 1   A rock breaks loose from the top of a tall cliff. What is its average speed (a) during the first 2 sec of fall? (b) during the 1-sec interval between second 1 and second 2?

Solution  The average speed of the rock during a given time interval is the change in distance, ∆y, divided by the length of the time interval, ∆t. (Increments like ∆y and ∆t are reviewed in Appendix 3, and pronounced “delta y” and “delta t.”) Measuring distance in feet and time in seconds, we have the following calculations: ∆y 16(2)2 = 2 ∆t ∆y 16(2)2 (b) From sec 1 to sec 2:   = 2 ∆t (a) For the first 2 sec:

16(0)2 ft = 32 sec 0 16(1)2 ft = 48 sec 1

We want a way to determine the speed of a falling object at a single instant t0, instead of using its average speed over an interval of time. To do this, we examine what happens when we calculate the average speed over shorter and shorter time intervals starting at t0 . The next example illustrates this process. Our discussion is informal here, but it will be made precise in Chapter 3.

Example 2  Find the speed of the falling rock in Example 1 at t = 1 and t = 2 sec. Solution  We can calculate the average speed of the rock over a time interval 3 t0, t0 + h4 , having length ∆t = h, as

∆y 16(t0 + h)2 - 16t0 2 = . (1) h ∆t

We cannot use this formula to calculate the “instantaneous” speed at the exact moment t0 by simply substituting h = 0, because we cannot divide by zero. But we can use it to calculate average speeds over increasingly short time intervals starting at t0 = 1 and t0 = 2. When we do so, by taking smaller and smaller values of h, we see a pattern (Table 2.1).

Table 2.1  Average speeds over short time intervals 3 t0, t0 + h4 Average speed:

Length of time interval h

∆y 16(t0 + h)2 - 16t0 2 = h ∆t

Average speed over interval of length h starting at t0 = 1

1 48 0.1 33.6 0.01 32.16 0.001 32.016 0.0001 32.0016

Average speed over interval of length h starting at t0 = 2 80 65.6 64.16 64.016 64.0016

The average speed on intervals starting at t0 = 1 seems to approach a limiting value of 32 as the length of the interval decreases. This suggests that the rock is falling at a speed of 32 ft > sec at t0 = 1 sec. Let’s confirm this algebraically.

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67

2.1  Rates of Change and Tangents to Curves

that

If we set t0 = 1 and then expand the numerator in Equation (1) and simplify, we find ∆y 16(1 + h)2 - 16(1)2 16(1 + 2h + h2) - 16 = = h h ∆t =

32h + 16h2 = 32 + 16h. h

For values of h different from 0, the expressions on the right and left are equivalent and the average speed is 32 + 16h ft>sec. We can now see why the average speed has the limiting value 32 + 16(0) = 32 ft>sec as h approaches 0. Similarly, setting t0 = 2 in Equation (1), the procedure yields ∆y = 64 + 16h ∆t for values of h different from 0. As h gets closer and closer to 0, the average speed has the limiting value 64 ft > sec when t0 = 2 sec, as suggested by Table 2.1.

The average speed of a falling object is an example of a more general idea which we discuss next.

Average Rates of Change and Secant Lines Given any function y = ƒ(x), we calculate the average rate of change of y with respect to x over the interval [x1, x2] by dividing the change in the value of y, ∆y = ƒ(x2) - ƒ(x1), by the length ∆x = x2 - x1 = h of the interval over which the change occurs. (We use the symbol h for ∆x to simplify the notation here and later on.)

y y = f (x) Q(x 2, f (x 2 ))

Secant

DEFINITION  The average rate of change of y = ƒ(x) with respect to x over the

Δy

P(x1, f(x1))

interval [x1, x2] is

Δx = h x2

x1

0

x

Figure 2.1  A secant to the graph y = ƒ(x). Its slope is ∆y> ∆x, the average rate of change of ƒ over the interval [x1, x2].

∆y ƒ(x2) - ƒ(x1) ƒ(x1 + h) - ƒ(x1) = = , x2 - x1 h ∆x

h ≠ 0.

Geometrically, the rate of change of ƒ over [x1, x2] is the slope of the line through the points P(x1, ƒ(x1)) and Q(x2, ƒ(x2)) (Figure 2.1). In geometry, a line joining two points of a curve is a secant to the curve. Thus, the average rate of change of ƒ from x1 to x2 is identical with the slope of secant PQ. Let’s consider what happens as the point Q approaches the point P along the curve, so the length h of the interval over which the change occurs approaches zero. We will see that this procedure leads to defining the slope of a curve at a point.

Defining the Slope of a Curve P L O

Figure 2.2  L is tangent to the circle at P if it passes through P perpendicular to radius OP.

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We know what is meant by the slope of a straight line, which tells us the rate at which it rises or falls—its rate of change as a linear function. But what is meant by the slope of a curve at a point P on the curve? If there is a tangent line to the curve at P—a line that just touches the curve like the tangent to a circle—it would be reasonable to identify the slope of the tangent as the slope of the curve at P. So we need a precise meaning for the tangent at a point on a curve. For circles, tangency is straightforward. A line L is tangent to a circle at a point P if L passes through P perpendicular to the radius at P (Figure 2.2). Such a line just touches the circle. But what does it mean to say that a line L is tangent to some other curve C at a point P?

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68

Chapter 2: Limits and Continuity

To define tangency for general curves, we need an approach that takes into account the behavior of the secants through P and nearby points Q as Q moves toward P along the curve (Figure 2.3). Here is the idea: 1. Start with what we can calculate, namely the slope of the secant PQ. 2. Investigate the limiting value of the secant slope as Q approaches P along the curve. (We clarify the limit idea in the next section.) 3. If the limit exists, take it to be the slope of the curve at P and define the tangent to the curve at P to be the line through P with this slope. HISTORICAL BIOGRAPHY Pierre de Fermat (1601–1665)

This procedure is what we were doing in the falling-rock problem discussed in Example 2. The next example illustrates the geometric idea for the tangent to a curve.

Secants

Tangent

P

P Q

Tangent

Secants

Q

Figure 2.3  The tangent to the curve at P is the line through P whose slope is the limit of the secant slopes as Q S P from either side.

Example 3  Find the slope of the parabola y = x2 at the point P(2, 4). Write an equation for the tangent to the parabola at this point.

Solution  We begin with a secant line through P(2, 4) and Q(2 + h, (2 + h)2) nearby. We then write an expression for the slope of the secant PQ and investigate what happens to the slope as Q approaches P along the curve: Secant slope =

∆y (2 + h)2 - 22 h2 + 4h + 4 - 4 = = h h ∆x h2 + 4h = h + 4. h

=

If h 7 0, then Q lies above and to the right of P, as in Figure 2.4. If h 6 0, then Q lies to the left of P (not shown). In either case, as Q approaches P along the curve, h approaches zero and the secant slope h + 4 approaches 4. We take 4 to be the parabola’s slope at P. y y = x2

Secant slope is

(2 + h) 2 − 4 = h + 4. h

Q(2 + h, (2 + h) 2) Tangent slope = 4 Δy = (2 + h)2 − 4 P(2, 4) Δx = h 0

2

2+h

x

NOT TO SCALE

Figure 2.4  Finding the slope of the parabola y = x2 at the point P(2, 4) as the limit of secant slopes (Example 3).

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2.1  Rates of Change and Tangents to Curves

69

The tangent to the parabola at P is the line through P with slope 4: y = 4 + 4(x - 2)

Point-slope equation

y = 4x - 4.

Instantaneous Rates of Change and Tangent Lines The rates at which the rock in Example 2 was falling at the instants t = 1 and t = 2 are called instantaneous rates of change. Instantaneous rates and slopes of tangent lines are closely connected, as we see in the following examples.

Example 4  Figure 2.5 shows how a population p of fruit flies (Drosophila) grew in a 50-day experiment. The number of flies was counted at regular intervals, the counted values plotted with respect to time t, and the points joined by a smooth curve (colored blue in Figure 2.5). Find the average growth rate from day 23 to day 45. Solution  There were 150 flies on day 23 and 340 flies on day 45. Thus the number of flies increased by 340 - 150 = 190 in 45 - 23 = 22 days. The average rate of change of the population from day 23 to day 45 was Average rate of change:

∆p 340 - 150 190 = = ≈ 8.6 flies>day. 22 45 - 23 ∆t

p

Number of flies

350

Q(45, 340)

300

Δp = 190

250 200

P(23, 150)

150

Δp ≈ 8.6 fliesday Δt Δt = 22

100 50 0

10

20 30 Time (days)

40

50

t

Figure 2.5  Growth of a fruit fly population in a controlled experiment. The average rate of change over 22 days is the slope ∆p> ∆t of the secant line (Example 4).

This average is the slope of the secant through the points P and Q on the graph in Figure 2.5. The average rate of change from day 23 to day 45 calculated in Example 4 does not tell us how fast the population was changing on day 23 itself. For that we need to examine time intervals closer to the day in question.

Example 5   How fast was the number of flies in the population of Example 4 growing on day 23? Solution  To answer this question, we examine the average rates of change over increasingly short time intervals starting at day 23. In geometric terms, we find these rates by calculating the slopes of secants from P to Q, for a sequence of points Q approaching P along the curve (Figure 2.6).

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Chapter 2: Limits and Continuity

Q

Slope of PQ = 𝚫p , 𝚫t (flies , day)

(45, 340)

340 - 150 ≈ 8.6 45 - 23

(40, 330)

330 - 150 ≈ 10.6 40 - 23

(35, 310)

310 - 150 ≈ 13.3 35 - 23

(30, 265)

265 - 150 ≈ 16.4 30 - 23

p B(35, 350)

350

Q(45, 340)

300 Number of flies

70

250 200

P(23, 150)

150 100 50 0

10 20 30 A(14, 0) Time (days)

40

50

t

Figure 2.6  The positions and slopes of four secants through the point P on the fruit fly graph (Example 5).

The values in the table show that the secant slopes rise from 8.6 to 16.4 as the t-coordinate of Q decreases from 45 to 30, and we would expect the slopes to rise slightly higher as t continued on toward 23. Geometrically, the secants rotate counterclockwise about P and seem to approach the red tangent line in the figure. Since the line appears to pass through the points (14, 0) and (35, 350), it has slope 350 - 0 = 16.7 flies>day (approximately). 35 - 14 On day 23 the population was increasing at a rate of about 16.7 flies > day.

The instantaneous rates in Example 2 were found to be the values of the average speeds, or average rates of change, as the time interval of length h approached 0. That is, the instantaneous rate is the value the average rate approaches as the length h of the interval over which the change occurs approaches zero. The average rate of change corresponds to the slope of a secant line; the instantaneous rate corresponds to the slope of the tangent line as the independent variable approaches a fixed value. In Example 2, the independent variable t approached the values t = 1 and t = 2. In Example 3, the independent variable x approached the value x = 2. So we see that instantaneous rates and slopes of tangent lines are closely connected. We investigate this connection thoroughly in the next chapter, but to do so we need the concept of a limit.

Exercises  2.1 Average Rates of Change In Exercises 1–6, find the average rate of change of the function over the given interval or intervals.

5. R(u) = 23u + 1;

1. ƒ(x) = 8x3 + 8

Slope of a Curve at a Point In Exercises 7–14, use the method in Example 3 to find (a) the slope of the curve at the given point P, and (b) an equation of the tangent line at P.

3- 5, 54 a. 35, 74 b.  2. g(x) = x2 - 2x

3- 2, 44 a. 31, 34 b.  3. h(t) = cot t

a. 33p>4, 5p>44 b.  3p>3, 3p>24 4. g(t) = 2 + cos t

a. 30, p4 b.  3- p, p4

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30, 54

6. P(u) = u 3 - 4u 2 + 5u;

31, 24

7. y = x2 - 5, P(2, - 1) 8. y = 1 - 5x2, P(2, - 19) 9. y = x2 - 2x - 3, P(2, - 3) 10. y = x2 - 4x, P(1, - 3) 11. y = x3, P(2, 8)

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2.1  Rates of Change and Tangents to Curves

b. What is the average rate of increase of the profits between 2012 and 2014?

12. y = 2 - x3, P(1, 1) 3

13. y = x - 12x, P(1, - 11)

c. Use your graph to estimate the rate at which the profits were changing in 2012.

14. y = x3 - 3x2 + 4, P(2, 0) Instantaneous Rates of Change 15. Speed of a car  The accompanying figure shows the time-todistance graph for a sports car accelerating from a standstill. P

650 600

Q3

T 19. Let g(x) = 2x for x Ú 0.

Q2

a. Find the average rate of change of g(x) with respect to x over the intervals 31, 24, 31, 1.54 and 31, 1 + h4.

300 Q1

200

b. Make a table of values of the average rate of change of g with respect to x over the interval 31, 1 + h4 for some values of h approaching zero, say h = 0.1, 0.01, 0.001, 0.0001, 0.00001, and 0.000001.

100 0

5 10 15 20 Elapsed time (sec)

t

c. What does your table indicate is the rate of change of g(x) with respect to x at x = 1?

a.  Estimate the slopes of secants PQ1, PQ2, PQ3, and PQ4, arranging them in order in a table like the one in Figure 2.6. What are the appropriate units for these slopes? b. Then estimate the car’s speed at time t = 20 sec. 16. The accompanying figure shows the plot of distance fallen versus time for an object that fell from the lunar landing module a distance 80 m to the surface of the moon. a. Estimate the slopes of the secants PQ1, PQ2, PQ3, and PQ4, arranging them in a table like the one in Figure 2.6. b. About how fast was the object going when it hit the surface? y

Distance fallen (m)

80 Q3

60

a. Find the average rate of change of ƒ with respect to t over the intervals (i) from t = 2 to t = 3, and (ii) from t = 2 to t = T. b. Make a table of values of the average rate of change of ƒ with respect to t over the interval 32, T4 , for some values of T approaching 2, say T = 2.1, 2.01, 2.001, 2.0001, 2.00001, and 2.000001. c. What does your table indicate is the rate of change of ƒ with respect to t at t = 2?

21. The accompanying graph shows the total distance s traveled by a bicyclist after t hours.

Q2

40

T 20. Let ƒ(t) = 1>t for t ≠ 0.

d. Calculate the limit as T approaches 2 of the average rate of change of ƒ with respect to t over the interval from 2 to T. You will have to do some algebra before you can substitute T = 2.

P

Q4

d. Calculate the limit as h approaches zero of the average rate of change of g(x) with respect to x over the interval 31, 1 + h4.

Q1

20

5

0

s 10

t

Elapsed time (sec)

T 17. The profits of a small company for each of the first five years of its operation are given in the following table: Year

Profit in \$1000s

2010   6 2011  27 2012  62 2013 111 2014 174

a. Plot points representing the profit as a function of year, and join them by as smooth a curve as you can.

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Distance traveled (mi)

Distance (m)

b. Extending the table if necessary, try to determine the rate of change of F(x) at x = 1.

Q4

500

T 18. Make a table of values for the function F(x) = (x + 2)>(x - 2) at the points x = 1.2, x = 11>10, x = 101>100, x = 1001>1000, x = 10001>10000, and x = 1. a. Find the average rate of change of F(x) over the intervals 31, x4 for each x ≠ 1 in your table.

s

400

71

40 30 20 10 0

1

2 3 Elapsed time (hr)

4

t

a. Estimate the bicyclist’s average speed over the time intervals 30, 14, 31, 2.54 , and 32.5, 3.54 .

b. Estimate the bicyclist’s instantaneous speed at the times t = 12, t = 2, and t = 3. c. Estimate the bicyclist’s maximum speed and the specific time at which it occurs.

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Chapter 2: Limits and Continuity

22. The accompanying graph shows the total amount of gasoline A in the gas tank of an automobile after being driven for t days.

a. Estimate the average rate of gasoline consumption over the time intervals 30, 34, 30, 54, and 37, 104 . b. Estimate the instantaneous rate of gasoline consumption at the times t = 1, t = 4, and t = 8.

Remaining amount (gal)

A 16

c. Estimate the maximum rate of gasoline consumption and the specific time at which it occurs.

12 8 4 0

1

2

3 4 5 6 7 8 Elapsed time (days)

9 10

t

2.2 Limit of a Function and Limit Laws In Section 2.1 we saw that limits arise when finding the instantaneous rate of change of a function or the tangent to a curve. Here we begin with an informal definition of limit and show how we can calculate the values of limits. A precise definition is presented in the next section. HISTORICAL ESSAY Limits

Limits of Function Values Frequently when studying a function y = ƒ(x), we find ourselves interested in the function’s behavior near a particular point c, but not at c. This might be the case, for instance, if c is an irrational number, like p or 22, whose values can only be approximated by “close” rational numbers at which we actually evaluate the function instead. Another situation occurs when trying to evaluate a function at c leads to division by zero, which is undefined. We encountered this last circumstance when seeking the instantaneous rate of change in y by considering the quotient function ∆y>h for h closer and closer to zero. Here’s a specific example in which we explore numerically how a function behaves near a particular point at which we cannot directly evaluate the function.

y

2 2 y = f (x) = x − 1 x− 1

1

−1

0

1

x

Example 1  How does the function ƒ(x) =

y

x2 - 1 x - 1

behave near x = 1? 2

Solution  The given formula defines ƒ for all real numbers x except x = 1 (we cannot divide by zero). For any x ≠ 1, we can simplify the formula by factoring the numerator and canceling common factors:

y=x+1 1

−1

0

1

Figure 2.7  The graph of ƒ is identical with the line y = x + 1 except at x = 1, where ƒ is not defined (Example 1).

M02_THOM9580_03_SE_C02_065-128.indd 72

x

ƒ(x) =

(x - 1)(x + 1) = x + 1 x - 1

for

x ≠ 1.

The graph of ƒ is the line y = x + 1 with the point (1, 2) removed. This removed point is shown as a “hole” in Figure 2.7. Even though ƒ(1) is not defined, it is clear that we can make the value of ƒ(x) as close as we want to 2 by choosing x close enough to 1 (Table 2.2).

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2.2  Limit of a Function and Limit Laws

Table 2.2  As x gets closer to 1, ƒ(x ) gets closer to 2.

ƒ(x) =

x

x2 −1 x−1

0.9 1.9 1.1 2.1 0.99 1.99 1.01 2.01 0.999 1.999 1.001 2.001 0.999999 1.999999 1.000001 2.000001

73

Generalizing the idea illustrated in Example 1, suppose ƒ(x) is defined on an open interval about c, except possibly at c itself. If ƒ(x) is arbitrarily close to the number L (as close to L as we like) for all x sufficiently close to c, we say that ƒ approaches the limit L as x approaches c, and write lim ƒ(x) = L,

xSc

which is read “the limit of ƒ(x) as x approaches c is L.” For instance, in Example 1 we would say that ƒ(x) approaches the limit 2 as x approaches 1, and write lim ƒ(x) = 2,

or

xS1

lim

xS1

x2 - 1 = 2. x - 1

Essentially, the definition says that the values of ƒ(x) are close to the number L whenever x is close to c (on either side of c). Our definition here is “informal” because phrases like arbitrarily close and sufficiently close are imprecise; their meaning depends on the context. (To a machinist manufacturing a piston, close may mean within a few thousandths of an inch. To an astronomer studying distant galaxies, close may mean within a few thousand light-years.) Nevertheless, the definition is clear enough to enable us to recognize and evaluate limits of many specific functions. We will need the precise definition given in Section 2.3, however, when we set out to prove theorems about limits or study complicated functions. Here are several more examples exploring the idea of limits.

Example 2   The limit value of a function does not depend on how the function is defined at the point being approached. Consider the three functions in Figure 2.8. The function ƒ has limit 2 as x S 1 even though ƒ is not defined at x = 1. The function g has limit 2 as x S 1 even though 2 ≠ g(1). The function h is the only one of the three functions in Figure 2.8 whose limit as x S 1 equals its value at x = 1. For h, we have limx S 1 h(x) = h(1). This equality of limit and function value is of special importance, and we return to it in Section 2.5. y y

y

y

2

2

2

1

1

1

y=x c −1

0

1

x

0

−1

x

c

2 (a) f (x) = x − 1 x− 1

(b) g(x) =

(a) Identity function

1

x

x2 − 1 , x ≠ 1 x− 1 1,

−1

0

1

x

(c) h(x) = x + 1

x=1

Figure 2.8  The limits of ƒ(x), g(x), and h(x) all equal 2 as x approaches 1. However, only h(x) has the same function value as its limit at x = 1 (Example 2).

y

y=k

k

Example 3 (a) If ƒ is the identity function ƒ(x) = x, then for any value of c (Figure 2.9a),

0

c

(b) Constant function

Figure 2.9  The functions in Example 3 have limits at all points c.

M02_THOM9580_03_SE_C02_065-128.indd 73

lim ƒ(x) = lim x = c.

x

xSc

xSc

(b) If ƒ is the constant function ƒ(x) = k (function with the constant value k), then for

any value of c (Figure 2.9b),

lim ƒ(x) = lim k = k.

xSc

xSc

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Chapter 2: Limits and Continuity

For instances of each of these rules we have lim x = 3

xS3

lim (4) = lim (4) = 4.

and

x S -7

xS2

We prove these rules in Example 3 in Section 2.3. A function may not have a limit at a particular point. Some ways that limits can fail to exist are illustrated in Figure 2.10 and described in the next example.

y

y y=

0, x < 0

y y=

1, x ≥ 0

1

0, x = 0

1

0

1, x≠0 x

x

x

0

x

0 y=

0,

x≤0

sin 1x , x > 0

–1 (a) Unit step function U(x)

(b) g(x)

(c) f(x)

Figure 2.10  None of these functions has a limit as x approaches 0 (Example 4).

Example 4   Discuss the behavior of the following functions, explaining why they have no limit as x S 0. (a) U(x) = e (b) g(x) = • (c) ƒ(x) = c

0, 1,

x 6 0 x Ú 0

1 x,

x≠0

0,

x = 0

0,

x … 0

1 sin x ,

x 7 0

Solution

(a) It jumps: The unit step function U(x) has no limit as x S 0 because its values jump

at x = 0. For negative values of x arbitrarily close to zero, U(x) = 0. For positive values of x arbitrarily close to zero, U(x) = 1. There is no single value L approached by U(x) as x S 0 (Figure 2.10a). (b) It grows too “large” to have a limit: g(x) has no limit as x S 0 because the values of g grow arbitrarily large in absolute value as x S 0 and do not stay close to any fixed real number (Figure 2.10b). We say the function is not bounded. (c) It oscillates too much to have a limit: ƒ(x) has no limit as x S 0 because the function’s values oscillate between +1 and -1 in every open interval containing 0. The values do not stay close to any one number as x S 0 (Figure 2.10c).

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75

2.2  Limit of a Function and Limit Laws

The Limit Laws To calculate limits of functions that are arithmetic combinations of functions having known limits, we can use several fundamental rules.

THEOREM 1—Limit Laws If L, M, c, and k are real numbers and lim ƒ(x) = L

xSc

1. Sum Rule: 2. Difference Rule: 3. Constant Multiple Rule: 4. Product Rule: 5. Quotient Rule: 6. Power Rule: 7. Root Rule:

lim g(x) = M, then

and

xSc

lim (ƒ(x) + g(x)) = L + M

xSc

lim (ƒ(x) - g(x)) = L - M

xSc

lim (k # ƒ(x)) = k # L

xSc

lim (ƒ(x) # g(x)) = L # M

xSc

lim

xSc

ƒ(x) L = , M ≠ 0 g(x) M

lim 3 ƒ(x)4 n = L n, n a positive integer

xSc

n

n

lim 2ƒ(x) = 2L = L 1>n, n a positive integer

xSc

(If n is even, we assume that lim ƒ(x) = L 7 0.) xSc

In words, the Sum Rule says that the limit of a sum is the sum of the limits. Similarly, the next rules say that the limit of a difference is the difference of the limits; the limit of a constant times a function is the constant times the limit of the function; the limit of a product is the product of the limits; the limit of a quotient is the quotient of the limits (provided that the limit of the denominator is not 0); the limit of a positive integer power (or root) of a function is the integer power (or root) of the limit (provided that the root of the limit is a real number). It is reasonable that the properties in Theorem 1 are true (although these intuitive arguments do not constitute proofs). If x is sufficiently close to c, then ƒ(x) is close to L and g(x) is close to M, from our informal definition of a limit. It is then reasonable that ƒ(x) + g(x) is close to L + M; ƒ(x) - g(x) is close to L - M; kƒ(x) is close to kL; ƒ(x)g(x) is close to LM; and ƒ(x)>g(x) is close to L>M if M is not zero. We prove the Sum Rule in Section 2.3, based on a precise definition of limit. Rules 2–5 are proved in Appendix 5. Rule 6 is obtained by applying Rule 4 repeatedly. Rule 7 is proved in more advanced texts. The Sum, Difference, and Product Rules can be extended to any number of functions, not just two.

Example 5   Use the observations limx S c k = k and limx S c x = c (Example 3) and the fundamental rules of limits to find the following limits. (a) lim ( x3 + 4x2 - 3 ) xSc

(b) lim

xSc

(c)

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x4 + x2 - 1 x2 + 5

lim 24x2 - 3

x S -2

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Chapter 2: Limits and Continuity

Solution (a) lim ( x3 + 4x2 - 3 ) = lim x3 + lim 4x2 - lim 3

Sum and Difference Rules

= c3 + 4c2 - 3

Power and Multiple Rules

xSc

xSc

lim ( x4 + x2 - 1 )

x4 + x2 - 1 = xSc x2 + 5

xSc

=

xSc

(b) lim

lim ( x2 + 5 )

(c)

xSc

Quotient Rule

xSc

lim x4 + lim x2 - lim 1 lim

xSc

=

xSc

xSc x2 +

xSc

lim 5

c4 + c2 - 1 c2 + 5

Power or Product Rule

lim 24x2 - 3 = 2 lim ( 4x2 - 3 )

x S -2

x S -2

= 2 lim 4x2 - lim 3 x S -2

Sum and Difference Rules

xSc

x S -2

= 24(-2)2 - 3

Root Rule with n = 2 Difference Rule Product and Multiple Rules

= 216 - 3        = 213

Theorem 1 simplifies the task of calculating limits of polynomials and rational functions. To evaluate the limit of a polynomial function as x approaches c, merely substitute c for x in the formula for the function. To evaluate the limit of a rational function as x approaches a point c at which the denominator is not zero, substitute c for x in the formula for the function. (See Examples 5a and 5b.) We state these results formally as theorems.

THEOREM 2—Limits of Polynomials If P(x) = an xn + an - 1 xn - 1 + g + a0, then lim P(x) = P(c) = an cn + an - 1 cn - 1 + g + a0.

xSc

THEOREM 3—Limits of Rational Functions If P(x) and Q(x) are polynomials and Q(c) ≠ 0, then lim

xSc

P(x) P(c) = . Q(x) Q(c)

Example 6   The following calculation illustrates Theorems 2 and 3: Identifying Common Factors It can be shown that if Q(x) is a polynomial and Q(c) = 0, then (x - c) is a factor of Q(x). Thus, if the numerator and denominator of a rational function of x are both zero at x = c, they have (x - c) as a common factor.

M02_THOM9580_03_SE_C02_065-128.indd 76

3 2 x3 + 4x2 - 3 (-1) + 4(-1) - 3 0 = = = 0 2 2 6 x S -1 x + 5 (-1) + 5

lim

Eliminating Common Factors from Zero Denominators Theorem 3 applies only if the denominator of the rational function is not zero at the limit point c. If the denominator is zero, canceling common factors in the numerator and denominator may reduce the fraction to one whose denominator is no longer zero at c. If this happens, we can find the limit by substitution in the simplified fraction.

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2.2  Limit of a Function and Limit Laws

Example 7  Evaluate

y 2 x−2 y= x + x2 − x (1, 3)

3

−2

0

x2 + x - 2 . xS1 x2 - x lim

x

1

Solution  We cannot substitute x = 1 because it makes the denominator zero. We test the numerator to see if it, too, is zero at x = 1. It is, so it has a factor of (x - 1) in common with the denominator. Canceling this common factor gives a simpler fraction with the same values as the original for x ≠ 1:

(a)

x2 + x - 2 (x - 1)(x + 2) x + 2 = = x , x(x - 1) x2 - x

y y=x+2 x (1, 3)

3

Using the simpler fraction, we find the limit of these values as x S 1 by Theorem 3: lim

xS1

−2

0

1

if x ≠ 1.

x

x2 + x - 2 x + 2 1 + 2 = lim x = = 3. 2 1 S x 1 x - x

See Figure 2.11.

Using Calculators and Computers to Estimate Limits (b)

Figure 2.11  The graph of ƒ(x) = ( x2 + x - 2 ) > ( x2 - x ) in part (a) is the same as the graph of g(x) = (x + 2)>x in part (b) except at x = 1, where ƒ is undefined. The functions have the same limit as x S 1 (Example 7).

When we cannot use the Quotient Rule in Theorem 1 because the limit of the denominator is zero, we can try using a calculator or computer to guess the limit numerically as x gets closer and closer to c. We used this approach in Example 1, but calculators and computers can sometimes give false values and misleading impressions for functions that are undefined at a point or fail to have a limit there. Usually the problem is associated with rounding errors, as we now illustrate.

Example 8  Estimate the value of lim

xS0

2x2 + 100 - 10

x2

.

Solution  Table 2.3 lists values of the function obtained on a calculator for several points approaching x = 0. As x approaches 0 through the points {1, {0.5, {0.10, and {0.01, the function seems to approach the number 0.05. As we take even smaller values of x, {0.0005, {0.0001, {0.00001, and {0.000001, the function appears to approach the number 0. Is the answer 0.05 or 0, or some other value? We resolve this question in the next example.

Table 2.3  Computed values of ƒ(x) =

M02_THOM9580_03_SE_C02_065-128.indd 77

2x 2 + 100 - 10

x

ƒ(x)

{1 {0.5 {0.1 {0.01

0.049876 0.049969 t approaches 0.05? 0.049999 0.050000

{0.0005 {0.0001 {0.00001 {0.000001

0.050000 0.000000 t approaches 0? 0.000000 0.000000

x2

near x = 0

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Chapter 2: Limits and Continuity

Using a computer or calculator may give ambiguous results, as in the last example. The calculator could not keep track of enough digits to avoid rounding errors in computing the values of ƒ(x) when x is very small. We cannot substitute x = 0 in the problem, and the numerator and denominator have no obvious common factors (as they did in Example 7). Sometimes, however, we can create a common factor algebraically.

Example 9  Evaluate lim

xS0

2x2 + 100 - 10

x2

.

Solution  This is the limit we considered in Example 8. We can create a common factor by multiplying both numerator and denominator by the conjugate radical expression 2x2 + 100 + 10 (obtained by changing the sign after the square root). The preliminary algebra rationalizes the numerator: 2x2 + 100 - 10

=

=

=

=

x

2

Therefore, lim

xS0

#

2x2 + 100 - 10 2x2 + 100 + 10

x2

2x2 + 100 + 10

x2 + 100 - 100 x ( 2x2 + 100 + 10 ) 2

x2 x ( 2x + 100 + 10) 2

2

1 . 2x + 100 + 10

2x2 + 100 - 10

x2

2

= lim

xS0

=

=

Common factor x2 Cancel x2 for x ≠ 0.

1 2x2 + 100 + 10

Denominator not 0 at 1       x = 0; substitute. 202 + 100 + 10

1 = 0.05. 20

This calculation provides the correct answer, in contrast to the ambiguous computer results in Example 8. We cannot always algebraically resolve the problem of finding the limit of a quotient where the denominator becomes zero. In some cases the limit might then be found with the aid of some geometry applied to the problem (see the proof of Theorem 7 in Section 2.4), or through methods of calculus (illustrated in Section 4.5). The next theorems give helpful tools by using function comparisons.

y h f

L

The Sandwich Theorem

g 0

c

Figure 2.12  The graph of ƒ is sandwiched between the graphs of g and h.

M02_THOM9580_03_SE_C02_065-128.indd 78

x

The following theorem enables us to calculate a variety of limits. It is called the Sandwich Theorem because it refers to a function ƒ whose values are sandwiched between the values of two other functions g and h that have the same limit L at a point c. Being trapped between the values of two functions that approach L, the values of ƒ must also approach L (Figure 2.12). You will find a proof in Appendix 5.

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2.2  Limit of a Function and Limit Laws

THEOREM 4—The Sandwich Theorem  Suppose that g(x) … ƒ(x) … h(x) for all x in some open interval containing c, except possibly at x = c itself. Suppose also that lim g(x) = lim h(x) = L. xSc

xSc

Then limx S c ƒ(x) = L. y

Example 10  Given that

2

y = u(x)

1 -

1

2

y=1− x 4

0

−1

The Sandwich Theorem is also called the Squeeze Theorem or the Pinching Theorem.

2 y=1+ x 2

x

1

xS0

u

y = −0 u 0

lim sin u = 0.

(b) From Section 1.3, 0 … 1 - cos u … 0 u 0 for all u (see Figure 2.14b), and we have

limu S 0 (1 - cos u) = 0 or

lim cos u = 1.

uS0

y = 0u0

1 0

xSc

Solution (a) In Section 1.3 we established that - 0 u 0 … sin u … 0 u 0 for all u (see Figure 2.14a). Since limu S 0 ( - 0 u 0 ) = limu S 0 0 u 0 = 0, we have uS0

y 2

uS0

xSc

(a)

−1

(b)  lim cos u = 1

(c) For any function ƒ, lim 0 ƒ(x) 0 = 0 implies lim ƒ(x) = 0.

y = sin u

−1

xS0

Example 11   The Sandwich Theorem helps us establish several important limit rules:

y = 0u0

p

lim ( 1 + ( x2 >2 ) ) = 1,

and

the Sandwich Theorem implies that limx S 0 u(x) = 1 (Figure 2.13).

uS0

−p

−2

Solution  Since

(a) lim sin u = 0

1

for all x ≠ 0,

find limx S 0 u(x), no matter how complicated u is. lim ( 1 - ( x2 >4 ) ) = 1

Figure 2.13  Any function u(x) whose graph lies in the region between y = 1 + (x2 >2) and y = 1 - (x2 >4) has limit 1 as x S 0 (Example 10).

y

x2 x2 … u(x) … 1 + 4 2

y = 1 − cos u 1

2

u

(b)

Figure 2.14  The Sandwich Theorem confirms the limits in Example 11.

(c) Since - 0 ƒ(x) 0 … ƒ(x) … 0 ƒ(x) 0 and - 0 ƒ(x) 0 and 0 ƒ(x) 0 have limit 0 as x S c, it

follows that limx S c ƒ(x) = 0.

Another important property of limits is given by the next theorem. A proof is given in the next section. THEOREM 5  If ƒ(x) … g(x) for all x in some open interval containing c, except possibly at x = c itself, and the limits of ƒ and g both exist as x approaches c, then lim ƒ(x) … lim g(x).

xSc

xSc

Caution  The assertion resulting from replacing the less than or equal to (… ) inequality by the strict less than (6 ) inequality in Theorem 5 is false. Figure 2.14a shows that for u ≠ 0, - 0 u 0 6 sin u 6 0 u 0 . So limu S 0 sin u = 0 = limu S 0 0 u 0 , not limu S 0 sin u 6 limu S 0 0 u 0 .

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Chapter 2: Limits and Continuity

Exercises  2.2 Limits from Graphs 1. For the function g(x) graphed here, find the following limits or explain why they do not exist.

a. lim g(x)  b.  lim g(x)  c.  lim g(x)  d.  lim g(x) xS1

xS2

xS3

x S 2.5

d. lim ƒ(x) exists at every point c in (-1, 1). xSc

e. lim ƒ(x) exists at every point c in (1, 3). xSc

y

y

1

−1

y = g(x) 1

2

x

3

−1 1

2

a. lim ƒ(t)  b.  lim ƒ(t)  c.  lim ƒ(t)  d.  lim ƒ(t) t S -1

tS0

t S -0.5

−1

−2

Existence of Limits In Exercises 5 and 6, explain why the limits do not exist. x 1 6. lim 5. lim xS0 0 x 0 xS1 x - 1

7. Suppose that a function ƒ(x) is defined for all real values of x except x = c. Can anything be said about the existence of limx S c ƒ(x)? Give reasons for your answer.

s

s = f (t)

−2

x

3

2. For the function ƒ(t) graphed here, find the following limits or explain why they do not exist. t S -2

y = f (x)

1

1 0

1

t

8. Suppose that a function ƒ(x) is defined for all x in 3- 1, 1]. Can anything be said about the existence of limx S 0 ƒ(x)? Give reasons for your answer. 9. If limx S 1 ƒ(x) = 5, must ƒ be defined at x = 1? If it is, must ƒ(1) = 5? Can we conclude anything about the values of ƒ at x = 1? Explain.

−1

3. Which of the following statements about the function y = ƒ(x) graphed here are true, and which are false? a. lim ƒ(x) exists. xS0

10. If ƒ(1) = 5, must limx S 1 ƒ(x) exist? If it does, then must limx S 1 ƒ(x) = 5? Can we conclude anything about limx S 1 ƒ(x)? Explain.

xS0

Calculating Limits Find the limits in Exercises 11–22.

xS0

11. lim ( x2 - 13 )

12. lim ( - x2 + 8x - 7 )

e. lim ƒ(x) = 0

13. lim 8(t - 5)(t - 7)

14. lim ( 2x3 - 5x2 + 3x + 5 )

f. lim ƒ(x) exists at every point c in (- 1, 1).

x + 2 15. lim xS2 x + 5

16. lim (8 - 3s)(2s - 1)

17. lim 16x(13x + 16)2

18. lim

19. lim (18 - y)7>3

20. lim 2z2 - 10

b. lim ƒ(x) = 0 c. lim ƒ(x) = 1 d. lim ƒ(x) = 1

x S -3

xS1

xS3

tS6

xS1 xSc

g. lim ƒ(x) does not exist. xS1

1

y = f (x)

1

−1

2

x

4. Which of the following statements about the function y = ƒ(x) graphed here are true, and which are false? xS2

b. lim ƒ(x) = 2 xS2

c. lim ƒ(x) does not exist. xS1

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21. lim

hS0

5 25h + 4 + 4

yS2

y + 2 y2 + 5y + 6

zS4

y S -9

−1

a. lim ƒ(x) does not exist.

s S 2>3

x S -1>4

y

x S -1

22. lim

hS0

24h + 1 - 1

h

Limits of quotients  Find the limits in Exercises 23–42. x - 9 x + 3 24. lim 2 23. lim 2 x S 9 x - 81 x S -3 x + 4x + 3 x2 - 2x - 48 x2 - 4x - 12 25. lim 26. lim x - 8 x - 6 xS 8 xS6 t2 + t - 2 t 2 + 3t + 2 27. lim 2 28. lim 2 tS1 t - 1 t S -1 t - t - 2 5y3 + 8y2 -2x - 4 29. lim 3 30. lim 4 2 x S -2 x + 2x y S 0 3y - 16y2

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2.2  Limit of a Function and Limit Laws

x-1 - 1 xS1 x - 1

31. lim

32. lim

x u - 81 34. lim 3 u S 3 u - 27 4x - x2 36. lim x S 4 2 - 2x

x - 87 x S 87 2x + 13 - 10

37. lim

2x2 + 12 - 4

x - 2

53. Suppose limx S 3 ƒ(x) = 3 and limx S 3 g(x) = -8. Find a. lim ƒ(x)g(x) b.  lim 3ƒ(x)g(x)

1 x + 1

xS3

40. lim

xS4

g(x) c. lim (g(x)) d.  lim S S ƒ(x) - 1 x 4 x 4

x + 1

55. Suppose limx S b ƒ(x) = 10 and limx S b g(x) = -3. Find

2x2 + 5 - 3

lim 6(x) a. lim (ƒ(x) + g(x)) b.

Limits with trigonometric functions  Find the limits in Exercises 43–50. 44. lim sin2 x

45. lim sec x

46. lim tan x

xS0

47. lim

xS0

48. lim (x2 - 1)(2 - cos x)

Using Limit Rules 51. Suppose limx S 0 ƒ(x) = 1 and limx S 0 g(x) = - 5. Name the rules in Theorem 1 that are used to accomplish steps (a), (b), and (c) of the following calculation.

(ƒ(x) + 7)2>3

=

lim (2ƒ(x) - g(x))

xS0

lim (ƒ(x) + 7)2>3

(a)

lim 2ƒ(x) - lim g(x)

xS0

xS0

a lim ( ƒ(x) + 7 ) b

2>3

(b)

xS0

=

2 lim ƒ(x) - lim g(x) xS0

xS0

a lim ƒ(x) + lim 7b xS0

=

2>3

(c)

(1 + 7)2>3

=

xS1

p(x)(4 - r(x))

=

lim 25h(x)

xS1

lim (p(x)(4 - r(x)))

(a)

5h(x) 4xlim S1 xS1

5lim h(x) 4 xS1

=

a lim p(x)b a lim 4 - lim r(x)b xS1

=

M02_THOM9580_03_SE_C02_065-128.indd 81

xS1

57. ƒ(x) = x2, x = 1 59. ƒ(x) = 3x - 4, x = 2 60. ƒ(x) = 1>x, x = - 2 62. ƒ(x) = 23x + 1, x = 0

64. If 2 - x2 … g(x) … 2 cos x for all x, find limx S 0 g(x). 65. a. It can be shown that the inequalities 1 -

xS1

lim

xS0

(c)

x2 x sin x 6 6 1 2 - 2 cos x 6

hold for all values of x close to zero. What, if anything, does this tell you about

(b)

a lim p(x)b a lim ( 4 - r(x) ) b xS1

occur frequently in calculus. In Exercises 57–62, evaluate this limit for the given value of x and function ƒ.

63. If 25 - 2x2 … ƒ(x) … 25 - x2 for - 1 … x … 1, find limx S 0 ƒ(x).

7 4

xS1

=

ƒ(x + h) - ƒ(x) h

Using the Sandwich Theorem

xS0

(2)(1) - (- 5)

lim

hS0

61. ƒ(x) = 2x, x = 7

52. Let limx S 1 h(x) = 5, limx S 1 p(x) = 1, and limx S 1 r(x) = 2. Name the rules in Theorem 1 that are used to accomplish steps (a), (b), and (c) of the following calculation. 25h(x)

x S -2

58. ƒ(x) = x2, x = - 2

xS0

=

lim

c. lim (- 4p(x) + 5r(x))>s(x) Limits of Average Rates of Change Because of their connection with secant lines, tangents, and instantaneous rates, limits of the form

xS0

xS0

a. lim (p(x) + r(x) + s(x))

xS0

x S -p

2ƒ(x) - g(x)

xSb

x S -2

49. lim 2x + 4 cos (x + p) 50. lim 27 + sec2 x

lim

xSb

56. Suppose that limx S -2 p(x) = 4, limx S -2 r(x) = 0, and limx S -2 s(x) = -3. Find x S -2

x S p>3

1 + x + sin x 3 cos x

xSb

b. lim p(x) # r(x) # s(x)

x S p>4

xS0

xSb

c. lim ƒ(x) # g(x) d.  lim ƒ(x)>g(x)

4 - x 42. lim x S 4 5 - 2x 2 + 9

43. lim (2 sin x - 1)

xS4

2

x + 2

x S -2

23 - 2x2 - 47 41. lim S x + 24 x -24

lim xƒ(x) a. lim (g(x) + 3) b.

2x + 8 - 3

x S -1

54. Suppose limx S 4 ƒ(x) = 0 and limx S 4 g(x) = -3. Find

2

38. lim

xS3

ƒ(x) c. lim (ƒ(x) + 5g(x)) d.  lim S S ƒ(x) - g(x) x 3 x 3

4

u - 16 33. lim 3 uS2 u - 8 2x - 5 3 5. lim x S 25 x - 25

xS2

+

xS0

4

39. lim

1 x - 1

81

x sin x ? 2 - 2 cos x

T b. Graph y = 1 - ( x2 >6 ) , y = (x sin x)>(2 - 2 cos x), and y = 1 together for -2 … x … 2. Comment on the behavior of the graphs as x S 0.

2(5)(5) 5 = (1)(4 - 2) 2

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Chapter 2: Limits and Continuity

66. a. Suppose that the inequalities x2 1 - cos x 1 1 6 6 2 24 2 x2 hold for values of x close to zero. (They do, as you will see in Section 9.9.) What, if anything, does this tell you about 1 - cos x lim ? xS0 x2 Give reasons for your answer. T b. G  raph the equations y = (1>2) - ( x2 >24 ) , y = (1 - cos x)>x2, and y = 1>2 together for -2 … x … 2. Comment on the behavior of the graphs as x S 0. Estimating Limits T You will find a graphing calculator useful for Exercises 67–76.

67. Let ƒ(x) = (x2 - 9)>(x + 3). a. M ake a table of the values of ƒ at the points x = -3.1, - 3.01, - 3.001, and so on as far as your calculator can go. Then estimate limx S -3 ƒ(x). What estimate do you arrive at if you evaluate ƒ at x = - 2.9, - 2.99, - 2.999, c instead? b. S upport your conclusions in part (a) by graphing ƒ near c = -3 and using Zoom and Trace to estimate y-values on the graph as x S - 3. c. Find limx S -3 ƒ(x) algebraically, as in Example 7.

68. Let g(x) = ( x2 - 2 ) >(x - 22).

a. M ake a table of the values of g at the points x = 1.4, 1.41, 1.414, and so on through successive decimal approximations of 22. Estimate limx S 22 g(x). b. S upport your conclusion in part (a) by graphing g near c = 22 and using Zoom and Trace to estimate y-values on the graph as x S 22.

c. Find limx S 22 g(x) algebraically.

69. Let G(x) = (x + 6)> ( x2 + 4x - 12 ) . a. M ake a table of the values of G at x = - 5.9, -5.99, - 5.999, and so on. Then estimate limx S -6 G(x). What estimate do you arrive at if you evaluate G at x = - 6.1, -6.01, - 6.001, cinstead?

b. Support your conclusion in part (a) by graphing ƒ near c = -1 and using Zoom and Trace to estimate y-values on the graph as x S -1. c. Find limx S -1 ƒ(x) algebraically.

72. Let F(x) = ( x2 + 3x + 2 ) > ( 2 - 0 x 0 ) .

a. Make tables of values of F at values of x that approach c = -2 from above and below. Then estimate limx S -2 F(x). b. Support your conclusion in part (a) by graphing F near c = -2 and using Zoom and Trace to estimate y-values on the graph as x S -2. c. Find limx S -2 F(x) algebraically.

73. Let g(u) = (sin u)>u. a. Make a table of the values of g at values of u that approach u0 = 0 from above and below. Then estimate limu S 0 g(u). b. Support your conclusion in part (a) by graphing g near u0 = 0. 74. Let G(t) = (1 - cos t)>t 2. a. Make tables of values of G at values of t that approach t0 = 0 from above and below. Then estimate limt S 0 G(t). b. Support your conclusion in part (a) by graphing G near t0 = 0. 75. Let ƒ(x) = x1>(1 - x). a. Make tables of values of ƒ at values of x that approach c = 1 from above and below. Does ƒ appear to have a limit as x S 1? If so, what is it? If not, why not? b. Support your conclusions in part (a) by graphing ƒ near c = 1. 76. Let ƒ(x) = (3x - 1)>x. a. Make tables of values of ƒ at values of x that approach c = 0 from above and below. Does ƒ appear to have a limit as x S 0? If so, what is it? If not, why not? b. Support your conclusions in part (a) by graphing ƒ near c = 0. Theory and Examples 77. If x4 … ƒ(x) … x2 for x in 3-1, 14 and x2 … ƒ(x) … x4 for x 6 - 1 and x 7 1, at what points c do you automatically know limx S c ƒ(x)? What can you say about the value of the limit at these points?

b. Support your conclusions in part (a) by graphing G and using Zoom and Trace to estimate y-values on the graph as x S - 6.

78. Suppose that g(x) … ƒ(x) … h(x) for all x ≠ 2 and suppose that

c. Find limx S -6 G(x) algebraically.

Can we conclude anything about the values of ƒ, g, and h at x = 2? Could ƒ(2) = 0? Could limx S 2 ƒ(x) = 0? Give reasons for your answers. ƒ(x) - 5 79. If lim = 1, find lim ƒ(x). xS4 x - 2 xS4 ƒ(x) 80. If lim 2 = 1, find x S -2 x ƒ(x) b. lim x a. lim ƒ(x) x S -2 x S -2

70. Let h(x) = ( x2 - 2x - 3 ) > ( x2 - 4x + 3 ) .

a. Make a table of the values of h at x = 2.9, 2.99, 2.999, and so on. Then estimate limx S 3 h(x). What estimate do you arrive at if you evaluate h at x = 3.1, 3.01, 3.001, c instead? b. Support your conclusions in part (a) by graphing h near c = 3 and using Zoom and Trace to estimate y-values on the graph as x S 3. c. Find limx S 3 h(x) algebraically.

71. Let ƒ(x) = ( x2 - 1 ) > ( 0 x 0 - 1 ) .

a. M ake tables of the values of ƒ at values of x that approach c = -1 from above and below. Then estimate limx S -1 ƒ(x).

M02_THOM9580_03_SE_C02_065-128.indd 82

lim g(x) = lim h(x) = - 5.

xS2

xS2

ƒ(x) - 5 = 3, find lim ƒ(x). x - 2 xS2 ƒ(x) - 5 b. If lim = 4, find lim ƒ(x). xS2 x - 2 xS2 81. a. If lim

xS2

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2.3  The Precise Definition of a Limit

82. If lim

xS0

ƒ(x) = 1, find x2

85. lim

xS2

x4 - 16 x - 2

x3 - x2 - 5x - 3 x S -1 (x + 1)2

a. lim ƒ(x)

86. lim

xS0

ƒ(x) b. lim x xS0

3

 raph g(x) = x sin (1>x) to estimate limx S 0 g(x), zooming in T 83. a. G on the origin as necessary. b. Confirm your estimate in part (a) with a proof.  raph h(x) = x2 cos (1>x3) to estimate limx S 0 h(x), zooming T 84. a. G in on the origin as necessary. b. Confirm your estimate in part (a) with a proof.

87. lim

xS0

21 + x - 1

x

x2 - 9 2x + 7 - 4 1 - cos x 89. lim x sin x xS0 88. lim

xS3

2

2x2 x S 0 3 - 3 cos x

90. lim

COMPUTER EXPLORATIONS Graphical Estimates of Limits In Exercises 85–90, use a CAS to perform the following steps:

a. Plot the function near the point c being approached. b. From your plot guess the value of the limit.

2.3  The Precise Definition of a Limit

We now turn our attention to the precise definition of a limit. We replace vague phrases like “gets arbitrarily close to” in the informal definition with specific conditions that can be applied to any particular example. With a precise definition, we can avoid misunderstandings, prove the limit properties given in the preceding section, and establish many important limits. To show that the limit of ƒ(x) as x S c equals the number L, we need to show that the gap between ƒ(x) and L can be made “as small as we choose” if x is kept “close enough” to c. Let us see what this would require if we specified the size of the gap between ƒ(x) and L.

Example 1  Consider the function y = 2x - 1 near x = 4. Intuitively it appears that y is close to 7 when x is close to 4, so limx S 4 (2x - 1) = 7. However, how close to x = 4 does x have to be so that y = 2x - 1 differs from 7 by, say, less than 2 units?

y y = 2x − 1 Upper bound: y=9

9 To satisfy this

7 5

0

Lower bound: y=5

3 4 5

x

Restrict to this

Figure 2.15 Keeping x within 1 unit of x = 4 will keep y within 2 units of y = 7 (Example 1).

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Solution  We are asked: For what values of x is 0 y - 7 0 6 2? To find the answer we first express 0 y - 7 0 in terms of x:

0 y - 7 0 = 0 (2x - 1) - 7 0 = 0 2x - 8 0 .

The question then becomes: what values of x satisfy the inequality 0 2x - 8 0 6 2? To find out, we solve the inequality:

0 2x - 8 0 6 2 -2 6 3 -1

6 6 6 6

2x - 8 6 2 2x 6 10 x 6 5 x - 4 6 1.

Solve for x. Solve for x - 4.

Keeping x within 1 unit of x = 4 will keep y within 2 units of y = 7 (Figure 2.15).

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Chapter 2: Limits and Continuity

In the previous example we determined how close x must be to a particular value c to ensure that the outputs ƒ(x) of some function lie within a prescribed interval about a limit value L. To show that the limit of ƒ(x) as x S c actually equals L, we must be able to show that the gap between ƒ(x) and L can be made less than any prescribed error, no matter how small, by holding x close enough to c.

y

L+

1 10

f(x)

L L−

f(x) lies in here

Definition of Limit

1 10

for all x ≠ c in here

d 0

d x c

c−d

x

c+d

Figure 2.16  How should we define d 7 0 so that keeping x within the interval (c - d, c + d) will keep ƒ(x) within the interval aL -

1 1 ,L + b? 10 10

Suppose we are watching the values of a function ƒ(x) as x approaches c (without taking on the value of c itself). Certainly we want to be able to say that ƒ(x) stays within onetenth of a unit from L as soon as x stays within some distance d of c (Figure 2.16). But that in itself is not enough, because as x continues on its course toward c, what is to prevent ƒ(x) from jittering about within the interval from L - (1>10) to L + (1>10) without tending toward L? We can be told that the error can be no more than 1>100 or 1>1000 or 1>100,000. Each time, we find a new d@interval about c so that keeping x within that interval satisfies the new error tolerance. And each time the possibility exists that ƒ(x) jitters away from L at some stage. The figures on the next page illustrate the problem. You can think of this as a quarrel between a skeptic and a scholar. The skeptic presents P@challenges to prove that the limit does not exist or, more precisely, that there is room for doubt. The scholar answers every challenge with a d@interval around c that keeps the function values within P of L. How do we stop this seemingly endless series of challenges and responses? We can do so by proving that for every error tolerance P that the challenger can produce, we can present a matching distance d that keeps x “close enough” to c to keep ƒ(x) within that P@tolerance of L (Figure 2.17). This leads us to the precise definition of a limit.

y

DEFINITION  Let ƒ(x) be defined on an open interval about c, except possibly at c

itself. We say that the limit of ƒ(x) as x approaches c is the number L, and write

L+P L

f(x)

lim ƒ(x) = L,

f(x) lies in here

xSc

if, for every number P 7 0, there exists a corresponding number d 7 0 such that for all x,

L−P

0 6 0x - c0 6 d

for all x ≠ c in here d x 0

c−d

d x c

c+d

Figure 2.17  The relation of d and P in the definition of limit.

1

0 ƒ(x) - L 0 6 P.

One way to think about the definition is to suppose we are machining a generator shaft to a close tolerance. We may try for diameter L, but since nothing is perfect, we must be satisfied with a diameter ƒ(x) somewhere between L - P and L + P. The d is the measure of how accurate our control setting for x must be to guarantee this degree of accuracy in the diameter of the shaft. Notice that as the tolerance for error becomes stricter, we may have to adjust d. That is, the value of d, how tight our control setting must be, depends on the value of P, the error tolerance.

Examples: Testing the Definition The formal definition of limit does not tell how to find the limit of a function, but it enables us to verify that a conjectured limit value is correct. The following examples show how the definition can be used to verify limit statements for specific functions. However, the real purpose of the definition is not to do calculations like this, but rather to prove general theorems so that the calculation of specific limits can be simplified, such as the theorems stated in the previous section.

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2.3  The Precise Definition of a Limit

y

L+

y = f (x)

1 10

L+

L−

L+

L−

1 10

x

c

0

L+

L 1 100

1 L+ 1000

L

L

1 1000

L−

1 1000

x

c

x

c

0

New challenge: P = 1 1000

Response: 0 x − c 0 < d1/1000

y

y y = f (x) 1 L+ 100,000

L

L

1 100,000

L−

c New challenge: 1 P = 100,000

y y = f (x)

1 100,000

0

c + d1/100 c − d1/100 Response: 0 x − c 0 < d1/100

y = f (x)

1 L+ 1000

0

L−

x

y y = f (x)

L+

c

0

New challenge: Make 0 f (x) − L 0 < P = 1 100

y

L−

x

c

0

c − d1/10 c + d1/10 Response: 0 x − c 0 < d1/10 (a number)

The challenge: Make 0 f (x) − L 0 < P = 1 10

1 100 L 1 L− 100

1 100

x

c

0

y = f(x)

y = f (x)

L

1 10

y

y = f (x)

1 10

L L−

y

y

85

x

y = f (x) L+P L L−P

1 100,000

0

c

x

c

0

Response: 0 x − c 0 < d1/100,000

x

New challenge: P = ...

Example 2  Show that lim (5x - 3) = 2.

xS1

Solution  Set c = 1, ƒ(x) = 5x - 3, and L = 2 in the definition of limit. For any given P 7 0, we have to find a suitable d 7 0 so that if x ≠ 1 and x is within distance d of c = 1, that is, whenever 0 6 0 x - 1 0 6 d,

it is true that ƒ(x) is within distance P of L = 2, so

0 ƒ(x) - 2 0 6 P.

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86

Chapter 2: Limits and Continuity

y

We find d by working backward from the P@inequality:

y = 5x − 3

0 (5x - 3) - 2 0 = 0 5x - 5 0 6 P 50x - 10 6 P 0 x - 1 0 6 P>5.

2+P 2 2−P x

1−P 1 1+P 5 5

0

Thus, we can take d = P>5 (Figure 2.18). If 0 6 0 x - 1 0 6 d = P>5, then

0 (5x - 3) - 2 0 = 0 5x - 5 0 = 5 0 x - 1 0 6 5(P>5) = P,

which proves that limx S 1(5x - 3) = 2. The value of d = P>5 is not the only value that will make 0 6 0 x - 1 0 6 d imply 0 5x - 5 0 6 P. Any smaller positive d will do as well. The definition does not ask for a “best” positive d, just one that will work.

Example 3   Prove the following results presented graphically in Section 2.2.

−3

(a) lim x = c

NOT TO SCALE

xSc

Figure 2.18 If ƒ(x) = 5x - 3, then 0 6 0 x - 1 0 6 P>5 guarantees that 0 ƒ(x) - 2 0 6 P (Example 2).

(b) lim k = k (k constant) xSc

Solution  (a) Let P 7 0 be given. We must find d 7 0 such that for all x

y

implies

0 6 0x - c0 6 d

implies

0 x - c 0 6 P.

The implication will hold if d equals P or any smaller positive number (Figure 2.19). This proves that limx S c x = c. (b) Let P 7 0 be given. We must find d 7 0 such that for all x

y=x c+P c+d c c−d

0 k - k 0 6 P.

Since k - k = 0, we can use any positive number for d and the implication will hold (Figure 2.20). This proves that limx S c k = k.

c−P

0

c−d c c+d

x

Finding Deltas Algebraically for Given Epsilons In Examples 2 and 3, the interval of values about c for which 0 ƒ(x) - L 0 was less than P was symmetric about c and we could take d to be half the length of that interval. When such symmetry is absent, as it usually is, we can take d to be the distance from c to the interval’s nearer endpoint.

Figure 2.19  For the function ƒ(x) = x, we find that 0 6 0 x - c 0 6 d will guarantee 0 ƒ(x) - c 0 6 P whenever d … P (Example 3a).

Example 4  For the limit limx S 5 2x - 1 = 2, find a d 7 0 that works for P = 1. That is, find a d 7 0 such that for all x

y

0 6 0x - 50 6 d

y=k

k+P k k−P

0

0 6 0x - c0 6 d

1

Solution  We organize the search into two steps.

0 2x - 1 - 2 0 6 1.

1. Solve the inequality 0 2x - 1 - 2 0 6 1 to find an interval containing x = 5 on which the inequality holds for all x ≠ 5. c−d

c

c+d

Figure 2.20  For the function ƒ(x) = k, we find that 0 ƒ(x) - k 0 6 P for any positive d (Example 3b).

M02_THOM9580_03_SE_C02_065-128.indd 86

x

0 2x - 1 - 2 0 6 1

-1 6 2x - 1 - 2 6 1 1 6 2x - 1 6 3 1 6 x - 1 6 9 2 6 x 6 10

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87

2.3  The Precise Definition of a Limit

(

3

3

2

5

)

x

10

8

Figure 2.21  An open interval of radius 3 about x = 5 will lie inside the open interval (2, 10).

y

y = "x − 1

3

0 6 0x - 50 6 3

0 2x - 1 - 2 0 6 1.

1

How to Find Algebraically a D for a Given ƒ, L, c, and E + 0 The process of finding a d 7 0 such that for all x

2

0 6 0x - c0 6 d

1

can be accomplished in two steps.

1 3 0

The inequality holds for all x in the open interval (2, 10), so it holds for all x ≠ 5 in this interval as well. 2. Find a value of d 7 0 to place the centered interval 5 - d 6 x 6 5 + d (centered at x = 5) inside the interval (2, 10). The distance from 5 to the nearer endpoint of (2, 10) is 3 (Figure 2.21). If we take d = 3 or any smaller positive number, then the inequality 0 6 0 x - 5 0 6 d will automatically place x between 2 and 10 to make 0 2x - 1 - 2 0 6 1 (Figure 2.22):

1 2

3 5

8

10

x

NOT TO SCALE

Figure 2.22  The function and intervals in Example 4.

0 ƒ(x) - L 0 6 P

1. Solve the inequality 0 ƒ(x) - L 0 6 P to find an open interval (a, b) containing c on which the inequality holds for all x ≠ c. 2. Find a value of d 7 0 that places the open interval (c - d, c + d) centered at c inside the interval (a, b). The inequality 0 ƒ(x) - L 0 6 P will hold for all x ≠ c in this d@interval.

Example 5  Prove that limx S 2 ƒ(x) = 4 if ƒ(x) = e

x2, 1,

x≠2 x = 2.

Solution  Our task is to show that given P 7 0 there exists a d 7 0 such that for all x y

0 6 0x - 20 6 d

y = x2

0 ƒ(x) - 4 0 6 P.

1. Solve the inequality 0 ƒ(x) - 4 0 6 P to find an open interval containing x = 2 on which the inequality holds for all x ≠ 2.

4+P

For x ≠ c = 2, we have ƒ(x) = x2, and the inequality to solve is 0 x2 - 4 0 6 P:

(2, 4)

4

0 x2 - 4 0 6 P

4−P (2, 1) 0

1

"4 − P

2

x "4 + P

Figure 2.23  An interval containing x = 2 so that the function in Example 5 satisfies 0 ƒ(x) - 4 0 6 P.

-P 6 x2 - 4 6 P 4 - P 6 x2 6 4 + P 24 - P 6 0 x 0 6 24 + P  Assumes P 24 - P 6 x 6 24 + P.

6 4; see below. An open interval about x = 2 that solves the inequality

The inequality 0 ƒ(x) - 4 0 6 P holds for all x ≠ 2 in the open interval ( 24 - P, 24 + P ) (Figure 2.23).

2. Find a value of d 7 0 that places the centered interval (2 - d, 2 + d) inside the interval ( 24 - P, 24 + P ) .

Take d to be the distance from x = 2 to the nearer endpoint of ( 24 - P, 24 + P ) . In other words, take d = min 5 2 - 24 - P, 24 + P - 2 6 , the minimum (the

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88

Chapter 2: Limits and Continuity

smaller) of the two numbers 2 - 24 - P and 24 + P - 2. If d has this or any smaller positive value, the inequality 0 6 0 x - 2 0 6 d will automatically place x between 24 - P and 24 + P to make 0 ƒ(x) - 4 0 6 P. For all x, 0 6 0x - 20 6 d

1

0 ƒ(x) - 4 0 6 P.

This completes the proof for P 6 4.   If P Ú 4, then we take d to be the distance from x = 2 to the nearer endpoint of the interval ( 0, 24 + P ) . In other words, take d = min 5 2, 24 + P - 2 6 . (See Figure 2.23.)

Using the Definition to Prove Theorems We do not usually rely on the formal definition of limit to verify specific limits such as those in the preceding examples. Rather, we appeal to general theorems about limits, in particular the theorems of Section 2.2. The definition is used to prove these theorems (Appendix 5). As an example, we prove part 1 of Theorem 1, the Sum Rule.

Example 6  Given that limx S c ƒ(x) = L and limx S c g(x) = M, prove that lim (ƒ(x) + g(x)) = L + M.

xSc

Solution  Let P 7 0 be given. We want to find a positive number d such that for all x 0 6 0x - c0 6 d

1

Regrouping terms, we get

0 ƒ(x) + g(x) - (L + M) 0 6 P.

0 ƒ(x) + g(x) - (L + M) 0 = 0 (ƒ(x) - L) + (g(x) - M) 0   Triangle Inequality: 0a + b0 … 0a0 + 0b0 … 0 ƒ(x) - L 0 + 0 g(x) - M 0 .

Since limx S c ƒ(x) = L, there exists a number d1 7 0 such that for all x 0 6 0 x - c 0 6 d1

1

0 ƒ(x) - L 0 6 P>2.

Similarly, since limx S c g(x) = M, there exists a number d2 7 0 such that for all x 0 6 0 x - c 0 6 d2

1

0 g(x) - M 0 6 P>2.

Let d = min 5d1, d2 6 , the smaller of d1 and d2. If 0 6 0 x - c 0 6 d then 0 x - c 0 6 d1, so 0 ƒ(x) - L 0 6 P>2, and 0 x - c 0 6 d2, so 0 g(x) - M 0 6 P>2. Therefore

0 ƒ(x) + g(x) - (L + M) 0 6 P + P = P. 2

2

This shows that limx S c (ƒ(x) + g(x)) = L + M. Next we prove Theorem 5 of Section 2.2.

Example 7  Given that limx S c ƒ(x) = L and limx S c g(x) = M, and that ƒ(x) … g(x) for all x in an open interval containing c (except possibly c itself), prove that L … M.

Solution  We use the method of proof by contradiction. Suppose, on the contrary, that L 7 M. Then by the limit of a difference property in Theorem 1, lim (g(x) - ƒ(x)) = M - L.

xSc

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2.3  The Precise Definition of a Limit

Therefore, for any P 7 0, there exists d 7 0 such that

0 (g(x) - ƒ(x)) - (M - L) 0 6 P

whenever 0 6 0 x - c 0 6 d.

Since L - M 7 0 by hypothesis, we take P = L - M in particular and we have a number d 7 0 such that

0 (g(x) - ƒ(x)) - (M - L) 0 6 L - M

whenever 0 6 0 x - c 0 6 d.

Since a … 0 a 0 for any number a, we have

whenever 0 6 0 x - c 0 6 d

(g(x) - ƒ(x)) - (M - L) 6 L - M

which simplifies to

whenever 0 6 0 x - c 0 6 d.

g(x) 6 ƒ(x)

But this contradicts ƒ(x) … g(x). Thus the inequality L 7 M must be false. Therefore L … M.

Exercises  2.3 Centering Intervals About a Point In Exercises 1–6, sketch the interval (a, b) on the x-axis with the point c inside. Then find a value of d 7 0 such that for all x, 0 6 0 x - c 0 6 d 1 a 6 x 6 b.

1. a = 1, b = 7, c = 5

2. a = 1, b = 7, c = 2

3. a = - 7>2, b = - 1>2, c = - 3 4. a = - 7>2, b = - 1>2, c = - 3>2

9. y 5 4 1 3 4

10.

f (x) = " x c=1 L=1 P = 1 y = "x 4

f(x) = 2" x + 1 c=3 L=4 P = 0.2

0

Finding Deltas Graphically In Exercises 7–14, use the graphs to find a d 7 0 such that for all x 0 6 0 x - c 0 6 d 1 0 ƒ(x) - L 0 6 P.

7.

y y = 2x − 4 6.2 6 5.8

0

f (x) = 2x − 4 c=5 L=6 P = 0.2

4.9

5

x

25 16

11.

12.

y

x

y = x2

7.65 7.5 7.35

y = 4 − x2

5 4

0

−2.9

0

NOT TO SCALE

3.25 3 2.75

3

NOT TO SCALE

−3

y f(x) = 4 − x 2 c = −1 L=3 P = 0.25

5.1

−3.1

x

2.61 3 3.41 NOT TO SCALE

f (x) = x 2 c=2 L=4 P=1

y = −3 x + 3 2

2

−1 0

y

f (x) = − 3 x + 3 2 c = −3 L = 7.5 P = 0.15

1

9 16

8.

y = 2" x + 1

4.2 4 3.8

5. a = 4>9, b = 4>7, c = 1>2 6. a = 2.7591, b = 3.2391, c = 3

y

x

"3

2

x "5

NOT TO SCALE

" 5 −1 " 3 − 2 2

0

x

NOT TO SCALE

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Chapter 2: Limits and Continuity

13.

14.

y 2 "−x c = −1 L=2 P = 0.5

y

f(x) =

f (x) = 1x c=1 2 L=2 P = 0.01

2.01

y= 2 "−x

Using the Formal Definition Each of Exercises 31–36 gives a function ƒ(x), a point c, and a positive number P. Find L = lim ƒ(x). Then find a number d 7 0 such xSc that for all x

0 6 0x - c0 6 d

31. ƒ(x) = 2 - 4x,

2.5

1.99

2

y = 1x

1.5

c = 4,

32. ƒ(x) = - 3x - 2,

2

x2 - 9 , x - 3

34. ƒ(x) =

x2 + 6x + 5 , x + 5

c = 3,

35. ƒ(x) = 21 - x,

16 − 9

−1

16 − 25

x

0

x

1 1 1 2 2.01 1.99

0

P = 0.03

c = -1,

33. ƒ(x) =

36. ƒ(x) = 4>x,

P = 0.03

c = -5, c = - 3,

c = 2,

Finding Deltas Algebraically Each of Exercises 15–30 gives a function ƒ(x) and numbers L, c, and P 7 0. In each case, find an open interval about c on which the inequality 0 ƒ(x) - L 0 6 P holds. Then give a value for d 7 0 such that for all x satisfying 0 6 0 x - c 0 6 d the inequality 0 ƒ(x) - L 0 6 P holds.

L = 5,

c = 4,

16. ƒ(x) = 2x - 2,

L = - 6,

17. ƒ(x) = 2x + 1,

L = 1,

18. ƒ(x) = 2x,

L = 1>2,

19. ƒ(x) = 219 - x,

20. ƒ(x) = 2x - 7, 21. ƒ(x) = 1>x,

L = 3, L = 4,

L = 1>4,

2

L = 3,

2

L = 4,

22. ƒ(x) = x , 23. ƒ(x) = x , 24. ƒ(x) = 1>x,

c = 1>4,

P = 0.1

c = 10,

P = 1

c = 23,

c = - 2,

c = - 1, c = 4,

L = 5,

P = 0.02 P = 0.1

c = 23,

L = 11,

26. ƒ(x) = 120>x,

c = - 2, c = 0,

c = 4,

L = - 1,

25. ƒ(x) = x2 - 5,

P = 0.01

c = 24,

P = 1 P = 0.05

P = 0.1

m 7 0,

L = 3m,

Prove the limit statements in Exercises 37–50. 37. lim (9 - x) = 5 38. lim (3x - 7) = 2 xS3

39. lim 2x - 5 = 2

40. lim 24 - x = 2

41. lim ƒ(x) = 1 if ƒ(x) = e xS1

x, 2,

42. lim ƒ(x) = 4 if ƒ(x) = e x S -2

1 43. lim x = 1 xS1 45. lim

x S -3

x ≠ 1 x = 1

x ≠ -2 x = -2

x 2, 1,

44. lim

x S 23

x2 - 9 = -6 x + 3

46. lim

xS1

47. lim ƒ(x) = 2 if ƒ(x) = e xS1

48. lim ƒ(x) = 0 if ƒ(x) = e xS0

x2 - 1 = 2 x - 1

4 - 2x, 6x - 4, 2x, x>2,

1 1 = 3 x2

x 6 1 x Ú 1 x 6 0 x Ú 0

1 49. lim x sin x = 0 xS0

y

P = 1 P = 1 c = 3,

L = (m>2) + b,

30. ƒ(x) = mx + b, P = 0.05

L = m + b,

M02_THOM9580_03_SE_C02_065-128.indd 90

xS0

2

P = 0.1

29. ƒ(x) = mx + b, m 7 0, c = 1>2, P = c 7 0 m 7 0,

P = 0.5

P = 0.5

27. ƒ(x) = mx, m 7 0, L = 2m, c = 2, P = 0.03 28. ƒ(x) = mx,

P = 0.05

P = 0.4

xS9

15. ƒ(x) = x + 1,

P = 0.03

xS4

NOT TO SCALE

0 ƒ(x) - L 0 6 P.

1

P = c 7 0

− 1 −p

1 2p

y = x sin 1x

1 2p 1 p

x

c = 1,

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When Is a Number L Not the Limit of ƒ(x) as x u c? Showing L is not a limit  We can prove that limx S c ƒ(x) ≠ L by providing an P 7 0 such that no possible d 7 0 satisfies the condition

1 50. lim x2 sin x = 0 xS0

y 1

for all x, 0 6 0 x - c 0 6 d

y = x2

0 ƒ(x) - L 0 6 P.

1

We accomplish this for our candidate P by showing that for each d 7 0 there exists a value of x such that y= −1

0

2 −p

x 2 sin

2 p

1 x x

1

0 6 0x - c0 6 d

0 ƒ(x) - L 0 Ú P.

and

y y = f(x) L+P

−1

L

y = −x 2

L−P f (x)

Theory and Examples 51. Define what it means to say that lim g(x) = k.

0

xS0

52. Prove that lim ƒ(x) = L if and only if lim ƒ(h + c) = L. xSc

53. A wrong statement about limits  Show by example that the following statement is wrong.

Explain why the function in your example does not have the given value of L as a limit as x S c. 54. Another wrong statement about limits  Show by example that the following statement is wrong.

57. Let ƒ(x) = e

x, x 6 1 x + 1, x 7 1.

y y=x+1

The number L is the limit of ƒ(x) as x approaches c if, given any P 7 0, there exists a value of x for which 0 ƒ(x) - L 0 6 P.

2

Explain why the function in your example does not have the given value of L as a limit as x S c. T 55. Grinding engine cylinders  Before contracting to grind engine cylinders to a cross-sectional area of 9 in2, you need to know how much deviation from the ideal cylinder diameter of c = 3.385 in. you can allow and still have the area come within 0.01 in2 of the required 9 in2. To find out, you let A = p(x>2)2 and look for the interval in which you must hold x to make 0 A - 9 0 … 0.01. What interval do you find?

56. Manufacturing electrical resistors  Ohm’s law for electrical circuits like the one shown in the accompanying figure states that V = RI. In this equation, V is a constant voltage, I is the current in amperes, and R is the resistance in ohms. Your firm has been asked to supply the resistors for a circuit in which V will be 120 volts and I is to be 5 { 0.1 amp. In what interval does R have to lie for I to be within 0.1 amp of the value I0 = 5?   − V +

I

R

x

c+ d

c

a value of x for which 0 < 0 x − c 0 < d and 0 f(x) − L 0 ≥ P

hS0

The number L is the limit of ƒ(x) as x approaches c if ƒ(x) gets closer to L as x approaches c.

c− d

y = f(x) 1 x

1 y=x

a. Let P = 1>2. Show that no possible d 7 0 satisfies the following condition: For all x, 0 6 0 x - 1 0 6 d

1

0 ƒ(x) - 2 0 6 1>2.

That is, for each d 7 0 show that there is a value of x such that 0 6 0x - 10 6 d

and

0 ƒ(x) - 2 0 Ú 1>2.

This will show that limx S 1 ƒ(x) ≠ 2. b. Show that limx S 1 ƒ(x) ≠ 1. c. Show that limx S 1 ƒ(x) ≠ 1.5.

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Chapter 2: Limits and Continuity

60. a. For the function graphed here, show that limx S -1 g(x) ≠ 2.

x 2, x 6 2 58. Let h(x) = c 3, x = 2 2, x 7 2.

b. Does limx S -1 g(x) appear to exist? If so, what is the value of the limit? If not, why not? y

y

2 y = h(x)

4

y = g(x) 3

1 y=2

2 1

y = x2

0

2

x

0

−1 x

COMPUTER EXPLORATIONS In Exercises 61–66, you will further explore finding deltas graphically. Use a CAS to perform the following steps:

Show that a. lim h(x) ≠ 4 xS2

a. Plot the function y = ƒ(x) near the point c being approached.

b. lim h(x) ≠ 3

b. Guess the value of the limit L and then evaluate the limit symbolically to see if you guessed correctly.

xS2

c. lim h(x) ≠ 2 xS2

c. Using the value P = 0.2, graph the banding lines y1 = L - P and y2 = L + P together with the function ƒ near c.

59. For the function graphed here, explain why a. lim ƒ(x) ≠ 4

d. From your graph in part (c), estimate a d 7 0 such that for all x

xS3

b. lim ƒ(x) ≠ 4.8

0 6 0x - c0 6 d

xS3

c. lim ƒ(x) ≠ 3 xS3

y

1

0 ƒ(x) - L 0 6 P.

Test your estimate by plotting ƒ, y1, and y2 over the interval 0 6 0 x - c 0 6 d. For your viewing window use c - 2d … x … c + 2d and L - 2P … y … L + 2P. If any function values lie outside the interval 3L - P, L + P], your choice of d was too large. Try again with a smaller estimate.

e. Repeat parts (c) and (d) successively for P = 0.1, 0.05, and 0.001.

4.8 4

y = f (x)

3

5x3 + 9x2 , c = 0 2x5 + 3x2 x(1 - cos x) 64. ƒ(x) = , c = 0 x - sin x

61. ƒ(x) =

x4 - 81 , c = 3 x - 3

63. ƒ(x) =

sin 2x , c = 0 3x

65. ƒ(x) =

2x - 1

66. ƒ(x) =

3x2 - (7x + 1) 2x + 5 , c = 1 x - 1

62. ƒ(x) =

3

0

3

x

x - 1

, c = 1

2.4  One-Sided Limits In this section we extend the limit concept to one-sided limits, which are limits as x approaches the number c from the left-hand side (where x 6 c) or the right-hand side (x 7 c) only.

Approaching a Limit from One Side To have a limit L as x approaches c, a function ƒ must be defined on both sides of c and its values ƒ(x) must approach L as x approaches c from either side. That is, ƒ must be defined in some open interval about c, but not necessarily at c. Because of this, ordinary limits are called two-sided.

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93

2.4  One-Sided Limits

y y= x 0x0

1

x

0

−1

If ƒ fails to have a two-sided limit at c, it may still have a one-sided limit, that is, a limit if the approach is only from one side. If the approach is from the right, the limit is a right-hand limit. From the left, it is a left-hand limit. The function ƒ(x) = x> 0 x 0 (Figure 2.24) has limit 1 as x approaches 0 from the right, and limit -1 as x approaches 0 from the left. Since these one-sided limit values are not the same, there is no single number that ƒ(x) approaches as x approaches 0. So ƒ(x) does not have a (two-sided) limit at 0. Intuitively, if ƒ(x) is defined on an interval (c, b), where c 6 b, and approaches arbitrarily close to L as x approaches c from within that interval, then ƒ has right-hand limit L at c. We write lim ƒ(x) = L.

x S c+

Figure 2.24  Different right-hand and left-hand limits at the origin.

The symbol “x S c+ ” means that we consider only values of x greater than c. Similarly, if ƒ(x) is defined on an interval (a, c), where a 6 c and approaches arbitrarily close to M as x approaches c from within that interval, then ƒ has left-hand limit M at c. We write lim ƒ(x) = M.

x S c-

The symbol “x S c- ” means that we consider only x-values less than c. These informal definitions of one-sided limits are illustrated in Figure 2.25. For the function ƒ(x) = x> 0 x 0 in Figure 2.24 we have lim ƒ(x) = 1

lim ƒ(x) = -1.

and

x S 0+

x S 0-

y

y

f (x)

L 0

c

M

f (x) x

x

0

(a) lim+ f (x) = L

x

c

x

(b) lim _ f(x) = M

x: c

x: c

Figure 2.25  (a) Right-hand limit as x approaches c. (b) Left-hand limit as x approaches c. y

Example 1   The domain of ƒ(x) = 24 - x2 is 3 -2, 24 ; its graph is the semicircle in Figure 2.26. We have

y = "4 − x 2

lim 24 - x2 = 0

x S -2 +

−2

0

2

x

Figure 2.26  The function ƒ(x) = 24 - x2 has right-hand limit 0 at x = - 2 and left-hand limit 0 at x = 2 (Example 1).

and

lim 24 - x2 = 0.

x S 2-

The function does not have a left-hand limit at x = -2 or a right-hand limit at x = 2. It does not have a two-sided limit at either -2 or 2 because each point does not belong to an open interval over which ƒ is defined. One-sided limits have all the properties listed in Theorem 1 in Section 2.2. The right-hand limit of the sum of two functions is the sum of their right-hand limits, and so on. The theorems for limits of polynomials and rational functions hold with one-sided limits, as do the Sandwich Theorem and Theorem 5. One-sided limits are related to limits in the following way.

THEOREM 6  A function ƒ(x) has a limit as x approaches c if and only if it has left-hand and right-hand limits there and these one-sided limits are equal: lim ƒ(x) = L

xSc

M02_THOM9580_03_SE_C02_065-128.indd 93

3

lim ƒ(x) = L

x S c-

and

lim ƒ(x) = L.

x S c+

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Chapter 2: Limits and Continuity

Example 2  For the function graphed in Figure 2.27,

y y = f (x)

2

At x = 0:

limx S 0+ ƒ(x) = 1, limx S 0- ƒ(x) and limx S 0 ƒ(x) do not exist. The function is not defined to the left of x = 0.

At x = 1:

limx S 1- ƒ(x) = 0 even though ƒ(1) = 1, limx S 1+ ƒ(x) = 1, limx S 1 ƒ(x) does not exist. The right- and left-hand limits are not equal.

At x = 2:

limx S 2- ƒ(x) = 1, limx S 2+ ƒ(x) = 1, limx S 2 ƒ(x) = 1 even though ƒ(2) = 2.

At x = 3:

limx S 3- ƒ(x) = limx S 3+ ƒ(x) = limx S 3 ƒ(x) = ƒ(3) = 2.

At x = 4:

limx S 4- ƒ(x) = 1 even though ƒ(4) ≠ 1, limx S 4+ ƒ(x) and limx S 4 ƒ(x) do not exist. The function is not defined to the right of x = 4.

1

0

1

2

3

x

4

Figure 2.27  Graph of the function in Example 2.

y

At every other point c in 3 0, 44, ƒ(x) has limit ƒ(c).

L+P

f(x) f(x) lies in here

L

Precise Definitions of One-Sided Limits The formal definition of the limit in Section 2.3 is readily modified for one-sided limits.

L−P for all x ≠ c in here

DEFINITIONS  We say that ƒ(x) has right-hand limit L at c, and write

d x 0

c

c+d

x

lim ƒ(x) = L  (see Figure 2.28)

x S c+

if for every number P 7 0 there exists a corresponding number d 7 0 such that for all x

Figure 2.28  Intervals associated with the definition of right-hand limit.

c 6 x 6 c + d

1

0 ƒ(x) - L 0 6 P.

We say that ƒ has left-hand limit L at c, and write y

lim ƒ(x) = L  (see Figure 2.29)

x S c-

if for every number P 7 0 there exists a corresponding number d 7 0 such that for all x L+P L

c - d 6 x 6 c f(x) f(x) lies in here

0 ƒ(x) - L 0 6 P.

Example 3  Prove that

L−P

lim 2x = 0.

x S 0+

for all x ≠ c in here

Solution  Let P 7 0 be given. Here c = 0 and L = 0, so we want to find a d 7 0 such that for all x

d x 0

1

c−d

c

Figure 2.29  Intervals associated with the definition of left-hand limit.

M02_THOM9580_03_SE_C02_065-128.indd 94

0 6 x 6 d

x

0 2x - 0 0 6 P,

1

or 0 6 x 6 d

1

2x 6 P.

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2.4  One-Sided Limits

y

Squaring both sides of this last inequality gives x 6 P2

f (x) = "x

0 6 x 6 d.

if

2

If we choose d = P we have

P

0 6 x 6 d = P2

f(x)

L=0

95

x

d=

P2

x

Figure 2.30  lim+ 1x = 0 in Example 3. xS0

1

2x 6 P,

or 0 6 x 6 P2

1

0 2x - 0 0 6 P.

According to the definition, this shows that limx S 0+ 2x = 0 (Figure 2.30).

The functions examined so far have had some kind of limit at each point of interest. In general, that need not be the case.

Example 4  Show that y = sin (1>x) has no limit as x approaches zero from either side (Figure 2.31).

y 1

x

0

y = sin 1x −1

Figure 2.31  The function y = sin (1>x) has neither a righthand nor a left-hand limit as x approaches zero (Example 4). The graph here omits values very near the y-axis.

Solution  As x approaches zero, its reciprocal, 1>x, grows without bound and the values of sin (1>x) cycle repeatedly from -1 to 1. There is no single number L that the function’s values stay increasingly close to as x approaches zero. This is true even if we restrict x to positive values or to negative values. The function has neither a right-hand limit nor a lefthand limit at x = 0.

Limits Involving (sin U) , U

A central fact about (sin u)>u is that in radian measure its limit as u S 0 is 1. We can see this in Figure 2.32 and confirm it algebraically using the Sandwich Theorem. You will see the importance of this limit in Section 3.5, where instantaneous rates of change of the trigonometric functions are studied. y 1

−3p

−2p

y = sin u (radians) u

−p

p

2p

3p

u

NOT TO SCALE

Figure 2.32  The graph of ƒ(u) = (sin u)>u suggests that the rightand left-hand limits as u approaches 0 are both 1.

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96

Chapter 2: Limits and Continuity

y

THEOREM 7—Limit of the Ratio sin U , U as U u 0

T

1

lim

uS0

sin u = 1 u

P

tan u

1 sin u

u

cos u

O

Q

A(1, 0)

x

1

Proof  The plan is to show that the right-hand and left-hand limits are both 1. Then we will know that the two-sided limit is 1 as well. To show that the right-hand limit is 1, we begin with positive values of u less than p>2 (Figure 2.33). Notice that Area ∆OAP 6 area sector OAP 6 area ∆OAT. We can express these areas in terms of u as follows:

Figure 2.33  The figure for the proof of Theorem 7. By definition, TA>OA = tan u, but OA = 1, so TA = tan u.

Area ∆OAP =

1 1 1 base * height = (1)(sin u) = sin u 2 2 2

Area sector OAP = Area ∆OAT =

u 1 2 1 r u = (1)2u = (2) 2 2 2 1 1 1 base * height = (1)(tan u) = tan u. 2 2 2

Thus, Equation (2) is where radian measure comes in: The area of sector OAP is u>2 only if u is measured in radians.

1 1 1 sin u 6 u 6 tan u. 2 2 2 This last inequality goes the same way if we divide all three terms by the number (1>2) sin u, which is positive, since 0 6 u 6 p>2: 1 6

u 1 6 . cos u sin u

Taking reciprocals reverses the inequalities: 1 7

sin u 7 cos u. u

Since limu S 0+ cos u = 1 (Example 11b, Section 2.2), the Sandwich Theorem gives lim

u S 0+

sin u = 1. u

To consider the left-hand limit, we recall that sin u and u are both odd functions (Section 1.1). Therefore, ƒ(u) = (sin u)>u is an even function, with a graph symmetric about the y-axis (see Figure 2.32). This symmetry implies that the left-hand limit at 0 exists and has the same value as the right-hand limit: lim

u S 0-

sin u sin u = 1 = lim+ , u u uS0

so limu S 0 (sin u)>u = 1 by Theorem 6.

Example 5  Show that (a) lim

hS0

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cos h - 1 sin 2x 2 = 0 and (b) lim = . h 5 x S 0 5x

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2.4  One-Sided Limits

Solution (a) Using the half-angle formula cos h = 1 - 2 sin2 (h>2), we calculate 2 sin2 (h>2) cos h - 1 = lim h h hS0 hS0 lim

= - lim

uS0

sin u sin u u

Let u = h>2. Eq. (1) and Example 11a in Section 2.2

= -(1)(0) = 0.

(b) Equation (1) does not apply to the original fraction. We need a 2x in the denominator,

not a 5x. We produce it by multiplying numerator and denominator by 2>5: (2>5) # sin 2x sin 2x = lim x S 0 5x x S 0 (2>5) # 5x lim

=

Now, Eq. (1) applies sin 2x 2 lim      with u = 2x. 5 x S 0 2x

=

2 2 (1) = 5 5

Example 6  Find lim

tS0

tan t sec 2t . 3t

Solution  From the definition of tan t and sec 2t, we have lim

tS0

tan t sec 2t 1 1 sin t 1 = lim # t # cos t # 3t cos 2t tS0 3 =

sin t # 1 # 1 1 lim 3 t S 0 t cos t cos 2t

=

1 1 (1)(1)(1) = . 3 3

Eq. (1) and Example 11b in Section 2.2

Exercises  2.4 Finding Limits Graphically 1. Which of the following statements about the function y = ƒ(x) graphed here are true, and which are false?

y y = f (x) 2

y y = f (x)

1

1

−1

0

1

2

x

−1

lim- ƒ(x) = 0 a. lim + ƒ(x) = 1 b. x S -1

xS0

xS0

xS0

c. lim- ƒ(x) = 1 d. lim- ƒ(x) = lim+ ƒ(x) xS0

e. lim ƒ(x) exists. f. lim ƒ(x) = 0 xS0

xS0

g. lim ƒ(x) = 1 h. lim ƒ(x) = 1 xS0

xS1

0

1

2

3

x

lim ƒ(x) does not exist. a. lim + ƒ(x) = 1 b. x S -1

xS2

xS2

xS1

c. lim ƒ(x) = 2 d. lim- ƒ(x) = 2 e. lim+ ƒ(x) = 1 f. lim ƒ(x) does not exist. xS1

xS1

g. lim+ ƒ(x) = lim- ƒ(x) xS0

xS0

i. lim ƒ(x) = 0 j. lim- ƒ(x) = 2

h. lim ƒ(x) exists at every c in the open interval (- 1, 1).

k. lim - ƒ(x) does not exist. l. lim+ ƒ(x) = 0

i. lim ƒ(x) exists at every c in the open interval (1, 3).

xS1

x S -1

xS2

xS2

2. Which of the following statements about the function y = ƒ(x) graphed here are true, and which are false?

M02_THOM9580_03_SE_C02_065-128.indd 97

xSc xSc

j.

lim ƒ(x) = 0 k. lim+ ƒ(x) does not exist.

x S -1-

xS3

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98

Chapter 2: Limits and Continuity

6. Let g(x) = 2x sin(1>x).

3 - x, x 6 2 3. Let ƒ(x) = c x + 1, x 7 2. 2

y 1

y = "x

y y = "x sin 1x

y=3−x 3

1 2p

y= x+1 2 0

0

x

4

2

1

2 p

1 p

x

a. Find limx S 2+ ƒ(x) and limx S 2- ƒ(x). b. Does limx S 2 ƒ(x) exist? If so, what is it? If not, why not?

−1

c. Find limx S 4- ƒ(x) and limx S 4+ ƒ(x).

a. Does limx S 0+ g(x) exist? If so, what is it? If not, why not?

d. Does limx S 4 ƒ(x) exist? If so, what is it? If not, why not?

4. Let ƒ(x) = d

b. Does limx S 0- g(x) exist? If so, what is it? If not, why not?

3 - x, x 6 2 2, x = 2 x , 2

c. Does limx S 0 g(x) exist? If so, what is it? If not, why not? x 3, x≠1 0, x = 1. b. Find limx S 1- ƒ(x) and limx S 1+ ƒ(x).

7. a. Graph ƒ(x) = e

x 7 2.

y

c. Does limx S 1 ƒ(x) exist? If so, what is it? If not, why not? 1 - x 2, x≠1 2, x = 1. b. Find limx S 1+ ƒ(x) and limx S 1- ƒ(x).

y=3−x

8. a. Graph ƒ(x) = e

3 y= x 2 0

−2

y = −"x

2

c. Does limx S 1 ƒ(x) exist? If so, what is it? If not, why not?

x

Graph the functions in Exercises 9 and 10. Then answer these questions. a. Find limx S 2+ ƒ(x), limx S 2- ƒ(x), and ƒ(2).

a. What are the domain and range of ƒ?

b. Does limx S 2 ƒ(x) exist? If so, what is it? If not, why not?

b. At what points c, if any, does limx S c ƒ(x) exist?

c. Find limx S -1- ƒ(x) and limx S -1+ ƒ(x).

c. At what points does only the left-hand limit exist?

d. Does limx S -1 ƒ(x) exist? If so, what is it? If not, why not?

d. At what points does only the right-hand limit exist?

0, 5. Let ƒ(x) = c

21 - x2,

x … 0

9. ƒ(x) = c 1, 2,

1 sin x , x 7 0.

y 1

x

0 y=

x≤0 1 sin x , x > 0

0,

−1

x, 10. ƒ(x) = c 1, 0,

b. Does limx S 0- ƒ(x) exist? If so, what is it? If not, why not? c. Does limx S 0 ƒ(x) exist? If so, what is it? If not, why not?

M02_THOM9580_03_SE_C02_065-128.indd 98

-1 … x 6 0, or 0 6 x … 1 x = 0 x 6 - 1 or x 7 1

Finding One-Sided Limits Algebraically Find the limits in Exercises 11–18.

11.

x + 2 + 1

lim

12. lim+

x S -0.5- A x

13. lim + a x S -2

a. Does limx S 0+ ƒ(x) exist? If so, what is it? If not, why not?

0 … x 6 1 1 … x 6 2 x = 2

14. lim- a xS1

15. lim+ hS0

xS1

x 2x + 5 ba b x + 1 x2 + x

x - 1 Ax + 2

x + 6 3 - x 1 ba x ba b 7 x + 1

2h2 + 6h + 13 - 213

h

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2.5  Continuity

16. limhS0

26 - 25h2 + 11h + 6

x S -2

xS1

uS0

h

17. a. lim +(x + 3) 18. a. lim+

41. lim

0x + 20 x + 2

22x (x - 1)

0x - 10

b. lim -(x + 3) x S -2

0x + 20 x + 2

22x (x - 1) b. limS x 1 0x - 10

Use the graph of the greatest integer function y = : x ; , Figure 1.10 in Section 1.1, to help you find the limits in Exercises 19 and 20. :u ; :u ; 19. a. lim+ b. limuS3 u uS3 u 20. a. lim+(t - : t ; ) b. lim-(t - : t ; ) tS4

tS4

sin U = 1 U Find the limits in Exercises 21–42. Using lim

Uu 0

2sin 29u 29u sin 19y 23. lim 7y yS0 tan 3x 2 5. lim x

22. lim

21. lim

uS0

tS0

h sin 3h 2t 26. lim tan t tS0 24. limhS0

xS0

27. 29. 31. 33. 35. 37.

x csc 5x lim x S 0 cos 12x 11x + 11x cos 11x lim x S 0 8sin 11x cos 11x 1 - cos u lim u S 0 sin 2u sin (1 - cos t) lim S 1 - cos t t 0 sin u lim u S 0 sin 2u lim u cos u

28. lim 6x2(cot x)(csc 2x) xS0

xS0

x2 - x + sin x 2x x - x cos x lim xS0 sin2 3x sin (sin h) lim sin h hS0 sin 5x lim x S 0 sin 4x lim sin u cot 2u

30. lim

xS0

32. 34. 36. 38.

uS0

39. lim

sin kt (k constant) t

tan 3x sin 8x

uS0

40. lim

yS0

sin 3y cot 5y y cot 4y

tan u u 2 cot 3u

42. lim

uS0

99

u cot 4u sin2 u cot2 2u

Theory and Examples 43. Once you know limx S a+ ƒ(x) and limx S a- ƒ(x) at an interior point of the domain of ƒ, do you then know limx S a ƒ(x)? Give reasons for your answer.

44. If you know that limx S c ƒ(x) exists, can you find its value by calculating limx S c+ ƒ(x)? Give reasons for your answer. 45. Suppose that ƒ is an odd function of x. Does knowing that limx S 0+ ƒ(x) = 3 tell you anything about limx S 0- ƒ(x)? Give reasons for your answer. 46. Suppose that ƒ is an even function of x. Does knowing that limx S 2- ƒ(x) = 7 tell you anything about either limx S -2- ƒ(x) or limx S -2+ ƒ(x)? Give reasons for your answer. Formal Definitions of One-Sided Limits 47. Given P 7 0, find an interval I = (5, 5 + d), d 7 0, such that if x lies in I, then 2x - 5 6 P. What limit is being verified and what is its value?

48. Given P 7 0, find an interval I = (4 - d, 4), d 7 0, such that if x lies in I, then 24 - x 6 P. What limit is being verified and what is its value? Use the definitions of right-hand and left-hand limits to prove the limit statements in Exercises 49 and 50. x x - 2 = -1 50. lim+ = 1 49. limxS0 0 x 0 xS2 0 x - 2 0 51. Greatest integer function Find (a) limx S 400+ : x ; and (b) limx S 400- : x ; ; then use limit definitions to verify your findings. (c) Based on your conclusions in parts (a) and (b), can you say anything about limx S 400 : x ; ? Give reasons for your answer.

52. One-sided limits Let ƒ(x) = e

x2 sin (1>x), x 6 0 2x, x 7 0.

Find (a) limx S 0+ ƒ(x) and (b) limx S 0- ƒ(x); then use limit definitions to verify your findings. (c) Based on your conclusions in parts (a) and (b), can you say anything about limx S 0 ƒ(x)? Give reasons for your answer.

2.5 Continuity y

Distance fallen (m)

500

Q4 Q3

375 Q2

250 125 0

When we plot function values generated in a laboratory or collected in the field, we often connect the plotted points with an unbroken curve to show what the function’s values are likely to have been at the points we did not measure (Figure 2.34). In doing so, we are assuming that we are working with a continuous function, so its outputs vary regularly and consistently with the inputs, and do not jump abruptly from one value to another without taking on the values in between. Intuitively, any function y = ƒ(x) whose graph can be sketched over its domain in one unbroken motion is an example of a continuous function. Such functions play an important role in the study of calculus and its applications.

Q1

5 Elapsed time (sec)

10

t

Figure 2.34  Connecting plotted points by an unbroken curve from experimental data Q1, Q2, Q3, c for a falling object.

M02_THOM9580_03_SE_C02_065-128.indd 99

Continuity at a Point To understand continuity, it helps to consider a function like that in Figure 2.35, whose limits we investigated in Example 2 in the last section.

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100

Chapter 2: Limits and Continuity

Example 1   At which numbers does the function ƒ in Figure 2.35 appear to be not continuous? Explain why. What occurs at other numbers in the domain? Solution  First we observe that the domain of the function is the closed interval 3 0, 44, so we will be considering the numbers x within that interval. From the figure, we notice right away that there are breaks in the graph at the numbers x = 1, x = 2, and x = 4. The breaks appear as jumps, which we identify later as “jump discontinuities.” These are numbers for which the function is not continuous, and we discuss each in turn.

y y = f (x)

2 1

0

1

2

3

Numbers at which the graph of ƒ has breaks:

x

4

Figure 2.35  The function is not continuous at x = 1, x = 2, and x = 4 (Example 1).

At x = 1, the function fails to have a limit. It does have both a left-hand limit, limx S 1- ƒ(x) = 0, as well as a right-hand limit, limx S 1+ ƒ(x) = 1, but the limit values are different, resulting in a jump in the graph. The function is not continuous at x = 1. At x = 2, the function does have a limit, limx S 2 ƒ(x) = 1, but the value of the function is ƒ(2) = 2. The limit and function values are not the same, so there is a break in the graph and ƒ is not continuous at x = 2. At x = 4, the function does have a left-hand limit at this right endpoint, limx S 4- ƒ(x) = 1, but again the value of the function ƒ(4) = 12 differs from the value of the limit. We see again a break in the graph of the function at this endpoint and the function is not continuous from the left. Numbers at which the graph of ƒ has no breaks: At x = 0, the function has a right-hand limit at this left endpoint, limx S 0+ ƒ(x) = 1, and the value of the function is the same, ƒ(0) = 1. So no break occurs in the graph of the function at this endpoint, and the function is continuous from the right at x = 0. At x = 3, the function has a limit, limx S 3 ƒ(x) = 2. Moreover, the limit is the same value as the function there, ƒ(3) = 2. No break occurs in the graph and the function is continuous at x = 3. At all other numbers x = c in the domain, which we have not considered, the function has a limit equal to the value of the function at the point, so limx S c ƒ(x) = ƒ(c). For example, limx S 5>2 ƒ(x) = ƒ1 52 2 = 32 . No breaks appear in the graph of the function at any of these remaining numbers and the function is continuous at each of them. The following definitions capture the continuity ideas we observed in Example 1.

DEFINITIONS Let c be a real number on the x-axis. The function ƒ is continuous at c if lim ƒ(x) = ƒ(c).

xSc

The function ƒ is right-continuous at c (or continuous from the right) if lim ƒ(x) = ƒ(c).

x S c+

Continuity from the right

Two-sided continuity

The function ƒ is left-continuous at c (or continuous from the left) if Continuity from the left

lim ƒ(x) = ƒ(c).

x S c-

y = f (x) a

c

b

x

Figure 2.36  Continuity at points a, b, and c.

M02_THOM9580_03_SE_C02_065-128.indd 100

From Theorem 6, it follows immediately that a function ƒ is continuous at an interior point c of its domain if and only if it is both right-continuous and left-continuous at c (Figure 2.36). We say that a function is continuous over a closed interval 3 a, b4 if it is rightcontinuous at a, left-continuous at b, and continuous at all interior points of the interval.

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2.5  Continuity

This definition applies to the infinite closed intervals 3 a, q) and (- q, b4 as well, but only one endpoint is involved. If a function is not continuous at an interior point c of its domain, we say that ƒ is discontinuous at c, and that c is a point of discontinuity of ƒ. Note that a function ƒ can be continuous, right-continuous, or left-continuous only at a point c for which ƒ(c) is defined.

y y = "4 − x 2

2

x

2

0

−2

101

Example 2  The function ƒ(x) = 24 - x2 is continuous over its domain 3 -2, 24

Figure 2.37  A function that is continuous over its domain (Example 2).

(Figure 2.37). It is right-continuous at x = -2, and left-continuous at x = 2.

Example 3  The unit step function U(x), graphed in Figure 2.38, is right-continuous at x = 0, but is neither left-continuous nor continuous there. It has a jump discontinuity at x = 0.

y

We summarize continuity at an interior point in the form of a test. y = U(x)

1

Continuity Test A function ƒ(x) is continuous at a point x = c if and only if it meets the following three conditions. 1. ƒ(c) exists (c lies in the domain of ƒ). 2. limx S c ƒ(x) exists (ƒ has a limit as x S c). 3. limx S c ƒ(x) = ƒ(c) (the limit equals the function value).

x

0

Figure 2.38  A function that has a jump discontinuity at the origin (Example 3).

For one-sided continuity and continuity at an endpoint of an interval, the limits in parts 2 and 3 of the test should be replaced by the appropriate one-sided limits.

Example 4  The function y = : x ; introduced in Section 1.1 is graphed in Figure 2.39. It is discontinuous at every integer because the left-hand and right-hand limits are not equal as x S n:

y 4

lim : x ; = n - 1   and   lim+ : x ; = n .

x S n-

3 2 1 1

−1

2

xSn

Since : n ; = n, the greatest integer function is right-continuous at every integer n (but not left-continuous). The greatest integer function is continuous at every real number other than the integers. For example,

y = :x;

3

4

lim : x ; = 1 = : 1.5 ; .

x

x S 1.5

In general, if n - 1 6 c 6 n, n an integer, then −2

Figure 2.39  The greatest integer function is continuous at every noninteger point. It is right-continuous, but not left-continuous, at every integer point (Example 4).

M02_THOM9580_03_SE_C02_065-128.indd 101

lim : x ; = n - 1 = : c ; .

xSc

Figure 2.40 displays several common types of discontinuities. The function in Figure 2.40a is continuous at x = 0. The function in Figure 2.40b would be continuous if it had ƒ(0) = 1. The function in Figure 2.40c would be continuous if ƒ(0) were 1 instead of 2. The discontinuity in Figure 2.40c is removable. The function has a limit as x S 0, and we can remove the discontinuity by setting ƒ(0) equal to this limit. The discontinuities in Figure 2.40d through f are more serious: limx S 0 ƒ(x) does not exist, and there is no way to improve the situation by changing ƒ at 0. The step function in Figure 2.40d has a jump discontinuity: The one-sided limits exist but have different values. The function ƒ(x) = 1>x2 in Figure 2.40e has an infinite discontinuity. The function in Figure 2.40f has an oscillating discontinuity: It oscillates too much to have a limit as x S 0.

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102

Chapter 2: Limits and Continuity

y = f (x)

0

2

y = f (x)

1

1 x

x

1 x

0

(b)

(c) y

y y = f(x) = 12 x

1

−1 0

y = f (x)

1

0

(a)

y

y

y

y

y = f(x)

0

x

(d) y = sin 1x

0

x

x −1 (f)

(e)

Figure 2.40  The function in (a) is continuous at x = 0; the functions in (b) through (f ) are not.

Continuous Functions Generally, we want to describe the continuity behavior of a function throughout its entire domain, not only at a single point. We know how to do that if the domain is a closed interval. In the same way, we define a continuous function as one that is continuous at every point in its domain. This is a property of the function. A function always has a specified domain, so if we change the domain, we change the function, and this may change its continuity property as well. If a function is discontinuous at one or more points of its domain, we say it is a discontinuous function.

Example 5   (a) The function y = 1>x (Figure 2.41) is a continuous function because it is continuous

at every point of its domain. It has a point of discontinuity at x = 0, however, because it is not defined there; that is, it is discontinuous on any interval containing x = 0. (b) The identity function ƒ(x) = x and constant functions are continuous everywhere by Example 3, Section 2.3.

y y = 1x

Algebraic combinations of continuous functions are continuous wherever they are defined. 0

x

Figure 2.41  The function y = 1>x is continuous over its natural domain. It has a point of discontinuity at the origin, so it is discontinuous on any interval containing x = 0 (Example 5).

THEOREM 8—Properties of Continuous Functions  If the functions ƒ and g are continuous at x = c, then the following algebraic combinations are continuous at x = c. 1. Sums: 2. Differences: 3. Constant multiples: 4. Products: 5. Quotients: 6. Powers:

ƒ + g ƒ - g k # ƒ, for any number k ƒ#g ƒ>g, provided g(c) ≠ 0 ƒn, n a positive integer n

7. Roots:  2ƒ, provided it is defined on an open interval containing c, where n is a positive integer

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2.5  Continuity

103

Most of the results in Theorem 8 follow from the limit rules in Theorem 1, Section 2.2. For instance, to prove the sum property we have lim (ƒ + g)(x) = lim (ƒ(x) + g(x))

xSc

xSc

= lim ƒ(x) + lim g(x)

Sum Rule, Theorem 1

= ƒ(c) + g(c) = (ƒ + g)(c).

Continuity of ƒ, g at c

xSc

xSc

This shows that ƒ + g is continuous.

Example 6

(a) Every polynomial P(x) = an xn + an - 1xn - 1 + g + a0 is continuous because

lim P(x) = P(c) by Theorem 2, Section 2.2.

xSc

(b) If P(x) and Q(x) are polynomials, then the rational function P(x)>Q(x) is continuous

wherever it is defined (Q(c) ≠ 0) by Theorem 3, Section 2.2.

Example 7  The function ƒ(x) = 0 x 0 is continuous. If x 7 0, we have ƒ(x) = x,

a polynomial. If x 6 0, we have ƒ(x) = -x, another polynomial. Finally, at the origin, limx S 0 0 x 0 = 0 = 0 0 0 .

The functions y = sin x and y = cos x are continuous at x = 0 by Example 11 of Section 2.2. Both functions are, in fact, continuous everywhere (see Exercise 70). It follows from Theorem 8 that all six trigonometric functions are then continuous wherever they are defined. For example, y = tan x is continuous on g ∪ (-p>2, p>2) ∪ (p>2, 3p>2) ∪ g.

Inverse Functions and Continuity The inverse function of any function continuous on an interval is continuous over its domain. This result is suggested by the observation that the graph of ƒ -1 , being the reflection of the graph of ƒ across the line y = x, cannot have any breaks in it when the graph of ƒ has no breaks. A rigorous proof that ƒ -1 is continuous whenever ƒ is continuous on an interval is given in more advanced texts. It follows that the inverse trigonometric functions are all continuous over their domains. We defined the exponential function y = ax in Section 1.5 informally by its graph. Recall that the graph was obtained from the graph of y = ax for x a rational number by “filling in the holes” at the irrational points x, so the function y = ax was defined to be continuous over the entire real line. The inverse function y = loga x is also continuous. In particular, the natural exponential function y = ex and the natural logarithm function y = ln x are both continuous over their domains.

Composites All composites of continuous functions are continuous. The idea is that if ƒ(x) is continuous at x = c and g(x) is continuous at x = ƒ(c), then g ∘ ƒ is continuous at x = c (Figure 2.42). In this case, the limit as x S c is g(ƒ(c)). g f ˚ Continuous at c

c

f

g

Continuous at c

Continuous at f(c) f (c)

g( f(c))

Figure 2.42  Composites of continuous functions are continuous.

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104

Chapter 2: Limits and Continuity

THEOREM 9—Composite of Continuous Functions  If ƒ is continuous at c and g is continuous at ƒ(c), then the composite g ∘ ƒ is continuous at c. Intuitively, Theorem 9 is reasonable because if x is close to c, then ƒ(x) is close to ƒ(c), and since g is continuous at ƒ(c), it follows that g(ƒ(x)) is close to g(ƒ(c)). The continuity of composites holds for any finite number of functions. The only requirement is that each function be continuous where it is applied. For an outline of a proof of Theorem 9, see Exercise 6 in Appendix 5.

Example 8   Show that the following functions are continuous on their natural domains. (a) y = 2x2 - 2x - 5 (c) y = `

y 0.4 0.3 0.2 0.1 −2p

−p

0

p

2p

Figure 2.43  The graph suggests that y = 0 (x sin x)>(x2 + 2) 0 is continuous (Example 8d).

x

x - 2 ` x2 - 2

(b) y =

x2>3 1 + x4

(d) y = `

x sin x ` x2 + 2

Solution (a) The square root function is continuous on 3 0, q) because it is a root of the continuous identity function ƒ(x) = x (Part 7, Theorem 8). The given function is then the composite of the polynomial ƒ(x) = x2 - 2x - 5 with the square root function g(t) = 2t , and is continuous on its natural domain. (b) The numerator is the cube root of the identity function squared; the denominator is an everywhere-positive polynomial. Therefore, the quotient is continuous. (c) The quotient (x - 2)>(x2 - 2) is continuous for all x ≠ { 22, and the function is the composition of this quotient with the continuous absolute value function (Example 7). (d) Because the sine function is everywhere-continuous (Exercise 70), the numerator term x sin x is the product of continuous functions, and the denominator term x2 + 2 is an everywhere-positive polynomial. The given function is the composite of a quotient of continuous functions with the continuous absolute value function (Figure 2.43). Theorem 9 is actually a consequence of a more general result, which we now state and prove. THEOREM 10—Limits of Continuous Functions  If g is continuous at the point b and limx S c ƒ(x) = b, then limx S c g(ƒ(x)) = g(b) = g(limx S c ƒ(x)).

Proof  Let P 7 0 be given. Since g is continuous at b, there exists a number d1 7 0 such that

0 g(y) - g(b) 0 6 P whenever 0 6 0 y - b 0 6 d1.

Since limx S c ƒ(x) = b, there exists a d 7 0 such that

0 ƒ(x) - b 0 6 d1 whenever 0 6 0 x - c 0 6 d.

If we let y = ƒ(x), we then have that

0 y - b 0 6 d1 whenever 0 6 0 x - c 0 6 d,

which implies from the first statement that 0 g(y) - g(b) 0 = 0 g(ƒ(x)) - g(b) 0 6 P whenever 0 6 0 x - c 0 6 d. From the definition of limit, this proves that limx S c g(ƒ(x)) = g(b).

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105

2.5  Continuity

Example 9  As an application of Theorem 10, we have the following calculations. (a)

lim cos a2x + sin a

x S p/2

(b) lim sin-1 a xS1

3p 3p + xb b = cos a lim 2x + lim sin a + xb b 2 2 x S p/2 x S p/2 = cos (p + sin 2p) = cos p = -1.

1 - x 1 - x b = sin-1 a lim b x S 1 1 - x2 1 - x2 = sin-1 a lim

xS1

= sin-1 We sometimes denote eu by exp u when u is a complicated mathematical expression.

(c)

Arcsine is continuous.

1 b 1 + x

Cancel common factor (1 - x).

1 p = 2 6

lim 2x + 1 etan x = lim 2x + 1 # exp a lim tan xb

xS0

xS0

= 1 # e0 = 1

xS0

Exponential is continuous.

Intermediate Value Theorem for Continuous Functions Functions that are continuous on intervals have properties that make them particularly useful in mathematics and its applications. One of these is the Intermediate Value Property. A function is said to have the Intermediate Value Property if whenever it takes on two values, it also takes on all the values in between. THEOREM 11—The Intermediate Value Theorem for Continuous Functions If ƒ is a continuous function on a closed interval 3 a, b4, and if y0 is any value between ƒ(a) and ƒ(b), then y0 = ƒ(c) for some c in 3 a, b4. y y = f (x) f (b)

y0 f (a)

y

0

a

c

b

x

3 2 1

0

1

2

3

4

x

Figure 2.44  The function 2x - 2, 1 … x 6 2 ƒ(x) = e 3, 2 … x … 4 does not take on all values between ƒ(1) = 0 and ƒ(4) = 3; it misses all the values between 2 and 3.

M02_THOM9580_03_SE_C02_065-128.indd 105

Theorem 11 says that continuous functions over finite closed intervals have the Intermediate Value Property. Geometrically, the Intermediate Value Theorem says that any horizontal line y = y0 crossing the y-axis between the numbers ƒ(a) and ƒ(b) will cross the curve y = ƒ(x) at least once over the interval 3 a, b4 . The proof of the Intermediate Value Theorem depends on the completeness property of the real number system (Appendix 7) and can be found in more advanced texts. The continuity of ƒ on the interval is essential to Theorem 11. If ƒ is discontinuous at even one point of the interval, the theorem’s conclusion may fail, as it does for the function graphed in Figure 2.44 (choose y0 as any number between 2 and 3). A Consequence for Graphing: Connectedness  Theorem 11 implies that the graph of a function continuous on an interval cannot have any breaks over the interval. It will be connected—a single, unbroken curve. It will not have jumps like the graph of the greatest integer function (Figure 2.39), or separate branches like the graph of 1>x (Figure 2.41).

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A Consequence for Root Finding  We call a solution of the equation ƒ(x) = 0 a root of the equation or zero of the function ƒ. The Intermediate Value Theorem tells us that if ƒ is continuous, then any interval on which ƒ changes sign contains a zero of the function. In practical terms, when we see the graph of a continuous function cross the horizontal axis on a computer screen, we know it is not stepping across. There really is a point where the function’s value is zero.

Example 10  Show that there is a root of the equation x3 - x - 1 = 0 between 1 and 2. Solution  Let ƒ(x) = x3 - x - 1. Since ƒ(1) = 1 - 1 - 1 = -1 6 0 and ƒ(2) = 23 - 2 - 1 = 5 7 0, we see that y0 = 0 is a value between ƒ(1) and ƒ(2). Since ƒ is continuous, the Intermediate Value Theorem says there is a zero of ƒ between 1 and 2. Figure 2.45 shows the result of zooming in to locate the root near x = 1.32. 5

1

1

1.6

2

−1 −2

−1 (a)

(b)

0.02

0.003

1.320

1.330

−0.02

1.3240

1.3248

−0.003 (c)

(d)

Figure 2.45  Zooming in on a zero of the function ƒ(x) = x3 - x - 1. The zero is near x = 1.3247 (Example 10).

Example 11  Use the Intermediate Value Theorem to prove that the equation y 4

has a solution (Figure 2.46).

y = 4 − x2

Solution  We rewrite the equation as

3

22x + 5 + x2 = 4,

2 y = " 2x + 5 1 0

22x + 5 = 4 - x2

c

2

Figure 2.46  The curves y = 22x + 5 and y = 4 - x2 have the same value at x = c where 22x + 5 = 4 - x2 (Example 11).

M02_THOM9580_03_SE_C02_065-128.indd 106

x

and set ƒ(x) = 22x + 5 + x2 . Now g(x) = 22x + 5 is continuous on the interval 3 -5>2, q) since it is the composite of the square root function with the nonnegative linear function y = 2x + 5. Then ƒ is the sum of the function g and the quadratic function y = x2, and the quadratic function is continuous for all values of x. It follows that ƒ(x) = 22x + 5 + x2 is continuous on the interval 3 -5>2, q). By trial and error, we find the function values ƒ(0) = 25 ≈ 2.24 and ƒ(2) = 29 + 4 = 7, and note that ƒ is also continuous on the finite closed interval 3 0, 24 ⊂ 3 -5>2, q). Since the value y0 = 4 is between the numbers 2.24 and 7, by the Intermediate Value Theorem there is a number c∊[0, 24 such that ƒ(c) = 4. That is, the number c solves the original equation.

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2.5  Continuity

Continuous Extension to a Point Sometimes the formula that describes a function ƒ does not make sense at a point x = c. It might nevertheless be possible to extend the domain of ƒ, to include x = c, creating a new function that is continuous at x = c. For example, the function y = ƒ(x) = (sin x)>x is continuous at every point except x = 0, since the origin is not in its domain. Since y = (sin x)>x has a finite limit as x S 0 (Theorem 7), we can extend the function’s domain to include the point x = 0 in such a way that the extended function is continuous at x = 0. We define the new function sin x , F(x) = • x 1,

x ≠ 0 x = 0.

The function F(x) is continuous at x = 0 because sin x lim x = F(0),

xS0

so it meets the requirements for continuity (Figure 2.47). y

y (0, 1)

p 2 a− , b 2 p −

p 2

(0, 1)

f (x) p 2 a , b 2 p

p 2 a− , b 2 p x

p 2

0

p 2

(a)

F(x) p 2 a , b 2 p

0

p 2

x

(b)

Figure 2.47  The graph (a) of ƒ(x) = (sin x)>x for - p>2 … x … p>2 does not include the point (0, 1) because the function is not defined at x = 0. (b) We can remove the discontinuity from the graph by defining the new function F(x) with F(0) = 1 and F(x) = ƒ(x) everywhere else. Note that F(0) = limx S 0 ƒ(x).

More generally, a function (such as a rational function) may have a limit at a point where it is not defined. If ƒ(c) is not defined, but limx S c ƒ(x) = L exists, we can define a new function F(x) by the rule y 2

y=

F(x) = e

2

x +x−6 x2 − 4

1 −1

0

1

−1

2

3

4

x

y=

The function F is continuous at x = c. It is called the continuous extension of ƒ to x = c. For rational functions ƒ, continuous extensions are often found by canceling common factors in the numerator and denominator.

ƒ(x) =

x+3 x+2

x2 + x - 6 ,  x ≠ 2 x2 - 4

has a continuous extension to x = 2, and find that extension.

1 0

if x is in the domain of ƒ if x = c.

Example 12  Show that

(a)

y 5 4

2

ƒ(x), L,

1

2

3

4

x

Solution  Although ƒ(2) is not defined, if x ≠ 2 we have

(b)

Figure 2.48  (a) The graph of ƒ(x) and (b) the graph of its continuous extension F(x) (Example 12).

M02_THOM9580_03_SE_C02_065-128.indd 107

ƒ(x) =

x2 + x - 6 (x - 2)(x + 3) x + 3 = = . (x - 2)(x + 2) x + 2 x2 - 4

The new function F(x) =

x + 3 x + 2

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Chapter 2: Limits and Continuity

is equal to ƒ(x) for x ≠ 2, but is continuous at x = 2, having there the value of 5>4. Thus F is the continuous extension of ƒ to x = 2, and x2 + x - 6 5 = lim ƒ(x) = . 4 xS2 xS2 x2 - 4 lim

The graph of ƒ is shown in Figure 2.48. The continuous extension F has the same graph except with no hole at (2, 5>4). Effectively, F is the function ƒ with its point of discontinuity at x = 2 removed.

Exercises  2.5 Continuity from Graphs In Exercises 1–4, say whether the function graphed is continuous on 3- 1, 34. If not, where does it fail to be continuous and why?

1.

2.

y

y

−1

2

c. Does limx S 1 ƒ(x) = ƒ(1)? d. Is ƒ continuous at x = 1?

1

0

1

2

x

3

0

−1

3.

1

3

x

y

2

2

1

1

0

1

2

3

10. To what new value should ƒ(1) be changed to remove the discontinuity?

y = k(x)

1

0

−1

2

Exercises 5–10 refer to the function

3

x

Applying the Continuity Test At which points do the functions in Exercises 11 and 12 fail to be continuous? At which points, if any, are the discontinuities removable? Not removable? Give reasons for your answers.

11. Exercise 1, Section 2.4

x2 - 1, 2x, ƒ(x) = e 1, - 2x + 4, 0,

-1 0 x 1 2

… x 6 x = 1 6 x 6 x

6 0 6 1 6 2 6 3

y

13. y =

1 - 3x x - 2

14. y =

1 + 4 (x + 2)2

15. y =

x + 1 x2 - 4x + 3

16. y =

x + 3 x2 - 3x - 10

y = −2x + 4

18. y =

21. y = csc 2x

22. y = tan

(1, 1) −1 y = x2 − 1

0

1

2

−1

The graph for Exercises 5–10.

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3

x

x2 1 0x0 + 1 2

17. y = 0 x - 1 0 + sin x

19. y =

(1, 2)

y = 2x

12. Exercise 2, Section 2.4

At what points are the functions in Exercises 13–30 continuous?

graphed in the accompanying figure.

y = f (x) 2

b. Is ƒ continuous at x = 2? 8. At what values of x is ƒ continuous?

x

7. a. Is ƒ defined at x = 2? (Look at the definition of ƒ.)

9. What value should be assigned to ƒ(2) to make the extended function continuous at x = 2?

y = h(x)

−1

2

4. y

d. Is ƒ continuous at x = -1? b. Does limx S 1 ƒ(x) exist?

y = g(x)

1

c. Does limx S -1+ ƒ(x) = ƒ(- 1)? 6. a. Does ƒ(1) exist?

y = f(x) 2

5. a. Does ƒ(-1) exist? b. Does limx S -1+ ƒ(x) exist?

23. y =

cos x x

x tan x x2 + 1

25. y = 22x + 3

27. y = (2x - 1)1>3

x + 2 20. y = cos x

24. y =

px 2

2x4 + 1

1 + sin2 x

4 26. y = 2 3x - 1

28. y = (2 - x)1>5

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2.5  Continuity

47. For what values of a and b is

x2 - x - 6 , x ≠ 3 29. g(x) = c x - 3 5, x = 3 x3 - 8 , x ≠ 2, x ≠ - 2 x2 - 4 30. ƒ(x) = d 3, x = 2 4, x = -2 Limits Involving Trigonometric Functions Find the limits in Exercises 31–38. Are the functions continuous at the point being approached? p 32. lim sina cos (tan t)b 31. lim sin (x - sin x) 2 xSp tS0

33. lim sec (y sec2 y - tan2 y - 1) yS1

34. lim tana xS0

p cos ( sin x1>3 ) b 4

35. lim cos a tS0

p

219 - 3 sec 2t

37. lim+ sin a xS0

p 2x e b 2

b 36. lim 2csc2 x + 5 13 tan x x S p/6

38. lim cos-1 1 ln 2x 2 xS1

Continuous Extensions 39. Define g(3) in a way that extends g(x) = (x2 - 9)>(x - 3) to be continuous at x = 3.

40. Define h(2) in a way that extends h(t) = (t 2 + 3t - 10)>(t - 2) to be continuous at t = 2. 41. Define ƒ(1) in a way that extends ƒ(s) = (s3 - 1)>(s2 - 1) to be continuous at s = 1. 42. Define g(4) in a way that extends g(x) = (x2 - 16)> (x2 - 3x - 4) to be continuous at x = 4. 43. For what value of a is ƒ(x) = e

x2 - 1, x 6 3 2ax, x Ú 3

continuous at every x? 44. For what value of b is

x, x 6 -2 bx2, x Ú - 2

continuous at every x?

a2x - 2a, x Ú 2 12, x 6 2

continuous at every x? 46. For what value of b is x - b , x 6 0 g(x) = c b + 1 x2 + b, x 7 0

M02_THOM9580_03_SE_C02_065-128.indd 109

continuous at every x? 48. For what values of a and b is ax + 2b, x … 0 g(x) = c x2 + 3a - b, 0 6 x … 2 3x - 5, x 7 2 continuous at every x? T In Exercises 49–52, graph the function ƒ to see whether it appears to have a continuous extension to the origin. If it does, use Trace and Zoom to find a good candidate for the extended function’s value at x = 0. If the function does not appear to have a continuous extension, can it be extended to be continuous at the origin from the right or from the left? If so, what do you think the extended function’s value(s) should be? 49. ƒ(x) = 51. ƒ(x) =

10 x - 1 x sin x 0x0

50. ƒ(x) =

10 0 x 0 - 1 x

52. ƒ(x) = (1 + 2x)1>x

Theory and Examples 53. A continuous function y = ƒ(x) is known to be negative at x = 0 and positive at x = 1. Why does the equation ƒ(x) = 0 have at least one solution between x = 0 and x = 1? Illustrate with a sketch.

54. Explain why the equation cos x = x has at least one solution. 55. Roots of a cubic  Show that the equation x3 - 15x + 1 = 0 has three solutions in the interval 3- 4, 4]. 56. A function value  Show that the function F(x) = (x - a)2 # (x - b)2 + x takes on the value (a + b)>2 for some value of x.

57. Solving an equation If ƒ(x) = x3 - 8x + 10, show that there are values c for which ƒ(c) equals (a) p; (b) - 23; (c) 5,000,000. 58. Explain why the following five statements ask for the same information. b. Find the x-coordinates of the points where the curve y = x3 crosses the line y = 3x + 1. c. Find all the values of x for which x3 - 3x = 1. d. Find the x-coordinates of the points where the cubic curve y = x3 - 3x crosses the line y = 1.

45. For what values of a is

continuous at every x?

-2, x … -1 ƒ(x) = c ax - b, - 1 6 x 6 1 3, x Ú 1

a. Find the roots of ƒ(x) = x3 - 3x - 1.

g(x) = e

ƒ(x) = b

109

e. Solve the equation x3 - 3x - 1 = 0. 59. Removable discontinuity  Give an example of a function ƒ(x) that is continuous for all values of x except x = 2, where it has a removable discontinuity. Explain how you know that ƒ is discontinuous at x = 2, and how you know the discontinuity is removable. 60. Nonremovable discontinuity  Give an example of a function g(x) that is continuous for all values of x except x = - 1, where it has a nonremovable discontinuity. Explain how you know that g is discontinuous there and why the discontinuity is not removable.

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Chapter 2: Limits and Continuity

61. A function discontinuous at every point a. Use the fact that every nonempty interval of real numbers contains both rational and irrational numbers to show that the function 1, if x is rational ƒ(x) = e 0, if x is irrational

is discontinuous at every point.

68. The sign-preserving property of continuous functions Let ƒ be defined on an interval (a, b) and suppose that ƒ(c) ≠ 0 at some c where ƒ is continuous. Show that there is an interval (c - d, c + d) about c where ƒ has the same sign as ƒ(c). 69. Prove that ƒ is continuous at c if and only if lim ƒ(c + h) = ƒ(c).

hS0

70. Use Exercise 69 together with the identities

b. Is ƒ right-continuous or left-continuous at any point? 62. If functions ƒ(x) and g(x) are continuous for 0 … x … 1, could ƒ(x)>g(x) possibly be discontinuous at a point of 30, 14? Give reasons for your answer.

63. If the product function h(x) = ƒ(x) # g(x) is continuous at x = 0, must ƒ(x) and g(x) be continuous at x = 0? Give reasons for your answer.

64. Discontinuous composite of continuous functions  Give an example of functions ƒ and g, both continuous at x = 0, for which the composite ƒ ∘ g is discontinuous at x = 0. Does this contradict Theorem 9? Give reasons for your answer. 65. Never-zero continuous functions  Is it true that a continuous function that is never zero on an interval never changes sign on that interval? Give reasons for your answer. 66. Stretching a rubber band  Is it true that if you stretch a rubber band by moving one end to the right and the other to the left, some point of the band will end up in its original position? Give reasons for your answer. 67. A fixed point theorem  Suppose that a function ƒ is continuous on the closed interval 30, 14 and that 0 … ƒ(x) … 1 for every x in 30, 14 . Show that there must exist a number c in 30, 14 such that ƒ(c) = c (c is called a fixed point of ƒ).

sin (h + c) = sin h cos c + cos h sin c, cos (h + c) = cos h cos c - sin h sin c to prove that both ƒ(x) = sin x and g(x) = cos x are continuous at every point x = c. Solving Equations Graphically T Use the Intermediate Value Theorem in Exercises 71–78 to prove that each equation has a solution. Then use a graphing calculator or computer grapher to solve the equations.

71. x3 - 3x - 1 = 0 72. 2x3 - 2x2 - 2x + 1 = 0 73. x(x - 1)2 = 1 (one root) 74. xx = 2 75. 2x + 21 + x = 4

76. x3 - 15x + 1 = 0 (three roots) 77. cos x = x (one root). Make sure you are using radian mode. 78. 2 sin x = x (three roots). Make sure you are using radian mode.

2.6 Limits Involving Infinity; Asymptotes of Graphs In this section we investigate the behavior of a function when the magnitude of the independent variable x becomes increasingly large, or x S { q . We further extend the concept of limit to infinite limits, which are not limits as before, but rather a new use of the term limit. Infinite limits provide useful symbols and language for describing the behavior of functions whose values become arbitrarily large in magnitude. We use these limit ideas to analyze the graphs of functions having horizontal or vertical asymptotes.

y 4 3 y = 1x

2 1 −1 0 −1

1

2

3

4

Figure 2.49  The graph of y = 1>x approaches 0 as x S q or x S - q .

M02_THOM9580_03_SE_C02_065-128.indd 110

x

Finite Limits as x u t H The symbol for infinity (q) does not represent a real number. We use q to describe the behavior of a function when the values in its domain or range outgrow all finite bounds. For example, the function ƒ(x) = 1>x is defined for all x ≠ 0 (Figure 2.49). When x is positive and becomes increasingly large, 1>x becomes increasingly small. When x is negative and its magnitude becomes increasingly large, 1>x again becomes small. We summarize these observations by saying that ƒ(x) = 1>x has limit 0 as x S q or x S - q, or that 0 is a limit of ƒ(x) = 1>x at infinity and negative infinity. Here are precise definitions.

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2.6  Limits Involving Infinity; Asymptotes of Graphs

111

DEFINITIONS 1. We say that ƒ(x) has the limit L as x approaches infinity and write lim ƒ(x) = L

xS q

if, for every number P 7 0, there exists a corresponding number M such that for all x 1

x 7 M

0 ƒ(x) - L 0 6 P.

2. We say that ƒ(x) has the limit L as x approaches minus infinity and write lim ƒ(x) = L

xS -q

if, for every number P 7 0, there exists a corresponding number N such that for all x x 6 N

y y = 1x

1 N = −P

0

y = –P

x S {q

x S {q

We prove the second result in Example 1, and leave the first to Exercises 87 and 88.

Example 1  Show that

y=P

M = 1P

0 ƒ(x) - L 0 6 P.

Intuitively, limx Sq ƒ(x) = L if, as x moves increasingly far from the origin in the positive direction, ƒ(x) gets arbitrarily close to L. Similarly, limx S -q ƒ(x) = L if, as x moves increasingly far from the origin in the negative direction, ƒ(x) gets arbitrarily close to L. The strategy for calculating limits of functions as x S { q is similar to the one for finite limits in Section 2.2. There we first found the limits of the constant and identity functions y = k and y = x. We then extended these results to other functions by applying Theorem 1 on limits of algebraic combinations. Here we do the same thing, except that the starting functions are y = k and y = 1>x instead of y = k and y = x. The basic facts to be verified by applying the formal definition are 1 and lim x = 0. (1) lim k = k

No matter what positive number P is, the graph enters 1 this band at x = P and stays.

P

1

1

x

–P

No matter what positive number P is, the graph enters 1 this band at x = − P and stays.

Figure 2.50  The geometry behind the argument in Example 1.

1

(a) lim x = 0 x Sq

(b) lim x = 0. x S -q

Solution (a) Let P 7 0 be given. We must find a number M such that for all x x 7 M

1

1 1 ` x - 0 ` = ` x ` 6 P.

The implication will hold if M = 1>P or any larger positive number (Figure 2.50). This proves limx Sq (1>x) = 0. (b) Let P 7 0 be given. We must find a number N such that for all x x 6 N

1

1 1 ` x - 0 ` = ` x ` 6 P.

The implication will hold if N = -1>P or any number less than -1>P (Figure 2.50). This proves limx S - q (1>x) = 0. Limits at infinity have properties similar to those of finite limits. THEOREM 12  All the Limit Laws in Theorem 1 are true when we replace limx S c by limx S q or limx S - q . That is, the variable x may approach a finite number c or { q.

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Chapter 2: Limits and Continuity

Example 2   The properties in Theorem 12 are used to calculate limits in the same way as when x approaches a finite number c. 1

1

(a) lim a5 + x b = lim 5 + lim x xS q xS q xS q

= 5 + 0 = 5

(b)

lim

xS - q

Sum Rule Known limits

p 23 1 1 = lim p 23 # x # x xS - q x2

1 1 = lim p 23 # lim x # lim x xS - q xS - q xS - q

= p 23 # 0 # 0 = 0

2

−5

1

Line y = 5 3

0

5

10

Known limits

Limits at Infinity of Rational Functions

2 y = 5x +2 8x − 3 3x + 2

y

Product Rule

To determine the limit of a rational function as x S { q, we first divide the numerator and denominator by the highest power of x in the denominator. The result then depends on the degrees of the polynomials involved. x

Example 3  These examples illustrate what happens when the degree of the numerator is less than or equal to the degree of the denominator.

−1

5 + (8>x) - (3>x2) Divide numerator and 5x2 + 8x - 3 = lim   denominator by x2 . 2 2 q S xS q x 3x + 2 3 + (2>x )

(a) lim −2

NOT TO SCALE

Figure 2.51  The graph of the function in Example 3a. The graph approaches the line y = 5>3 as 0 x 0 increases. y 8 y= 6

0

(11>x2) + (2>x3) 11x + 2 = lim x S - q 2x 3 - 1 xS - q 2 - (1>x3)

Divide numerator and denominator by x3 .

lim

=

0 + 0 = 0 2 - 0

See Fig. 2.52.

Horizontal Asymptotes

2 −2

(b)

5 + 0 - 0 5 = See Fig. 2.51. 3 3 + 0

Cases for which the degree of the numerator is greater than the degree of the denominator are illustrated in Examples 10 and 14.

11x + 2 2x 3 − 1

4

−4

=

2

4

6

x

If the distance between the graph of a function and some fixed line approaches zero as a point on the graph moves increasingly far from the origin, we say that the graph approaches the line asymptotically and that the line is an asymptote of the graph. Looking at ƒ(x) = 1>x (see Figure 2.49), we observe that the x-axis is an asymptote of the curve on the right because

−2

1 lim x = 0

xS q

−4 −6

and on the left because 1 lim x = 0.

−8

Figure 2.52  The graph of the function in Example 3b. The graph approaches the x-axis as 0 x 0 increases.

M02_THOM9580_03_SE_C02_065-128.indd 112

xS - q

We say that the x-axis is a horizontal asymptote of the graph of ƒ(x) = 1>x. DEFINITION  A line y = b is a horizontal asymptote of the graph of a function y = ƒ(x) if either lim ƒ(x) = b

xS q

or

lim ƒ(x) = b.

xS - q

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2.6  Limits Involving Infinity; Asymptotes of Graphs

113

The graph of the function ƒ(x) =

5x2 + 8x - 3 3x2 + 2

sketched in Figure 2.51 (Example 3a) has the line y = 5>3 as a horizontal asymptote on both the right and the left because lim ƒ(x) =

xS q

5 3

xS - q

5 . 3

Example 4  Find the horizontal asymptotes of the graph of

y 2

ƒ(x) =

y=1

y = −1

lim ƒ(x) =

and

x

0

x3 − 2 0x03 + 1

f(x) = −2

Solution  We calculate the limits as x S { q. For x Ú 0:

Figure 2.53  The graph of the function in Example 4 has two horizontal asymptotes.

x3 - 2 . 0x03 + 1

For x 6 0:

1 - (2>x3) x3 - 2 x3 - 2 = lim = lim = 1. xS q 0 x 0 3 + 1 x S q x3 + 1 x S q 1 + (1>x 3) lim

1 - ( 2>x3 ) x3 - 2 x3 - 2 = lim = lim = -1. x S - q ( -x ) 3 + 1 x S - q -1 + ( 1>x 3 ) xS - q 0 x 0 3 + 1 lim

The horizontal asymptotes are y = -1 and y = 1. The graph is displayed in Figure 2.53. Notice that the graph crosses the horizontal asymptote y = -1 for a positive value of x.

Example 5  The x-axis (the line y = 0) is a horizontal asymptote of the graph of

y

y = ex because

lim ex = 0.

xS - q

y = ex

To see this, we use the definition of a limit as x approaches - q . So let P 7 0 be given, but arbitrary. We must find a constant N such that for all x,

1

x 6 N P N = ln P

x

Figure 2.54  The graph of y = ex approaches the x-axis as x S - q (Example 5).

1

0 ex - 0 0 6 P.

Now 0 ex - 0 0 = ex , so the condition that needs to be satisfied whenever x 6 N is ex 6 P.

Let x = N be the number where ex = P. Since ex is an increasing function, if x 6 N , then ex 6 P. We find N by taking the natural logarithm of both sides of the equation eN = P, so N = ln P (see Figure 2.54). With this value of N the condition is satisfied, and we conclude that limx S - q ex = 0.

Example 6  Find (a) limq sin (1>x) and (b) limq x sin (1>x). xS{

xS

Solution (a) We introduce the new variable t = 1>x. From Example 1, we know that t S 0+ as x S q (see Figure 2.49). Therefore, 1 lim sin x = lim+ sin t = 0. tS0

xS q

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Chapter 2: Limits and Continuity

(b) We calculate the limits as x S q and x S - q:

y

sin t 1 lim x sin x = lim+ t = 1

1

xS q

x

1

Example 7  Find lim-e1>x . xS0

Solution  We let t = 1>x. From Figure 2.49, we can see that t S - q as x S 0- . (We make this idea more precise further on.) Therefore,

y

y=

−3

−2

lim e1>x = lim et = 0  Example 5

1 0.8 0.6 0.4 0.2

tS - q

x S 0-

(Figure 2.56). The Sandwich Theorem also holds for limits as x S { q. You must be sure, though, that the function whose limit you are trying to find stays between the bounding functions at very large values of x in magnitude consistent with whether x S q or x S - q.

x

0

−1

tS0

Likewise, we can investigate the behavior of y = ƒ(1>x) as x S 0 by investigating y = ƒ(t) as t S { q , where t = 1>x.

Figure 2.55  The line y = 1 is a horizontal asymptote of the function graphed here (Example 6b).

e1x

sin t 1 lim x sin x = lim- t = 1.

xS - q

The graph is shown in Figure 2.55, and we see that the line y = 1 is a horizontal asymptote.

1 y = x sin x −1

and

tS0

Figure 2.56  The graph of y = e1>x for x 6 0 shows limx S 0- e1>x = 0 (Example 7).

Example 8  Using the Sandwich Theorem, find the horizontal asymptote of the curve y = 2 +

sin x x .

Solution  We are interested in the behavior as x S { q. Since y

sin x 1 0 … ` x ` … `x`

y = 2 + sinx x

and limx S {q 0 1>x 0 = 0, we have limx S {q (sin x)>x = 0 by the Sandwich Theorem. Hence,

2 1 −3p −2p −p

0

p

2p

3p

lim a2 +

x S {q

x

Figure 2.57  A curve may cross one of its asymptotes infinitely often (Example 8).

sin x x b = 2 + 0 = 2,

and the line y = 2 is a horizontal asymptote of the curve on both left and right (Figure 2.57). This example illustrates that a curve may cross one of its horizontal asymptotes many times.

Example 9  Find limq 1 x - 2x2 + 16 2. xS

Solution  Both of the terms x and 2x2 + 16 approach infinity as x S q, so what happens to the difference in the limit is unclear (we cannot subtract q from q because the symbol does not represent a real number). In this situation we can multiply the numerator and the denominator by the conjugate radical expression to obtain an equivalent algebraic result: lim 1 x - 2x2 + 16 2 = lim 1 x - 2x2 + 16 2

xS q

xS q

= lim

x2 - (x2 + 16)

xS q x

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+ 2x2 + 16

x + 2x2 + 16 x + 2x2 + 16

= lim

xS q x

-16 . + 2x2 + 16

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2.6  Limits Involving Infinity; Asymptotes of Graphs

115

As x S q, the denominator in this last expression becomes arbitrarily large, so we see that the limit is 0. We can also obtain this result by a direct calculation using the Limit Laws:

-16 = lim x S q x + 2x 2 + 16 xS q lim

Oblique Asymptotes 2 y= x −3=x+1+ 1 2x − 4 2 2x − 4

y

The vertical distance between curve and line goes to zero as x : ∞

6 5 4 3

y= x+1 2

2

1

Example 10  Find the oblique asymptote of the graph of

2

3

4

ƒ(x) =

x2 - 3 2x - 4

in Figure 2.58.

1 −1 0 −1

If the degree of the numerator of a rational function is 1 greater than the degree of the denominator, the graph has an oblique or slant line asymptote. We find an equation for the asymptote by dividing numerator by denominator to express ƒ as a linear function plus a remainder that goes to zero as x S { q.

Oblique asymptote

x=2

16 - x 0 = = 0. 2 1 + 2 1 + 0 16 x 1 + + 2 x A x2

x

x

Solution  We are interested in the behavior as x S { q . We divide (2x - 4) into (x2 - 3):

−2

x + 1 2 2 2x - 4) x - 3 x2 - 2x 2x - 3 2x - 4 1

−3

Figure 2.58  The graph of the function in Example 10 has an oblique asymptote.

This tells us that ƒ(x) =

x2 - 3 x 1 ≤. = ¢ + 1≤ + ¢ 2x - 4 2()* 2x - 4 (1)1*

linear g(x) remainder y

B

You can get as high as you want by taking x close enough to 0. No matter how high B is, the graph goes higher. y = 1x

x 0

You can get as low as you want by taking x close enough to 0.

x

x No matter how low −B is, the graph goes lower.

−B

Figure 2.59  One-sided infinite limits: 1 1 and lim = - q. lim = q x S 0+ x x S 0- x

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As x S { q , the remainder, whose magnitude gives the vertical distance between the graphs of ƒ and g, goes to zero, making the slanted line g(x) =

x + 1 2

an asymptote of the graph of ƒ (Figure 2.58). The line y = g(x) is an asymptote both to the right and to the left. The next subsection will confirm that the function ƒ(x) grows arbitrarily large in absolute value as x S 2 (where the denominator is zero), as shown in the graph. Notice in Example 10 that if the degree of the numerator in a rational function is greater than the degree of the denominator, then the limit as 0 x 0 becomes large is + q or - q, depending on the signs assumed by the numerator and denominator.

Infinite Limits Let us look again at the function ƒ(x) = 1>x. As x S 0+, the values of ƒ grow without bound, eventually reaching and surpassing every positive real number. That is, given any positive real number B, however large, the values of ƒ become larger still (Figure 2.59).

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Chapter 2: Limits and Continuity

Thus, ƒ has no limit as x S 0+. It is nevertheless convenient to describe the behavior of ƒ by saying that ƒ(x) approaches q as x S 0+. We write 1 lim ƒ(x) = lim+ x = q. xS0

x S 0+

In writing this equation, we are not saying that the limit exists. Nor are we saying that there is a real number q, for there is no such number. Rather, we are saying that limx S 0+ (1>x) does not exist because 1>x becomes arbitrarily large and positive as x S 0+. As x S 0 -, the values of ƒ(x) = 1>x become arbitrarily large and negative. Given any negative real number -B, the values of ƒ eventually lie below -B. (See Figure 2.59.) We write 1 lim ƒ(x) = lim- x = - q. xS0

x S 0-

Again, we are not saying that the limit exists and equals the number - q. There is no real number - q. We are describing the behavior of a function whose limit as x S 0- does not exist because its values become arbitrarily large and negative.

y

y=

1 x−1

Example 11  Find lim+

1 −1

0

xS1

1

2

3

x

Geometric Solution  The graph of y = 1>(x - 1) is the graph of y = 1>x shifted 1 unit to the right (Figure 2.60). Therefore, y = 1>(x - 1) behaves near 1 exactly the way y = 1>x behaves near 0: lim

x S 1+

Figure 2.60 Near x = 1, the function y = 1>(x - 1) behaves the way the function y = 1>x behaves near x = 0. Its graph is the graph of y = 1>x shifted 1 unit to the right (Example 11).

1 1  and  lim. x - 1 xS1 x - 1

1 = q x - 1

and

lim

x S 1-

1 = - q. x - 1

Analytic Solution  Think about the number x - 1 and its reciprocal. As x S 1+, we have (x - 1) S 0+ and 1>(x - 1) S q. As x S 1-, we have (x - 1) S 0 - and 1>(x - 1) S - q.

Example 12  Discuss the behavior of ƒ(x) =

as

x S 0.

Solution  As x approaches zero from either side, the values of 1>x2 are positive and become arbitrarily large (Figure 2.61). This means that

y

lim ƒ(x) = lim

No matter how high B is, the graph goes higher.

B

xS0

x

1

x S 0 x2

= q.

The function y = 1>x shows no consistent behavior as x S 0. We have 1>x S q if S x 0+, but 1>x S - q if x S 0-. All we can say about limx S 0 (1>x) is that it does not exist. The function y = 1>x2 is different. Its values approach infinity as x approaches zero from either side, so we can say that limx S 0 (1>x2) = q.

f(x) = 12 x

x 0

1 x2

x

Figure 2.61  The graph of ƒ(x) in Example 12 approaches infinity as x S 0.

Example 13  These examples illustrate that rational functions can behave in various ways near zeros of the denominator. (x - 2)2 (x - 2)2 x - 2 = lim = lim = 0 x S 2 x2 - 4 x S 2 (x - 2)(x + 2) xS2 x + 2

(a) lim (b) lim

xS2

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x - 2 x - 2 1 1 = lim = lim = x2 - 4 x S 2 (x - 2)(x + 2) x S 2 x + 2 4

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2.6  Limits Involving Infinity; Asymptotes of Graphs

(c) lim+

x - 3 x - 3 = lim+ = - q 2 (x 2)(x + 2) S x 2 x - 4

The values are negative for x 7 2, x near 2.

(d) lim-

x - 3 x - 3 = lim = q x2 - 4 x S 2- (x - 2)(x + 2)

The values are positive for x 6 2, x near 2.

xS2

xS2

(e) lim

x - 3 x - 3 = lim does not exist. x2 - 4 x S 2 (x - 2)(x + 2)

(f ) lim

-(x - 2) 2 - x -1 = lim = lim = -q x S 2 (x - 2)2 (x - 2)3 x S 2 (x - 2)3

xS2

xS2

117

See parts (c) and (d).

In parts (a) and (b) the effect of the zero in the denominator at x = 2 is canceled because the numerator is zero there also. Thus a finite limit exists. This is not true in part (f ), where cancelation still leaves a zero factor in the denominator. y

Example 14  Find lim

xS - q

2x5 - 6x4 + 1 . 3x2 + x - 7

Solution  We are asked to find the limit of a rational function as x S - q , so we divide the numerator and denominator by x2 , the highest power of x in the denominator:

y = f (x)

2x5 - 6x4 + 1 = x S - q 3x 2 + x - 7 lim

B

2x3 - 6x2 + x-2 x S - q 3 + x -1 - 7x-2 lim

2x2 (x - 3) + x-2 x S - q 3 + x -1 - 7x-2 = - q,   x-n S 0, x - 3 S - q

= lim 0

c−d

c

x c+d

because the numerator tends to - q while the denominator approaches 3 as x S - q .

Figure 2.62 For c - d 6 x 6 c + d, the graph of ƒ(x) lies above the line y = B.

y c−d

c

c+d x

0

Precise Definitions of Infinite Limits Instead of requiring ƒ(x) to lie arbitrarily close to a finite number L for all x sufficiently close to c, the definitions of infinite limits require ƒ(x) to lie arbitrarily far from zero. Except for this change, the language is very similar to what we have seen before. Figures 2.62 and 2.63 accompany these definitions.

DEFINITIONs 1. We say that ƒ(x) approaches infinity as x approaches c, and write lim ƒ(x) = q,

xSc

if for every positive real number B there exists a corresponding d 7 0 such that for all x

−B

y = f (x)

0 6 0x - c0 6 d

1

ƒ(x) 7 B.

2. We say that ƒ(x) approaches minus infinity as x approaches c, and write lim ƒ(x) = - q,

xSc

Figure 2.63 For c - d 6 x 6 c + d, the graph of ƒ(x) lies below the line y = -B.

if for every negative real number -B there exists a corresponding d 7 0 such that for all x 0 6 0x - c0 6 d

1

ƒ(x) 6 -B.

The precise definitions of one-sided infinite limits at c are similar and are stated in the exercises.

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Chapter 2: Limits and Continuity

Example 15  Prove that lim 12 = q. xS0

x

Solution  Given B 7 0, we want to find d 7 0 such that 0 6  x - 0  6 d implies

1 7 B. x2

1 7 B x2

x2 6

Now, if and only if

1 B

or, equivalently,

0x0 6 1 . 2B

Thus, choosing d = 1> 2B (or any smaller positive number), we see that

0 x 0 6 d implies

Therefore, by definition,

lim

xS0

1 1 7 2 Ú B. x2 d

1 = q. x2

Vertical Asymptotes Notice that the distance between a point on the graph of ƒ(x) = 1>x and the y-axis approaches zero as the point moves vertically along the graph and away from the origin (Figure 2.64). The function ƒ(x) = 1>x is unbounded as x approaches 0 because

y Vertical asymptote

Horizontal asymptote

1 0

lim

1

Horizontal asymptote, y=0

1

x S 0+ x

y = 1x x

Vertical asymptote, x=0

and

lim

1

x S 0- x

= - q.

We say that the line x = 0 (the y-axis) is a vertical asymptote of the graph of ƒ(x) = 1>x. Observe that the denominator is zero at x = 0 and the function is undefined there. DEFINITION  A line x = a is a vertical asymptote of the graph of a function y = ƒ(x) if either lim ƒ(x) = { q

x S a+

Figure 2.64  The coordinate axes are asymptotes of both branches of the hyperbola y = 1>x.

= q

or

lim ƒ(x) = { q.

x S a-

Example 16  Find the horizontal and vertical asymptotes of the curve y =

x + 3 . x + 2

Solution  We are interested in the behavior as x S { q and the behavior as x S -2, where the denominator is zero. The asymptotes are quickly revealed if we recast the rational function as a polynomial with a remainder, by dividing (x + 2) into (x + 3): 1 x + 2) x + 3   x + 2 1

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2.6  Limits Involving Infinity; Asymptotes of Graphs

This result enables us to rewrite y as:

y Vertical asymptote, x = −2

6 5

Horizontal asymptote, y=1

4

x+3 y= x+2

3

=1+

1 x+2

1

3

y = 1 +

2 1

−5 −4 −3 −2 −1 0 −1

119

2

x

−2

1 . x + 2

As x S { q , the curve approaches the horizontal asymptote y = 1; as x S -2, the curve approaches the vertical asymptote x = -2. We see that the curve in question is the graph of ƒ(x) = 1>x shifted 1 unit up and 2 units left (Figure 2.65). The asymptotes, instead of being the coordinate axes, are now the lines y = 1 and x = -2.

Example 17  Find the horizontal and vertical asymptotes of the graph of

−3 −4

ƒ(x) = -

Figure 2.65  The lines y = 1 and x = -2 are asymptotes of the curve in Example 16.

8 . x2 - 4

Solution  We are interested in the behavior as x S { q and as x S {2, where the denominator is zero. Notice that ƒ is an even function of x, so its graph is symmetric with respect to the y-axis. (a) The behavior as x S { q. Since lim x Sq ƒ(x) = 0, the line y = 0 is a horizontal

asymptote of the graph to the right. By symmetry it is an asymptote to the left as well (Figure 2.66). Notice that the curve approaches the x-axis from only the negative side (or from below). Also, ƒ(0) = 2. (b) The behavior as x S {2. Since

y 8 7 6 5 4 3 2 1

Vertical asymptote, x = −2

y=−

8 x2 − 4

Vertical asymptote, x = 2

lim ƒ(x) = - q

Horizontal asymptote, y = 0

−4−3−2−1 0

1 2 3 4

and

x S 2+

x

lim ƒ(x) = q,

x S 2-

the line x = 2 is a vertical asymptote both from the right and from the left. By symmetry, the line x = -2 is also a vertical asymptote. There are no other asymptotes because ƒ has a finite limit at all other points.

Example 18   The graph of the natural logarithm function has the y-axis (the line Figure 2.66  Graph of the function in Example 17. Notice that the curve approaches the x-axis from only one side. Asymptotes do not have to be two-sided.

y

x = 0) as a vertical asymptote. We see this from the graph sketched in Figure 2.67 (which is the reflection of the graph of the natural exponential function across the line y = x) and the fact that the x-axis is a horizontal asymptote of y = ex (Example 5). Thus, lim ln x = - q.

x S 0+

The same result is true for y = loga x whenever a 7 1.

Example 19  The curves y = ex

1 y = sec x = cos x

4 3 2

and

sin x y = tan x = cos x

both have vertical asymptotes at odd-integer multiples of p>2, where cos x = 0 (Figure 2.68).

y = ln x

1 −1 −1

1

2

3

4

x

Figure 2.67  The line x = 0 is a vertical asymptote of the natural logarithm function (Example 18).

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Dominant Terms In Example 10 we saw that by long division we could rewrite the function ƒ(x) =

x2 - 3 2x - 4

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Chapter 2: Limits and Continuity y

y

y = sec x

1 −

3p −p p − 2 2

0

y = tan x

1 p 2

p

3p 2

x −

0 p 3p −p p − −1 2 2 2

p

3p 2

x

Figure 2.68  The graphs of sec x and tan x have infinitely many vertical asymptotes (Example 19).

as a linear function plus a remainder term:

y

y

This tells us immediately that

20

500,000

ƒ(x) ≈

15

300,000

ƒ(x) ≈

10 f (x)

5 −2

−1

g(x) = 3x 4

0

1

2

x

−5 (a) y

100,000 x

−20

−10

0

For 0 x 0 large,

1 is near 0. 2x - 4

1   For x near 2, this term is very large in absolute value. 2x - 4

100,000 If we want to know how ƒ behaves, this is the way to find out. It behaves like y = (x>2) + 1 when |x| is large xand the contribution of 1>(2x - 4) to the total value of ƒ −20 −10 0 It behaves 10 20 1>(2x - 4) when x is so close to 2 that 1>(2x - 4) makes is insignificant. like −100,000 the dominant contribution. We say that (x>2) + 1 dominates when x is numerically large, and we say that (b) 1>(2x - 4) dominates when x is near 2. Dominant terms like these help us predict a function’s behavior.

although ƒ and g are quite different for numerically small values of x, they are virtually identical for 0 x 0 very large, in the sense that their ratios approach 1 as x S q or x S - q .

300,000

2

x + 1    2

x 1 b. + 1b + a 2 2x - 4

Example 20  Let ƒ(x) = 3x4 - 2x3 + 3x2 - 5x + 6 and g(x) = 3x4. Show that

500,000

g(x) = 3x 4

ƒ(x) = a

10

20

x

Solution  The graphs of ƒ and g behave quite differently near the origin (Figure 2.69a), but appear as virtually identical on a larger scale (Figure 2.69b). We can test that the term 3x4 in ƒ, represented graphically by g, dominates the polynomial ƒ for numerically large values of x by examining the ratio of the two functions as x S { q. We find that

−100,000

ƒ(x) 3x4 - 2x3 + 3x2 - 5x + 6 = lim x S {q g(x) x S {q 3x4 lim

(b)

Figure 2.69  The graphs of ƒ and g are (a) distinct for 0 x 0 small, and (b) nearly identical for 0 x 0 large (Example 20).

= lim a1 x S {q

= 1,

5 2 1 2 + + b 3x x2 3x3 x4

which means that ƒ and g appear nearly identical when 0 x 0 is large.

Summary

In this chapter we presented several important calculus ideas that are made meaningful and precise by the concept of the limit. These include the three ideas of the exact rate of change of a function, the slope of the graph of a function at a point, and the continuity of a function. The primary methods used for calculating limits of many functions are captured in the algebraic

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2.6  Limits Involving Infinity; Asymptotes of Graphs

121

Limit Laws of Theorem 1 and in the Sandwich Theorem, all of which are proved from the precise definition of the limit. We saw that these computational rules also apply to one-sided limits and to limits at infinity. Moreover, we can sometimes apply these rules when calculating limits of simple transcendental functions, as illustrated by our examples or in cases like the following: lim

xS0

ex - 1 ex - 1 1 1 1 = lim x = lim = = . 2x (e 1)(ex + 1) x S 0 ex + 1 1 + 1 2 e - 1 xS0

However, calculating more complicated limits involving transcendental functions such as lim

xS0

x , e2x - 1

ln x 1 x lim x , and  lim a1 + x b xS0 xS0

requires more than simple algebraic techniques. The derivative is exactly the tool we need to calculate limits such as these (see Section 4.5), and this notion is the main subject of our next chapter.

Exercises  2.6 Finding Limits 1. For the function ƒ whose graph is given, determine the following limits.

a. lim ƒ(x) xS2

b. lim + ƒ(x) c. lim - ƒ(x) x S -3

2 3. ƒ(x) = x - 3

x S -3

d. lim ƒ(x) e. lim+ ƒ(x) f. lim- ƒ(x) x S -3

xS0

In Exercises 3–8, find the limit of each function (a) as x S q and (b) as x S - q. (You may wish to visualize your answer with a graphing calculator or computer.)

xS0

g. lim ƒ(x) h. lim ƒ(x) i. lim ƒ(x)

5. g(x) =

7. f(x) =

xS0

x S -q

x Sq

y 3 2

−6 −5 −4 −3 −2 −1 −1

1

2

3

4 5

6

x

−2

2. For the function ƒ whose graph is given, determine the following limits. a. lim ƒ(x) b. lim+ ƒ(x) c. lim- ƒ(x) xS2

xS2

d. lim ƒ(x) e. lim + ƒ(x) f. lim - ƒ(x) xS2

x S -3

x S -3

g. lim ƒ(x) h. lim+ ƒ(x) i. lim- ƒ(x) x S -3

xS0

x S -q

x Sq

2x + 3 5x + 7 x + 1 15. ƒ(x) = 2 x + 3

8. h(x) =

3

f

2 1

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4 + ( 22>x2)

1

2

3

4 5

6

x

2x3 + 7 x - x2 + x + 7 3x + 7 16. ƒ(x) = 2 x - 2 14. ƒ(x) =

3

7x3 x3 - 3x2 + 6x

18. h(x) =

9x4 + x 2x4 + 5x2 - x + 6

19. g(x) =

10x5 + x4 + 31 x6

20. g(x) =

x3 + 7x2 - 2 x2 - x + 1

21. f(x) =

3x7 + 5x2 - 1 6x3 - 7x + 3

22. h(x) =

5x8 - 2x3 + 9 3 + x - 4x5

Limits as x u H or x u − H The process by which we determine limits of rational functions applies equally well to ratios containing noninteger or negative powers of x: Divide numerator and denominator by the highest power of x in the denominator and proceed from there. Find the limits in Exercises 23–36.

−2 −3

3 - (2>x)

17. h(x) =

y

−6 −5 −4 −3 −2 −1 −1

4 - (3>x2)

13. ƒ(x) =

xS0

j. lim ƒ(x) k. lim ƒ(x) l. lim ƒ(x) xS0

-3 + (4>x)

Limits of Rational Functions In Exercises 13–22, find the limit of each rational function (a) as x S q and (b) as x S - q.

−3

xS4

1 5 + (1>x)

Find the limits in Exercises 9–12. cos u sin 2x 10. lim 9. lim x x Sq u S -q 3u 2 - t + sin t r + sin r 12. lim 11. lim t S -q t + cos t r Sq 2r + 7 - 5 sin r

f

1

2 x2 1 6. g(x) = 8 - (5>x2)

4. ƒ(x) = p -

8x2 - 3 x Sq A 2x 2 + x

23. lim

24. lim ¢ x S -q

x2 + x - 1 1>3 ≤ 8x2 - 3

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Chapter 2: Limits and Continuity

25. lim ¢ xS - q

1 - x3 5 ≤ x2 + 7x

2 2x + x-1 3x - 7 xS q

27. lim

3

29. lim

xS - q

28. lim

xS q

5

2x - 2x 3

5

2x + 2x

2x5>3 - x1>3 + 7 x S q x 8>5 + 3x + 2x

31. lim 33. lim

xS q

2x2 + 1

x + 1

x2 - 5x xS q A x + x - 2

26. lim

x - 3 35. lim x S q 24x 2 + 25

3

2 + 2x 2 - 2x

x-1 + x-4 x S q x -2 - x -3

30. lim

3

32. lim

xS - q

34. lim

xS - q

2x - 5x + 3

2x + x2>3 - 4 2x2 + 1

x + 1

4 - 3x3 36. lim x S - q 2x 6 + 9

Infinite Limits Find the limits in Exercises 37–48. 1 37. lim+ 38. x S 0 3x 3 39. lim 40. xS2 x - 2 2x 41. lim + 42. x S -8 x + 8 4 43. lim 44. x S 7 (x - 7)2

lim

x S 0-

5 2x

1 x - 3 3x lim x S -5- 2x + 10 -1 lim x S 0 x 2(x + 1) lim

x S 3+

51. lim- (1 + csc u) uS0

50.

lim

x S (-p>2) +

sec x

52. lim (2 - cot u) uS0

Find the limits in Exercises 53–58. 1 53. lim 2 as x - 4 a. x S 2+ b. x S 2+ c. x S - 2 d. x S - 2x 54. lim 2 as x - 1 a. x S 1+ b. x S 1 c. x S - 1+ d. x S - 1x2 1 55. lim a - x b as 2 a. x S 0+ b. x S 03 c. x S 2 2 d. x S -1

x2 - 1 56. lim as 2x + 4 a. x S - 2+ b. x S -2-

c. x S 1+ d. x S 0-

M02_THOM9580_03_SE_C02_065-128.indd 122

e. What, if anything, can be said about the limit as x S 0? Find the limits in Exercises 59–62. 3 b as t 1>3 a. t S 0+ b. t S 0-

59. lima2 -

1 2 + b as x2>3 (x - 1)2>3 a. x S 0+ b. x S 0 c. x S 1+ d. x S 1 61. lima

1 1 b as x1>3 (x - 1)4>3 a. x S 0+ b. x S 0+ c. x S 1 d. x S 1 62. lima

Graphing Simple Rational Functions Graph the rational functions in Exercises 63–68. Include the graphs and equations of the asymptotes and dominant terms.

Find the limits in Exercises 49–52. lim tan x

x2 - 3x + 2 as x3 - 4x a. x S 2+ b. x S - 2+ S c. x 0 d. x S 1+ 58. lim

1 + 7b as t 3>5 a. t S 0+ b. t S 0-

2 2 b. lim x S 0 3x 1>3 x S 0- 3x 1>3 2 2 46. a. lim+ 1>5 b. lim xS0 x x S 0- x 1>5 4 1 47. lim 2>5 48. lim 2>3 xS0 x xS0 x

x S (p>2)-

e. What, if anything, can be said about the limit as x S 0?

60. lima

45. a. lim+

49.

x2 - 3x + 2 as x3 - 2x2 a. x S 0+ b. x S 2+ S c. x 2 d. xS2 57. lim

1 x - 1 1 65. y = 2x + 4 x + 3 6 7. y = x + 2

1 x + 1 -3 66. y = x - 3 2x 68. y = x + 1

63. y =

64. y =

Inventing Graphs and Functions In Exercises 69–72, sketch the graph of a function y = ƒ(x) that satisfies the given conditions. No formulas are required—just label the coordinate axes and sketch an appropriate graph. (The answers are not unique, so your graphs may not be exactly like those in the answer section.)

69. ƒ(0) = 0, ƒ(1) = 2, ƒ(- 1) = -2, lim ƒ(x) = -1, and x S -q

lim ƒ(x) = 1 x Sq

70. ƒ(0) = 0, lim ƒ(x) = 0, lim+ ƒ(x) = 2, and x S {q

lim- ƒ(x) = - 2

xS0

xS0

71. ƒ(0) = 0, lim ƒ(x) = 0, lim- ƒ(x) = lim + ƒ(x) = q, x S {q xS1 x S -1 lim ƒ(x) = - q, and lim ƒ(x) = - q xS1 +

x S -1-

72. ƒ(2) = 1, ƒ(-1) = 0, lim ƒ(x) = 0, lim+ ƒ(x) = q, x Sq xS0 lim ƒ(x) = - q, and lim ƒ(x) = 1 x S 0-

x S -q

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2.6  Limits Involving Infinity; Asymptotes of Graphs

In Exercises 73–76, find a function that satisfies the given conditions and sketch its graph. (The answers here are not unique. Any function that satisfies the conditions is acceptable. Feel free to use formulas defined in pieces if that will help.) 73. lim ƒ(x) = 0, lim ƒ(x) = q, and lim ƒ(x) = q x S {q

x S 2-

x S {q

xS3

x S 2+

74. lim g(x) = 0, lim- g(x) = - q, and lim+ g(x) = q xS3

75. lim h(x) = - 1, lim h(x) = 1, lim- h(x) = - 1, and xS - q

lim+ h(x) = 1

xS q

xS0

xS0

76. lim k(x) = 1, lim- k(x) = q, and lim+ k(x) = - q x S {q

xS1

xS1

77. Suppose that ƒ(x) and g(x) are polynomials in x and that limx S q (ƒ(x)>g(x)) = 2. Can you conclude anything about limx S - q (ƒ(x)>g(x))? Give reasons for your answer. 78. Suppose that ƒ(x) and g(x) are polynomials in x. Can the graph of ƒ(x)>g(x) have an asymptote if g(x) is never zero? Give reasons for your answer. 79. How many horizontal asymptotes can the graph of a given rational function have? Give reasons for your answer. Finding Limits of Differences When x u t H Find the limits in Exercises 80–86. xS q

81. lim ( 2x2 + 25 - 2x2 - 1 ) xS q

82. lim

xS - q

83. lim

xS - q

x S c-

b. lim+ ƒ(x) = - q xSc

c. lim- ƒ(x) = - q xSc

Use the formal definitions from Exercise 93 to prove the limit statements in Exercises 94–98. 1 94. lim+ x = q xS0 1 95. lim- x = - q xS0 1 96. lim= -q xS2 x - 2 1 97. lim+ = q xS2 x - 2 1 98. lim= q x S 1 1 - x2 Oblique Asymptotes Graph the rational functions in Exercises 99–104. Include the graphs and equations of the asymptotes.

100.  y =

( 2x + 3 + x ) 2

( 2x +

Modify the definition to cover the following cases. a. lim ƒ(x) = q

99.  y =

80. lim ( 2x + 9 - 2x + 4 )

101.  y =

24x2 + 3x - 2 )

102.  y =

84. lim ( 29x2 - x - 3x ) xS q

85. lim ( 2x2 + 3x - 2x2 - 2x )

103.  y =

xS q

86. lim ( 2x + x - 2x - x ) 2

2

104.  y =

xS q

Using the Formal Definitions Use the formal definitions of limits as x S { q to establish the limits in Exercises 87 and 88.

87. If ƒ has the constant value ƒ(x) = k, then lim ƒ(x) = k. x Sq

88. If ƒ has the constant value ƒ(x) = k, then lim ƒ(x) = k. x S -q

Use formal definitions to prove the limit statements in Exercises 89–92. -1 = - q x2 -2 = - q 91. lim x S 3 (x - 3)2 89. lim

90. lim

xS0

xS0

92. lim

x S -5

1

0x0

= q

1 = q (x + 5)2

93. Here is the definition of infinite right-hand limit. We say that ƒ(x) approaches infinity as x approaches c from the right, and write lim ƒ(x) = q,

if, for every positive real number B, there exists a corresponding number d 7 0 such that for all x

M02_THOM9580_03_SE_C02_065-128.indd 123

1

x2 x - 1 x2 + 1 x - 1 x2 - 4 x - 1 x2 - 1 2x + 4 x2 - 1 x x3 + 1 x2

Additional Graphing Exercises T Graph the curves in Exercises 105–108. Explain the relationship between the curve’s formula and what you see. x 105.  y = 24 - x2 -1 106.  y = 24 - x2 1 107.  y = x2>3 + 1>3 x p 108.  y = sin a 2 b x + 1

T Graph the functions in Exercises 109 and 110. Then answer the following questions. a. How does the graph behave as x S 0+? b. How does the graph behave as x S { q? c. How does the graph behave near x = 1 and x = - 1?

x S c+

c 6 x 6 c + d

123

ƒ(x) 7 B.

3 1 2>3 ax - x b 2

110. y =

2>3 x 3 a b 2 x - 1

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124

Chapter 2: Limits and Continuity

Chapter  2

1. What is the average rate of change of the function g(t) over the interval from t = a to t = b? How is it related to a secant line?

11. What conditions must be satisfied by a function if it is to be continuous at an interior point of its domain? At an endpoint?

2. What limit must be calculated to find the rate of change of a function g(t) at t = t0?

12. How can looking at the graph of a function help you tell where the function is continuous?

3. Give an informal or intuitive definition of the limit

13. What does it mean for a function to be right-continuous at a point? Left-continuous? How are continuity and one-sided continuity related?

lim ƒ(x) = L.

xSc

Why is the definition “informal”? Give examples. 4. Does the existence and value of the limit of a function ƒ(x) as x approaches c ever depend on what happens at x = c? Explain and give examples. 5. What function behaviors might occur for which the limit may fail to exist? Give examples. 6. What theorems are available for calculating limits? Give examples of how the theorems are used. 7. How are one-sided limits related to limits? How can this relationship sometimes be used to calculate a limit or prove it does not exist? Give examples. 8. What is the value of limu S 0 ((sin u)>u)? Does it matter whether u is measured in degrees or radians? Explain. 9. What exactly does limx S c ƒ(x) = L mean? Give an example in which you find a d 7 0 for a given ƒ, L, c, and P 7 0 in the precise definition of limit. 10. Give precise definitions of the following statements. a. limx S 2- ƒ(x) = 5 c. limx S 2 ƒ(x) = q

Chapter  2

b. limx S 2+ ƒ(x) = 5 d. limx S 2 ƒ(x) = - q

14. What does it mean for a function to be continuous on an interval? Give examples to illustrate the fact that a function that is not continuous on its entire domain may still be continuous on selected intervals within the domain. 15. What are the basic types of discontinuity? Give an example of each. What is a removable discontinuity? Give an example. 16. What does it mean for a function to have the Intermediate Value Property? What conditions guarantee that a function has this property over an interval? What are the consequences for graphing and solving the equation ƒ(x) = 0? 17. Under what circumstances can you extend a function ƒ(x) to be continuous at a point x = c? Give an example. 18. What exactly do limx Sq ƒ(x) = L and limx S -q ƒ(x) = L mean? Give examples. 19. What are limx S {q k (k a constant) and limx S {q (1>x)? How do you extend these results to other functions? Give examples. 20. How do you find the limit of a rational function as x S { q? Give examples. 21. What are horizontal and vertical asymptotes? Give examples.

Practice Exercises

Limits and Continuity 1. Graph the function

1, - x, ƒ(x) = e 1, - x, 1,

x -1 x 0 x

… 6 = 6 Ú

-1 x 6 0 0 x 6 1 1.

Then discuss, in detail, limits, one-sided limits, continuity, and one-sided continuity of ƒ at x = - 1, 0, and 1. Are any of the discontinuities removable? Explain. 2. Repeat the instructions of Exercise 1 for 0, 1>x, ƒ(x) = d 0, 1,

M02_THOM9580_03_SE_C02_065-128.indd 124

x 0 x x

… 6 = 7

-1 0x0 6 1 1 1.

3. Suppose that ƒ(x) and g(x) are defined for all x and that limx S x0 ƒ(x) = - 5 and limx S x0 g(x) = 1. Find the limit as x S x0 of the following functions. (ƒ(x))3 a. 5ƒ(x) b. ƒ(x) c. ƒ(x) # g(x) d. g(x) - 6

0 g(x) 0 e. log g(x) f.

g. ƒ(x) - g(x) h. 1>ƒ(x)

4. Suppose the functions ƒ(x) and g(x) are defined for all x and that limx S 0 ƒ(x) = 1>2 and limx S 0 g(x) = 22. Find the limits as x S 0 of the following functions.

g(x) # ƒ(x) a. -g(x) b. c. ƒ(x) + g(x) d. 1>ƒ(x)

ƒ(x) # cos x e. x + ƒ(x) f. x - 1

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Chapter 2  Practice Exercises

In Exercises 5 and 6, find the value that limx S 0 g(x) must have if the given limit statements hold. 4 - g(x) 5. lim a b = 1 6. lim ax lim g(x)b = 2 x xS0 x S -4 xS0

7. On what intervals are the following functions continuous? g(x) = x3>4 a. ƒ(x) = x1>3 b. c. h(x) = x-2>3 d. k(x) = x-1>6

125

c. It can be shown that the exact value of the solution in part (b) is a

269 1>3 269 1>3 1 1 + b + a b . 2 18 2 18

Evaluate this exact answer and compare it with the value you found in part (b). T 34. Let ƒ(u) = u 3 - 2u + 2.

8. On what intervals are the following functions continuous?

a. Use the Intermediate Value Theorem to show that ƒ has a zero between -2 and 0.

g(x) = csc x a. ƒ(x) = tan x b. cos x sin x c. h(x) = x - p d. k(x) = x

b. Solve the equation ƒ(u) = 0 graphically with an error of magnitude at most 10-4. c. It can be shown that the exact value of the solution in part (b) is

Finding Limits In Exercises 9–28, find the limit or explain why it does not exist.

x2 + 5x + 5 x + 3x2 - 11x a. as x S 0 b. as x S 3 9. lim

3

3

x + 1 x7 + 4x6 + x5 a. as x S 2 b. as x S -2 10. lim

11. lim a xS0

21 + x - 1

x

3

(x + h) - x3 13. lim h hS0

b

1 1 3 + x 3 15. lim x xS0 x2 - 2x x S 1 2x - 1

x5 - 25 x S 2 x 3 - 23 (x + h)2 - x2 14. lim h xS0 12. lim

(3 + x)4 - 81 x xS0

16. lim

x2>3 - 162>3 2x - 4

17. lim

18. lim

sin mx 19. lim tan nx xS0

20. lim sec x = -1

21. lim cos a xSp

x + sin xb 2

7x x S 0 8 tan x - x

x S 16

xSp

22. lim sin2 (x - cot x) xSp

x2 - tan 2x tan x xS0

23. lim

24. lim

25. lim (1 + x)1>x

26. lim t 3 ln ( 3 - 2t )

27. lim 2ue-cos (p>u)

28. lim

xS0

uS1

tS4

xS4

ex - e4 x - 4

In Exercises 29–32, find the limit of g(x) as x approaches the indicated value. 1 = 5 30. lim 29. lim+ [9g(x)]1>3 = 3 xS0 x S 27 x + g(x) 31. lim

xS2

5x4 + 2 q = g(x)

32. lim

x S -3

10 - x2 = 0 (g(x))1>3

T Roots 33. Let ƒ(x) = x3 - x - 1.

a

1>3 1>3 19 19 - 1b - a + 1b . 27 27 A A

Evaluate this exact answer and compare it with the value you found in part (b). Continuous Extension 35. Can ƒ(x) = x(x2 - 1)> 0 x2 - 1 0 be extended to be continuous at x = 1 or -1? Give reasons for your answers. (Graph the function—you will find the graph interesting.)

36. Explain why the function ƒ(x) = sin (1>x) has no continuous extension to x = 0. T In Exercises 37–40, graph the function to see whether it appears to have a continuous extension to the given point a. If it does, use Trace and Zoom to find a good candidate for the extended function’s value at a. If the function does not appear to have a continuous extension, can it be extended to be continuous from the right or left? If so, what do you think the extended function’s value should be? x - 1 37. ƒ(x) = , a = 1 4 x - 2 x 5 cos u , a = p>2 38. g(u) = 4u - 2p 39. h(t) = ( 1 + 0 t 0 ) 1>t, a = 0 x 40. k(x) = , a = 0 1 - 20 x 0

Limits at Infinity Find the limits in Exercises 41–54. 5x + 8 3x4 + 8 41. lim 42. lim x S q 7x + 4 x S q 5x 4 + 9 x2 - 7x + 9 1 43. lim 44. lim 2 q xS x S q 2x - 9x + 5 5x3

x2 + 11x xS - q x - 1

45. lim 47. lim

xS q

sin x :x ;

(If you have a grapher, try graphing ƒ(x) = x (cos (1>x) - 1) near the origin to “see” the limit at infinity.) x + tan x + 3 2x x5>7 + x-3 49. lim 50. lim 5>7 x + tan x q q xS xS x + cos x cos u - 1 u uS q

48. lim

a. Use the Intermediate Value Theorem to show that ƒ has a zero between - 1 and 2.

2 51. lim e2>x sin x

b. Solve the equation ƒ(x) = 0 graphically with an error of magnitude at most 10-8.

53. lim sin-1 x

M02_THOM9580_03_SE_C02_065-128.indd 125

x5 + x4 x S q 9x 4 + 108 (If you have a grapher, try graphing the function for - 5 … x … 5.) 46. lim

xS q

xS - 1

2 52. lim ln a1 - t b tS q

1 54. lim e3t sin-1 t tS - q

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126

Chapter 2: Limits and Continuity

Horizontal and Vertical Asymptotes 55. Use limits to determine the equations for all vertical asymptotes.

56. Use limits to determine the equations for all horizontal asymptotes.

x2 + 4 x2 - x - 2 a. y = b. ƒ(x) = 2 x - 3 x - 2x + 1 x2 + x - 6 c. y = 2 x + 2x - 8

a. y =

Chapter  2

2x + 4 1 - x2 b. ƒ(x) = x2 + 1 2x + 4

c. g(x) =

2x2 + 4

x

d. y =

x2 + 9 B 9x2 + 1

0 T 1. Assigning a value to 0   The rules of exponents tell us that a0 = 1 if a is any number different from zero. They also tell us that 0n = 0 if n is any positive number. If we tried to extend these rules to include the case 00, we would get conflicting results. The first rule would say 00 = 1, whereas the second would say 00 = 0. We are not dealing with a question of right or wrong here. Neither rule applies as it stands, so there is no contradiction. We could, in fact, define 00 to have any value we wanted as long as we could persuade others to agree. What value would you like 00 to have? Here is an example that might help you to decide. (See Exercise 2 below for another example.)

a. C  alculate xx for x = 0.1, 0.01, 0.001, and so on as far as your calculator can go. Record the values you get. What pattern do you see? b. G  raph the function y = xx for 0 6 x … 1. Even though the function is not defined for x … 0, the graph will approach the y-axis from the right. Toward what y-value does it seem to be headed? Zoom in to further support your idea. T 2. A reason you might want 00 to be something other than 0 or 1 As the number x increases through positive values, the numbers 1>x and 1 > (ln x) both approach zero. What happens to the number 1 1>(ln x) ƒ(x) = a x b

as x increases? Here are two ways to find out. a. E  valuate ƒ for x = 10, 100, 1000, and so on as far as your calculator can reasonably go. What pattern do you see? b. G  raph ƒ in a variety of graphing windows, including windows that contain the origin. What do you see? Trace the y-values along the graph. What do you find? 3. Lorentz contraction  In relativity theory, the length of an object, say a rocket, appears to an observer to depend on the speed at which the object is traveling with respect to the observer. If the observer measures the rocket’s length as L 0 at rest, then at speed y the length will appear to be L = L0

B

1 -

y2 . c2

This equation is the Lorentz contraction formula. Here, c is the speed of light in a vacuum, about 3 * 108 m>sec. What happens to L as y increases? Find limy S c- L. Why was the left-hand limit needed?

M02_THOM9580_03_SE_C02_065-128.indd 126

4. Controlling the flow from a draining tank  Torricelli’s law says that if you drain a tank like the one in the figure shown, the rate y at which water runs out is a constant times the square root of the water’s depth x. The constant depends on the size and shape of the exit valve.

x Exit rate y ft3min

Suppose that y = 2x>2 for a certain tank. You are trying to maintain a fairly constant exit rate by adding water to the tank with a hose from time to time. How deep must you keep the water if you want to maintain the exit rate a. within 0.2 ft3 >min of the rate y0 = 1 ft3 >min?

b. within 0.1 ft3 >min of the rate y0 = 1 ft3 >min?

5. Thermal expansion in precise equipment  As you may know, most metals expand when heated and contract when cooled. The dimensions of a piece of laboratory equipment are sometimes so critical that the shop where the equipment is made must be held at the same temperature as the laboratory where the equipment is to be used. A typical aluminum bar that is 10 cm wide at 70°F will be y = 10 + (t - 70) * 10-4 centimeters wide at a nearby temperature t. Suppose that you are using a bar like this in a gravity wave detector, where its width must stay within 0.0005 cm of the ideal 10 cm. How close to t0 = 70°F must you maintain the temperature to ensure that this tolerance is not exceeded? 6. Stripes on a measuring cup  The interior of a typical 1-L measuring cup is a right circular cylinder of radius 6 cm (see accompanying figure). The volume of water we put in the cup is therefore a function of the level h to which the cup is filled, the formula being V = p62h = 36ph. How closely must we measure h to measure out 1 L of water (1000 cm3) with an error of no more than 1% (10 cm3)?

16/11/15 4:36 PM

17. A function continuous at only one point Let ƒ(x) = e

b. U  se the fact that every nonempty open interval of real numbers contains both rational and irrational numbers to show that ƒ is not continuous at any nonzero value of x. 18. The Dirichlet ruler function If x is a rational number, then x can be written in a unique way as a quotient of integers m>n where n 7 0 and m and n have no common factors greater than 1. (We say that such a fraction is in lowest terms. For example, 6>4 written in lowest terms is 3>2.) Let ƒ(x) be defined for all x in the interval 30, 14 by

(a) r = 6 cm

Liquid volume V = 36ph

h

A 1-L measuring cup (a), modeled as a right circular cylinder (b) of radius r = 6 cm Precise Definition of Limit In Exercises 7–10, use the formal definition of limit to prove that the function is continuous at c.

7. ƒ(x) = x2 - 7, c = 1

8. g(x) = 1>(2x), c = 1>4

9. h(x) = 22x - 3, c = 2 10. F(x) = 29 - x, c = 5

11. Uniqueness of limits  Show that a function cannot have two different limits at the same point. That is, if limx S c ƒ(x) = L 1 and limx S c ƒ(x) = L 2, then L 1 = L 2.

12. Prove the limit Constant Multiple Rule: lim kƒ(x) = k lim ƒ(x) for any constant k. xSc

13. One-sided limits If limx S 0+ ƒ(x) = A and limx S 0- ƒ(x) = B, find limx S 0- ƒ(x3 - x) a. limx S 0+ ƒ(x3 - x) b. limx S 0- ƒ(x2 - x4) c. limx S 0+ ƒ(x2 - x4) d. 14. Limits and continuity  Which of the following statements are true, and which are false? If true, say why; if false, give a counterexample (that is, an example confirming the falsehood). a. If limx S c ƒ(x) exists but limx S c g(x) does not exist, then limx S c(ƒ(x) + g(x)) does not exist. b. If neither limx S c ƒ(x) nor limx S c g(x) exists, then limx S c (ƒ(x) + g(x)) does not exist. c. If ƒ is continuous at x, then so is 0 ƒ 0 .

d. If 0 ƒ 0 is continuous at c, then so is ƒ.

In Exercises 15 and 16, use the formal definition of limit to prove that the function has a continuous extension to the given value of x. x2 - 1 , x = - 1 x + 1

M02_THOM9580_03_SE_C02_065-128.indd 127

ƒ(x) = e

1>n, if x = m>n is a rational number in lowest terms 0, if x is irrational.

For instance, ƒ(0) = ƒ(1) = 1, ƒ(1>2) = 1>2, ƒ(1>3) = ƒ(2>3) = 1>3, ƒ(1>4) = ƒ(3>4) = 1>4, and so on. a. Show that ƒ is discontinuous at every rational number in 30, 14 .

(b)

15. ƒ(x) =

x, if x is rational 0, if x is irrational.

a. Show that ƒ is continuous at x = 0.

xSc

127

16. g(x) =

x2 - 2x - 3 , x = 3 2x - 6

b. S  how that ƒ is continuous at every irrational number in 30, 14 . (Hint: If P is a given positive number, show that there are only finitely many rational numbers r in 30, 14 such that ƒ(r) Ú P.)

c. Sketch the graph of ƒ. Why do you think ƒ is called the “ruler function”?

19. Antipodal points  Is there any reason to believe that there is always a pair of antipodal (diametrically opposite) points on Earth’s equator where the temperatures are the same? Explain. 20. If limx S c (ƒ(x) + g(x)) = 3 and limx S c (ƒ(x) - g(x)) = -1, find limx S c ƒ(x)g(x). 21. Roots of a quadratic equation that is almost linear  The equation ax2 + 2x - 1 = 0, where a is a constant, has two roots if a 7 -1 and a ≠ 0, one positive and one negative: r+(a) =

-1 + 21 + a , a

r-(a) =

- 1 - 21 + a , a

a. What happens to r+(a) as a S 0? As a S - 1+? b. What happens to r-(a) as a S 0? As a S - 1+? c. S  upport your conclusions by graphing r+(a) and r-(a) as functions of a. Describe what you see. d. For added support, graph ƒ(x) = ax2 + 2x - 1 simultaneously for a = 1, 0.5, 0.2, 0.1, and 0.05. 22. Root of an equation  Show that the equation x + 2 cos x = 0 has at least one solution. 23. Bounded functions  A real-valued function ƒ is bounded from above on a set D if there exists a number N such that ƒ(x) … N for all x in D. We call N, when it exists, an upper bound for ƒ on D and say that ƒ is bounded from above by N. In a similar manner, we say that ƒ is bounded from below on D if there exists a number M such that ƒ(x) Ú M for all x in D. We call M, when it exists, a lower bound for ƒ on D and say that ƒ is bounded from below by M. We say that ƒ is bounded on D if it is bounded from both above and below. a. S  how that ƒ is bounded on D if and only if there exists a number B such that 0 ƒ(x) 0 … B for all x in D.

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Chapter 2: Limits and Continuity

b. Suppose that ƒ is bounded from above by N. Show that if limx S c ƒ(x) = L, then L … N. c. Suppose that ƒ is bounded from below by M. Show that if limx S c ƒ(x) = L, then L Ú M. 24. Max 5a, b6  and min 5a, b6 a. Show that the expression

max 5a, b6 =

0a - b0 a + b + 2 2

equals a if a Ú b and equals b if b Ú a. In other words, max 5a, b6  gives the larger of the two numbers a and b.

b. Find a similar expression for min 5a, b6, the smaller of a and b.

sin U U The formula limu S 0 (sin u)>u = 1 can be generalized. If limx S c ƒ(x) = 0 and ƒ(x) is never zero in an open interval containing the point x = c, except possibly c itself, then Generalized Limits Involving

lim

xSc

sin ƒ(x) = 1. ƒ(x)

Here are several examples. sin x2 = 1 x S 0 x2

a. lim b. lim

xS0

sin x2 sin x2 x2 lim = 1 # 0 = 0 x = xlim S 0 x2 x S 0 x

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sin (x2 - x - 2) sin (x2 - x - 2) # = lim x + 1 x S -1 x S -1 (x 2 - x - 2)

c. lim

(x2 - x - 2) (x + 1)(x - 2) = 1 # lim = -3 x + 1 x + 1 x S -1 x S -1

lim d. lim

xS1

sin 1 1 - 2x 2 sin 1 1 - 2x 2 1 - 2x = lim = x - 1 x - 1 xS1 1 - 2x

1 # lim

xS1

11

- 2x 21 1 + 2x 2

(x - 1)1 1 + 2x 2

= lim

x S 1 (x

1 - x 1 = 2 - 1)1 1 + 2x 2

Find the limits in Exercises 25–30. sin (1 - cos x) sin x 25. lim 26. lim+ x xS0 x S 0 sin 2x sin (sin x) x

28. lim

sin ( x2 - 4 ) x - 2 xS2

30. lim

27. lim

xS0

29. lim

xS0

sin ( x2 + x ) x

sin 1 2x - 3 2 x - 9 xS9

Oblique Asymptotes Find all possible oblique asymptotes in Exercises 31–34.

31. y =

2x3>2 + 2x - 3 2x + 1

33. y = 2x2 + 1

1 32. y = x + x sin x 34. y = 2x2 + 2x

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chapter

3

Derivatives

Overview  In the beginning of Chapter 2, we discussed how to determine the slope of a

curve at a point and how to measure the rate at which a function changes. Now that we have studied limits, we can define these ideas precisely and see that both are interpretations of the derivative of a function at a point. We then extend this concept from a single point to the derivative function, and we develop rules for finding this derivative function easily, without having to calculate any limits directly. These rules are used to find derivatives of most of the common functions reviewed in Chapter 1, as well as various combinations of them.

3.1  Tangents and the Derivative at a Point In this section we define the slope and tangent to a curve at a point, and the derivative of a function at a point. The derivative gives a way to find both the slope of a graph and the instantaneous rate of change of a function.

Finding a Tangent to the Graph of a Function y y = f (x) Q(x 0 + h, f (x 0 + h)) f (x 0 + h) − f (x 0)

To find a tangent to an arbitrary curve y = ƒ(x) at a point P(x0, ƒ(x0)), we use the procedure introduced in Section 2.1. We calculate the slope of the secant through P and a nearby point Q(x0 + h, ƒ(x0 + h)). We then investigate the limit of the slope as h S 0 (Figure 3.1). If the limit exists, we call it the slope of the curve at P and define the tangent at P to be the line through P having this slope.

P(x 0, f(x 0)) h 0

x0

x0 + h

Figure 3.1  The slope of the tangent ƒ(x0 + h) - ƒ(x0) line at P is lim . h hS0

x

Definitions The slope of the curve y = ƒ(x) at the point P(x0, ƒ(x0)) is the number m = lim

hS0

ƒ(x0 + h) - ƒ(x0) h

(provided the limit exists).

The tangent line to the curve at P is the line through P with this slope.

In Section 2.1, Example 3, we applied these definitions to find the slope of the parabola ƒ(x) = x2 at the point P(2, 4) and the tangent line to the parabola at P. Let’s look at another example.

129

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Chapter 3: Derivatives

EXAMPLE 1

y

(a) Find the slope of the curve y = 1>x at any point x = a ≠ 0. What is the slope at the point x = -1? (b) Where does the slope equal -1>4? (c) What happens to the tangent to the curve at the point (a, 1>a) as a changes?

y = 1x slope is −

0

1 a2

x

a

Solution (a) Here ƒ(x) = 1>x. The slope at (a, 1>a) is 1 1 a + h a ƒ(a + h) - ƒ(a) 1 a - (a + h) = lim lim = lim h hS0 hS0 h S 0 h a(a + h) h

slope is −1 at x = −1

Figure 3.2  The tangent slopes, steep near the origin, become more gradual as the point of tangency moves away (Example 1). y y = 1x

slope is −

1 4

hS0

-

1 slope is − 4

Figure 3.3  The two tangent lines to y = 1>x having slope - 1>4 (Example 1).

-h -1 1 = lim = - 2. ha(a + h) h S 0 a(a + h) a

Notice how we had to keep writing “limh S 0” before each fraction until the stage at which we could evaluate the limit by substituting h = 0. The number a may be positive or negative, but not 0. When a = -1, the slope is -1>(-1)2 = -1 (Figure 3.2). (b) The slope of y = 1>x at the point where x = a is -1>a2. It will be -1>4 provided that

1 a2, b 2 x

1 a−2, − 2b

= lim

1 1 = - . 4 a2

This equation is equivalent to a2 = 4, so a = 2 or a = -2. The curve has slope -1>4 at the two points (2, 1>2) and (-2, -1>2) (Figure 3.3). (c) The slope -1>a2 is always negative if a ≠ 0. As a S 0+, the slope approaches - q and the tangent becomes increasingly steep (Figure 3.2). We see this situation again as a S 0-. As a moves away from the origin in either direction, the slope approaches 0 and the tangent levels off becoming more and more horizontal.

Rates of Change: Derivative at a Point The expression ƒ(x0 + h) - ƒ(x0) , h ≠ 0 h is called the difference quotient of ƒ at x0 with increment h. If the difference quotient has a limit as h approaches zero, that limit is given a special name and notation. Definition The derivative of a function ƒ at a point x0 , denoted ƒ′(x0), is

The notation ƒ′(x0) is read “ƒ prime of x0.”

ƒ′(x0) = lim

hS0

ƒ(x0 + h) - ƒ(x0) h

provided this limit exists.

If we interpret the difference quotient as the slope of a secant line, then the derivative gives the slope of the curve y = ƒ(x) at the point P(x0, ƒ(x0)). If we interpret the difference quotient as an average rate of change (Section 2.1), the derivative gives the function’s instantaneous rate of change with respect to x at the point x = x0. We study this interpretation in Section 3.4.

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3.1  Tangents and the Derivative at a Point

EXAMPLE 2  In Examples 1 and 2 in Section 2.1, we studied the speed of a rock falling freely from rest near the surface of the earth. What was the rock’s exact speed at the instant t = 1? Solution  We let ƒ(t) = 16t 2. The average speed of the rock over the interval between t = 1 and t = 1 + h seconds, for h 7 0, was found to be 32 + 16h. The rock’s speed at the instant t = 1 is then ƒ′(1) = lim (32 + 16h) = 32 ft>sec.

hS0

Summary We have been discussing slopes of curves, lines tangent to a curve, the rate of change of a function, and the derivative of a function at a point. All of these ideas refer to the same limit.

The following are all interpretations for the limit of the difference quotient, ƒ(x0 + h) - ƒ(x0) . h hS0 lim

1. The slope of the graph of y = ƒ(x) at x = x0 2. The slope of the tangent to the curve y = ƒ(x) at x = x0 3. The rate of change of ƒ(x) with respect to x at x = x0 4. The derivative ƒ′(x0) at a point

In the next sections, we allow the point x0 to vary across the domain of the function ƒ.

Exercises  3.1 Slopes and Tangent Lines In Exercises 1–4, use the grid and a straight edge to make a rough estimate of the slope of the curve (in y-units per x-unit) at the points P1 and P2.

1.

2. y

3.

4.

y

y 4

2

3

y

2

2

P2

−2

−1

0

P1 −1

P1 0

1

x −2

2

P1

P2

1

1 1

P2

P1

1

P2

0 1

2

1

2

x

x

−2

−1

0

1

2

x

In Exercises 5–10, find an equation for the tangent to the curve at the given point. Then sketch the curve and tangent together. 5. y = 4 - x2, (- 1, 3)

6. y = (x - 1)2 + 1, (1, 1)

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Chapter 3: Derivatives

7. y = 2 2x, (1, 2)

8. y =

9. y = x3, (- 2, - 8)

10. y =

1 , (- 1, 1) x2 1 a- 2, - b 8

1 , x3

In Exercises 11–18, find the slope of the function’s graph at the given point. Then find an equation for the line tangent to the graph there. 11. ƒ(x) = x2 + 1, (2, 5)

12. ƒ(x) = x - 2x2, (1, -1)

At what points do the graphs of the functions in Exercises 25 and 26 have horizontal tangents? 25. ƒ(x) = x2 + 4x - 1

26. g(x) = x3 - 3x

27. Find equations of all lines having slope -1 that are tangent to the curve y = 1>(x - 1). 28. Find an equation of the straight line having slope 1>4 that is tangent to the curve y = 2x.

15. h(t) = t 3, (2, 8)

16. h(t) = t 3 + 3t, (1, 4)

Rates of Change 29. Object dropped from a tower  An object is dropped from the top of a 660-m-high tower. Its height above ground after t sec is 660 - 4.9t 2 m. How fast is it falling 2 sec after it is dropped?

17. ƒ(x) = 2x, (4, 2)

18. ƒ(x) = 2x + 1, (8, 3)

30. Speed of a rocket At t sec after liftoff, the height of a rocket is 4t 2 ft. How fast is the rocket climbing 7 sec after liftoff?

19. y = 5x - 3x2, x = 1

20. y = x3 - 2x + 7, x = -2

13. g(x) =

x , (3, 3) x - 2

14. g(x) =

8 , (2, 2) x2

In Exercises 19–22, find the slope of the curve at the point indicated.

21. y =

1 , x = 3 x - 1

22. y =

x - 1 , x = 0 x + 1

Interpreting Derivative Values 23. Growth of yeast cells  In a controlled laboratory experiment, yeast cells are grown in an automated cell culture system that counts the number P of cells present at hourly intervals. The number after t hours is shown in the accompanying figure. p

32. Ball’s changing volume  What is the rate of change of the volume of a ball ( V = (4>3)pr 3 ) with respect to the radius when the radius is r = 1? 33. Show that the line y = mx + b is its own tangent line at any point (x0, mx0 + b). 34. Find the slope of the tangent to the curve f(x) = 4> 2x at the point where x = 1/4. Testing for Tangents 35. Does the graph of

250 200 150 100 50 0

31. Circle’s changing area  What is the rate of change of the area of a circle ( A = pr 2 ) with respect to the radius when the radius is r = 5?

1 2 3 4 5 6 7

t

a. Explain what is meant by the derivative P′(5). What are its units? b. Which is larger, P′(2) or P′(3)? Give a reason for your answer. c. The quadratic curve capturing the trend of the data points (see Section 1.4) is given by P(t) = 6.10t 2 - 9.28t + 16.43. Find the instantaneous rate of growth when t = 5 hours. 24. Effectiveness of a drug  On a scale from 0 to 1, the effectiveness E of a pain-killing drug t hours after entering the bloodstream is displayed in the accompanying figure.

ƒ(x) = e

have a tangent at the origin? Give reasons for your answer. 36. Does the graph of g(x) = e

x sin (1>x), x ≠ 0 0, x = 0

have a tangent at the origin? Give reasons for your answer. Vertical Tangents We say that a continuous curve y = ƒ(x) has a vertical tangent at the point where x = x0 if the limit of the difference quotient is q or - q. For example, y = x1>3 has a vertical tangent at x = 0 (see accompanying figure):

lim

E

x2 sin (1>x), x ≠ 0 0, x = 0

hS0

ƒ(0 + h) - ƒ(0) h1>3 - 0 = lim h h hS0

1.0 0.8 0.6 0.4 0.2 0

= lim y

1

2

3

4

5

b. At what time would you estimate that the drug reaches its maximum effectiveness? What is true about the derivative at that time? What is true about the derivative as time increases in the 1 hour before your estimated time?

= q.

y = f (x) = x 13

t

a. At what times does the effectiveness appear to be increasing? What is true about the derivative at those times?

1

h S 0 h2>3

0

x

VERTICAL TANGENT AT ORIGIN

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3.2  The Derivative as a Function

However, y = x2>3 has no vertical tangent at x = 0 (see next figure): lim

hS0

g(0 + h) - g(0) h2>3 - 0 = lim S h h h 0 1 = lim 1>3 hS0 h

b.  Confirm your findings in part (a) with limit calculations. But before you do, read the introduction to Exercises 37 and 38. 39. y = x2>5 41. y = x

1>5

40. y = x4>5 42. y = x3>5

43. y = 4x2>5 - 2x

does not exist, because the limit is q from the right and - q from the left. y

133

45. y = x

2>3

47. y = e

44. y = x5>3 - 5x2>3

- (x - 1)

1>3

46. y = x1>3 + (x - 1)1>3

- 20x0, x … 0 2x, x 7 0

48. y = 2 0 4 - x 0

Computer Explorations Use a CAS to perform the following steps for the functions in Exercises 49–52:

y = g(x) = x 23

a. Plot y = ƒ(x) over the interval (x0 - 1>2) … x … (x0 + 3). b. Holding x0 fixed, the difference quotient

x

0 NO VERTICAL TANGENT AT ORIGIN

q(h) =

37. Does the graph of

at x0 becomes a function of the step size h. Enter this function into your CAS workspace. c. Find the limit of q as h S 0.

- 1, x 6 0 ƒ(x) = c 0, x = 0 1, x 7 0 have a vertical tangent at the origin? Give reasons for your answer. 38. Does the graph of U(x) = e

ƒ(x0 + h) - ƒ(x0) h

0, x 6 0 1, x Ú 0

d.  Define the secant lines y = ƒ(x0) + q # (x - x0) for h = 3, 2, and 1. Graph them together with ƒ and the tangent line over the interval in part (a). 49. ƒ(x) = x3 + 2x, x0 = 0

have a vertical tangent at the point (0, 1)? Give reasons for your answer.

5 50. ƒ(x) = x + x , x0 = 1

51. ƒ(x) = x + sin (2x), x0 = p>2 52. ƒ(x) = cos x + 4 sin (2x), x0 = p

T Graph the curves in Exercises 39–48. a. Where do the graphs appear to have vertical tangents?

3.2  The Derivative as a Function In the last section we defined the derivative of y = ƒ(x) at the point x = x0 to be the limit HISTORICAL ESSAY The Derivative

ƒ′(x0) = lim

hS0

ƒ(x0 + h) - ƒ(x0) . h

We now investigate the derivative as a function derived from ƒ by considering the limit at each point x in the domain of ƒ. Definition The derivative of the function ƒ(x) with respect to the variable x is the function ƒ′ whose value at x is ƒ(x + h) - ƒ(x) , h hS0

ƒ′(x) = lim provided the limit exists.

We use the notation ƒ(x) in the definition to emphasize the independent variable x with respect to which the derivative function ƒ′(x) is being defined. The domain of ƒ′ is the set of points in the domain of ƒ for which the limit exists, which means that the domain may be the same as or smaller than the domain of ƒ. If ƒ′ exists at a particular x, we say

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134

Chapter 3: Derivatives y = f(x) Secant slope is f (z) − f (x) z−x

Q(z, f (z))

f (z) − f (x)

P(x, f(x)) h=z−x x

that ƒ is differentiable (has a derivative) at x. If ƒ′ exists at every point in the domain of ƒ, we call ƒ differentiable. If we write z = x + h, then h = z - x and h approaches 0 if and only if z approaches x. Therefore, an equivalent definition of the derivative is as follows (see Figure 3.4). This formula is sometimes more convenient to use when finding a derivative function, and focuses on the point z that approaches x.

Alternative Formula for the Derivative z=x+h

ƒ′(x) = lim

zSx

Derivative of f at x is f (x + h) − f (x) f '(x) = lim h h:0 = lim

z :x

f (z) − f (x) z−x

Figure 3.4  Two forms for the difference quotient.

ƒ(z) - ƒ(x) z - x

Calculating Derivatives from the Definition The process of calculating a derivative is called differentiation. To emphasize the idea that differentiation is an operation performed on a function y = ƒ(x), we use the notation d ƒ(x) dx

Derivative of the Reciprocal Function

as another way to denote the derivative ƒ′(x). Example 1 of Section 3.1 illustrated the differentiation process for the function y = 1>x when x = a. For x representing any point in the domain, we get the formula

d 1 1 a b = - 2, x ≠ 0 dx x x

d 1 1 a b = - 2. dx x x

Here are two more examples in which we allow x to be any point in the domain of ƒ.

EXAMPLE 1  Differentiate ƒ(x) =

x . x - 1

Solution  We use the definition of derivative, which requires us to calculate ƒ(x + h) and then subtract ƒ(x) to obtain the numerator in the difference quotient. We have (x + h) x , so and ƒ(x + h) = x - 1 (x + h) - 1 ƒ(x + h) - ƒ(x) ƒ′(x) = lim Definition h hS0 x + h x x + h - 1 x - 1 = lim hS0 h (x + h) (x - 1) - x(x + h - 1) 1 a c ad - cb = lim # - = b d bd (x + h - 1) (x - 1) hS0 h ƒ(x) =

= lim

= lim

1#

hS0 h

h S 0 (x

-h (x + h - 1) (x - 1)

Simplify.

-1 -1 = . + h - 1) (x - 1) (x - 1)2

Cancel h ≠ 0.

EXAMPLE 2 (a) Find the derivative of ƒ(x) = 2x for x 7 0. (b) Find the tangent line to the curve y = 2x at x = 4.

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3.2  The Derivative as a Function

Solution (a) We use the alternative formula to calculate ƒ′:

Derivative of the Square Root Function d 1 2x = , x 7 0 dx 2 2x

ƒ′(x) = lim

zSx

= lim

zSx

y

= lim

y = 1x + 1 4 (4, 2)

zSx

= lim

zSx

y = "x

1

(b) The slope of the curve at x = 4 is

0

135

x

4

ƒ(z) - ƒ(x) z - x 1z - 1x

z - x

1 1z

- 1x 21 1z + 1x 2

1

1z + 1x

ƒ′(4) =

Figure 3.5  The curve y = 2x and its tangent at (4, 2). The tangent’s slope is found by evaluating the derivative at x = 4 (Example 2).

1z - 1x

=

1 . 2 1x

1 1 = . 2 24 4

The tangent is the line through the point (4, 2) with slope 1>4 (Figure 3.5): y = 2 + y =

y

1 (x - 4) 4

1 x + 1. 4



Notations Slope 0 A 10

Slope −1 B C

Slope − 4 3 E D Slope 0

5

0

5

There are many ways to denote the derivative of a function y = ƒ(x), where the independent variable is x and the dependent variable is y. Some common alternative notations for the derivative are

y = f (x)

≈8

≈ 4 x-units 10 15

ƒ′(x) = y′ = x

(a) Slope

4

A'

3

y = f '(x)

2 1

E′

−1 −2

5 C′

D′ 10

The symbols d>dx and D indicate the operation of differentiation. We read dy>dx as “the derivative of y with respect to x,” and dƒ>dx and (d>dx) ƒ(x) as “the derivative of ƒ with respect to x.” The “prime” notations y′ and ƒ′ come from notations that Newton used for derivatives. The d>dx notations are similar to those used by Leibniz. The symbol dy>dx should not be regarded as a ratio (until we introduce the idea of “differentials” in Section 3.11). To indicate the value of a derivative at a specified number x = a, we use the notation ƒ′(a) =

15

B′ Vertical coordinate −1 (b)

Figure 3.6  We made the graph of y = ƒ′(x) in (b) by plotting slopes from the graph of y = ƒ(x) in (a). The vertical coordinate of B′ is the slope at B and so on. The slope at E is approximately 8>4 = 2. In (b) we see that the rate of change of ƒ is negative for x between A′ and D′; the rate of change is positive for x to the right of D′.

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dy dƒ d = = ƒ(x) = D(ƒ)(x) = Dx ƒ(x). dx dx dx

x

For instance, in Example 2 ƒ′(4) =

Graphing the Derivative

df dy d ` = ` = ƒ(x) ` . dx x = a dx x = a dx x=a

d 1 1 1 2x ` = ` = = . 4 dx 2 1 x 2 24 x=4 x=4

We can often make a reasonable plot of the derivative of y = ƒ(x) by estimating the slopes on the graph of ƒ. That is, we plot the points (x, ƒ′(x)) in the xy-plane and connect them with a smooth curve, which represents y = ƒ′(x).

EXAMPLE 3   Graph the derivative of the function y = ƒ(x) in Figure 3.6a. Solution  We sketch the tangents to the graph of ƒ at frequent intervals and use their slopes to estimate the values of ƒ′(x) at these points. We plot the corresponding (x, ƒ′(x)) pairs and connect them with a smooth curve as sketched in Figure 3.6b.

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Chapter 3: Derivatives

What can we learn from the graph of y = ƒ′(x)? At a glance we can see

Slope = f(a + h) − f(a) lim + h h:0

1. where the rate of change of ƒ is positive, negative, or zero; 2. the rough size of the growth rate at any x and its size in relation to the size of ƒ(x); 3. where the rate of change itself is increasing or decreasing.

Slope = f (b + h) − f (b) lim h h:0−

Differentiable on an Interval; One-Sided Derivatives A function y = ƒ(x) is differentiable on an open interval (finite or infinite) if it has a derivative at each point of the interval. It is differentiable on a closed interval 3 a, b4 if it is differentiable on the interior (a, b) and if the limits

y = f (x)

a

a+h h>0

b+h hx) near the origin, where it is discontinuous (see Figure 2.31).

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Chapter 3: Derivatives

Differentiable Functions Are Continuous A function is continuous at every point where it has a derivative.

THEOREM 1—Differentiability Implies Continuity  If ƒ has a derivative at x = c, then ƒ is continuous at x = c.

Proof  Given that ƒ′(c) exists, we must show that limx S c ƒ(x) = ƒ(c), or equivalently, that limh S 0 ƒ(c + h) = ƒ(c). If h ≠ 0, then ƒ(c + h) = ƒ(c) + (ƒ(c + h) - ƒ(c)) ƒ(c + h) - ƒ(c) # = ƒ(c) + h. h Now take limits as h S 0. By Theorem 1 of Section 2.2,

ƒ(c + h) - ƒ(c) # lim h h hS0 hS0

lim ƒ(c + h) = lim ƒ(c) + lim

hS0

hS0

= ƒ(c) + ƒ′(c) # 0 = ƒ(c) + 0 = ƒ(c).



Similar arguments with one-sided limits show that if ƒ has a derivative from one side (right or left) at x = c, then ƒ is continuous from that side at x = c. Theorem 1 says that if a function has a discontinuity at a point (for instance, a jump discontinuity), then it cannot be differentiable there. The greatest integer function y = : x ; fails to be differentiable at every integer x = n (Example 4, Section 2.5). Caution  The converse of Theorem 1 is false. A function need not have a derivative at a point where it is continuous, as we saw with the absolute value function in Example 4.

Exercises  3.2 Finding Derivative Functions and Values Using the definition, calculate the derivatives of the functions in Exercises 1–6. Then find the values of the derivatives as specified.

1. ƒ(x) = 2 - x2; ƒ′(- 7), ƒ′(0), ƒ′(5) 2. F(x) = (x - 1)2 + 1; F′(- 1), F′(0), F′(2) 3. g(t) = 4. k(z) =

5 ; g′(- 4), g′(2), g′1 26 2 t4

1 - z ; k′(- 1), k′(1), k′1 22 2 2z

11.

dy if y = x3>2 dx

12.

dz 1 if z = dw 2w2 - 1

Slopes and Tangent Lines In Exercises 13–16, differentiate the functions and find the slope of the tangent line at the given value of the independent variable.

2 13. ƒ(x) = 3x + x , x = -1

14. k(x) =

15. s = 9t 3 - 8t 2, t = 6

16. y =

1 , x = 2 2 + x

x + 3 , x = -2 1 - x

5. p(u) = 23u ; p′(1), p′(3), p′(2>3)

In Exercises 17–18, differentiate the functions. Then find an equation of the tangent line at the indicated point on the graph of the function.

In Exercises 7–12, find the indicated derivatives.

17. y = ƒ(x) =

6. r (s) = 22s + 1 ; r′(0), r′(1), r′(1>2) 7.

dy if y = 4x3 dx

9.

ds t if s = 4t + 9 dt

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8. 10.

dr if r = s3 - 2s2 + 3 ds dy 1 if y = t - t dt

8 2x - 2

, (x, y) = (6, 4)

18. y = f(x) = 1 + 26 - x, (x, y) = (2, 3)

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31. a.  The graph in the accompanying figure is made of line segments joined end to end. At which points of the interval 3-4, 64 is ƒ′ not defined? Give reasons for your answer.

In Exercises 19–22, find the values of the derivatives. dy dy 1 if y = 5 - 2x2 20. ` if y = 1 - x 19. ` dx x = -4 dx x = 23

y

dw dr 2 0 21. ` if r = 22. if w = z + 1z du u = 0 dz z = 4 24 - u

y = f(x)

Using the Alternative Formula for Derivatives Use the formula

ƒ′(x) = lim

zSx

0

(−4, 0)

ƒ(z) - ƒ(x) z - x

1 x + 2

25. g(x) =

x x - 1

24. ƒ(x) = x2 - 3x + 4

x

6

(4, −2)

b. Graph the derivative of ƒ. The graph should show a step function.

26. g(x) = 1 + 1x

32. Recovering a function from its derivative

Graphs Match the functions graphed in Exercises 27–30 with the derivatives graphed in the accompanying figures (a)–(d). y′

1

(1, −2)

to find the derivative of the functions in Exercises 23–26. 23. ƒ(x) =

(6, 2)

(0, 2)

a. Use the following information to graph the function ƒ over the closed interval 3-2, 54.

i) The graph of ƒ is made of closed line segments joined end to end.

y′

ii) The graph starts at the point (- 2, 3).

iii) The derivative of ƒ is the step function in the figure shown here. y′ x

0

y′ = f ′(x)

x

0

1

(a)

(b)

y′

y′

−2

0

1

3

5

x

−2 x

0

(c)

27.

x

0

33. Growth in the economy  The graph in the accompanying figure shows the average annual percentage change y = ƒ(t) in the U.S. gross national product (GNP) for the years 2005–2011. Graph dy>dt (where defined).

(d)

28.

y

y

y = f 2 (x)

y = f1(x) x

0

x

0

29.

b. Repeat part (a), assuming that the graph starts at (- 2, 0) instead of (-2, 3).

7% 6 5 4 3 2 1 0 2005 2006 2007 2008 2009 2010 2011

30. y

y y = f3(x)

0

y = f4(x) x

0

x

34. Fruit flies  (Continuation of Example 4, Section 2.1.) Populations starting out in closed environments grow slowly at first, when there are relatively few members, then more rapidly as the number of reproducing individuals increases and resources are still abundant, then slowly again as the population reaches the carrying capacity of the environment.

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Chapter 3: Derivatives

a. Use the graphical technique of Example 3 to graph the derivative of the fruit fly population. The graph of the population is reproduced here.

Number of flies

p

i)  t = 1  ii)  t = 4  iii)  t = 11 b. When does Jared lose weight most rapidly and what is this rate of weight loss? c. Use the graphical technique of Example 3 to graph the derivative of weight W.

350 300 250 200 150 100 50 0

a. Estimate Jared’s rate of weight loss when

One-Sided Derivatives Compute the right-hand and left-hand derivatives as limits to show that the functions in Exercises 37–40 are not differentiable at the point P.

37. 10

20 30 Time (days)

40

38.

y

y = f (x)

t

50

35. Temperature  The given graph shows the temperature T in °F at Davis, CA, on April 18, 2008, between 6 a.m. and 6 p.m. T

y=2 2

Temperature (°F)

P(1, 2)

y=x

1 x

P(0, 0)

39.

80

y = 2x

y = f (x)

y = x2

b. During what days does the population seem to be increasing fastest? Slowest?

0

40.

y

70

1

60

y = f (x) y = 2x − 1

P(1, 1)

1

y = 1x

50 1

40 6 A.M.

3

9 A.M.

6

12 NOON Time (hr)

9

3 P.M.

12

t

P(1, 1) y = "x

0

6 P.M.

x

1

y=x x

1

In Exercises 41 and 42, determine if the piecewise-defined function is differentiable at the origin.

a. Estimate the rate of temperature change at the times i) 7 a.m.  ii) 9 a.m.  iii) 2 p.m.  iv) 4 p.m. b. At what time does the temperature increase most rapidly? Decrease most rapidly? What is the rate for each of those times? c. Use the graphical technique of Example 3 to graph the derivative of temperature T versus time t. 36. Weight loss  Jared Fogle, also known as the “Subway Sandwich Guy,” weighed 425 lb in 1997 before losing more than 240 lb in 12 months (http://en.wikipedia.org/wiki/Jared_Fogle). A chart showing his possible dramatic weight loss is given in the accompanying figure.

41. ƒ(x) = e 42. g(x) = e

2x - 1, x2 + 2x + 7, x2>3, x1>3,

x Ú 0 x 6 0

x Ú 0 x 6 0

Differentiability and Continuity on an Interval Each figure in Exercises 43–48 shows the graph of a function over a closed interval D. At what domain points does the function appear to be

a. differentiable? b. continuous but not differentiable? c. neither continuous nor differentiable?

W 500

425

43.

44.

y y = f (x) D: −3 ≤ x ≤ 2 2

300 200

1 2 3 4 5 6 7 8 9 10 11 12 Time (months)

t

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2

x

0

−2 −1

1

2

3

x

−1

−1 −2

y = f (x) D: −2 ≤ x ≤ 3

1 0 1

−3 −2 −1

y 2

1

100 0

x

2

y

y = f (x)

0

Weight (lb)

y

−2

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3.2  The Derivative as a Function

45.

46.

y

y = f (x) D: −2 ≤ x ≤ 3

−3 −2 −1 0 −1

1

2

3

1 −2 −1

1

0

2

3

x

y

y = f (x) D: −1 ≤ x ≤ 2

4

y = f (x) D: −3 ≤ x ≤ 3

2

1

0

1

2

x

−3 −2 −1 0

1 2

3

x

Theory and Examples In Exercises 49–52,

(x + h)3 - x3 h

for h = 2, 1, 0.2. Then try h = - 2, - 1, - 0.2. Explain what is going on. 61. Derivative of y = ∣ x ∣   Graph the derivative of ƒ(x) = 0 x 0 . Then graph y = ( 0 x 0 - 0 ) >(x - 0) = 0 x 0 >x. What can you conclude?

T 62. Weierstrass’s nowhere differentiable continuous function  The sum of the first eight terms of the Weierstrass function q ƒ(x) = g n = 0 (2>3)n cos (9npx) is g(x) = cos (px) + (2>3)1 cos (9px) + (2>3)2 cos (92px)

a. Find the derivative ƒ′(x) of the given function y = ƒ(x). b. Graph y = ƒ(x) and y = ƒ′(x) side by side using separate sets of coordinate axes, and answer the following questions. c. For what values of x, if any, is ƒ′ positive? Zero? Negative? d. Over what intervals of x-values, if any, does the function y = ƒ(x) increase as x increases? Decrease as x increases? How is this related to what you found in part (c)? (We will say more about this relationship in Section 4.3.) 49. y = - x2

50. y = - 1>x

3

4

51. y = x >3

52. y = x >4

53. Tangent to a parabola  Does the parabola y = 2x2 - 13x + 5 have a tangent whose slope is - 1? If so, find an equation for the line and the point of tangency. If not, why not? 54. Tangent to y = 2x  Does any tangent to the curve y = 2x cross the x-axis at x = -1? If so, find an equation for the line and the point of tangency. If not, why not? 55. Derivative of −ƒ  Does knowing that a function ƒ(x) is differentiable at x = x0 tell you anything about the differentiability of the function -ƒ at x = x0? Give reasons for your answer. 56. Derivative of multiples  Does knowing that a function g(t) is differentiable at t = 7 tell you anything about the differentiability of the function 3g at t = 7? Give reasons for your answer. 57. Limit of a quotient  Suppose that functions g(t) and h(t) are defined for all values of t and g(0) = h(0) = 0. Can limt S 0 (g(t))>(h(t)) exist? If it does exist, must it equal zero? Give reasons for your answers. 58. a.  Let ƒ(x) be a function satisfying 0 ƒ(x) 0 … x2 for -1 … x … 1. Show that ƒ is differentiable at x = 0 and find ƒ′(0). b. Show that

1 x sin x , x ≠ 0 2

ƒ(x) = c

h

T 60. Graph y = 3x2 in a window that has - 2 … x … 2, 0 … y … 3. Then, on the same screen, graph y =

48.

y

2x + h - 2x

for h = 1, 0.5, 0.1. Then try h = - 1, -0.5, - 0.1. Explain what is going on.

2

x

−2

−1

y =

3

1

47.

T 59. Graph y = 1> 1 2 2x 2 in a window that has 0 … x … 2. Then, on the same screen, graph

y

y = f (x) D: −3 ≤ x ≤ 3

141

0,

+ (2>3)3 cos (93px) + g + (2>3)7 cos (97px). Graph this sum. Zoom in several times. How wiggly and bumpy is this graph? Specify a viewing window in which the displayed portion of the graph is smooth. COMPUTER EXPLORATIONS Use a CAS to perform the following steps for the functions in Exercises 63–68.

a. Plot y = ƒ(x) to see that function’s global behavior. b. Define the difference quotient q at a general point x, with general step size h. c. Take the limit as h S 0. What formula does this give? d. Substitute the value x = x0 and plot the function y = ƒ(x) together with its tangent line at that point. e. Substitute various values for x larger and smaller than x0 into the formula obtained in part (c). Do the numbers make sense with your picture? f. Graph the formula obtained in part (c). What does it mean when its values are negative? Zero? Positive? Does this make sense with your plot from part (a)? Give reasons for your answer. 63. ƒ(x) = x3 + x2 - x, x0 = 1 64. ƒ(x) = x1>3 + x2>3, x0 = 1 65. ƒ(x) =

4x , x0 = 2 x2 + 1

66. ƒ(x) =

x - 1 , x0 = - 1 3x2 + 1

67. ƒ(x) = sin 2x, x0 = p>2 68. ƒ(x) = x2 cos x, x0 = p>4

x = 0

is differentiable at x = 0 and find ƒ′(0).

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Chapter 3: Derivatives

3.3  Differentiation Rules This section introduces several rules that allow us to differentiate constant functions, power functions, polynomials, exponential functions, rational functions, and certain combinations of them, simply and directly, without having to take limits each time.

Powers, Multiples, Sums, and Differences A simple rule of differentiation is that the derivative of every constant function is zero. Derivative of a Constant Function If ƒ has the constant value ƒ(x) = c, then

y c

(x, c)

0

x

(x + h, c)

y=c

h x+h

Figure 3.9  The rule (d>dx)(c) = 0 is another way to say that the values of constant functions never change and that the slope of a horizontal line is zero at every point.

x

dƒ d = (c) = 0. dx dx

Proof   We apply the definition of the derivative to ƒ(x) = c, the function whose outputs have the constant value c (Figure 3.9). At every value of x, we find that ƒ′(x) = lim

hS0

ƒ(x + h) - ƒ(x) c - c = lim = lim 0 = 0. h h hS0 hS0

From Section 3.1, we know that d 1 1 a b = - 2, dx x x

or

d -1 ( x ) = -x - 2. dx

From Example 2 of the last section we also know that d 1 2x 2 = 1 , dx 2 2x

or

d 1>2 ( x ) = 12 x - 1>2 . dx

These two examples illustrate a general rule for differentiating a power xn . We first prove the rule when n is a positive integer. Derivative of a Positive Integer Power If n is a positive integer, then d n x = nxn - 1. dx HISTORICAL BIOGRAPHY Richard Courant (1888–1972)

Proof of the Positive Integer Power Rule   The formula zn - xn = (z - x)(zn - 1 + zn - 2 x + g + zxn - 2 + xn - 1) can be verified by multiplying out the right-hand side. Then from the alternative formula for the definition of the derivative,

ƒ(z) - ƒ(x) zn - x n = lim z - x z x zSx zSx

ƒ′(x) = lim

= lim (zn - 1 + zn - 2x + g + zxn - 2 + xn - 1)  n terms S z

= nx

x

n-1

.



The Power Rule is actually valid for all real numbers n. We have seen examples for a negative integer and fractional power, but n could be an irrational number as well. Here we state the general version of the rule, but postpone its proof until Section 3.8.

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3.3  Differentiation Rules

143

Power Rule (General Version)

If n is any real number, then d n x = nxn - 1, dx for all x where the powers xn and xn - 1 are defined.

EXAMPLE 1   Differentiate the following powers of x. (a) x3   (b)  x2/3   (c)  x 22   (d)

Applying the Power Rule Subtract 1 from the exponent and multiply the result by the original exponent.

Solution d 3 ( x ) = 3x3 - 1 = 3x2 (a) dx (b) (c) (d) (e) (f)

1   (e)  x-4>3   (f)  2x2 + p x4

d 2>3 ( x ) = 23 x(2>3) - 1 = 23 x-1>3 dx d 22 1 x 2 = 22x 22 - 1 dx

d 1 d -4 ( x ) = -4x-4 - 1 = -4x-5 = - 45 a b = dx x4 dx x

d -4>3 ( x ) = - 43 x-(4>3) - 1 = - 43 x-7>3 dx

d d 1 + (p>2) 1 2x2 + p 2 = dx 1x 2 = a1 + p2 b x1 + (p>2) - 1 = 21 (2 + p) 2xp  dx

The next rule says that when a differentiable function is multiplied by a constant, its derivative is multiplied by the same constant. Derivative Constant Multiple Rule If u is a differentiable function of x, and c is a constant, then

y y = 3x 2

3

d du (cu) = c . dx dx Slope = 3(2x) = 6x Slope = 6(1) = 6

(1, 3)

y=

x2

2

1

Proof

Slope = 2x Slope = 2(1) = 2 (1, 1)

cu(x + h) - cu(x) d cu = lim dx h hS0

= clim

u(x + h) - u(x) h

= c

du dx

hS0

Derivative definition with ƒ(x) = cu(x) Constant Multiple Limit Property u is differentiable.

EXAMPLE 2 0

1

x

2

Figure 3.10  The graphs of y = x and y = 3x2. Tripling the y-coordinate triples the slope (Example 2).

2

M03_THOM9580_03_SE_C03_129-222.indd 143

(a) The derivative formula d (3x2) = 3 # 2x = 6x dx says that if we rescale the graph of y = x2 by multiplying each y-coordinate by 3, then we multiply the slope at each point by 3 (Figure 3.10).

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Chapter 3: Derivatives

(b) Negative of a function The derivative of the negative of a differentiable function u is the negative of the function’s derivative. The Constant Multiple Rule with c = -1 gives d d d du (-u) = (-1 # u) = -1 # (u) = - . dx dx dx dx

Denoting Functions by u and Y The functions we are working with when we need a differentiation formula are likely to be denoted by letters like ƒ and g. We do not want to use these same letters when stating general differentiation rules, so instead we use letters like u and y that are not likely to be already in use.

The next rule says that the derivative of the sum of two differentiable functions is the sum of their derivatives. Derivative Sum Rule If u and y are differentiable functions of x, then their sum u + y is differentiable at every point where u and y are both differentiable. At such points, d du dy (u + y) = + . dx dx dx

Proof   We apply the definition of the derivative to ƒ(x) = u(x) + y(x): 3 u(x + h) + y(x + h)4 - 3 u(x) + y(x)4 d 3 u(x) + y(x) 4 = lim dx h hS0

= lim c hS0

= lim

hS0

u(x + h) - u(x) y(x + h) - y(x) + d h h

u(x + h) - u(x) y(x + h) - y(x) du dy + lim = + . h h dx dx S h 0

Combining the Sum Rule with the Constant Multiple Rule gives the Difference Rule, which says that the derivative of a difference of differentiable functions is the difference of their derivatives: d d du dy du dy (u - y) = 3 u + (-1)y 4 = + (-1) = . dx dx dx dx dx dx

The Sum Rule also extends to finite sums of more than two functions. If u1, u2, c, un are differentiable at x, then so is u1 + u2 + g + un , and dun du1 du2 d (u + u2 + g + un) = + + + . dx 1 dx dx g dx A proof by mathematical induction for any finite number of terms is given in Appendix 2.

EXAMPLE 3   Find the derivative of the polynomial y = x3 + 4 x2 - 5x + 1. 3

Solution

dy d 3 d d 4 2 d = (5x) + (1)  Sum and Difference Rules x + a x b dx dx dx 3 dx dx = 3x2 +

8 4# 2x - 5 + 0 = 3x2 + x - 5 3 3

We can differentiate any polynomial term by term, the way we differentiated the polynomial in Example 3. All polynomials are differentiable at all values of x.

EXAMPLE 4   Does the curve y = x4 - 2x2 + 2 have any horizontal tangents? If so, where?

Solution  The horizontal tangents, if any, occur where the slope dy>dx is zero. We have dy d 4 = (x - 2x2 + 2) = 4x3 - 4x. dx dx

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3.3  Differentiation Rules

y

y = x 4 − 2x 2 + 2

Now solve the equation

dy = 0 for x: dx 4x3 - 4x = 0 4x(x2 - 1) = 0 x = 0, 1, -1.

(0, 2)

1

(−1, 1)

(1, 1)

0

−1

145

1

The curve y = x4 - 2x2 + 2 has horizontal tangents at x = 0, 1, and -1. The corresponding points on the curve are (0, 2), (1, 1), and (-1, 1). See Figure 3.11.

x

Figure 3.11  The curve in Example 4 and its horizontal tangents.

Derivatives of Exponential Functions We briefly reviewed exponential functions in Section 1.5. When we apply the definition of the derivative to ƒ(x) = ax , we get

d x ax + h - ax (a ) = lim dx h S h 0

ax # ah - ax h hS0

ax + h = ax # ah

ah - 1 h

Factoring out ax

ah - 1 h hS0

ax is constant as h S 0.

= lim

= lim ax #

= ax # lim

Derivative definition

hS0

ah - 1 # x = a lim b a .(1) h hS0 (1+)1+* a fixed number L

Thus we see that the derivative of ax is a constant multiple L of ax . The constant L is a limit unlike any we have encountered before. Note, however, that it equals the derivative of ƒ(x) = ax at x = 0: ƒ′(0) = lim

hS0

y

a = 3 a = e a = 2.5

a=2

L = 1.0 1.1 0.92 0.69

h y = a − 1, a > 0 h

0

h

Figure 3.12  The position of the curve y = (ah - 1)>h, a 7 0, varies continuously with a. The limit L of y as h S 0 changes with different values of a. The number for which L = 1 as h S 0 is the number e between a = 2 and a = 3.

ah - a0 ah - 1 = lim = L. h h S h 0

The limit L is therefore the slope of the graph of ƒ(x) = ax where it crosses the y-axis. In Chapter 7, where we carefully develop the logarithmic and exponential functions, we prove that the limit L exists and has the value ln a. For now we investigate values of L by graphing the function y = (ah - 1)>h and studying its behavior as h approaches 0. Figure 3.12 shows the graphs of y = (ah - 1)>h for four different values of a. The limit L is approximately 0.69 if a = 2, about 0.92 if a = 2.5, and about 1.1 if a = 3. It appears that the value of L is 1 at some number a chosen between 2.5 and 3. That number is given by a = e ≈ 2.718281828. With this choice of base we obtain the natural exponential function ƒ(x) = ex as in Section 1.5, and see that it satisfies the property eh - 1 = 1(2) h hS0

ƒ′(0) = lim

because it is the exponential function whose graph has slope 1 when it crosses the y-axis. That the limit is 1 implies an important relationship between the natural exponential function ex and its derivative:

d x eh - 1 # x (e ) = lim ¢ ≤ e   Eq. (1) with a dx h hS0 = 1 # ex = ex.

= e

Eq. (2)

Therefore the natural exponential function is its own derivative.

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Chapter 3: Derivatives

Derivative of the Natural Exponential Function d x (e ) = ex dx

EXAMPLE 5   Find an equation for a line that is tangent to the graph of y = ex and

goes through the origin. y 6

y = ex

4 (a, e a )

2 −1

a

x

Figure 3.13  The line through the origin is tangent to the graph of y = ex when a = 1 (Example 5).

Solution  Since the line passes through the origin, its equation is of the form y = mx, where m is the slope. If it is tangent to the graph at the point (a, ea), the slope is m = (ea - 0)>(a - 0). The slope of the natural exponential at x = a is ea . Because these slopes are the same, we then have that ea = ea >a. It follows that a = 1 and m = e, so the equation of the tangent line is y = ex. See Figure 3.13. We might ask if there are functions other than the natural exponential function that are their own derivatives. The answer is that the only functions that satisfy the property that ƒ′(x) = ƒ(x) are functions that are constant multiples of the natural exponential function, ƒ(x) = c # ex , c any constant. We prove this fact in Section 7.2. Note from the Constant Multiple Rule that indeed d # x d (c e ) = c # (ex) = c # ex. dx dx

Products and Quotients While the derivative of the sum of two functions is the sum of their derivatives, the derivative of the product of two functions is not the product of their derivatives. For instance, d # d 2 (x x) = (x ) = 2x, dx dx

while

d d (x) # (x) = 1 # 1 = 1. dx dx

The derivative of a product of two functions is the sum of two products, as we now explain.

Derivative Product Rule If u and y are differentiable at x, then so is their product uy, and d dy du (uy) = u + y . dx dx dx

Equation (3) is equivalent to saying that (ƒg)′ = ƒ′g + ƒg′. This form of the Product Rule is useful and applies to dot products and cross products of vector-valued functions, studied in Chapter 12.

The derivative of the product uy is u times the derivative of y plus y times the derivative of u. In prime notation, (uy)′ = uy′ + yu′. In function notation,

d 3 ƒ(x)g(x) 4 = ƒ(x)g′(x) + g(x)ƒ′(x).(3) dx

EXAMPLE 6   Find the derivative of (a)  y = 1x 1 x2 + ex 2 , (b) y = e2x, (c)  (x2 + 1)(x3 + 3).

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3.3  Differentiation Rules

147

Solution (a) We apply the Product Rule with u = 1>x and y = x2 + ex: d

(uy) d 1 2 1 1 c ( x + ex ) d = x ( 2x + ex ) + ( x2 + ex ) a- 2 b    dx dx x x d

= u

dy du + y , and dx dx

1 1 a b = - 2 x

dx x ex ex = 2 + x - 1 - 2 x ex = 1 + (x - 1) 2 . x d 2x d x# x d d (b) (e ) = (e e ) = ex # (ex) + ex # (ex) = 2ex # ex = 2e2x dx dx dx dx

(c) This particular product can be differentiated by multiplying out the original expression for y and differentiating the resulting polynomial: y = ( x2 + 1 ) ( x3 + 3 ) = x5 + x3 + 3x2 + 3

dy = 5x4 + 3x2 + 6x. dx

Picturing the Product Rule Suppose u(x) and y(x) are positive and increase when x increases, and h 7 0. y(x + h) Δy

Proof of the Derivative Product Rule u(x + h)y(x + h) - u(x)y(x) d (uy) = lim dx h hS0 To change this fraction into an equivalent one that contains difference quotients for the derivatives of u and y, we subtract and add u(x + h)y(x) in the numerator:

u(x + h) Δy

y(x) u(x)y(x)

0



u(x + h)y(x + h) - u(x + h)y(x) + u(x + h)y(x) - u(x)y(x) d (uy) = lim dx h hS0

y(x) Δu

u(x)

= lim c u(x + h) hS0

Δu u(x + h)

Then the change in the product uy is the difference in areas of the larger and smaller “squares,” which is the sum of the upper and right-hand reddish-shaded rectangles. That is, ∆(uy) = u(x + h)y(x + h) - u(x)y(x) = u(x + h)∆y + y(x)∆u.

y(x + h) - y(x) u(x + h) - u(x) + y(x) d h h

= lim u(x + h) # lim hS0

hS0

y(x + h) - y(x) u(x + h) - u(x) + y(x) # lim . h h hS0

As h approaches zero, u(x + h) approaches u(x) because u, being differentiable at x, is continuous at x. The two fractions approach the values of dy>dx at x and du>dx at x. Therefore, d dy du (uy) = u + y . dx dx dx

The derivative of the quotient of two functions is given by the Quotient Rule.

Division by h gives ∆(uy) ∆u ∆y = u(x + h) + y(x) . h h h The limit as h S 0 gives the Product Rule. +

Derivative Quotient Rule If u and y are differentiable at x and if y(x) ≠ 0, then the quotient u>y is differentiable at x, and d u a b = dx y

y

du dy - u dx dx y2

.

In function notation, g(x)ƒ′(x) - ƒ(x)g′(x) d ƒ(x) . c d = dx g(x) g2(x)

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EXAMPLE 7   Find the derivative of (a)  y =

t2 - 1 ,  (b)  y = e-x. t3 + 1

Solution (a) We apply the Quotient Rule with u = t 2 - 1 and y = t 3 + 1: (b)

dy (t 3 + 1) # 2t - (t 2 - 1) # 3t 2 d u =    dt ayb dt (t 3 + 1)2 2t 4 + 2t - 3t 4 + 3t 2 = (t 3 + 1)2 4 -t + 3t 2 + 2t = . (t 3 + 1)2

=

y(du>dt) - u(dy>dt) y2

d -x d 1 ex # 0 - 1 # ex -1 (e ) = a xb = = x = -e - x  e e dx dx (ex)2

Proof of the Derivative Quotient Rule d u a b = lim dx y hS0

u(x + h) u(x) y(x + h) y(x) h

y(x)u(x + h) - u(x)y(x + h) = lim hy(x + h)y(x) hS0

To change the last fraction into an equivalent one that contains the difference quotients for the derivatives of u and y, we subtract and add y(x)u(x) in the numerator. We then get y(x)u(x + h) - y(x)u(x) + y(x)u(x) - u(x)y(x + h) d u a b = lim dx y hy(x + h)y(x) hS0 y(x)

= lim

u(x + h) - u(x) y(x + h) - y(x) - u(x) h h y(x + h)y(x)

hS0

.

Taking the limits in the numerator and denominator now gives the Quotient Rule. Exercise 74 outlines another proof. The choice of which rules to use in solving a differentiation problem can make a difference in how much work you have to do. Here is an example.

EXAMPLE 8   Find the derivative of y =

(x - 1)(x2 - 2x) . x4

Solution  Using the Quotient Rule here will result in a complicated expression with many terms. Instead, use some algebra to simplify the expression. First expand the numerator and divide by x4: y =

(x - 1)(x2 - 2x) x3 - 3x2 + 2x = = x-1 - 3x-2 + 2x-3 . x4 x4

Then use the Sum and Power Rules:

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dy = -x-2 - 3(-2)x-3 + 2(-3)x-4 dx 6 6 1 = - 2 + 3 - 4.  x x x

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149

Second- and Higher-Order Derivatives If y = ƒ(x) is a differentiable function, then its derivative ƒ′(x) is also a function. If ƒ′ is also differentiable, then we can differentiate ƒ′ to get a new function of x denoted by ƒ″. So ƒ″ = (ƒ′)′. The function ƒ″ is called the second derivative of ƒ because it is the derivative of the first derivative. It is written in several ways: ƒ″(x) =

d 2y dy′ d dy = y″ = D2(ƒ)(x) = Dx2 ƒ(x). = a b = 2 dx dx dx dx

The symbol D2 means the operation of differentiation is performed twice. If y = x6, then y′ = 6x5 and we have dy′ d = (6x5) = 30x4. dx dx

y″ =

Thus D2 ( x6 ) = 30x4. If y″ is differentiable, its derivative, y‴ = dy″>dx = d 3y>dx3, is the third derivative of y with respect to x. The names continue as you imagine, with

How to Read the Symbols for Derivatives y′ “y prime”

y(n) =

y″ “y double prime” d 2y “d squared y dx squared” dx2 y‴ “y triple prime” y(n) “y super n” d ny “d to the n of y by dx to the n” dxn

n d (n - 1) d y y = n = Dny dx dx

denoting the nth derivative of y with respect to x for any positive integer n. We can interpret the second derivative as the rate of change of the slope of the tangent to the graph of y = ƒ(x) at each point. You will see in the next chapter that the second derivative reveals whether the graph bends upward or downward from the tangent line as we move off the point of tangency. In the next section, we interpret both the second and third derivatives in terms of motion along a straight line.

EXAMPLE 9   The first four derivatives of y = x3 - 3x2 + 2 are

Dn “D to the n”

First derivative: Second derivative: Third derivative: Fourth derivative:

y′ = 3x2 - 6x y″ = 6x - 6 y‴ = 6 y(4) = 0.

All polynomial functions have derivatives of all orders. In this example, the fifth and later derivatives are all zero.

Exercises  3.3 Derivative Calculations In Exercises 1–12, find the first and second derivatives.

1. y = - 3x9 - 1

2. y = x2 + x + 8

3. s = 7t 4 - 4t 7

4. w = 3z7 - 7z3 + 21z2

5. y =

4x3 - x + 2ex 3

6 7. y = 8x-7 - x 9. y = 5x2 - 15x - 4x-2 11. r =

5 1 - 2s 3s2

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6. y =

x3 x2 + + e-x 3 2

8. s = - 2t -1 +

4 t2

10. y = 4 - 2x - x-3 12. r =

12 4 1 - 3 + 4 u u u

In Exercises 13–16, find y′ (a) by applying the Product Rule and (b) by multiplying the factors to produce a sum of simpler terms to differentiate. 13. y = ( 3 - x2 ) ( x3 - x + 1 ) 14. y = (2x + 3) ( 5x2 - 4x ) 1 15. y = ( x2 + 1 ) ax + 5 + x b 16. y = ( 1 + x2 )( x3>4 - x-3 ) Find the derivatives of the functions in Exercises 17–40. 3x - 1 4 - 3x 17. y = 18. z = 2 7x + 2 3x + x 19. y =

9x2 + 2 x2 + 3

20. ƒ(t) =

t2 - 1 t + t - 2 2

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21. s = (5 - t) ( 1 + t 2 ) -1 1s - 3 1s + 3

23. ƒ(s) =

22. w = (2x - 7)-1(x + 5) 24. u =

5x + 1 2 1x

25. y =

1 + x - 4 2x x

27. y =

(x + 1) (x + 2) 1 28. y = 2 (x - 1) (x - 2) (x - 1) (x + x + 1)

26. r = 2a

1

2u

+ 2ub

2

x2 + 3ex 2ex - x

29. y = 2e-x + e3x

30. y =

31. y = x3ex

32. w = re-r

33. y = x9>4 + e-2x

34. y = x-3>5 + p3>2

35. s = 5t 8>5 + 5e4

36. w =

7 2 37. y = 2 x - xe

3 9.6 38. y = 2 x + 2e1.3

es 39. r = s

1 + u -p>2 b u2

Find the derivatives of all orders of the functions in Exercises 41–44. 41. y =

3 x4 - x2 - x 2 2

42. y =

x5 120

43. y = (x - 1) (x + 2)(x + 3) 44. y = (4x2 + 3)(2 - x) x Find the first and second derivatives of the functions in Exercises 45–52. x3 + 7 t 2 + 5t - 1 45. y = 46. s = x t2 2

47. r =

(u - 1)(u + u + 1) u3

1 + 3z 49. w = a b(3 - z) 3z 51. w = 3z2e2z

2

48. u =

2

(x + x)(x - x + 1) x4

q2 + 3 50. p = (q - 1)3 + (q + 1)3 52. w = ez(z - 1)(z2 + 1)

53. Suppose u and y are functions of x that are differentiable at x = 0 and that u(0) = 5, u′(0) = - 3, y(0) = - 1, y′(0) = 2. Find the values of the following derivatives at x = 0. a.

57. Find the tangents to Newton’s serpentine (graphed here) at the origin and the point (1, 2). y y = 24x x +1 2 1

p 1 + z1.4 2z

40. r = eu a

b. Smallest slope  What is the smallest slope on the curve? At what point on the curve does the curve have this slope? Find an equation for the line that is perpendicular to the curve’s tangent at this point.

d d d u d y (uy)  b.  a y b   c.  a u b   d.  (7y - 2u) dx dx dx dx

0

(1, 2)

58. Find the tangent to the Witch of Agnesi (graphed here) at the point (2, 1). y y= 2 1

8 x2 + 4 (2, 1)

Find the values of the following derivatives at x = 1. a.

d d d u d y (uy)  b.  a y b   c.  a u b   d.  (7y - 2u) dx dx dx dx

Slopes and Tangents 55. a.  Normal to a curve  Find an equation for the line perpendicular to the tangent to the curve y = x3 - 4x + 1 at the point (2, 1).

x

1 2 3

0

59. Quadratic tangent to identity function  The curve y = ax2 + bx + c passes through the point (1, 2) and is tangent to the line y = x at the origin. Find a, b, and c. 60. Quadratics having a common tangent  The curves y = x2 + ax + b and y = cx - x2 have a common tangent line at the point (1, 0). Find a, b, and c. 61. Find all points (x, y) on the graph of ƒ(x) = 3x2 - 4x with tangent lines parallel to the line y = 8x + 5. 62. Find all points (x, y) on the graph of g(x) = 13 x3 - 32 x2 + 1 with tangent lines parallel to the line 8x - 2y = 1. 63. Find all points (x, y) on the graph of y = x>(x - 2) with tangent lines perpendicular to the line y = 2x + 3. 64. Find all points (x, y) on the graph of ƒ(x) = x2 with tangent lines passing through the point (3, 8). y 10

f (x) = x 2 (3, 8)

54. Suppose u and y are differentiable functions of x and that u(1) = 2, u′(1) = 0, y(1) = 5, y′(1) = -1.

x

1 2 3 4

6 (x, y) 2 −2

2

4

x

b. Smallest slope  What is the smallest slope on the curve? At what point on the curve does the curve have this slope?

65. a.  Find an equation for the line that is tangent to the curve y = x3 - x at the point (-1, 0).

c. Tangents having specified slope  Find equations for the tangents to the curve at the points where the slope of the curve is 8.

T b. Graph the curve and tangent line together. The tangent intersects the curve at another point. Use Zoom and Trace to estimate the point’s coordinates.

56. a.  Horizontal tangents  Find equations for the horizontal tangents to the curve y = x3 - 3x - 2. Also find equations for the lines that are perpendicular to these tangents at the points of tangency.

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T c. Confirm your estimates of the coordinates of the second intersection point by solving the equations for the curve and tangent simultaneously (Solver key). 66. a.  Find an equation for the line that is tangent to the curve y = x3 - 6x2 + 5x at the origin. T b. Graph the curve and tangent together. The tangent intersects the curve at another point. Use Zoom and Trace to estimate the point’s coordinates. T c. Confirm your estimates of the coordinates of the second intersection point by solving the equations for the curve and tangent simultaneously (Solver key). Theory and Examples For Exercises 67 and 68 evaluate each limit by first converting each to a derivative at a particular x-value.

67. lim

xS1

x50 - 1 x - 1

68. lim

x S -1

x2>9 - 1 x + 1

69. Find the value of a that makes the following function differentiable for all x-values. g(x) = e

ax, x2 - 3x,

if x 6 0 if x Ú 0

70. Find the values of a and b that make the following function differentiable for all x-values. ƒ(x) = e

ax + b, bx2 - 3,

x 7 -1 x … -1

71. The general polynomial of degree n has the form P(x) = an xn + an - 1 xn - 1 + g + a2 x2 + a1 x + a0

151

b. Show that the Reciprocal Rule and the Derivative Product Rule together imply the Derivative Quotient Rule. 75. Generalizing the Product Rule  The Derivative Product Rule gives the formula d dy du (uy) = u + y dx dx dx for the derivative of the product uy of two differentiable functions of x. a. What is the analogous formula for the derivative of the product uyw of three differentiable functions of x? b. What is the formula for the derivative of the product u1 u2 u3 u4 of four differentiable functions of x? c. What is the formula for the derivative of a product u1 u2 u3 gun of a finite number n of differentiable functions of x? 76. Power Rule for negative integers  Use the Derivative Quotient Rule to prove the Power Rule for negative integers, that is, d -m (x ) = -mx-m - 1 dx where m is a positive integer. 77. Cylinder pressure  If gas in a cylinder is maintained at a constant temperature T, the pressure P is related to the volume V by a formula of the form P =

an2 nRT - 2, V - nb V

in which a, b, n, and R are constants. Find dP>dV . (See accompanying figure.)

where an ≠ 0. Find P′(x). 72. The body’s reaction to medicine  The reaction of the body to a dose of medicine can sometimes be represented by an equation of the form R = M2 a

C M - b, 2 3

where C is a positive constant and M is the amount of medicine absorbed in the blood. If the reaction is a change in blood pressure, R is measured in millimeters of mercury. If the reaction is a change in temperature, R is measured in degrees, and so on.   Find dR>dM. This derivative, as a function of M, is called the sensitivity of the body to the medicine. In Section 4.5, we will see how to find the amount of medicine to which the body is most sensitive. 73. Suppose that the function y in the Derivative Product Rule has a constant value c. What does the Derivative Product Rule then say? What does this say about the Derivative Constant Multiple Rule? 74. The Reciprocal Rule a. The Reciprocal Rule says that at any point where the function y(x) is differentiable and different from zero, d 1 1 dy a b = - 2 . dx y y dx

78. The best quantity to order  One of the formulas for inventory management says that the average weekly cost of ordering, paying for, and holding merchandise is hq km A(q) = q + cm + , 2 where q is the quantity you order when things run low (shoes, TVs, brooms, or whatever the item might be); k is the cost of placing an order (the same, no matter how often you order); c is the cost of one item (a constant); m is the number of items sold each week (a constant); and h is the weekly holding cost per item (a constant that takes into account things such as space, utilities, insurance, and security). Find dA>dq and d 2A>dq2.

Show that the Reciprocal Rule is a special case of the Derivative Quotient Rule.

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3.4  The Derivative as a Rate of Change In Section 2.1 we introduced average and instantaneous rates of change. In this section we study further applications in which derivatives model the rates at which things change. It is natural to think of a quantity changing with respect to time, but other variables can be treated in the same way. For example, an economist may want to study how the cost of producing steel varies with the number of tons produced, or an engineer may want to know how the power output of a generator varies with its temperature.

Instantaneous Rates of Change If we interpret the difference quotient (ƒ(x + h) - ƒ(x))>h as the average rate of change in ƒ over the interval from x to x + h, we can interpret its limit as h S 0 as the rate at which ƒ is changing at the point x.

Definition The instantaneous rate of change of ƒ with respect to x at x0 is the derivative ƒ′(x0) = lim

hS0

ƒ(x0 + h) - ƒ(x0) , h

provided the limit exists.

Thus, instantaneous rates are limits of average rates. It is conventional to use the word instantaneous even when x does not represent time. The word is, however, frequently omitted. When we say rate of change, we mean instantaneous rate of change.

EXAMPLE 1   The area A of a circle is related to its diameter by the equation A =

p 2 D. 4

How fast does the area change with respect to the diameter when the diameter is 10 m? Solution  The rate of change of the area with respect to the diameter is p pD dA = # 2D = . 4 2 dD When D = 10 m, the area is changing with respect to the diameter at the rate of (p>2)10 = 5p m2 >m ≈ 15.71 m2 >m.

Motion Along a Line: Displacement, Velocity, Speed, Acceleration, and Jerk

Position at time t …

and at time t + Δt Δs

s = f(t)

s + Δs = f (t + Δt)

Figure 3.14  The positions of a body moving along a coordinate line at time t and shortly later at time t + ∆t. Here the coordinate line is horizontal.

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s

Suppose that an object (or body, considered as a whole mass) is moving along a coordinate line (an s-axis), usually horizontal or vertical, so that we know its position s on that line as a function of time t: s = ƒ(t). The displacement of the object over the time interval from t to t + ∆t (Figure 3.14) is ∆s = ƒ(t + ∆t) - ƒ(t),

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153

and the average velocity of the object over that time interval is yay =

displacement ∆s ƒ(t + ∆t) - ƒ(t) = = . travel time ∆t ∆t

To find the body’s velocity at the exact instant t, we take the limit of the average velocity over the interval from t to t + ∆t as ∆t shrinks to zero. This limit is the derivative of ƒ with respect to t. Definition  Velocity (instantaneous velocity) is the derivative of position with respect to time. If a body’s position at time t is s = ƒ(t), then the body’s velocity at time t is y(t) = s s = f (t) ds > 0 dt

t

0 (a) s increasing: positive slope so moving upward

ƒ(t + ∆t) - ƒ(t) ds = lim . dt ∆t ∆t S 0

Besides telling how fast an object is moving along the horizontal line in Figure 3.14, its velocity tells the direction of motion. When the object is moving forward (s increasing), the velocity is positive; when the object is moving backward (s decreasing), the velocity is negative. If the coordinate line is vertical, the object moves upward for positive velocity and downward for negative velocity. The blue curves in Figure 3.15 represent position along the line over time; they do not portray the path of motion, which lies along the vertical s-axis. If we drive to a friend’s house and back at 30 mph, say, the speedometer will show 30 on the way over but it will not show -30 on the way back, even though our distance from home is decreasing. The speedometer always shows speed, which is the absolute value of velocity. Speed measures the rate of progress regardless of direction. Definition  Speed is the absolute value of velocity.

s

Speed =  y(t)  = `

s = f (t) ds < 0 dt

EXAMPLE 2  Figure 3.16 shows the graph of the velocity y = ƒ′(t) of a particle moving

t

0 (b) s decreasing: negative slope so moving downward

Figure 3.15  For motion s = ƒ(t) along a straight line (the vertical axis), y = ds>dt is (a) positive when s increases and (b) negative when s decreases.

HISTORICAL BIOGRAPHY Bernard Bolzano (1781–1848)

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ds ` dt

along a horizontal line (as opposed to showing a position function s = ƒ(t) such as in Figure 3.15). In the graph of the velocity function, it’s not the slope of the curve that tells us if the particle is moving forward or backward along the line (which is not shown in the figure), but rather the sign of the velocity. Looking at Figure 3.16, we see that the particle moves forward for the first 3 sec (when the velocity is positive), moves backward for the next 2 sec (the velocity is negative), stands motionless for a full second, and then moves forward again. The particle is speeding up when its positive velocity increases during the first second, moves at a steady speed during the next second, and then slows down as the velocity decreases to zero during the third second. It stops for an instant at t = 3 sec (when the velocity is zero) and reverses direction as the velocity starts to become negative. The particle is now moving backward and gaining in speed until t = 4 sec, at which time it achieves its greatest speed during its backward motion. Continuing its backward motion at time t = 4, the particle starts to slow down again until it finally stops at time t = 5 (when the velocity is once again zero). The particle now remains motionless for one full second, and then moves forward again at t = 6 sec, speeding up during the final second of the forward motion indicated in the velocity graph. The rate at which a body’s velocity changes is the body’s acceleration. The acceleration measures how quickly the body picks up or loses speed. In Chapter 12 we will study motion in the plane and in space, where acceleration of an object may also lead to a change in direction. A sudden change in acceleration is called a jerk. When a ride in a car or a bus is jerky, it is not that the accelerations involved are necessarily large but that the changes in acceleration are abrupt.

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Definitions  Acceleration is the derivative of velocity with respect to time. If a body’s position at time t is s = ƒ(t), then the body’s acceleration at time t is a(t) =

dy d 2s = 2. dt dt

Jerk is the derivative of acceleration with respect to time: j(t) =

da d 3s = 3. dt dt

y MOVES FORWARD

FORWARD AGAIN

(y > 0)

(y > 0)

Velocity y = f ′(t) Speeds up

Slows down

Speeds up Stands still (y = 0)

0

1

2

3

4

5

6

7

t (sec)

Greatest speed

Speeds up

Slows down

MOVES BACKWARD

(y < 0)

Figure 3.16  The velocity graph of a particle moving along a horizontal line, discussed in Example 2.

Near the surface of Earth all bodies fall with the same constant acceleration. Galileo’s experiments with free fall (see Section 2.1) lead to the equation s =

1 2 gt , 2

where s is the distance fallen and g is the acceleration due to Earth’s gravity. This equation holds in a vacuum, where there is no air resistance, and closely models the fall of dense, heavy objects, such as rocks or steel tools, for the first few seconds of their fall, before the effects of air resistance are significant. The value of g in the equation s = (1>2)gt 2 depends on the units used to measure t and s. With t in seconds (the usual unit), the value of g determined by measurement at sea level is approximately 32 ft>sec2 (feet per second squared) in English units, and g = 9.8 m>sec2 (meters per second squared) in metric units. (These gravitational constants depend on the distance from Earth’s center of mass, and are slightly lower on top of Mt. Everest, for example.) The jerk associated with the constant acceleration of gravity (g = 32 ft>sec2) is zero: j =

d (g) = 0. dt

An object does not exhibit jerkiness during free fall.

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t (seconds) t=0

s (meters)

t=1

5

EXAMPLE 3  Figure 3.17 shows the free fall of a heavy ball bearing released from rest at time t = 0 sec.

0

(a) How many meters does the ball fall in the first 3 sec? (b) What is its velocity, speed, and acceleration when t = 3?

10 15 t=2

Solution (a) The metric free-fall equation is s = 4.9t 2. During the first 3 sec, the ball falls

20 25

s(3) = 4.9(3)2 = 44.1 m.

30

(b) At any time t, velocity is the derivative of position:

35 40 t=3

155

y(t) = s′(t) =

45

d ( 4.9t2 ) = 9.8t. dt

At t = 3, the velocity is Figure 3.17  A ball bearing falling from rest (Example 3).

y(3) = 29.4 m>sec in the downward (increasing s) direction. The speed at t = 3 is speed = 0 y(3) 0 = 29.4 m>sec.

s

The acceleration at any time t is

smax

a(t) = y′(t) = s″(t) = 9.8 m>sec2.

y=0

Height (ft)

At t = 3, the acceleration is 9.8 m>sec2. 256

EXAMPLE 4   A dynamite blast blows a heavy rock straight up with a launch velocity of 160 ft > sec (about 109 mph) (Figure 3.18a). It reaches a height of s = 160t - 16t 2 ft after t sec.

t=?

(a) How high does the rock go? (b) What are the velocity and speed of the rock when it is 256 ft above the ground on the way up? On the way down? (c) What is the acceleration of the rock at any time t during its flight (after the blast)? (d) When does the rock hit the ground again?

s=0 (a)

Solution (a) In the coordinate system we have chosen, s measures height from the ground up, so the velocity is positive on the way up and negative on the way down. The instant the rock is at its highest point is the one instant during the flight when the velocity is 0. To find the maximum height, all we need to do is to find when y = 0 and evaluate s at this time. At any time t during the rock’s motion, its velocity is

s, y 400

s = 160t − 16t 2

160

0 −160

5

10

t

y = ds = 160 − 32t dt (b)

Figure 3.18  (a) The rock in Example 4. (b) The graphs of s and y as functions of time; s is largest when y = ds>dt = 0. The graph of s is not the path of the rock: It is a plot of height versus time. The slope of the plot is the rock’s velocity, graphed here as a straight line.

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y =

ds d = (160t - 16t 2) = 160 - 32t ft>sec. dt dt

The velocity is zero when 160 - 32t = 0

or

t = 5 sec.

The rock’s height at t = 5 sec is smax = s(5) = 160(5) - 16(5)2 = 800 - 400 = 400 ft. See Figure 3.18b. (b) To find the rock’s velocity at 256 ft on the way up and again on the way down, we first find the two values of t for which s(t) = 160t - 16t 2 = 256.

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To solve this equation, we write 16t 2 - 160t + 256 16(t 2 - 10t + 16) (t - 2)(t - 8) t

= = = =

0 0 0 2 sec, t = 8 sec.

The rock is 256 ft above the ground 2 sec after the explosion and again 8 sec after the explosion. The rock’s velocities at these times are y(2) = 160 - 32(2) = 160 - 64 = 96 ft>sec. y(8) = 160 - 32(8) = 160 - 256 = -96 ft>sec. At both instants, the rock’s speed is 96 ft > sec. Since y(2) 7 0, the rock is moving upward (s is increasing) at t = 2 sec; it is moving downward (s is decreasing) at t = 8 because y(8) 6 0. (c) At any time during its flight following the explosion, the rock’s acceleration is a constant a = Cost y (dollars) Slope = marginal cost

0

y = c (x)

x+h x Production (tons/week)

The acceleration is always downward and is the effect of gravity on the rock. As the rock rises, it slows down; as it falls, it speeds up. (d) The rock hits the ground at the positive time t for which s = 0. The equation 160t - 16t 2 = 0 factors to give 16t(10 - t) = 0, so it has solutions t = 0 and t = 10. At t = 0, the blast occurred and the rock was thrown upward. It returned to the ground 10 sec later.

x

Derivatives in Economics and Sensitivity to Change

Figure 3.19  Weekly steel production: c(x) is the cost of producing x tons per week. The cost of producing an additional h tons is c(x + h) - c(x).

y y = c(x)

Δc

dy d = (160 - 32t) = -32 ft>sec2. dt dt

Engineers use the terms velocity and acceleration to refer to the derivatives of functions describing motion. Economists, too, have a specialized vocabulary for rates of change and derivatives. They call them marginals. In a manufacturing operation, the cost of production c(x) is a function of x, the number of units produced. The marginal cost of production is the rate of change of cost with respect to level of production, so it is dc>dx. Suppose that c(x) represents the dollars needed to produce x tons of steel in one week. It costs more to produce x + h tons per week, and the cost difference, divided by h, is the average cost of producing each additional ton: average cost of each of the additional c(x + h) - c(x) = h tons of steel produced. h

dc dx

The limit of this ratio as h S 0 is the marginal cost of producing more steel per week when the current weekly production is x tons (Figure 3.19): c(x + h) - c(x) dc = lim = marginal cost of production. dx h S 0 h

Δx = 1

Sometimes the marginal cost of production is loosely defined to be the extra cost of producing one additional unit:

0

x

x+1

x

Figure 3.20  The marginal cost dc>dx is approximately the extra cost ∆c of producing ∆x = 1 more unit.

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∆c c(x + 1) - c(x) = , 1 ∆x which is approximated by the value of dc>dx at x. This approximation is acceptable if the slope of the graph of c does not change quickly near x. Then the difference quotient will be close to its limit dc>dx, which is the rise in the tangent line if ∆x = 1 (Figure 3.20). The approximation works best for large values of x.

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157

Economists often represent a total cost function by a cubic polynomial c(x) = ax3 + bx2 + gx + d where d represents fixed costs, such as rent, heat, equipment capitalization, and management costs. The other terms represent variable costs, such as the costs of raw materials, taxes, and labor. Fixed costs are independent of the number of units produced, whereas variable costs depend on the quantity produced. A cubic polynomial is usually adequate to capture the cost behavior on a realistic quantity interval.

The additional cost will be about \$195. The marginal revenue is r′(x) =

dydp 2 y

The marginal revenue function estimates the increase in revenue that will result from selling one additional unit. If you currently sell 10 radiators a day, you can expect your revenue to increase by about

dy = 2 − 2p dp

1 y = 2p − p 2 0

1

d 3 (x - 3x2 + 12x) = 3x2 - 6x + 12. dx

r′(10) = 3(100) - 6(10) + 12 = +252

if you increase sales to 11 radiators a day. p

0

(a)

p

When a small change in x produces a large change in the value of a function ƒ(x), we say that the function is relatively sensitive to changes in x. The derivative ƒ′(x) is a mea(b) sure of this sensitivity. 1

dydp

EXAMPLE 6   The Austrian monk Gregor Johann Mendel (1822–1884), working with garden peas and other plants, provided the first scientific explanation of hybridization. His careful records showed that if p (a number between 0 and 1) is the frequency of the gene for smooth skin in peas (dominant) and (1 - p) is the frequency of the gene for wrinkled skin in peas, then the proportion of smooth-skinned peas in the next generation will be

2

dy = 2 − 2p dp y = 2p − p 2 1 (a)

y = 2p(1 - p) + p2 = 2p - p2. p

0

1

p

(b)

Figure 3.21  (a) The graph of y = 2p - p2, describing the proportion of smooth-skinned peas in the next generation. (b) The graph of dy>dp (Example 6).

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The graph of y versus p in Figure 3.21a suggests that the value of y is more sensitive to a change in p when p is small than when p is large. Indeed, this fact is borne out by the derivative graph in Figure 3.21b, which shows that dy>dp is close to 2 when p is near 0 and close to 0 when p is near 1. The implication for genetics is that introducing a few more smooth skin genes into a population where the frequency of wrinkled skin peas is large will have a more dramatic effect on later generations than will a similar increase when the population has a large proportion of smooth skin peas.

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Exercises  3.4 Motion Along a Coordinate Line Exercises 1–6 give the positions s = ƒ(t) of a body moving on a coordinate line, with s in meters and t in seconds.

a. Find the body’s displacement and average velocity for the given time interval. b. Find the body’s speed and acceleration at the endpoints of the interval. c. When, if ever, during the interval does the body change direction? 1. s = t 2 - 3t + 2, 0 … t … 2

12. Speeding bullet  A 45-caliber bullet shot straight up from the surface of the moon would reach a height of s = 832t - 2.6t 2 ft after t sec. On Earth, in the absence of air, its height would be s = 832t - 16t 2 ft after t sec. How long will the bullet be aloft in each case? How high will the bullet go? 13. Free fall from the Tower of Pisa  Had Galileo dropped a cannonball from the Tower of Pisa, 179 ft above the ground, the ball’s height above the ground t sec into the fall would have been s = 179 - 16t 2.

2. s = 6t - t 2, 0 … t … 6

a. What would have been the ball’s velocity, speed, and acceleration at time t?

3. s = - t 3 + 3t 2 - 3t, 0 … t … 3

b. About how long would it have taken the ball to hit the ground?

4. s = ( t >4 ) - t + t , 0 … t … 3

c. What would have been the ball’s velocity at the moment of impact?

4

3

2

5. s =

25 5 - t, 1 … t … 5 t2

6. s =

25 , -4 … t … 0 t + 5

7. Particle motion  At time t, the position of a body moving along the s-axis is s = t 3 - 6t 2 + 9t m. a. Find the body’s acceleration each time the velocity is zero. b. Find the body’s speed each time the acceleration is zero. c. Find the total distance traveled by the body from t = 0 to t = 2. 8. Particle motion  At time t Ú 0, the velocity of a body moving along the horizontal s-axis is y = t 2 - 4t + 3. a. Find the body’s acceleration each time the velocity is zero.

14. Galileo’s free-fall formula  Galileo developed a formula for a body’s velocity during free fall by rolling balls from rest down increasingly steep inclined planks and looking for a limiting formula that would predict a ball’s behavior when the plank was vertical and the ball fell freely; see part (a) of the accompanying figure. He found that, for any given angle of the plank, the ball’s velocity t sec into motion was a constant multiple of t. That is, the velocity was given by a formula of the form y = kt. The value of the constant k depended on the inclination of the plank.    In modern notation—part (b) of the figure—with distance in meters and time in seconds, what Galileo determined by experiment was that, for any given angle u, the ball’s velocity t sec into the roll was y = 9.8(sin u)t m>sec.

b. When is the body moving forward? Backward? c. When is the body’s velocity increasing? Decreasing? Free-Fall Applications 9. Free fall on Mars and Jupiter  The equations for free fall at the surfaces of Mars and Jupiter (s in meters, t in seconds) are s = 1.86t 2 on Mars and s = 11.44t 2 on Jupiter. How long does it take a rock falling from rest to reach a velocity of 27.8 m > sec (about 100 km > h) on each planet?

10. Lunar projectile motion  A rock thrown vertically upward from the surface of the moon at a velocity of 24 m > sec (about 86 km > h) reaches a height of s = 24t - 0.8t 2 m in t sec. a. Find the rock’s velocity and acceleration at time t. (The acceleration in this case is the acceleration of gravity on the moon.) b. How long does it take the rock to reach its highest point? c. How high does the rock go? d. How long does it take the rock to reach half its maximum height? e. How long is the rock aloft?

11. Finding g on a small airless planet  Explorers on a small airless planet used a spring gun to launch a ball bearing vertically upward from the surface at a launch velocity of 15 m > sec. Because the acceleration of gravity at the planet’s surface was gs m>sec2, the explorers expected the ball bearing to reach a height of s = 15t - (1>2)gs t 2 m t sec later. The ball bearing reached its maximum height 20 sec after being launched. What was the value of gs?

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Free-fall position

?

u (b)

(a)

a. What is the equation for the ball’s velocity during free fall? b. Building on your work in part (a), what constant acceleration does a freely falling body experience near the surface of Earth? Understanding Motion from Graphs 15. The accompanying figure shows the velocity y = ds>dt = ƒ(t) (m > sec) of a body moving along a coordinate line. y (m/sec)

y = f (t)

3 0

2

4

6

8 10

t (sec)

−3

a. When does the body reverse direction? b. When (approximately) is the body moving at a constant speed?

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3.4  The Derivative as a Rate of Change

c. Graph the body’s speed for 0 … t … 10.

18. The accompanying figure shows the velocity y = ƒ(t) of a particle moving on a horizontal coordinate line.

d. Graph the acceleration, where defined. 16. A particle P moves on the number line shown in part (a) of the accompanying figure. Part (b) shows the position of P as a function of time t. P

s (cm)

0

159

y

y = f(t) 0

1 2 3 4 5 6 7 8 9

t (sec)

(a) s (cm)

a. When does the particle move forward? Move backward? Speed up? Slow down?

s = f (t)

2 0

1

2

3

4

5

6

t (sec)

b. When is the particle’s acceleration positive? Negative? Zero? c. When does the particle move at its greatest speed? d. When does the particle stand still for more than an instant?

−2 (6, −4)

−4 (b)

a. When is P moving to the left? Moving to the right? Standing still?

19. Two falling balls  The multiflash photograph in the accompanying figure shows two balls falling from rest. The vertical rulers are marked in centimeters. Use the equation s = 490t 2 (the freefall equation for s in centimeters and t in seconds) to answer the following questions.

b. Graph the particle’s velocity and speed (where defined). 17. Launching a rocket  When a model rocket is launched, the propellant burns for a few seconds, accelerating the rocket upward. After burnout, the rocket coasts upward for a while and then begins to fall. A small explosive charge pops out a parachute shortly after the rocket starts down. The parachute slows the rocket to keep it from breaking when it lands. The figure here shows velocity data from the flight of the model rocket. Use the data to answer the following. a. How fast was the rocket climbing when the engine stopped? b. For how many seconds did the engine burn? 200

Velocity (ftsec)

150 100 50 0 −50 −100 0

2

4 6 8 10 Time after launch (sec)

12

c. When did the rocket reach its highest point? What was its velocity then? d. When did the parachute pop out? How fast was the rocket falling then?

a. How long did it take the balls to fall the first 160 cm? What was their average velocity for the period?

e. How long did the rocket fall before the parachute opened?

b. How fast were the balls falling when they reached the 160-cm mark? What was their acceleration then?

f. When was the rocket’s acceleration greatest?

c. About how fast was the light flashing (flashes per second)?

g. When was the acceleration constant? What was its value then (to the nearest integer)?

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Chapter 3: Derivatives

20. A traveling truck  The accompanying graph shows the position s of a truck traveling on a highway. The truck starts at t = 0 and returns 15 h later at t = 15.

Economics 23. Marginal cost  Suppose that the dollar cost of producing x washing machines is c(x) = 2000 + 100x - 0.1x2.

a. Use the technique described in Section 3.2, Example 3, to graph the truck’s velocity y = ds>dt for 0 … t … 15. Then repeat the process, with the velocity curve, to graph the truck’s acceleration dy>dt.

a. Find the average cost per machine of producing the first 100 washing machines.

b. Suppose that s = 15t 2 - t 3. Graph ds>dt and d 2s>dt 2 and compare your graphs with those in part (a).

c. Show that the marginal cost when 100 washing machines are produced is approximately the cost of producing one more washing machine after the first 100 have been made, by calculating the latter cost directly.

Position, s (km)

500

b. Find the marginal cost when 100 washing machines are produced.

24. Marginal revenue  Suppose that the revenue from selling x washing machines is

400 300

dollars.

200

1 r(x) = 20,000a1 - x b

a. Find the marginal revenue when 100 machines are produced. 100 0

5 10 Elapsed time, t (hr)

15

21. The graphs in the accompanying figure show the position s, velocity y = ds>dt, and acceleration a = d 2s>dt 2 of a body moving along a coordinate line as functions of time t. Which graph is which? Give reasons for your answers. y A

B

b. Use the function r′(x) to estimate the increase in revenue that will result from increasing production from 100 machines a week to 101 machines a week. c. Find the limit of r′(x) as x S q. How would you interpret this number? Additional Applications 25. Bacterium population  When a bactericide was added to a nutrient broth in which bacteria were growing, the bacterium population continued to grow for a while, but then stopped growing and began to decline. The size of the population at time t (hours) was b = 106 + 104t - 103t 2. Find the growth rates at

a. t = 0 hours.

C

b. t = 5 hours. c. t = 10 hours. t

0

22. The graphs in the accompanying figure show the position s, the velocity y = ds>dt, and the acceleration a = d 2s>dt 2 of a body moving along a coordinate line as functions of time t. Which graph is which? Give reasons for your answers. y

26. Body surface area  A typical male’s body surface area S in 1 square meters is often modeled by the formula S = 60 2wh, where h is the height in cm, and w the weight in kg, of the person. Find the rate of change of body surface area with respect to weight for males of constant height h = 180 cm (roughly 5′9″). Does S increase more rapidly with respect to weight at lower or higher body weights? Explain. T 27. Draining a tank  It takes 12 hours to drain a storage tank by opening the valve at the bottom. The depth y of fluid in the tank t hours after the valve is opened is given by the formula y = 6a1 -

A

t 2 b m. 12

a. Find the rate dy>dt (m > h) at which the tank is draining at time t.

b. When is the fluid level in the tank falling fastest? Slowest? What are the values of dy>dt at these times?

0

B

C

M03_THOM9580_03_SE_C03_129-222.indd 160

t

c. Graph y and dy>dt together and discuss the behavior of y in relation to the signs and values of dy>dt. 28. Draining a tank  The number of gallons of water in a tank t minutes after the tank has started to drain is Q(t) = 200(30 - t)2. How fast is the water running out at the end of 10 min? What is the average rate at which the water flows out during the first 10 min?

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3.5  Derivatives of Trigonometric Functions

29. Vehicular stopping distance  Based on data from the U.S. Bureau of Public Roads, a model for the total stopping distance of a moving car in terms of its speed is

vent in the crater’s floor, which at one point shot lava 1900 ft straight into the air (a Hawaiian record). What was the lava’s exit velocity in feet per second? In miles per hour? (Hint: If y0 is the exit velocity of a particle of lava, its height t sec later will be s = y0 t - 16t 2 ft. Begin by finding the time at which ds>dt = 0. Neglect air resistance.)

s = 1.1y + 0.054y2, where s is measured in ft and y in mph. The linear term 1.1y models the distance the car travels during the time the driver perceives a need to stop until the brakes are applied, and the quadratic term 0.054y2 models the additional braking distance once they are applied. Find ds>dy at y = 35 and y = 70 mph, and interpret the meaning of the derivative. 30. Inflating a balloon  The volume V = (4>3)pr 3 of a spherical balloon changes with the radius.

Analyzing Motion Using Graphs T Exercises 33–36 give the position function s = ƒ(t) of an object moving along the s-axis as a function of time t. Graph ƒ together with the velocity function y(t) = ds>dt = ƒ′(t) and the acceleration function a(t) = d 2s>dt 2 = ƒ″(t). Comment on the object’s behavior in relation to the signs and values of y and a. Include in your commentary such topics as the following:

a. At what rate (ft3 >ft) does the volume change with respect to the radius when r = 2 ft?

a. When is the object momentarily at rest? b. When does it move to the left (down) or to the right (up)?

b. By approximately how much does the volume increase when the radius changes from 2 to 2.2 ft? 31. Airplane takeoff  Suppose that the distance an aircraft travels along a runway before takeoff is given by D = (10>9)t 2, where D is measured in meters from the starting point and t is measured in seconds from the time the brakes are released. The aircraft will become airborne when its speed reaches 200 km>h. How long will it take to become airborne, and what distance will it travel in that time? 32. Volcanic lava fountains  Although the November 1959 Kilauea Iki eruption on the island of Hawaii began with a line of fountains along the wall of the crater, activity was later confined to a single

161

c. When does it change direction? d. When does it speed up and slow down? e. When is it moving fastest (highest speed)? Slowest? f. When is it farthest from the axis origin? 33. s = 200t - 16t 2, 0 … t … 12.5 (a heavy object fired straight up from Earth’s surface at 200 ft > sec)

34. s = t 2 - 3t + 2, 0 … t … 5

35. s = t 3 - 6t 2 + 7t, 0 … t … 4

36. s = 4 - 7t + 6t 2 - t 3, 0 … t … 4

3.5  Derivatives of Trigonometric Functions Many phenomena of nature are approximately periodic (electromagnetic fields, heart rhythms, tides, weather). The derivatives of sines and cosines play a key role in describing periodic changes. This section shows how to differentiate the six basic trigonometric functions.

Derivative of the Sine Function To calculate the derivative of ƒ(x) = sin x, for x measured in radians, we combine the limits in Example 5a and Theorem 7 in Section 2.4 with the angle sum identity for the sine function: sin (x + h) = sin x cos h + cos x sin h. If ƒ(x) = sin x, then

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ƒ(x + h) - ƒ(x) sin (x + h) - sin x = lim h h S h 0 (sin x cos h + cos x sin h) - sin x = lim h hS0 sin x (cos h - 1) + cos x sin h = lim h hS0

ƒ′(x) = lim

hS0

= lim asin x # hS0

Derivative definition

cos h - 1 sin h b + lim acos x # b h h hS0

cos h - 1 sin h + cos x # lim = sin x # 0 + cos x # 1 = cos x. = sin x # lim h hS0 hS0 h (++)++* (11)11* Example 5a and limit 0

limit 1

Theorem 7, Section 2.4

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The derivative of the sine function is the cosine function: d (sin x) = cos x. dx

EXAMPLE 1   We find derivatives of the sine function involving differences, products, and quotients. dy d = 2x (sin x) (a) y = x2 - sin x: Difference Rule dx dx (b) y = ex sin x: sin x (c) y = x :

= 2x - cos x dy d d x = ex (sin x) + (e ) sin x dx dx dx = ex cos x + ex sin x

Product Rule

= ex (cos x + sin x)

#d # dy x dx (sin x) - sin x 1 = dx x2 =

Quotient Rule

x cos x - sin x  x2

Derivative of the Cosine Function With the help of the angle sum formula for the cosine function, cos (x + h) = cos x cos h - sin x sin h, we can compute the limit of the difference quotient: cos (x + h) - cos x d (cos x) = lim   dx h hS0

y y = cos x

1 −p

0 −1

p y′

y′ = −sin x

0 −1

p

x

Figure 3.22  The curve y′ = - sin x as the graph of the slopes of the tangents to the curve y = cos x.

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(cos x cos h - sin x sin h) - cos x h hS0

= lim

= lim

= lim cos x #

cos h - 1 sin h - lim sin x # h h hS0

= cos x # lim

cos h - 1 sin h - sin x # lim h hS0 h

= cos x # 0 - sin x # 1 = -sin x.

hS0

Cosine angle sum identity

cos x (cos h - 1) - sin x sin h h

hS0

hS0

Example 5a and Theorem 7, Section 2.4

x

1 −p

Derivative definition

The derivative of the cosine function is the negative of the sine function: d (cos x) = -sin x. dx

Figure 3.22 shows a way to visualize this result in the same way we did for graphing derivatives in Section 3.2, Figure 3.6.

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3.5  Derivatives of Trigonometric Functions

163

EXAMPLE 2   We find derivatives of the cosine function in combinations with other functions. (a) y = 5ex + cos x: dy d ( 5ex ) + d (cos x) = dx dx dx

Sum Rule

= 5ex - sin x

(b) y = sin x cos x:

dy d d = sin x (cos x) + cos x (sin x) dx dx dx

= sin x (-sin x) + cos x (cos x) = cos2 x - sin2 x

(c) y =

Product Rule

cos x : 1 - sin x d d dy (1 - sin x) dx (cos x) - cos x dx (1 - sin x) = dx (1 - sin x)2 (1 - sin x)(-sin x) - cos x(0 - cos x) (1 - sin x)2 1 - sin x = (1 - sin x)2 1 =  1 - sin x

Quotient Rule

=

sin2 x + cos2 x = 1

Simple Harmonic Motion The motion of an object or weight bobbing freely up and down with no resistance on the end of a spring is an example of simple harmonic motion. The motion is periodic and repeats indefinitely, so we represent it using trigonometric functions. The next example describes a case in which there are no opposing forces such as friction to slow the motion.

EXAMPLE 3   A weight hanging from a spring (Figure 3.23) is stretched down 5 units beyond its rest position and released at time t = 0 to bob up and down. Its position at any later time t is s = 5 cos t.

−5

What are its velocity and acceleration at time t? 0

Rest position

5

Position at t=0

s

Figure 3.23  A weight hanging from a vertical spring and then displaced oscillates above and below its rest position (Example 3).

M03_THOM9580_03_SE_C03_129-222.indd 163

Solution  We have Position: s = 5 cos t ds d Velocity: y = = (5 cos t) = -5 sin t dt dt Acceleration: a =

dy d = (-5 sin t) = -5 cos t. dt dt

Notice how much we can learn from these equations: 1. As time passes, the weight moves down and up between s = -5 and s = 5 on the s-axis. The amplitude of the motion is 5. The period of the motion is 2p, the period of the cosine function. 2. The velocity y = -5 sin t attains its greatest magnitude, 5, when cos t = 0, as the graphs show in Figure 3.24. Hence, the speed of the weight, 0 y 0 = 5 0 sin t 0 , is greatest

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164

Chapter 3: Derivatives

s, y 5

y = −5 sin t

0

p 2

p

s = 5 cos t

3p 2

2p 5p 2

t

−5

Figure 3.24  The graphs of the position and velocity of the weight in Example 3.

when cos t = 0, that is, when s = 0 (the rest position). The speed of the weight is zero when sin t = 0. This occurs when s = 5 cos t = {5, at the endpoints of the interval of motion. 3. The weight is acted on by the spring and by gravity. When the weight is below the rest position, the combined forces pull it up, and when it is above the rest position, they pull it down. The weight’s acceleration is always proportional to the negative of its displacement. This property of springs is called Hooke’s Law, and is studied further in Section 6.5. 4. The acceleration, a = -5 cos t, is zero only at the rest position, where cos t = 0 and the force of gravity and the force from the spring balance each other. When the weight is anywhere else, the two forces are unequal and acceleration is nonzero. The acceleration is greatest in magnitude at the points farthest from the rest position, where cos t = {1.

EXAMPLE 4   The jerk associated with the simple harmonic motion in Example 3 is j =

da d = (-5 cos t) = 5 sin t. dt dt

It has its greatest magnitude when sin t = {1, not at the extremes of the displacement but at the rest position, where the acceleration changes direction and sign.

Derivatives of the Other Basic Trigonometric Functions Because sin x and cos x are differentiable functions of x, the related functions sin x tan x = cos x ,

cot x =

cos x , sin x

1 sec x = cos x ,

and

csc x =

1 sin x

are differentiable at every value of x at which they are defined. Their derivatives, calculated from the Quotient Rule, are given by the following formulas. Notice the negative signs in the derivative formulas for the cofunctions. The derivatives of the other trigonometric functions:

d (tan x) = sec2 x dx d (sec x) = sec x tan x dx

d (cot x) = -csc2 x dx d (csc x) = -csc x cot x dx

To show a typical calculation, we find the derivative of the tangent function. The other derivations are left to Exercise 60.

EXAMPLE 5  Find d(tan x)>dx. Solution  We use the Derivative Quotient Rule to calculate the derivative:

d d (sin x) - sin x (cos x) dx dx cos2 x

=

cos x cos x - sin x (-sin x) cos2 x

=

cos2 x + sin2 x cos2 x

=

1 = sec2 x. cos2 x

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d d sin x (tan x) = a b = dx dx cos x

cos x

Quotient Rule



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3.5  Derivatives of Trigonometric Functions

165

The differentiability of the trigonometric functions throughout their domains gives another proof of their continuity at every point in their domains (Theorem 1, Section 3.2). So we can calculate limits of algebraic combinations and composites of trigonometric functions by direct substitution.

EXAMPLE 6   We can use direct substitution in computing limits provided there is no division by zero, which is algebraically undefined.

lim

xS0

22 + sec x 22 + sec 0 22 + 1 23 = = = = - 23 cos (p - tan x) cos (p - tan 0) cos (p - 0) -1

Exercises  3.5 8 2. y = x + 3 sin x

Tangent Lines In Exercises 35–38, graph the curves over the given intervals, together with their tangents at the given values of x. Label each curve and tangent with its equation.

4. y = 2x sec x + 3

x = - p, 0, 3p>2

Derivatives In Exercises 1–18, find dy>dx.

1. y = - 16x + 9 cos x 3. y = x5 cos x 5. y = csc x - 4 1x +

7 ex

7. ƒ(x) = sin x tan x

8. g(x) =

-x

9. y = xe sec x 11. y =

6. y = x2 cot x -

tan x 1 + tan x

4 1 13. y = cos x + tan x

35. y = sin x, - 3p>2 … x … 2p

1 x2

36. y = tan x, - p>2 6 x 6 p>2 x = - p>3, 0, p>3

cos x sin2 x

37. y = sec x, - p>2 6 x 6 p>2

10. y = (sin x + cos x) sec x

x = - p>3, p>4

12. y =

4 cos x 1 + sin x

x = - p>3, 3p>2

14. y =

cos x x x + cos x

38. y = 1 + cos x, - 3p>2 … x … 2p T Do the graphs of the functions in Exercises 39–42 have any horizontal tangents in the interval 0 … x … 2p? If so, where? If not, why not? Visualize your findings by graphing the functions with a grapher.

15. y = 3(sec x + tan x) (sec x - tan x) 16. y = x2 cos x - 2x sin x - 2 cos x

39. y = x + sin x

40. y = 2x + sin x

17. ƒ(x) = x3 sin x cos x

41. y = x - cot x

42. y = x + 2 cos x

18. g(x) = (2 - x) tan2 x

In Exercises 19–22, find ds>dt. -t

2

19. s = tan t - e 21. s =

t

20. s = t - sec t + 5e

1 + csc t 1 - csc t

22. s =

sin t 1 - cos t

In Exercises 23–26, find dr>du. 23. r = 4 - u 2 sin u

24. r = u sin u + cos u

25. r = sec u csc u

26. r = (1 + sec u) sin u

In Exercises 27–32, find dp>dq. 1 27. p = 5 + cot q 28. p = (1 + csc q) cos q 29. p =

sin q + cos q cos q

30. p =

tan q 1 + tan q

31. p =

q sin q q2 - 1

32. p =

3q + tan q q sec q

33. Find y″ if y = sec x. a. y = csc x. b. 34. Find y

(4)

4

4

= d y>dx if

y = 9 cos x. a. y = - 2 sin x. b.

43. Find all points on the curve y = tan x, - p>2 6 x 6 p>2, where the tangent line is parallel to the line y = 2x. Sketch the curve and tangent(s) together, labeling each with its equation. 44. Find all points on the curve y = cot x, 0 6 x 6 p, where the tangent line is parallel to the line y = - x. Sketch the curve and tangent(s) together, labeling each with its equation. In Exercises 45 and 46, find an equation for (a) the tangent to the curve at P and (b) the horizontal tangent to the curve at Q. 45. y

46. y

Q p P a , 2b 2

2

1 4

0

p 2 2 y = 4 + cot x − 2csc x 1

p P a , 4b 4

x Q

0

p1 4

2

3

x

y = 1 + " 2 csc x + cot x

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Chapter 3: Derivatives

Trigonometric Limits Find the limits in Exercises 47–54.

1 1 47. lim sin a x - b 2 xS2 49.

lim

u S p>6

sin u u -

p 6

1 2

a. Find the spring’s displacement when t = 0, t = p>3, and t = 3p>4.

48. 50.

51. lim secc ex + p tan a xS0

lim

x S - p>6

lim

u S p>4

21 + cos (p csc x)

tan u - 1 u - p4

p b - 1d 4 sec x

p + tan x 52. lim sin a b tan x - 2 sec x xS0 sin t 53. lim tan a1 - t b tS0

pu 54. lim cos a b sin u uS0

Theory and Examples The equations in Exercises 55 and 56 give the position s = ƒ(t) of a body moving on a coordinate line (s in meters, t in seconds). Find the body’s velocity, speed, acceleration, and jerk at time t = p>4 sec.

55. s = 2 - 2 sin t

56. s = sin t + cos t

b. Find the spring’s velocity when t = 0, t = p>3, and t = 3p>4. 62. Assume that a particle’s position on the x-axis is given by x = 3 cos t + 4 sin t, where x is measured in feet and t is measured in seconds. a. Find the particle’s position when t = 0, t = p>2, and t = p. b. Find the particle’s velocity when t = 0, t = p>2, and t = p. T 63. Graph y = cos x for -p … x … 2p. On the same screen, graph y =

sin (x + h) - sin x h

for h = 1, 0.5, 0.3, and 0.1. Then, in a new window, try h = -1, -0.5, and -0.3. What happens as h S 0+? As h S 0-? What phenomenon is being illustrated here? T 64. Graph y = -sin x for -p … x … 2p. On the same screen, graph

57. Is there a value of c that will make sin2 3x , x≠0 2 ƒ(x) = • x c, x = 0

continuous at x = 0? Give reasons for your answer. 58. Is there a value of b that will make

y =

cos (x + h) - cos x h

for h = 1, 0.5, 0.3, and 0.1. Then, in a new window, try h = -1, -0.5, and -0.3. What happens as h S 0+? As h S 0-? What phenomenon is being illustrated here? T 65. Centered difference quotients The centered difference quotient

x + b, x 6 0 g(x) = e cos x, x Ú 0

continuous at x = 0? Differentiable at x = 0? Give reasons for your answers. 59. By computing the first few derivatives and looking for a pattern, find d 999 >dx999 (cos x).

ƒ(x + h) - ƒ(x - h) 2h is used to approximate ƒ′(x) in numerical work because (1) its limit as h S 0 equals ƒ′(x) when ƒ′(x) exists, and (2) it usually gives a better approximation of ƒ′(x) for a given value of h than the difference quotient

60. Derive the formula for the derivative with respect to x of

ƒ(x + h) - ƒ(x) . h

a. sec x.  b. csc x.  c.  cot x.

61. A weight is attached to a spring and reaches its equilibrium position (x = 0). It is then set in motion resulting in a displacement of

See the accompanying figure. y

x = 10 cos t,

Slope = f ′(x)

where x is measured in centimeters and t is measured in seconds. See the accompanying figure.

Slope =

C

B

f(x + h) − f(x) h

A Slope = y = f (x)

−10

0

f(x + h) − f(x − h) 2h

Equilibrium position at x = 0

h 0

x−h

h x

x+h

x

10 x

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3.6  The Chain Rule

may have a limit as h S 0 when ƒ has no derivative at x. As a case in point, take ƒ(x) = 0 x 0 and calculate

a. To see how rapidly the centered difference quotient for ƒ(x) = sin x converges to ƒ′(x) = cos x, graph y = cos x together with

lim

sin (x + h) - sin (x - h) y = 2h

hS0

00 + h0 - 00 - h0 2h

.

As you will see, the limit exists even though ƒ(x) = 0 x 0 has no derivative at x = 0. Moral: Before using a centered difference quotient, be sure the derivative exists.

over the interval 3- p, 2p4 for h = 1, 0.5, and 0.3. Compare the results with those obtained in Exercise 63 for the same values of h. b. To see how rapidly the centered difference quotient for ƒ(x) = cos x converges to ƒ′(x) = - sin x, graph y = -sin x together with y =

167

cos (x + h) - cos (x - h) 2h

over the interval 3- p, 2p4 for h = 1, 0.5, and 0.3. Compare the results with those obtained in Exercise 64 for the same values of h. 66. A caution about centered difference quotients (Continuation of Exercise 65.) The quotient

T 67. Slopes on the graph of the tangent function Graph y = tan x and its derivative together on (- p>2, p>2). Does the graph of the tangent function appear to have a smallest slope? A largest slope? Is the slope ever negative? Give reasons for your answers. T 68. Exploring (sin kx) , x Graph y = (sin x)>x, y = (sin 2x)>x, and y = (sin 4x)>x together over the interval - 2 … x … 2. Where does each graph appear to cross the y-axis? Do the graphs really intersect the axis? What would you expect the graphs of y = (sin 5x)>x and y = (sin (-3x))>x to do as x S 0? Why? What about the graph of y = (sin kx)>x for other values of k? Give reasons for your answers.

ƒ(x + h) - ƒ(x - h) 2h

3.6  The Chain Rule How do we differentiate F(x) = sin (x2 - 4)? This function is the composite ƒ ∘ g of two functions y = ƒ(u) = sin u and u = g(x) = x2 - 4 that we know how to differentiate. The answer, given by the Chain Rule, says that the derivative is the product of the derivatives of ƒ and g. We develop the rule in this section. 2

3 1

C: y turns B: u turns A: x turns

Figure 3.25  When gear A makes x turns, gear B makes u turns and gear C makes y turns. By comparing circumferences or counting teeth, we see that y = u>2 (C turns one-half turn for each B turn) and u = 3x (B turns three times for A’s one), so y = 3x>2. Thus, dy>dx = 3>2 = (1>2)(3) = (dy>du)(du>dx).

Derivative of a Composite Function The function y =

3 1 1 x = (3x) is the composite of the functions y = u and u = 3x. 2 2 2

We have dy 3 = , dx 2 Since

dy 1 = , du 2

and

du = 3. dx

3 1# = 3, we see in this case that 2 2 dy dy du # . = dx du dx

If we think of the derivative as a rate of change, our intuition allows us to see that this relationship is reasonable. If y = ƒ(u) changes half as fast as u and u = g(x) changes three times as fast as x, then we expect y to change 3>2 times as fast as x. This effect is much like that of a multiple gear train (Figure 3.25). Let’s look at another example.

EXAMPLE 1   The function y = ( 3x2 + 1 ) 2

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Chapter 3: Derivatives

is the composite of y = ƒ(u) = u2 and u = g(x) = 3x2 + 1. Calculating derivatives, we see that dy du # = 2u # 6x du dx

= 2(3x2 + 1) # 6x  Substitute for u = 36x3 + 12x.

Calculating the derivative from the expanded formula (3x2 + 1)2 = 9x4 + 6x2 + 1 gives the same result: dy d ( 9x4 + 6x2 + 1 ) = dx dx

= 36x3 + 12x.



The derivative of the composite function ƒ(g(x)) at x is the derivative of ƒ at g(x) times the derivative of g at x. This is known as the Chain Rule (Figure 3.26). Composite f ˚ g Rate of change at x is f ′(g(x)) · g′(x).

x

g

f

Rate of change at x is g′(x).

Rate of change at g(x) is f ′( g(x)).

u = g(x)

y = f (u) = f(g(x))

Figure 3.26  Rates of change multiply: The derivative of ƒ ∘ g at x is the derivative of ƒ at g(x) times the derivative of g at x.

THEOREM 2—The Chain Rule If ƒ(u) is differentiable at the point u = g(x) and g(x) is differentiable at x, then the composite function (ƒ ∘ g) (x) = ƒ(g(x)) is differentiable at x, and (ƒ ∘ g)′(x) = ƒ′(g(x)) # g′(x).

In Leibniz’s notation, if y = ƒ(u) and u = g(x), then dy dy du # , = dx du dx where dy>du is evaluated at u = g(x).

A Proof of One Case of the Chain Rule: Let ∆u be the change in u when x changes by ∆x, so that ∆u = g(x + ∆x) - g(x). Then the corresponding change in y is ∆y = ƒ(u + ∆u) - ƒ(u). If ∆u ≠ 0, we can write the fraction ∆y> ∆x as the product

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∆y ∆y ∆u # (1) = ∆x ∆u ∆x

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3.6  The Chain Rule

169

and take the limit as ∆x S 0:

dy ∆y = lim dx S ∆x ∆x 0

= lim

= lim

= lim

=

∆y ∆u # ∆x S 0 ∆u ∆x ∆y # lim ∆u ∆x S 0 ∆u ∆x S 0 ∆x ∆y

#

lim

∆u S 0 ∆u ∆x S 0

(Note that ∆u S 0 as ∆x S 0 ∆u   since g is continuous.) ∆x

dy du # . du dx

The problem with this argument is that if the function g(x) oscillates rapidly near x, then ∆u can be zero even when ∆x ≠ 0, so the cancelation of ∆u in Equation (1) would be invalid. A complete proof requires a different approach that avoids this problem, and we give one such proof in Section 3.11.

EXAMPLE 2   An object moves along the x-axis so that its position at any time t Ú 0 is given by x(t) = cos(t 2 + 1). Find the velocity of the object as a function of t.

Solution  We know that the velocity is dx>dt. In this instance, x is a composite function: x = cos(u) and u = t 2 + 1. We have dx = -sin(u)   x du

= cos(u)

du = 2t. dt

= t2 + 1

u

By the Chain Rule, dx dx # du = dt du dt

= -sin (u) # 2t = -sin (t 2 + 1) # 2t = -2t sin (t 2 + 1).

dx evaluated at u du

Ways to Write the Chain Rule

“Outside-Inside” Rule

(ƒ ∘ g)′(x) = ƒ′(g(x)) # g′(x)

A difficulty with the Leibniz notation is that it doesn’t state specifically where the derivatives in the Chain Rule are supposed to be evaluated. So it sometimes helps to think about the Chain Rule using functional notation. If y = ƒ(g(x)), then

dy dy du # = dx du dx dy = ƒ′(g(x)) # g′(x) dx d du ƒ(u) = ƒ′(u) dx dx

dy = ƒ′(g(x)) # g′(x). dx In words, differentiate the “outside” function ƒ and evaluate it at the “inside” function g(x) left alone; then multiply by the derivative of the “inside function.”

EXAMPLE 3  Differentiate sin ( x2 + ex ) with respect to x. Solution  We apply the Chain Rule directly and find d sin ( x2 + ex ) = cos ( x2 + ex ) # ( 2x + ex ) . dx (1)1* (1)1* (1)1*

inside

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inside left alone

derivative of the inside

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Chapter 3: Derivatives

EXAMPLE 4  Differentiate y = ecos x. Solution  Here the inside function is u = g (x) = cos x and the outside function is the exponential function ƒ(x) = ex. Applying the Chain Rule, we get dy d cos x ( e ) = ecos x d (cos x) = ecos x(-sin x) = -ecos x sin x. = dx dx dx

Generalizing Example 4, we see that the Chain Rule gives the formula

d u du e = eu . dx dx

For example, d kx d (e ) = ekx # (kx) = kekx, dx dx

for any constant k

and 2 d x2 d 2 1 e 2 = ex2 # dx (x ) = 2xex . dx

Repeated Use of the Chain Rule We sometimes have to use the Chain Rule two or more times to find a derivative.

HISTORICAL BIOGRAPHY Johann Bernoulli (1667–1748)

EXAMPLE 5   Find the derivative of g(t) = tan (5 - sin 2t). Solution  Notice here that the tangent is a function of 5 - sin 2t, whereas the sine is a function of 2t, which is itself a function of t. Therefore, by the Chain Rule,

g′(t) =

d (tan (5 - sin 2t)) dt

= sec2 (5 - sin 2t) #

Derivative of tan u with u = 5 - sin 2t

d (5 - sin 2t) dt

= sec2 (5 - sin 2t) # a0 - cos 2t #

= sec2 (5 - sin 2t) # (-cos 2t) # 2 = -2(cos 2t) sec2 (5 - sin 2t).

d (2t)b dt

Derivative of 5 - sin u with u = 2t

The Chain Rule with Powers of a Function If n is any real number and ƒ is a power function, ƒ(u) = un, the Power Rule tells us that ƒ′(u) = nun - 1. If u is a differentiable function of x, then we can use the Chain Rule to extend this to the Power Chain Rule:

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d n du (u ) = nun - 1 . dx dx

d n (u ) = nun - 1 du

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3.6  The Chain Rule

171

EXAMPLE 6   The Power Chain Rule simplifies computing the derivative of a power of an expression. Power Chain Rule with d ( 5x3 - x4 ) 7 = 7 ( 5x3 - x4 ) 6 d ( 5x3 - x4 )    u = 5x3 - x4, n = 7 (a) dx dx = 7 ( 5x3 - x4 ) 6 ( 15x2 - 4x3 ) (b)

d d 1 a b = (3x - 2)-1 dx 3x - 2 dx

d (3x - 2) dx = -1(3x - 2)-2(3) 3 = (3x - 2)2 = -1(3x - 2)-2

Power Chain Rule with u = 3x - 2, n = - 1

In part (b) we could also find the derivative with the Derivative Quotient Rule. (c)

Power Chain Rule with u = sin x, n = 5, d ( sin5 x ) = 5 sin4 x # d sin x  because sinn x means (sin x)n, n ≠ - 1. dx dx

= 5 sin4 x cos x d d (d) 1 e23x + 1 2 = e23x + 1 # 1 23x + 1 2 dx dx

1 = e23x + 1 # (3x + 1)-1>2 # 3  Power Chain Rule with u 2 =

Derivative of the Absolute Value Function d x (0x0) = , x≠0 dx 0x0 = e

1, x 7 0 - 1, x 6 0

3 e23x + 1 2 23x + 1

= 3x + 1, n = 1>2



EXAMPLE 7  In Section 3.2, we saw that the absolute value function y = 0 x 0 is not differentiable at x = 0. However, the function is differentiable at all other real numbers, as we now show. Since 0 x 0 = 2x2 , we can derive the following formula:

d ( 0 x 0 ) = d 2x2 dx dx 1

=

2

#

Power Chain Rule with d 2 (x )   u = x2, n = 1>2, x ≠ 0 dx

2 2x 1 # = 2x   2x2 = 0 x 0 20x0 =

x

0x0

, x ≠ 0.

EXAMPLE 8   The formulas for the derivatives of both sin x and cos x were obtained under the assumption that x is measured in radians, not degrees. The Chain Rule gives us new insight into the difference between the two. Since 180° = p radians, x° = px>180 radians where x° is the size of the angle measured in degrees. By the Chain Rule, d p p d px px sin (x°) = sin a b = cos a b = cos (x°). 180 180 180 180 dx dx

See Figure 3.27. Similarly, the derivative of cos (x°) is -(p>180) sin (x°). The factor p>180 would compound with repeated differentiation, showing an advantage for the use of radian measure in computations.

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172

Chapter 3: Derivatives y

y = sin(x°) = sin px 180

1

x

180

y = sin x

Figure 3.27  The function sin (x°) oscillates only p>180 times as often as sin x oscillates. Its maximum slope is p>180 at x = 0 (Example 8).

Exercises  3.6 Derivative Calculations In Exercises 1–8, given y = ƒ(u) and u = g(x), find dy>dx = ƒ′(g(x))g′(x).

1. y = 6u - 9, u = (1>2)x4 2. y = 2u3, u = 8x - 1 3. y = sin u, u = 3x + 1

4. y = cos u, u = e-x

5. y = 2u, u = sin x

6. y = sin u, u = x - cos x 1 8. y = - sec u, u = x + 7x

7. y = tan u, u = px2

In Exercises 9–22, write the function in the form y = ƒ(u) and u = g(x). Then find dy>dx as a function of x. 9. y = (2x + 1)5 11. y = a1 -

12. y = a

x2 1 4 13. y = a + x - x b 8

2x

2

- 1b

-10

18. y = 5 cos-4 x

19. y = e-5x

20. y = e2x>3

45. r = sin (u 2) cos (2u) 47. q = sin a

t 2t + 1 2

x 1 30. y = x sin-5 x - cos3 x 3 1 -1 1 (3x - 2)6 + a4 - 2 b 31. y = 18 2x 29. y = x2 sin4 x + x cos-2 x

32. y = (5 - 2x)

4

-3

(pt - 1)

33. y = (4x + 3) (x + 1) 3

34. y = (2x - 5)

( x - 5x ) 2

35. y = xe-x + ex

36. y = (1 + 2x)e-2x

37. y = ( x2 - 2x + 2 ) e5x>2

38. y = (9x2 - 6x + 2)ex

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sin t 48. q = cota t b

50. y = u 3e-2u cos 5u

58. y = ( esin (t>2) ) 3

3

2

t b t - 4t 3

3

63. y = a1 + tan4 a

3 t bb 12

65. y = 21 + cos (t 2)

3t - 4 -5 b 5t + 2

t 62. y = cos a5 sin a b b 3

64. y =

1 1 1 + cos2 (7t) 23 6

66. y = 4 sin 1 21 + 1t 2

68. y = cos4 ( sec2 3t )

69. y = 3t ( 2t 2 - 5 ) 4

70. y = 43t + 32 + 21 - t

3

1 3 71. y = a1 + x b 6

60. y = a

67. y = tan ( sin t ) 2

Second Derivatives Find y″ in Exercises 71–78.

4

-1

1 46. r = sec 2u tan a b u

56. y = (t -3>4 sin t)4>3

61. y = sin (cos (2t - 5))

28. r = 6 (sec u - tan u)3>2

1 2 + a x + 1b 8

1 + sin 3t -1 b 3 - 2t

54. y = (1 + cot (t>2))-2

59. y = a

3pt 3pt 4 4 sin 3t + cos 5t 26. s = sin a b + cos a b 3p 2 2 5p

-3

44. g(t) = a

53. y = (1 + cos 2t)-4 57. y = ecos

3 24. q = 2 2r - r 2

27. r = (csc u + cot u)-1

tan 3x (x + 7)4

52. y = sec2 pt

Find the derivatives of the functions in Exercises 23–50.

25. s =

42. g(x) =

51. y = sin2 (pt - 2)

2

2

23. p = 23 - t

b

55. y = (t tan t)

22. y = e142x + x 2

21. y = e5 - 7x

2 sin u b 1 + cos u

10

1 16. y = cot ap - x b

17. y = tan3 x

43. ƒ(u) = a

1 40. k(x) = x2 sec a x b

In Exercises 51–70, find dy>dt.

14. y = 23x2 - 4x + 6

15. y = sec (tan x)

41. ƒ(x) = 27 + x sec x

49. y = cos 1 e-u 2

10. y = (4 - 3x)9

x -7 b 7

39. h(x) = x tan 1 2 1x 2 + 7

73. y =

1 cot (3x - 1) 9

75. y = x (2x + 1)4 2

77. y = ex + 5x

72. y = 1 1 - 1x 2-1

x 74. y = 9 tan a b 3

76. y = x2 ( x3 - 1 ) 5

78. y = sin ( x2ex )

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3.6  The Chain Rule

Finding Derivative Values In Exercises 79–84, find the value of (ƒ ∘ g)′ at the given value of x.

79. ƒ(u) = u5 + 1, u = g(x) = 1x, x = 1

1 1 80. ƒ(u) = 1 - u , u = g(x) = , x = -1 1 - x 81. ƒ(u) = cot

83. ƒ(u) =

a. y = (u>5) + 7 and u = 5x - 35

1 , u = g(x) = px, x = 1>4 cos2 u

2u , u = g(x) = 10x2 + x + 1, x = 0 u2 + 1

u - 1 2 1 84. ƒ(u) = a b , u = g(x) = 2 - 1, x = -1 u + 1 x

85. Assume that ƒ′(3) = - 1, g′(2) = 5, g(2) = 3, and y = ƒ(g(x)). What is y′ at x = 2? 86. If r = sin (ƒ(t)), ƒ(0) = p>3, and ƒ′(0) = 4, then what is dr>dt at t = 0? 87. Suppose that functions ƒ and g and their derivatives with respect to x have the following values at x = 2 and x = 3. x

ƒ(x)

g(x)

2 8 2 3 3 - 4

ƒ′(x)

g′(x)

1>3 2p

-3 5

Find the derivatives with respect to x of the following combinations at the given value of x. ƒ(x) + g(x), x = 3 a. 2ƒ(x), x = 2 b.

c. ƒ(x) # g(x), x = 3 d. ƒ(x)>g(x), x = 2 e. ƒ(g(x)), x = 2 f. 2ƒ(x), x = 2

2ƒ2(x) + g2(x), x = 2 g. 1>g2(x), x = 3 h.

88. Suppose that the functions ƒ and g and their derivatives with respect to x have the following values at x = 0 and x = 1. x

ƒ(x)

g(x)

0 1 1 1 3 - 4

b. y = 1 + (1>u) and u = 1>(x - 1). 92. Find dy>dx if y = x3>2 by using the Chain Rule with y as a composite of a. y = u3 and u = 1x

b. y = 1u and u = x3.

93. Find the tangent to y = ((x - 1)>(x + 1))2 at x = 0. 94. Find the tangent to y = 2x2 - x + 7 at x = 2.

ƒ′(x)

g′(x)

97. Running machinery too fast  Suppose that a piston is moving straight up and down and that its position at time t sec is

5 - 1>3

1>3 - 8>3

s = A cos (2pbt),

Find the derivatives with respect to x of the following combinations at the given value of x. ƒ(x)g3(x), x = 0 a. 5ƒ(x) - g(x), x = 1 b. c.

Theory and Examples What happens if you can write a function as a composite in different ways? Do you get the same derivative each time? The Chain Rule says you should. Try it with the functions in Exercises 91 and 92.

91. Find dy>dx if y = x by using the Chain Rule with y as a compos­ ite of

pu , u = g(x) = 5 1x, x = 1 10

82. ƒ(u) = u +

173

ƒ(x) , x = 1 d. ƒ(g(x)), x = 0 g(x) + 1

e. g(ƒ(x)), x = 0 f. (x11 + ƒ(x))-2, x = 1 g. ƒ(x + g(x)), x = 0 89. Find ds>dt when u = 3p>2 if s = cos u and du>dt = 5. 90. Find dy>dt when x = 1 if y = x2 + 7x - 5 and dx>dt = 1>3.

with A and b positive. The value of A is the amplitude of the motion, and b is the frequency (number of times the piston moves up and down each second). What effect does doubling the frequency have on the piston’s velocity, acceleration, and jerk? (Once you find out, you will know why some machinery breaks when you run it too fast.) 98. Temperatures in Fairbanks, Alaska  The graph in the accompanying figure shows the average Fahrenheit temperature in Fairbanks, Alaska, during a typical 365-day year. The equation that approximates the temperature on day x is y = 37 sin c

2p (x - 101) d + 25 365

and is graphed in the accompanying figure.

M03_THOM9580_03_SE_C03_129-222.indd 173

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174

Chapter 3: Derivatives

a. On what day is the temperature increasing the fastest? b. About how many degrees per day is the temperature increasing when it is increasing at its fastest?

T 105. The derivative of sin 2x  Graph the function y = 2 cos 2x for -2 … x … 3.5. Then, on the same screen, graph y =

y

Temperature (°F)

60 40 20 0

.. ..... ... .. ......

.. .. ...

... .... . ..... .... .... .... . .. .... .. .. .... . .. ... ... ... ... ... ..

for h = 1.0, 0.5, and 0.2. Experiment with other values of h, including negative values. What do you see happening as h S 0? Explain this behavior.

.... .... .... ............ ..... ..

106. The derivative of cos (x2) Graph y = - 2x sin (x2) for - 2 … x … 3. Then, on the same screen, graph

. ... x

y =

Ja n Fe b M ar A pr M ay Ju n Ju l A ug Se p O ct N ov D ec Ja n Fe b M ar

−20

99. Particle motion  The position of a particle moving along a coordinate line is s = 21 + 4t, with s in meters and t in seconds. Find the particle’s velocity and acceleration at t = 6 sec. 100. Constant acceleration  Suppose that the velocity of a falling body is y = k 1s m>sec (k a constant) at the instant the body has fallen s m from its starting point. Show that the body’s acceleration is constant. 101. Falling meteorite  The velocity of a heavy meteorite entering Earth’s atmosphere is inversely proportional to 2s when it is s km from Earth’s center. Show that the meteorite’s acceleration is inversely proportional to s2. 102. Particle acceleration  A particle moves along the x-axis with velocity dx>dt = ƒ(x). Show that the particle’s acceleration is ƒ(x)ƒ′(x). 103. Temperature and the period of a pendulum  For oscillations of small amplitude (short swings), we may safely model the relationship between the period T and the length L of a simple pendulum with the equation T = 2p

L , Ag

where g is the constant acceleration of gravity at the pendulum’s location. If we measure g in centimeters per second squared, we measure L in centimeters and T in seconds. If the pendulum is made of metal, its length will vary with temperature, either increasing or decreasing at a rate that is roughly proportional to L. In symbols, with u being temperature and k the proportionality constant, dL = kL. du Assuming this to be the case, show that the rate at which the period changes with respect to temperature is kT>2. 104. Chain Rule  Suppose that ƒ(x) = x2 and g(x) =  x . Then the composites (ƒ ∘ g)(x) =  x  2 = x2 and (g ∘ ƒ)(x) =  x2  = x2 are both differentiable at x = 0 even though g itself is not differentiable at x = 0. Does this contradict the Chain Rule? Explain.

M03_THOM9580_03_SE_C03_129-222.indd 174

sin 2(x + h) - sin 2x h

cos ((x + h)2) - cos (x2) h

for h = 1.0, 0.7, and 0.3. Experiment with other values of h. What do you see happening as h S 0? Explain this behavior. Using the Chain Rule, show that the Power Rule (d>dx)xn = nxn - 1 holds for the functions xn in Exercises 107 and 108. 107. x1>4 = 2 1x

108. x3>4 = 2x 1x

COMPUTER EXPLORATIONS Trigonometric Polynomials 109. As the accompanying figure shows, the trigonometric “polynomial”

s = ƒ(t) = 0.78540 - 0.63662 cos 2t - 0.07074 cos 6t - 0.02546 cos 10t - 0.01299 cos 14t gives a good approximation of the sawtooth function s = g(t) on the interval 3- p, p4. How well does the derivative of ƒ approximate the derivative of g at the points where dg>dt is defined? To find out, carry out the following steps. a. Graph dg>dt (where defined) over 3- p, p4.

b. Find dƒ>dt.

c. Graph dƒ>dt. Where does the approximation of dg>dt by dƒ>dt seem to be best? Least good? Approximations by trigonometric polynomials are important in the theories of heat and oscillation, but we must not expect too much of them, as we see in the next exercise. s p 2

−p

0

s = g(t) s = f (t)

p

t

110. (Continuation of Exercise 109.) In Exercise 109, the trigonometric polynomial ƒ(t) that approximated the sawtooth function g(t) on 3- p, p4 had a derivative that approximated the derivative of the sawtooth function. It is possible, however, for a trigonometric polynomial to approximate a function in a reasonable way without its derivative approximating the function’s derivative at all well. As a case in point, the trigonometric “polynomial” s = h(t) = 1.2732 sin 2t + 0.4244 sin 6t + 0.25465 sin 10t + 0.18189 sin 14t + 0.14147 sin 18t

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3.7  Implicit Differentiation

graphed in the accompanying figure approximates the step function s = k(t) shown there. Yet the derivative of h is nothing like the derivative of k. s s = k(t)

p − 2

a. Graph dk>dt (where defined) over 3- p, p4. b. Find dh>dt.

c. Graph dh>dt to see how badly the graph fits the graph of dk>dt. Comment on what you see.

s = h(t)

1 −p

175

p 2

0

p

t

−1

3.7 Implicit Differentiation Most of the functions we have dealt with so far have been described by an equation of the form y = ƒ(x) that expresses y explicitly in terms of the variable x. We have learned rules for differentiating functions defined in this way. Another situation occurs when we encounter equations like

y 5

y = f1(x) (x 0, y 1) A

x3 + y3 - 9xy = 0,

x 3 + y 3 − 9xy = 0 y = f2(x)

(x 0, y 2) x0

0

5

x

y2 - x = 0,

or

x2 + y2 - 25 = 0.

(See Figures 3.28, 3.29, and 3.30.) These equations define an implicit relation between the variables x and y. In some cases we may be able to solve such an equation for y as an explicit function (or even several functions) of x. When we cannot put an equation F(x, y) = 0 in the form y = ƒ(x) to differentiate it in the usual way, we may still be able to find dy>dx by implicit differentiation. This section describes the technique.

Implicitly Defined Functions (x 0, y3)

y = f3 (x)

Figure 3.28  The curve x3 + y3 - 9xy = 0 is not the graph of any one function of x. The curve can, however, be divided into separate arcs that are the graphs of functions of x. This particular curve, called a folium, dates to Descartes in 1638.

We begin with examples involving familiar equations that we can solve for y as a function of x to calculate dy>dx in the usual way. Then we differentiate the equations implicitly, and find the derivative to compare the two methods. Following the examples, we summarize the steps involved in the new method. In the examples and exercises, it is always assumed that the given equation determines y implicitly as a differentiable function of x so that dy>dx exists.

EXAMPLE 1  Find dy>dx if y2 = x. Solution  The equation y2 = x defines two differentiable functions of x that we can actually find, namely y1 = 2x and y2 = - 2x (Figure 3.29). We know how to calculate the derivative of each of these for x 7 0: dy1 1 = dx 2 1x

and

dy2 1 = . dx 2 1x

But suppose that we knew only that the equation y2 = x defined y as one or more differentiable functions of x for x 7 0 without knowing exactly what these functions were. Could we still find dy>dx? The answer is yes. To find dy>dx, we simply differentiate both sides of the equation y2 = x with respect to x, treating y = ƒ(x) as a differentiable function of x:

M03_THOM9580_03_SE_C03_129-222.indd 175

y2 = x   The Chain Rule gives d ( y2 ) = dx dy d 2y = 1      3 ƒ(x) 4 2 = 2ƒ(x)ƒ′(x) dx dx dy 1 = . dx 2y

= 2y

dy . dx

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176 y

Chapter 3: Derivatives

y2 = x Slope = 1 = 1 2y1 2 "x

y1 = "x

P(x, "x ) x

0 Q(x, − "x )

y 2 = − "x

Slope = 1 = − 1 2y 2 2 "x

Figure 3.29  The equation y2 - x = 0, or y2 = x as it is usually written, defines two differentiable functions of x on the interval x 7 0. Example 1 shows how to find the derivatives of these functions without solving the equation y2 = x for y.

This one formula gives the derivatives we calculated for both explicit solutions y1 = 2x and y2 = - 2x: dy1 1 1 = = 2y1 2 1x dx

Solution  The circle is not the graph of a single function of x. Rather, it is the combined graphs of two differentiable functions, y1 = 225 - x2 and y2 = - 225 - x2 (Figure 3.30). The point (3, -4) lies on the graph of y2, so we can find the slope by calculating the derivative directly, using the Power Chain Rule: d 1 - ( 25 - x2 ) 1>2 2 = dx dy2 3 -2x -6 ` = ` = = .   1 dx x = 3     - ( 25 - x2 ) -1>2(- 2x) 2 225 - x2 x = 3 2 225 - 9 4 2

We can solve this problem more easily by differentiating the given equation of the circle implicitly with respect to x:

−5

2x + 2y

0

5

x2

x

Slope = − xy = 3 4

Figure 3.30  The circle combines the graphs of two functions. The graph of y2 is the lower semicircle and passes through (3, - 4).

dy = 0 dx

See Example 1.

dy x = - y. dx

(3, −4) y2 = −"25 − x 2

d 2 ( x ) + d ( y2 ) = d (25) dx dx dx

y1 = "25 −

dy2 1 1 1 = = = . 2y2 21 - 1x 2 dx 2 1x

EXAMPLE 2   Find the slope of the circle x2 + y2 = 25 at the point (3, -4).

y

and

x The slope at (3, -4) is - y `

(3, -4)

= -

3 3 = . -4 4

Notice that unlike the slope formula for dy2 >dx, which applies only to points below the x-axis, the formula dy>dx = -x>y applies everywhere the circle has a slope; that is, at all circle points (x, y) where y ≠ 0. Notice also that the derivative involves both variables x and y, not just the independent variable x. To calculate the derivatives of other implicitly defined functions, we proceed as in Examples 1 and 2: We treat y as a differentiable implicit function of x and apply the usual rules to differentiate both sides of the defining equation.

Implicit Differentiation 1.  Differentiate both sides of the equation with respect to x, treating y as a differentiable function of x. 2. Collect the terms with dy>dx on one side of the equation and solve for dy>dx.

EXAMPLE 3  Find dy>dx if y2 = x2 + sin xy (Figure 3.31). Solution  We differentiate the equation implicitly.

M03_THOM9580_03_SE_C03_129-222.indd 176

y2 = x2 + sin xy d 2 d 2 d 1 y 2 = dx 1 x 2 + dx 1 sin xy 2 dx 2y

dy d = 2x + (cos xy) (xy) dx dx

Differentiate both sides with respect to x c ctreating y as a function of x and using the Chain Rule.

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3.7  Implicit Differentiation

y

y2 = x2 + sin xy

4

2y 2

−4

0

−2

2

4

x

−2

2y

dy dy = 2x + (cos xy)ay + x b dx dx

dy dy - (cos xy) ax b = 2x + (cos xy)y dx dx (2y - x cos xy)

177

Treat xy as a product.

Collect terms with dy>dx.

Solve for dy>dx.

dy = 2x + y cos xy dx

dy 2x + y cos xy = dx 2y - x cos xy

Notice that the formula for dy>dx applies everywhere that the implicitly defined curve has a slope. Notice again that the derivative involves both variables x and y, not just the independent variable x.

−4

Figure 3.31  The graph of the equation in Example 3.

Derivatives of Higher Order Implicit differentiation can also be used to find higher derivatives.

EXAMPLE 4  Find d 2y>dx2 if 2x3 - 3y2 = 8. Solution  To start, we differentiate both sides of the equation with respect to x in order to find y′ = dy>dx.

d ( 2x3 - 3y2 ) = d (8) dx dx 2 6x - 6yy′ = 0 2

x y′ = y ,

Treat y as a function of x. when y ≠ 0  Solve for y′.

We now apply the Quotient Rule to find y″. y″ =

2xy - x2y′ 2x x2 d x2 ayb = = y - 2 # y′ dx y2 y

Finally, we substitute y′ = x2 >y to express y″ in terms of x and y.

Light ray

Curve of lens surface A P

Point of entry B

Figure 3.32  The profile of a lens, showing the bending (refraction) of a ray of light as it passes through the lens surface.

M03_THOM9580_03_SE_C03_129-222.indd 177

when y ≠ 0

Lenses, Tangents, and Normal Lines

Tangent

Normal line

2x x2 x2 2x x4 y″ = y - 2 a y b = y - 3 , y y

In the law that describes how light changes direction as it enters a lens, the important angles are the angles the light makes with the line perpendicular to the surface of the lens at the point of entry (angles A and B in Figure 3.32). This line is called the normal to the surface at the point of entry. In a profile view of a lens like the one in Figure 3.32, the normal is the line perpendicular (also said to be orthogonal) to the tangent of the profile curve at the point of entry.

EXAMPLE 5   Show that the point (2, 4) lies on the curve x3 + y3 - 9xy = 0. Then find the tangent and normal to the curve there (Figure 3.33). Solution  The point (2, 4) lies on the curve because its coordinates satisfy the equation given for the curve: 23 + 43 - 9(2) (4) = 8 + 64 - 72 = 0.

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178

Chapter 3: Derivatives y

To find the slope of the curve at (2, 4), we first use implicit differentiation to find a formula for dy>dx:

t

n ge

n

Ta

x3 + y3 - 9xy = 0

4 No

d 3 d 3 ( y ) - d (9xy) = d (0) (x ) + dx dx dx dx

al

rm

x 3 + y 3 − 9xy = 0

0

2

x

3x2 + 3y2

Figure 3.33  Example 5 shows how to find equations for the tangent and normal to the folium of Descartes at (2, 4).

dy dy dx - 9ax + y b = 0 dx dx dx

Differentiate both sides with respect to x.

Treat xy as a product and y as a function of x.

dy

( 3y2 - 9x ) + 3x2 - 9y = 0 dx 3 ( y2 - 3x )

dy = 9y - 3x2 dx dy 3y - x2 = . dx y2 - 3x

Solve for dy>dx.

We then evaluate the derivative at (x, y) = (2, 4): 3y - x2 dy 3(4) - 22 8 4 = ` = 2 ` = 2 = . dx (2, 4) y - 3x (2, 4) 4 - 3(2) 10 5

The tangent at (2, 4) is the line through (2, 4) with slope 4>5: y = 4 +

4 (x - 2) 5

4 12 x + . 5 5

y =

The normal to the curve at (2, 4) is the line perpendicular to the tangent there, the line through (2, 4) with slope -5>4: 5 (x - 2) 4

y = 4 -

5 13 y = - x + . 4 2



Exercises  3.7 Differentiating Implicitly Use implicit differentiation to find dy>dx in Exercises 1–16.

1. x2y + xy2 = 6

2. x3 + y3 = 18xy

3. 2xy + y2 = x + y 2

2

2

4. x3 - xy + y3 = 1 2

5. x (x - y) = x - y 7. y2 =

x - 1 x + 1

2

6. (3xy + 7) = 6y 8. x3 =

2x - y x + 3y

9. x = sec y

10. xy = cot (xy)

11. x + tan (xy) = 0

12. x4 + sin y = x3y2

1 13. y sin a y b = 1 - xy 15. e2x = sin (x + 3y)

M03_THOM9580_03_SE_C03_129-222.indd 178

14. x cos (2x + 3y) = y sin x 2

16. ex y = 2x + 2y

Find dr>du in Exercises 17–20. 17. u 1>2 + r 1>2 = 1 19. sin (r u) =

1 2

18. r - 2 2u =

3 2>3 4 3>4 u + u 2 3

20. cos r + cot u = er u

Second Derivatives In Exercises 21–26, use implicit differentiation to find dy>dx and then d 2y>dx2.

21. x2 + y2 = 1 x2

2

22. x2>3 + y2>3 = 1

23. y = e + 2x

24. y2 - 2x = 1 - 2y

25. 2 1y = x - y

26. xy + y2 = 1

3

3

27. If x + y = 16, find the value of d 2y>dx2 at the point (2, 2). 28. If xy + y2 = 1, find the value of d 2y>dx2 at the point (0, -1).

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3.7  Implicit Differentiation

In Exercises 29 and 30, find the slope of the curve at the given points. 2

2

4

29. y + x = y - 2x at (- 2, 1) and (- 2, -1) 30. (x2 + y2)2 = (x - y)2 at (1, 0) and (1, -1)

179

45. The devil’s curve (Gabriel Cramer, 1750)  Find the slopes of the devil’s curve y4 - 4y2 = x4 - 9x2 at the four indicated points. y

Slopes, Tangents, and Normals In Exercises 31–40, verify that the given point is on the curve and find the lines that are (a) tangent and (b) normal to the curve at the given point. 2

y 4 − 4y 2 = x 4 − 9x 2

(−3, 2)

2

(3, 2)

2

31. x + xy - y = 1, (2, 3) 32. x2 + y2 = 25, (3, - 4)

−3 (−3, −2)

33. x2y2 = 9, (- 1, 3)

x 3 (3, −2)

−2

34. y2 - 2x - 4y - 1 = 0, (- 2, 1) 35. 6x2 + 3xy + 2y2 + 17y - 6 = 0, (- 1, 0) 36. x2 - 23xy + 2y2 = 5,

1 23, 2 2

46. The folium of Descartes  (See Figure 3.28.) a. Find the slope of the folium of Descartes x3 + y3 - 9xy = 0 at the points (4, 2) and (2, 4).

37. 2xy + p sin y = 2p, (1, p>2)

38. x sin 2y = y cos 2x, (p>4, p>2)

b. At what point other than the origin does the folium have a horizontal tangent?

39. y = 2 sin (px - y), (1, 0) 40. x2 cos2 y - sin y = 0, (0, p) 41. Parallel tangents  Find the two points where the curve x2 + xy + y2 = 7 crosses the x-axis, and show that the tangents to the curve at these points are parallel. What is the common slope of these tangents? 42. Normals parallel to a line  Find the normals to the curve xy + 2x - y = 0 that are parallel to the line 2x + y = 0. 43. The eight curve  Find the slopes of the curve y4 = y2 - x2 at the two points shown here.

c. Find the coordinates of the point A in Figure 3.28 where the folium has a vertical tangent. Theory and Examples 47. Intersecting normal  The line that is normal to the curve x2 + 2xy - 3y2 = 0 at (1, 1) intersects the curve at what other point?

48. Power rule for rational exponents Let p and q be integers with q 7 0. If y = x p>q, differentiate the equivalent equation yq = xp implicitly and show that, for y ≠ 0, d p>q p (p>q) - 1 x = qx . dx

y a" 3 , " 3b 4 2

1

a" 3 , 1b 4 2 x

0

49. Normals to a parabola  Show that if it is possible to draw three normals from the point (a, 0) to the parabola x = y2 shown in the accompanying diagram, then a must be greater than 1>2. One of the normals is the x-axis. For what value of a are the other two normals perpendicular?

y4 = y2 − x2

y x = y2

−1

44. The cissoid of Diocles (from about 200 b.c.)  Find equations for the tangent and normal to the cissoid of Diocles y2(2 - x) = x3 at (1, 1).

0

(a, 0)

x

y y 2(2 − x) = x 3

(1, 1)

1

0

M03_THOM9580_03_SE_C03_129-222.indd 179

1

x

16/11/15 4:40 PM

180

Chapter 3: Derivatives

50. Is there anything special about the tangents to the curves y2 = x3 and 2x2 + 3y2 = 5 at the points (1, {1)? Give reasons for your answer. y

How do dy>dx and dx>dy seem to be related? Explain the relationship geometrically in terms of the graphs. 53. xy3 + x2y = 6 54. x3 + y2 = sin2 y

y2 = x3

55. Derivative of arcsine  Assume that y = sin-1 x is a differentiable function of x. By differentiating the equation x = sin y implicitly, show that dy>dx = 1> 21 - x2 .

2x 2 + 3y 2 = 5 (1, 1)

56. Use the formula in Exercise 55 to find dy>dx if x

0 (1, −1)

1 y = sin-1 a x b . a. y = ( sin-1 x ) 2 b.

COMPUTER EXPLORATIONS Use a CAS to perform the following steps in Exercises 57–64.

51. Verify that the following pairs of curves meet orthogonally. a. x2 + y2 = 4, x2 = 3y2 b. x = 1 - y2, x =

b. Using implicit differentiation, find a formula for the derivative dy>dx and evaluate it at the given point P.

1 2 y 3

52. The graph of y2 = x3 is called a semicubical parabola and is shown in the accompanying figure. Determine the constant b so that the line y = - 13 x + b meets this graph orthogonally. y

a. Plot the equation with the implicit plotter of a CAS. Check to see that the given point P satisfies the equation.

y2 = x3

c. Use the slope found in part (b) to find an equation for the tangent line to the curve at P. Then plot the implicit curve and tangent line together on a single graph. 57. x3 - xy + y3 = 7, P (2, 1) 58. x5 + y3x + yx2 + y4 = 4, P (1, 1) 59. y2 + y =

1 y=− x+b 3 0

2 + x , P (0, 1) 1 - x

60. y3 + cos xy = x2, P (1, 0) y p 61. x + tan a x b = 2, P a1, b 4

x

p 62. xy3 + tan (x + y) = 1, P a , 0b 4 T In Exercises 53 and 54, find both dy>dx (treating y as a differentiable function of x) and dx>dy (treating x as a differentiable function of y).

63. 2y2 + (xy)1>3 = x2 + 2, P (1, 1) 64. x 21 + 2y + y = x2, P (1, 0)

3.8  Derivatives of Inverse Functions and Logarithms In Section 1.6 we saw how the inverse of a function undoes, or inverts, the effect of that function. We defined there the natural logarithm function ƒ-1(x) = ln x as the inverse of the natural exponential function ƒ(x) = ex. This is one of the most important function-inverse pairs in mathematics and science. We learned how to differentiate the exponential function in Section 3.3. Here we learn a rule for differentiating the inverse of a differentiable function and we apply the rule to find the derivative of the natural logarithm function.

Derivatives of Inverses of Differentiable Functions We calculated the inverse of the function ƒ(x) = (1>2)x + 1 as ƒ-1(x) = 2x - 2 in Example 3 of Section 1.6. Figure 3.34 shows again the graphs of both functions. If we calculate their derivatives, we see that d d 1 1 ƒ(x) = a x + 1b = 2 dx dx 2 d -1 d ƒ (x) = (2x - 2) = 2. dx dx

M03_THOM9580_03_SE_C03_129-222.indd 180

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3.8  Derivatives of Inverse Functions and Logarithms

y

y = 2x − 2

y=x

y= 1x+1 2 1 1

−2

181

The derivatives are reciprocals of one another, so the slope of one line is the reciprocal of the slope of its inverse line. (See Figure 3.34.) This is not a special case. Reflecting any nonhorizontal or nonvertical line across the line y = x always inverts the line’s slope. If the original line has slope m ≠ 0, the reflected line has slope 1 > m.

x

y

y y = f (x)

−2 b = f (a)

(a, b)

Figure 3.34  Graphing a line and its inverse together shows the graphs’ symmetry with respect to the line y = x. The slopes are reciprocals of each other.

(b, a)

a = f –1(b)

0

a

x

y = f –1(x) b

0

The slopes are reciprocal: ( f –1)′(b) =

x

1 1 or ( f –1)′(b) = f ′(a) f ′( f –1(b))

Figure 3.35  The graphs of inverse functions have reciprocal slopes at corresponding points.

The reciprocal relationship between the slopes of ƒ and ƒ -1 holds for other functions as well, but we must be careful to compare slopes at corresponding points. If the slope of y = ƒ(x) at the point (a, ƒ(a)) is ƒ′(a) and ƒ′(a) ≠ 0, then the slope of y = ƒ -1(x) at the point (ƒ(a), a) is the reciprocal 1>ƒ′(a) (Figure 3.35). If we set b = ƒ(a), then (ƒ -1)′(b) =

1 1 . = ƒ′(a) ƒ′(ƒ -1(b))

If y = ƒ(x) has a horizontal tangent line at (a, ƒ(a)), then the inverse function ƒ -1 has a vertical tangent line at (ƒ(a), a), and this infinite slope implies that ƒ -1 is not differentiable at ƒ(a). Theorem 3 gives the conditions under which ƒ -1 is differentiable in its domain (which is the same as the range of ƒ).

THEOREM 3—The Derivative Rule for Inverses  If ƒ has an interval I as domain and ƒ′(x) exists and is never zero on I, then ƒ -1 is differentiable at every point in its domain (the range of ƒ). The value of (ƒ -1)′ at a point b in the domain of ƒ -1 is the reciprocal of the value of ƒ′ at the point a = ƒ -1(b): or

( ƒ -1 ) ′(b) =

ƒ′ ( ƒ

1

-1

(b) )

(1)

dƒ -1 1 2 = . dx x = b dƒ 2 dx x = ƒ -1(b)

Theorem 3 makes two assertions. The first of these has to do with the conditions under which ƒ -1 is differentiable; the second assertion is a formula for the derivative of

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ƒ -1 when it exists. While we omit the proof of the first assertion, the second one is proved in the following way:

ƒ ( ƒ -1 ( x ) ) = x

d ƒ ( ƒ -1 ( x ) ) = 1 dx

ƒ′ ( ƒ -1 ( x ) ) #

Slope 4 (2, 4) Slope 1– 4

2

(4, 2)

y = "x

1 0

1

2

3

x

4

Figure 3.36  The derivative of ƒ-1(x) = 1x at the point (4, 2) is the reciprocal of the derivative of ƒ(x) = x2 at (2, 4) (Example 1).

6

(2, 6)

( ƒ -1 ) ′(4) =

y = x3 − 2 Slope 3x 2 = 3(2)2 = 12

Reciprocal slope: 1 12 (6, 2)

−2

0

6

x

ƒ′(x) = 2x with x replaced by ƒ -1(x)

1 1 1 1 = ` = . = ƒ′(2) 2x x = 2 4 ƒ′ ( ƒ -1 ( 4 ) )

We will use the procedure illustrated in Example 1 to calculate formulas for the derivatives of many inverse functions throughout this chapter. Equation (1) sometimes enables us to find specific values of dƒ -1 >dx without knowing a formula for ƒ -1.

EXAMPLE 2  Let ƒ(x) = x3 - 2, x 7 0. Find the value of dƒ -1 >dx at x = 6 = ƒ(2) without finding a formula for ƒ -1(x). See Figure 3.37.

Solution  We apply Theorem 3 to obtain the value of the derivative of ƒ -1 at x = 6:

−2

Figure 3.37  The derivative of ƒ(x) = x3 - 2 at x = 2 tells us the derivative of ƒ -1 at x = 6 (Example 2).

1 ƒ′ ( ƒ -1 ( x ) ) 1 = 2 ( ƒ -1 ( x ) ) 1 = . 2 ( 1x )

( ƒ -1 ) ′(x) =

Theorem 3 gives a derivative that agrees with the known derivative of the square root function. Let’s examine Theorem 3 at a specific point. We pick x = 2 (the number a) and ƒ(2) = 4 (the value b). Theorem 3 says that the derivative of ƒ at 2, which is ƒ′(2) = 4, and the derivative of ƒ -1 at ƒ(2), which is ( ƒ -1 ) ′(4), are reciprocals. It states that

See Figure 3.36. y

d -1 ƒ (x) = 1   Chain Rule dx d -1 1 ƒ (x) = .  Solving for the derivative dx ƒ′ ( ƒ -1 ( x ) )

derivatives ƒ′(x) = 2x and (ƒ -1)′(x) = 1> 1 2 2x 2. Let’s verify that Theorem 3 gives the same formula for the derivative of ƒ -1(x):

y = x 2, x > 0

3

Differentiating both sides

EXAMPLE 1   The function ƒ(x) = x2, x 7 0 and its inverse ƒ -1(x) = 2x have

y

4

Inverse function relationship

dƒ 2 = 3x2 ` = 12 dx x = 2 x=2

dƒ -1 1 1 2 = = .  Eq. (1) 12 dx x = ƒ(2) dƒ 2 dx x = 2 

Derivative of the Natural Logarithm Function Since we know the exponential function ƒ(x) = ex is differentiable everywhere, we can apply Theorem 3 to find the derivative of its inverse ƒ -1(x) = ln x:

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3.8  Derivatives of Inverse Functions and Logarithms

( ƒ -1 ) ′(x) =

=

=

1 = x.

ƒ′ ( ƒ

1 -1

(x) )

1 eƒ

-1

(x)

1 eln x

Theorem 3

ƒ′(u) = eu

x 7 0

Inverse function relationship

183

Alternate Derivation  Instead of applying Theorem 3 directly, we can find the derivative of y = ln x using implicit differentiation, as follows:

y = ln x ey = x

d y d (e ) = (x) dx dx

ey

dy = 1 dx

dy 1 1 = = x . dx ey

x 7 0 Inverse function relationship Differentiate implicitly.

Chain Rule

ey = x

No matter which derivation we use, the derivative of y = ln x with respect to x is d 1 (ln x) = x , x 7 0. dx The Chain Rule extends this formula to positive functions u(x):

d 1 du ln u = , dx u dx

u 7 0.(2)

EXAMPLE 3   We use Equation (2) to find derivatives. (a)

d 1 d 1 1 ln 2x = (2x) = (2) = x , x 7 0 2x dx 2x dx

(b) Equation (2) with u = x2 + 3gives d 2x 1 # d 2 1 # (x + 3) = 2 ln (x2 + 3) = 2 2x = 2 . dx x + 3 dx x + 3 x + 3 (c) Equation (2) with u = 0 x 0 gives an important derivative:

d d du ln 0 x 0 = ln u #    u = 0 x 0 , x ≠ 0 dx du dx 1# x d ( 0x0 ) = x   = u    dx 0x0 0x0

Derivative of ln ∣ x ∣ d 1 ln 0 x 0 = x , x ≠ 0 dx

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1 # x   Substitute for u.   = 0x0 0x0

x   = 2 x 1   = x.

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So 1>x is the derivative of ln x on the domain x 7 0, and the derivative of ln (-x) on the domain x 6 0. Notice from Example 3a that the function y = ln 2x has the same derivative as the function y = ln x. This is true of y = ln bx for any constant b, provided that bx 7 0: d 1 # d 1 1 ln bx = (bx) = (b) = x .(3) dx bx dx bx

d 1 ln bx = x , bx 7 0 dx

EXAMPLE 4   A line with slope m passes through the origin and is tangent to the graph of y = ln x. What is the value of m? Solution  Suppose the point of tangency occurs at the unknown point x = a 7 0. Then we know that the point (a, ln a) lies on the graph and that the tangent line at that point has slope m = 1>a (Figure 3.38). Since the tangent line passes through the origin, its slope is

y

2 1 0

(a, ln a)

1 2 y = ln x

m =

1 Slope = a 3

4

5

x

ln a - 0 ln a = a . a - 0

Setting these two formulas for m equal to each other, we have ln a a ln a eln a a

1 = a = 1 = e1 = e 1 m = e .

Figure 3.38  The tangent line intersects the curve at some point (a, ln a), where the slope of the curve is 1>a (Example 4).

The Derivatives of au and log a u x

We start with the equation ax = eln (a ) = e x ln a , a 7 0, which was seen in Section 1.6:

d x d x ln a a = e dx dx

= e x ln a #

= ax ln a.

d d u (x ln a)   dx e dx

= eu

du dx

That is, if a 7 0, then ax is differentiable and d x a = ax ln a.(4) dx

This equation shows why ex is the preferred exponential function in calculus. If a = e, then ln a = 1 and the derivative of ax simplifies to d x e = ex ln e = ex. dx With the Chain Rule, we get a more general form for the derivative of a general exponential function au. If a 7 0 and u is a differentiable function of x, then au is a differentiable function of x and

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d u du a = au ln a .(5) dx dx

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185

EXAMPLE 5   Here are some derivatives of general exponential functions. d x 3 = 3x ln 3 dx d -x d (b) 3 = 3-x(ln 3) (-x) = -3-x ln 3 dx dx d sin x d (c) 3 = 3sin x(ln 3) (sin x) = 3sin x(ln 3) cos x dx dx

(a)

Eq. (5) with a = 3, u = x Eq. (5) with a = 3, u = - x c, u = sin x

In Section 3.3 we looked at the derivative ƒ′(0) for the exponential functions ƒ(x) = ax at various values of the base a. The number ƒ′(0) is the limit, limh S 0 (ah - 1)>h, and gives the slope of the graph of ax when it crosses the y-axis at the point (0, 1). We now see from Equation (4) that the value of this slope is ah - 1 = ln a.(6) h hS0

lim

In particular, when a = e we obtain lim

hS0

eh - 1 = ln e = 1. h

However, we have not fully justified that these limits actually exist. While all of the arguments given in deriving the derivatives of the exponential and logarithmic functions are correct, they do assume the existence of these limits. In Chapter 7 we will give another development of the theory of logarithmic and exponential functions which fully justifies that both limits do in fact exist and have the values derived above. To find the derivative of loga u for an arbitrary base (a 7 0, a ≠ 1), we start with the change-of-base formula for logarithms (reviewed in Section 1.6) to express loga u in terms of natural logarithms, and then take derivatives:

d d ln x loga x = a b    loga x dx dx ln a

=

ln x ln a

=

1 # d ln x  ln a is a constant. ln a dx

=

1 #1 ln a x

=

1 . x ln a

If u is a differentiable function of x and u 7 0, the Chain Rule gives a more general formula.

For a 7 0 and a ≠ 1,

d 1 du loga u = .(7) dx u ln a dx

Logarithmic Differentiation The derivatives of positive functions given by formulas that involve products, quotients, and powers can often be found more quickly if we take the natural logarithm of both sides before differentiating. This enables us to use the laws of logarithms to simplify the formulas before differentiating. The process, called logarithmic differentiation, is illustrated in the next example.

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Chapter 3: Derivatives

EXAMPLE 6  Find dy>dx if y =

(x2 + 1)(x + 3)1>2 , x - 1

x 7 1.

Solution  We take the natural logarithm of both sides and simplify the result with the algebraic properties of logarithms from Theorem 1 in Section 1.6:

ln y = ln

(x2 + 1)(x + 3)1>2 x - 1

= ln ((x2 + 1)(x + 3)1>2) - ln (x - 1) 2

= ln (x + 1) + ln (x + 3)

= ln (x2 + 1) +

1>2

Rule 2

- ln (x - 1)  Rule 1

1 ln (x + 3) - ln (x - 1).  Rule 4 2

We then take derivatives of both sides with respect to x, using Equation (2) on the left: 1 1 1 dy 1# 1 # y dx = x2 + 1 2x + 2 x + 3 - x - 1 . Next we solve for dy>dx: dy 2x 1 1 = ya 2 + b. dx x + 1 2x + 6 x - 1

Finally, we substitute for y:

dy (x2 + 1)(x + 3)1>2 2x 1 1 = + a 2 b . x - 1 x 1 dx 2x + 6 x + 1

Irrational Exponents and the Power Rule (General Version) The definition of the general exponential function enables us to raise any positive number to any real power n, rational or irrational. That is, we can define the power function y = xn for any exponent n. DEFINITION  For any x 7 0 and for any real number n, xn = en ln x. Because the logarithm and exponential functions are inverses of each other, the definition gives ln xn = n ln x, for all real numbers n. That is, the rule for taking the natural logarithm of any power holds for all real exponents n, not just for rational exponents. The definition of the power function also enables us to establish the derivative Power Rule for any real power n, as stated in Section 3.3. General Power Rule for Derivatives For x 7 0 and any real number n, d n x = nxn - 1. dx If x … 0, then the formula holds whenever the derivative, xn, and xn - 1 all exist.

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3.8  Derivatives of Inverse Functions and Logarithms

187

Proof  Differentiating xn with respect to x gives

d n d n ln x x = e dx dx

Definition of xn, x 7 0

= en ln x #

n = xn # x

Definition and derivative of ln x

= nxn - 1.

xn # x-1 = xn - 1

d (n ln x) dx

Chain Rule for eu

In short, whenever x 7 0, d n x = nxn - 1. dx For x 6 0, if y = xn, y′, and xn - 1 all exist, then ln 0 y 0 = ln 0 x 0 n = n ln 0 x 0 .

Using implicit differentiation (which assumes the existence of the derivative y′) and Example 3(c), we have y′ n y = x. Solving for the derivative, y xn y′ = n x = n x = nxn - 1.   y

= xn

It can be shown directly from the definition of the derivative that the derivative equals 0 when x = 0 and n Ú 1 (see Exercise 99). This completes the proof of the general version of the Power Rule for all values of x.

EXAMPLE 7  Differentiate ƒ(x) = xx, x 7 0. Solution  We note that ƒ(x) = xx = ex ln x, so differentiation gives ƒ′(x) =

d x ln x (e ) dx

= ex ln x

d (x ln x) dx

d u dx e ,

1 = ex ln x aln x + x # x b

= xx (ln x + 1).

u = x ln x

x 7 0

We can also find the derivative of y = xx using logarithmic differentiation, assuming y′ exists.

The Number e Expressed as a Limit In Section 1.5 we defined the number e as the base value for which the exponential function y = ax has slope 1 when it crosses the y-axis at (0, 1). Thus e is the constant that satisfies the equation eh - 1 = ln e = 1.  Slope equals ln e from Eq. (6). h hS0 lim

We now prove that e can be calculated as a certain limit.

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Chapter 3: Derivatives

THEOREM 4—The Number e as a Limit  The number e can be calculated as the limit e = lim (1 + x)1>x. xS0

Proof  If ƒ(x) = ln x, then ƒ′(x) = 1>x, so ƒ′(1) = 1. But, by the definition of derivative, ƒ′(1) = lim

= lim

= lim ln (1 + x)1>x = ln c lim (1 + x)1>x d .

y 3

hS0

xS0

ln (1 + x) - ln 1 1 = lim x ln (1 + x) x xS0

xS0

xS0

ln 1 = 0 ln is continuous, Theorem 10 in Chapter 2.

Because ƒ′(1) = 1, we have

e

ln c lim (1 + x)1>x d = 1.

y = (1 + x)1x

2

ƒ(1 + h) - ƒ(1) ƒ(1 + x) - ƒ(1) = lim x h xS0

xS0

Therefore, exponentiating both sides we get 1

lim (1 + x)1>x = e.

xS0

x

0

Figure 3.39  The number e is the limit of the function graphed here as x S 0.

See Figure 3.39. Approximating the limit in Theorem 4 by taking x very small gives approximations to e. Its value is e ≈ 2.718281828459045 to 15 decimal places.

Exercises  3.8 Derivatives of Inverse Functions In Exercises 1–4:

c. Find the slopes of the tangents to the graphs at h and k at (2, 2) and (-2, -2).

a. Find ƒ -1(x).

d. What lines are tangent to the curves at the origin?

b. Graph ƒ and ƒ -1 together. c.  Evaluate dƒ>dx at x = a and dƒ -1 >dx at x = ƒ(a) to show that at these points dƒ -1 >dx = 1>(dƒ>dx). 1. ƒ(x) = 2x + 3, a = - 1

2. ƒ(x) = (1>5)x + 7, a = -1

3. ƒ(x) = 5 - 4x, a = 1>2

4. ƒ(x) = 2x2, x Ú 0, a = 5

3 5. a.  Show that ƒ(x) = x3 and g(x) = 1 x are inverses of one another.

b. Graph ƒ and g over an x-interval large enough to show the graphs intersecting at (1, 1) and (- 1, - 1). Be sure the picture shows the required symmetry about the line y = x.

c. Find the slopes of the tangents to the graphs of ƒ and g at (1, 1) and (- 1, - 1) (four tangents in all). d. What lines are tangent to the curves at the origin? 6. a.  Show that h(x) = x3 >4 and k(x) = (4x)1>3 are inverses of one another. b. Graph h and k over an x-interval large enough to show the graphs intersecting at (2, 2) and (- 2, - 2). Be sure the picture shows the required symmetry about the line y = x.

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7. Let ƒ(x) = x3 - 3x2 - 1, x Ú 2. Find the value of dƒ -1 >dx at the point x = -1 = ƒ(3).

8. Let ƒ(x) = x2 - 4x - 5, x 7 2. Find the value of dƒ -1 >dx at the point x = 0 = ƒ(5).

9. Suppose that the differentiable function y = ƒ(x) has an inverse and that the graph of ƒ passes through the point (2, 4) and has a slope of 1>3 there. Find the value of dƒ -1 >dx at x = 4. 10. Suppose that the differentiable function y = g(x) has an inverse and that the graph of g passes through the origin with slope 2. Find the slope of the graph of g-1 at the origin. Derivatives of Logarithms In Exercises 11–40, find the derivative of y with respect to x, t, or u, as appropriate. 1 12. y = 11. y = ln 3x + x ln 3x

13. y = ln ( t 2 ) 3 15. y = ln x

14. y = ln ( t 3>2 ) + 2t

16. y = ln (sin x)

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17. y = ln (u + 1) - eu 3

19. y = ln x 21. y = t (ln t)

3

x4 x4 ln x 4 16

ln x 1 + ln x

28. y =

29. y = ln (ln x)

t 2ln t

35. y =

1 + ln t 1 - ln t

39. y = ln a

(x2 + 1)5 21 - x

1 1 + x ln 2 1 - x

b

43. y =

t At + 1

38. y = ln a 40. y = ln

2sin u cos u

1 + 2 ln u

(x + 1)5 C(x + 2)20

45. y = (sin u) 2u + 3 47. y = t(t + 1)(t + 2) u + 5 49. y = u cos u x 2x2 + 1 51. y = (x + 1)2>3 53. y =

x(x - 2) B x2 + 1 3

2

b

-t

57. y = ln (3te ) 59. y = ln a

eu b 1 + eu

61. y = e(cos t + ln t)

2

1 A t(t + 1)

1 48. y = t(t + 1)(t + 2)

(x + 1)10 52. y = C(2x + 1)5

x(x + 1)(x - 2) (x B 2 + 1)(2x + 3) 3

56. y = ln (3ue-u) 58. y = ln (2e-t sin t) 60. y = ln a

2u

1 + 2u

63. ln y = e sin x

64. ln xy = e

65. xy = yx

66. tan y = ex + ln x

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b

62. y = esin t (ln t 2 + 1)

x+y

78. y = log3 r # log9 r

ln 5 7x b 3x + 2 B sin u cos u 82. y = log7 a b eu 2u

80. y = log5

84. y = log2 a

a

x2e2 b 2 2x + 1

86. y = 3 log8 (log2 t)

88. y = t log3 1 e(sin t)(ln 3) 2

90. y = x(x + 1)

91. y =

( 1t )

92. y = t 2t

t

94. y = xsin x

93. y = (sin x)x 95. y = x

ln x

96. y = (ln x)ln x

Theory and Applications 97. If we write g(x) for ƒ-1(x), Equation (1) can be written as

g′(ƒ(a)) =

1 , or g′(ƒ(a)) # ƒ′(a) = 1. ƒ′(a)

If we then write x for a, we get g′(ƒ(x)) # ƒ′(x) = 1.

u sin u 50. y = 2sec u

In Exercises 63–66, find dy>dx. y

x + 1 ln 3 b b x - 1

89. y = (x + 1)x

Finding Derivatives In Exercises 55–62, find the derivative of y with respect to x, t, or u, as appropriate.

55. y = ln (cos2 u)

76. y = log25 ex - log5 1x

log2 r # log4 r

Logarithmic Differentiation with Exponentials In Exercises 89–96, use logarithmic differentiation to find the derivative of y with respect to the given independent variable.

46. y = (tan u) 22u + 1

54. y =

74. y = log3 (1 + u ln 3)

75. y = log4 x + log4 x2

87. y = log2 (8t ln 2)

42. y = 2(x + 1)(x - 1)

44. y =

73. y = log2 5u

85. y = 3log2 t

Logarithmic Differentiation In Exercises 41–54, use logarithmic differentiation to find the derivative of y with respect to the given independent variable.

41. y = 2x(x + 1)

72. y = t 1 - e

83. y = log5 ex

36. y = 2ln 1t

37. y = ln (sec (ln u))

71. y = xp

81. y = u sin (log7 u)

32. y = ln (sec u + tan u) 34. y =

70. y = 2(s )

79. y = log3 a a

30. y = ln (ln (ln x))

1 x 2x + 1

2

69. y = 5

77. y =

x ln x 1 + ln x

31. y = u(sin (ln u) + cos (ln u)) 33. y = ln

68. y = 3-x

2s

24. y = (x2 ln x)4 26. y =

In Exercises 67–88, find the derivative of y with respect to the given independent variable. 67. y = 2x

22. y = t ln 2t

ln t 25. y = t 27. y =

18. y = (cos u) ln (2u + 2) 20. y = (ln x)

2

23. y =

189

3.8  Derivatives of Inverse Functions and Logarithms

The latter equation may remind you of the Chain Rule, and indeed there is a connection.    Assume that ƒ and g are differentiable functions that are inverses of one another, so that (g ∘ ƒ)(x) = x. Differentiate both sides of this equation with respect to x, using the Chain Rule to express (g ∘ ƒ)′(x) as a product of derivatives of g and ƒ. What do you find? (This is not a proof of Theorem 3 because we assume here the theorem’s conclusion that g = ƒ -1 is differentiable.) x n 98. Show that limn S q a1 + n b = ex for any x 7 0.

99. If ƒ(x) = xn, n Ú 1, show from the definition of the derivative that ƒ′(0) = 0. 100. Using mathematical induction, show that for n 7 1 (n - 1)! dn ln x = (- 1)n - 1 . xn dxn COMPUTER EXPLORATIONS In Exercises 101–108, you will explore some functions and their inverses together with their derivatives and tangent line approxima-

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tions at specified points. Perform the following steps using your CAS:

103. y =

4x , -1 … x … 1, x0 = 1>2 x2 + 1

a.  Plot the function y = ƒ(x) together with its derivative over the given interval. Explain why you know that ƒ is one-to-one over the interval.

104. y =

x3 , -1 … x … 1, x0 = 1>2 x + 1

b.  Solve the equation y = ƒ(x) for x as a function of y, and name the resulting inverse function g. c.  Find the equation for the tangent line to ƒ at the specified point (x0, ƒ(x0)).

27 10

106. y = 2 - x - x3, -2 … x … 2, x0 =

3 2

108. y = sin x, -

p p … x … , x0 = 1 2 2

In Exercises 109 and 110, repeat the steps above to solve for the functions y = ƒ(x) and x = ƒ -1(y) defined implicitly by the given equations over the interval.

e.  Plot the functions ƒ and g, the identity, the two tangent lines, and the line segment joining the points (x0, ƒ(x0)) and (ƒ(x0), x0). Discuss the symmetries you see across the main diagonal.

102. y =

105. y = x3 - 3x2 - 1, 2 … x … 5, x0 =

107. y = ex, -3 … x … 5, x0 = 1

d.  Find the equation for the tangent line to g at the point (ƒ(x0), x0) located symmetrically across the 45° line y = x (which is the graph of the identity function). Use Theorem 3 to find the slope of this tangent line.

101. y = 23x - 2,

2

109. y1>3 - 1 = (x + 2)3, - 5 … x … 5, x0 = - 3>2 110. cos y = x1>5, 0 … x … 1, x0 = 1>2

2 … x … 4, x0 = 3 3

3x + 2 , - 2 … x … 2, x0 = 1>2 2x - 11

3.9 Inverse Trigonometric Functions We introduced the six basic inverse trigonometric functions in Section 1.6, but focused there on the arcsine and arccosine functions. Here we complete the study of how all six inverse trigonometric functions are defined, graphed, and evaluated, and how their derivatives are computed.

Inverses of tan x, cot x, sec x, and csc x The graphs of these four basic inverse trigonometric functions are shown again in Figure 3.40. We obtain these graphs by reflecting the graphs of the restricted trigonometric functions (as discussed in Section 1.6) through the line y = x. Let’s take a closer look at the arctangent, arccotangent, arcsecant, and arccosecant functions.

Domain: −∞ < x < ∞ Range: − p < y < p 2 2 y p 2 −2 −1 −

Domain: x ≤ −1 or x ≥ 1 Range: 0 ≤ y ≤ p, y ≠ p 2 y

Domain: −∞ < x < ∞ 02, p>2) for which tan y = x. cot−1 x is the number in (0, p) for which cot y = x. sec−1 x is the number in 3 0, p/2) ∪ (p/2, p4  for which sec y = x. csc−1 x is the number in 3 -p/2, 0) ∪ (0, p/24 for which csc y = x.

We use open or half-open intervals to avoid values for which the tangent, cotangent, secant, and cosecant functions are undefined. (See Figure 3.40.) The graph of y = tan-1 x is symmetric about the origin because it is a branch of the graph x = tan y that is symmetric about the origin (Figure 3.40a). Algebraically this means that tan-1 (-x) = -tan-1 x; the arctangent is an odd function. The graph of y = cot-1 x has no such symmetry (Figure 3.40b). Notice from Figure 3.40a that the graph of the arctangent function has two horizontal asymptotes: one at y = p>2 and the other at y = -p>2. The inverses of the restricted forms of sec x and csc x are chosen to be the functions graphed in Figures 3.40c and 3.40d. Caution  There is no general agreement about how to define sec-1 x for negative values of x. We chose angles in the second quadrant between p>2 and p. This choice makes sec-1 x = cos-1 (1>x). It also makes sec-1 x an increasing function on each interval of its domain. Some tables choose sec-1 x to lie in 3 -p, -p>2) for x 6 0 and some texts choose it to lie in 3 p, 3p>2) (Figure 3.41). These choices simplify the formula for the derivative (our formula needs absolute value signs) but fail to satisfy the computational equation sec-1 x = cos-1 (1>x). From this, we can derive the identity p 1 1 sec-1 x = cos-1 a x b = - sin-1 a x b (1) 2

by applying Equation (5) in Section 1.6.

EXAMPLE 1   The accompanying figures show two values of tan-1 x. y

tan–1 1 = tan–1 " 3 = p 3 6 "3 p 6 2 1 x 0 "3 tan p = 1 6 "3

y

tan–1 a−" 3b = − p 3 − 1

0 2

p 3 x

−" 3 tan a− pb = −" 3 3

x

tan-1 x

23 p>3 1 p>4 23>3 p>6 - 23>3 - p>6 - 1 - p>4 - 23 - p>3

The angles come from the first and fourth quadrants because the range of tan-1 x is (-p>2, p>2).

The Derivative of y = sin−1 u We know that the function x = sin y is differentiable in the interval -p>2 6 y 6 p>2 and that its derivative, the cosine, is positive there. Theorem 3 in Section 3.8 therefore assures us that the inverse function y = sin-1 x is differentiable throughout the interval

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192

Chapter 3: Derivatives y

y = sin–1x Domain: −1 ≤ x ≤ 1 Range: −p2 ≤ y ≤ p2

p 2

1

−1

-1 6 x 6 1. We cannot expect it to be differentiable at x = 1 or x = -1 because the tangents to the graph are vertical at these points (see Figure 3.42). We find the derivative of y = sin-1 x by applying Theorem 3 with ƒ(x) = sin x and ƒ -1(x) = sin-1 x:

( ƒ -1 ) ′(x) =

=

=

x

p − 2

-1

Figure 3.42  The graph of y = sin x has vertical tangents at x = - 1 and x = 1.

1 ƒ′ ( ƒ -1(x) ) 1

cos ( sin-1 x )

Theorem 3

ƒ′(u) = cos u

1 21 - sin2 ( sin-1 x ) 1 . = 21 - x2

cos u = 21 - sin2 u sin (sin-1 x) = x

If u is a differentiable function of x with 0 u 0 6 1, we apply the Chain Rule to get the general formula d du 1 (sin-1 u) = , dx 21 - u2 dx

0 u 0 6 1.

EXAMPLE 2   Using the Chain Rule, we calculate the derivative d 1 # d (x2) = 2x 4 . (sin-1 x2) = dx 21 - (x2)2 dx 21 - x

The Derivative of y = tan−1 u We find the derivative of y = tan-1 x by applying Theorem 3 with ƒ(x) = tan x and ƒ -1(x) = tan-1 x. Theorem 3 can be applied because the derivative of tan x is positive for -p>2 6 x 6 p>2: 1 ƒ′ ( ƒ -1 ( x ) )

Theorem 3

=

1 sec2 (tan-1 x)

ƒ′(u)

=

1    sec2 u 1 + tan2 (tan-1 x)

=

1 . 1 + x2

( ƒ -1 ) ′(x) =

= sec2 u = 1 + tan2 u

tan (tan-1 x)

= x

The derivative is defined for all real numbers. If u is a differentiable function of x, we get the Chain Rule form:

d ( tan-1 u ) = 1 2 du . dx 1 + u dx

The Derivative of y = sec−1 u Since the derivative of sec x is positive for 0 6 x 6 p>2 and p>2 6 x 6 p, Theorem 3 says that the inverse function y = sec-1 x is differentiable. Instead of applying the formula

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3.9  Inverse Trigonometric Functions

193

in Theorem 3 directly, we find the derivative of y = sec-1 x, 0 x 0 7 1, using implicit differentiation and the Chain Rule as follows: y = sec-1 x sec y = x

d d (sec y) = x dx dx sec y tan y

dy = 1 dx

Inverse function relationship

Differentiate both sides.

Chain Rule

dy 1 = . dx sec y tan y

Since 0 x 0 7 1, y lies in (0, p>2) ∪ (p>2, p) and sec y tan y ≠ 0.

To express the result in terms of x, we use the relationships sec y = x

tan y = { 2sec2 y - 1 = { 2x2 - 1

and

to get

y

dy 1 = { . dx x 2x2 - 1

Can we do anything about the { sign? A glance at Figure 3.43 shows that the slope of the graph y = sec-1 x is always positive. Thus,

y = sec –1x

p

1 d x 2x2 - 1 sec-1 x = d dx 1 x 2x2 - 1

p 2 −1

0

1

+

x

Figure 3.43  The slope of the curve y = sec-1 x is positive for both x 6 - 1 and x 7 1.

if x 7 1 if x 6 -1.

With the absolute value symbol, we can write a single expression that eliminates the “ {” ambiguity: d 1 sec-1 x = . dx 0 x 0 2x2 - 1

If u is a differentiable function of x with 0 u 0 7 1, we have the formula d du 1 ( sec-1 u ) = , dx 0 u 0 2u2 - 1 dx

0 u 0 7 1.

EXAMPLE 3   Using the Chain Rule and derivative of the arcsecant function, we find

d d 1 ( 5x4 ) sec-1 ( 5x4 ) = dx 0 5x4 0 2 ( 5x4 ) 2 - 1 dx = =

1 ( 20x3 ) 5x4 225x8 - 1 4 . x 225x8 - 1

5x4 7 1 7 0



Derivatives of the Other Three Inverse Trigonometric Functions We could use the same techniques to find the derivatives of the other three inverse trigonometric functions—arccosine, arccotangent, and arccosecant—but there is an easier way, thanks to the following identities.

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Chapter 3: Derivatives

Inverse Function–Inverse Cofunction Identities cos-1 x = p>2 - sin-1 x cot-1 x = p>2 - tan-1 x csc-1 x = p>2 - sec-1 x

We saw the first of these identities in Equation (5) of Section 1.6. The others are derived in a similar way. It follows easily that the derivatives of the inverse cofunctions are the negatives of the derivatives of the corresponding inverse functions. For example, the derivative of cos-1 x is calculated as follows: d ( cos-1 x ) = d a p - sin-1 xb   Identity dx dx 2

= -

d ( sin-1 x ) dx

= -

1 21 - x2

.

Derivative of arcsine

The derivatives of the inverse trigonometric functions are summarized in Table 3.1.

Table 3.1  Derivatives of the inverse trigonometric functions

1. 2. 3. 4. 5. 6.

d ( sin-1 u ) du 1 = , 2 dx dx 21 - u

d ( cos-1 u ) du 1 = , 2 dx dx 21 - u

0u0 6 1

d ( tan-1 u ) 1 du = dx 1 + u2 dx

0u0 6 1

d ( cot-1 u ) 1 du = dx 1 + u2 dx d ( sec-1 u ) du 1 = , 2 dx 0 u 0 2u - 1 dx

d ( csc-1 u ) du 1 = , dx 0 u 0 2u2 - 1 dx

0u0 7 1

0u0 7 1

Exercises  3.9 Common Values Use reference triangles in an appropriate quadrant, as in Example 1, to find the angles in Exercises 1–8. 1 1. a. tan-1 1 b. tan-11 - 23 2 c. tan-1 a b 23

-1 2. a. tan-1(- 1) b. tan-1 23 c. tan-1 a b 23

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3. a. sin-1 a

- 23 -1 1 b b. sin-1 a b c. sin-1 a b 2 2 22

23 1 -1 4. a. sin-1 a b b. sin-1 a b c. sin-1 a b 2 2 22

23 1 -1 5. a. cos-1 a b b. cos-1 a b c. cos-1 a b 2 2 22

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6. a. csc-1 22 b. csc-1 a 7. a. sec-11 - 22 2 b. sec-1 a

-2 23

2

23

b c. csc-1 2

b c. sec-1(-2)

12′

-1 8. a. cot (-1) b. cot 1 23 2 c. cot a b 23 -1

-1

-1

You

Evaluations Find the values in Exercises 9–12.

9. sin acos-1 a

22

2

bb

1 11. tan asin-1 a- b b 2

1 10. sec acos-1 b 2

12. cot asin-1 a-

a

3′

Wall x

44. Two derivations of the identity sec−1 ( −x) = P − sec−1 x 23

2

a. (Geometric) Here is a pictorial proof that sec-1 (- x) = p - sec-1 x. See if you can tell what is going on.

bb

y

Limits Find the limits in Exercises 13–20. (If in doubt, look at the function’s graph.) -1

14. lim + cos x

15. lim tan-1 x

16. lim tan-1 x

17. lim sec-1 x

18. lim sec-1 x

19. lim csc-1 x

20. lim csc-1 x

x S -1

p 2 −x

−1

0

1

x

x

x S -q

x Sq

x S -q

x Sq

Finding Derivatives In Exercises 21–42, find the derivative of y with respect to the appropriate variable.

21. y = cos-1 (x2)

22. y = cos-1 (1>x)

23. y = sin-1 22 t

24. y = sin-1 (1 - t)

25. y = sec-1 (2s + 1)

26. y = sec-1 5s

33. y = ln (tan-1 x)

34. y = tan-1 (ln x)

35. y = csc-1 (et)

36. y = cos-1 (e-t)

37. y = s 21 - s2 + cos-1 s

38. y = 2s2 - 1 - sec-1 s

39. y = tan-1 2x2 - 1 + csc-1 x, x 7 1 -1

40. y = cot x - tan x

-1

41. y = x sin x + 21 - x

x 42. y = ln (x2 + 4) - x tan-1 a b 2

if you are x ft from the front wall.

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cos-1 (- x) = p - cos-1 x  Eq. (4), Section 1.6

sec-1 x = cos-1 (1>x)   Eq. (1)

cos-1 2 45. a. tan-1 2 b. 46. a. csc-1 (1>2) b. csc-1 2 47. a. sec-1 0 b. sin-1 22 48. a. cot-1 (- 1>2) b. cos-1 (- 5) 49. Use the identity csc-1 u =

p - sec-1 u 2

to derive the formula for the derivative of csc-1 u in Table 3.1 from the formula for the derivative of sec-1 u. 50. Derive the formula dy 1 = dx 1 + x2

2

Theory and Examples 43. You are sitting in a classroom next to the wall looking at the blackboard at the front of the room. The blackboard is 12 ft long and starts 3 ft from the wall you are sitting next to. Show that your viewing angle is

x x a = cot-1 - cot-1 3 15

b. (Algebraic) Derive the identity sec-1 (- x) = p - sec-1 x by combining the following two equations from the text:

Which of the expressions in Exercises 45–48 are defined, and which are not? Give reasons for your answers.

27. y = csc-1 (x2 + 1), x 7 0 x 28. y = csc-1 2 3 1 29. y = sec-1 t , 0 6 t 6 1 30. y = sin-1 2 t 31. y = cot-1 2t 32. y = cot-1 2t - 1

-1 1

p

x S -q

x Sq

y = sec –1x

-1

13. lim- sin x xS1

195

Blackboard

3.9  Inverse Trigonometric Functions

for the derivative of y = tan-1 x by differentiating both sides of the equivalent equation tan y = x. 51. Use the Derivative Rule in Section 3.8, Theorem 3, to derive d 1 sec-1 x = , dx 0 x 0 2x2 - 1

52. Use the identity

cot-1 u =

0 x 0 7 1.

p - tan-1 u 2

to derive the formula for the derivative of cot-1 u in Table 3.1 from the formula for the derivative of tan-1 u.

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Chapter 3: Derivatives

53. What is special about the functions x - 1 ƒ(x) = sin-1 , x Ú 0, and g(x) = 2 tan-1 1x? x + 1

T Use your graphing utility for Exercises 58–62. 58. Graph y = sec (sec-1 x) = sec (cos-1(1>x)). Explain what you see.

Explain.

59. Newton’s serpentine  Graph Newton’s serpentine, y = 4x>(x2 + 1). Then graph y = 2 sin (2 tan-1 x) in the same graphing window. What do you see? Explain.

54. What is special about the functions 1 1 ƒ(x) = sin-1 and g(x) = tan-1 x ? 2x2 + 1 Explain. T In Exercises 55–57, find the domain and range of each composite function. Then graph the composites on separate screens. Do the graphs make sense in each case? Give reasons for your answers. Comment on any differences you see. 55. a. y = tan-1 (tan x) b. y = tan (tan-1 x) y = sin (sin-1 x) 56. a. y = sin-1 (sin x) b.

60. Graph the rational function y = (2 - x2)>x2. Then graph y = cos (2 sec-1 x) in the same graphing window. What do you see? Explain. 61. Graph ƒ(x) = sin-1 x together with its first two derivatives. Comment on the behavior of ƒ and the shape of its graph in relation to the signs and values of ƒ′ and ƒ″. 62. Graph ƒ(x) = tan-1 x together with its first two derivatives. Comment on the behavior of ƒ and the shape of its graph in relation to the signs and values of ƒ′ and ƒ″.

y = cos (cos-1 x) 57. a. y = cos-1 (cos x) b.

3.10 Related Rates In this section we look at problems that ask for the rate at which some variable changes when it is known how the rate of some other related variable (or perhaps several variables) changes. The problem of finding a rate of change from other known rates of change is called a related rates problem.

Related Rates Equations Suppose we are pumping air into a spherical balloon. Both the volume and radius of the balloon are increasing over time. If V is the volume and r is the radius of the balloon at an instant of time, then V =

4 3 pr . 3

Using the Chain Rule, we differentiate both sides with respect to t to find an equation relating the rates of change of V and r, dV dV dr dr = = 4pr 2 . dt dr dt dt So if we know the radius r of the balloon and the rate dV>dt at which the volume is increasing at a given instant of time, then we can solve this last equation for dr>dt to find how fast the radius is increasing at that instant. Note that it is easier to directly measure the rate of increase of the volume (the rate at which air is being pumped into the balloon) than it is to measure the increase in the radius. The related rates equation allows us to calculate dr>dt from dV>dt. Very often the key to relating the variables in a related rates problem is drawing a picture that shows the geometric relations between them, as illustrated in the following example.

dV = 9 ft3min dt 5 ft

dy =? dt when y = 6 ft

x 10 ft y

EXAMPLE 1   Water runs into a conical tank at the rate of 9 ft3 >min. The tank stands

point down and has a height of 10 ft and a base radius of 5 ft. How fast is the water level rising when the water is 6 ft deep? Solution  Figure 3.44 shows a partially filled conical tank. The variables in the problem are

Figure 3.44  The geometry of the conical tank and the rate at which water fills the tank determine how fast the water level rises (Example 1).

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V = volume (ft3) of the water in the tank at time t (min) x = radius (ft) of the surface of the water at time t y = depth (ft) of the water in the tank at time t.

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3.10  Related Rates

197

We assume that V, x, and y are differentiable functions of t. The constants are the dimensions of the tank. We are asked for dy>dt when y = 6 ft

and

The water forms a cone with volume V =

dV = 9 ft3 >min. dt 1 2 px y. 3

This equation involves x as well as V and y. Because no information is given about x and dx>dt at the time in question, we need to eliminate x. The similar triangles in Figure 3.44 give us a way to express x in terms of y: y x 5 or x = . y = 10 2 Therefore, we find V = to give the derivative

y 2 p 3 1 pa b y = y 3 2 12

dV p # 2 dy p 2 dy = 3y = y . 12 4 dt dt dt Finally, use y = 6 and dV>dt = 9 to solve for dy>dt. dy p 9 = (6)2 4 dt dy 1 = p ≈ 0.32 dt At the moment in question, the water level is rising at about 0.32 ft>min.

Related Rates Problem Strategy 1.  Draw a picture and name the variables and constants. Use t for time. Assume that all variables are differentiable functions of t. 2.  Write down the numerical information (in terms of the symbols you have chosen). 3. Write down what you are asked to find (usually a rate, expressed as a derivative). 4.  Write an equation that relates the variables. You may have to combine two or more equations to get a single equation that relates the variable whose rate you want to the variables whose rates you know. 5.  Differentiate with respect to t. Then express the rate you want in terms of the rates and variables whose values you know. 6. Evaluate. Use known values to find the unknown rate.

Balloon

du = 0.14 radmin dt when u = p4

Range finder

dy =? y dt when u = p4

u 150 m

Figure 3.45  The rate of change of the balloon’s height is related to the rate of change of the angle the range finder makes with the ground (Example 2).

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EXAMPLE 2   A hot air balloon rising straight up from a level field is tracked by a range finder 150 m from the liftoff point. At the moment the range finder’s elevation angle is p>4, the angle is increasing at the rate of 0.14 rad > min. How fast is the balloon rising at that moment? Solution  We answer the question in the six strategy steps.

1. Draw a picture and name the variables and constants (Figure 3.45). The variables in the picture are u = the angle in radians the range finder makes with the ground. y = the height in meters of the balloon above the ground.

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Chapter 3: Derivatives

We let t represent time in minutes and assume that u and y are differentiable functions of t. The one constant in the picture is the distance from the range finder to the liftoff point (150 m). There is no need to give it a special symbol. 2. Write down the additional numerical information. du = 0.14 rad>min dt

when

u =

p 4

3. Write down what we are to find. We want dy>dt when u = p>4. 4. Write an equation that relates the variables y and u. y = tan u or y = 150 tan u 150 5. Differentiate with respect to t using the Chain Rule. The result tells how dy>dt (which we want) is related to du>dt (which we know). dy du = 150 (sec2 u) dt dt 6. Evaluate with u = p>4 and du>dt = 0.14 to find dy>dt. dy = 1501 22 22(0.14) = 42   sec p4 dt

= 22

At the moment in question, the balloon is rising at the rate of 42 m>min.

EXAMPLE 3   A police cruiser, approaching a right-angled intersection from the north, is chasing a speeding car that has turned the corner and is now moving straight east. When the cruiser is 0.6 mi north of the intersection and the car is 0.8 mi to the east, the police determine with radar that the distance between them and the car is increasing at 20 mph. If the cruiser is moving at 60 mph at the instant of measurement, what is the speed of the car?

y Situation when x = 0.8, y = 0.6 y dy = −60 dt 0

ds = 20 dt

dx = ? dt

x

Figure 3.46  The speed of the car is related to the speed of the police cruiser and the rate of change of the distance s between them (Example 3).

x

Solution  We picture the car and cruiser in the coordinate plane, using the positive x-axis as the eastbound highway and the positive y-axis as the southbound highway (Figure 3.46). We let t represent time and set x = position of car at time t y = position of cruiser at time t s = distance between car and cruiser at time t. We assume that x, y, and s are differentiable functions of t. We want to find dx>dt when x = 0.8 mi,

dy = -60 mph, dt

y = 0.6 mi,

ds = 20 mph. dt

Note that dy>dt is negative because y is decreasing. We differentiate the distance equation between the car and the cruiser, s2 = x2 + y2 (we could also use s = 2x2 + y2 ), and obtain 2s

dy ds dx = 2x + 2y dt dt dt dy ds 1 dx = s ax + y b dt dt dt =

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dy dx 1 + y b. ax 2 dt dt 2x + y 2

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3.10  Related Rates

199

Finally, we use x = 0.8, y = 0.6, dy>dt = -60, ds>dt = 20, and solve for dx>dt. 20 =

1 2(0.8)2 + (0.6)2

a0.8

dx + (0.6)(-60)b dt

2 2 dx 20 2(0.8) + (0.6) + (0.6)(60) = = 70 0.8 dt

At the moment in question, the car’s speed is 70 mph. y P 10 Q

u 0

(x, 0)

Figure 3.47  The particle P travels clockwise along the circle (Example 4).

x

EXAMPLE 4   A particle P moves clockwise at a constant rate along a circle of radius 10 m centered at the origin. The particle’s initial position is (0, 10) on the y-axis, and its final destination is the point (10, 0) on the x-axis. Once the particle is in motion, the tangent line at P intersects the x-axis at a point Q (which moves over time). If it takes the particle 30 sec to travel from start to finish, how fast is the point Q moving along the x-axis when it is 20 m from the center of the circle? Solution  We picture the situation in the coordinate plane with the circle centered at the origin (see Figure 3.47). We let t represent time and let u denote the angle from the x-axis to the radial line joining the origin to P. Since the particle travels from start to finish in 30 sec, it is traveling along the circle at a constant rate of p>2 radians in 1>2 min, or p rad>min. In other words, du>dt = -p, with t being measured in minutes. The negative sign appears because u is decreasing over time. Setting x(t) to be the distance at time t from the point Q to the origin, we want to find dx>dt when x = 20 m

and

To relate the variables x and u, we see from Figure 3.47 that x cos u = 10, or x = 10 sec u. Differentiation of this last equation gives du dx = 10 sec u tan u = -10p sec u tan u. dt dt Note that dx>dt is negative because x is decreasing (Q is moving toward the origin). When x = 20, cos u = 1>2 and sec u = 2. Also, tan u = 2sec2 u - 1 = 23. It follows that dx = (-10p)(2)1 23 2 = -20 23p. dt

At the moment in question, the point Q is moving toward the origin at the speed of 20 23p ≈ 109 m>min.

A

12,000 u R

x

Figure 3.48  Jet airliner A traveling at constant altitude toward radar station R (Example 5).

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EXAMPLE 5   A jet airliner is flying at a constant altitude of 12,000 ft above sea level as it approaches a Pacific island. The aircraft comes within the direct line of sight of a radar station located on the island, and the radar indicates the initial angle between sea level and its line of sight to the aircraft is 30°. How fast (in miles per hour) is the aircraft approaching the island when first detected by the radar instrument if it is turning upward (counterclockwise) at the rate of 2>3 deg>sec in order to keep the aircraft within its direct line of sight? Solution  The aircraft A and radar station R are pictured in the coordinate plane, using the positive x-axis as the horizontal distance at sea level from R to A, and the positive y-axis as the vertical altitude above sea level. We let t represent time and observe that y = 12,000 is a constant. The general situation and line-of-sight angle u are depicted in Figure 3.48. We want to find dx>dt when u = p>6 rad and du>dt = 2>3 deg>sec.

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200

Chapter 3: Derivatives

From Figure 3.48, we see that 12,000 = tan u x

or

x = 12,000 cot u.

Using miles instead of feet for our distance units, the last equation translates to x =

12,000 cot u. 5280

Differentiation with respect to t gives du dx 1200 = csc2 u . dt dt 528 When u = p>6, sin2 u = 1>4, so csc2 u = 4. Converting du>dt = 2>3 deg>sec to radians per hour, we find du 2 p = a b (3600) rad>hr.   1 hr 3 180 dt

= 3600 sec, 1 deg = p>180 rad

Substitution into the equation for dx>dt then gives dx 1200 p 2 = ab (4)a b a b (3600) ≈ -380. 3 180 dt 528

The negative sign appears because the distance x is decreasing, so the aircraft is approaching the island at a speed of approximately 380 mi>hr when first detected by the radar.

EXAMPLE 6  Figure 3.49a shows a rope running through a pulley at P and bearing a

P

W

dx = 4 ftsec dt M

O

5 ft

x (a)

P

20 ft

weight W at one end. The other end is held 5 ft above the ground in the hand M of a worker. Suppose the pulley is 25 ft above ground, the rope is 45 ft long, and the worker is walking rapidly away from the vertical line PW at the rate of 4 ft>sec. How fast is the weight being raised when the worker’s hand is 21 ft away from PW? Solution  We let OM be the horizontal line of length x ft from a point O directly below the pulley to the worker’s hand M at any instant of time (Figure 3.49). Let h be the height of the weight W above O, and let z denote the length of rope from the pulley P to the worker’s hand. We want to know dh>dt when x = 21 given that dx>dt = 4. Note that the height of P above O is 20 ft because O is 5 ft above the ground. We assume the angle at O is a right angle. At any instant of time t we have the following relationships (see Figure 3.49b): 20 - h + z = 45  Total length of rope is 45 ft. 202 + x2 = z2.  Angle at O is a right angle.

z W h

If we solve for z = 25 + h in the first equation, and substitute into the second equation, we have

dh = ? dt

O

x

M

(b)

Figure 3.49  A worker at M walks to the right, pulling the weight W upward as the rope moves through the pulley P (Example 6).

202 + x2 = (25 + h)2.(1)

Differentiating both sides with respect to t gives 2x

dx dh = 2(25 + h) , dt dt

and solving this last equation for dh>dt we find

dh x dx = .(2) dt 25 + h dt

Since we know dx>dt, it remains only to find 25 + h at the instant when x = 21. From Equation (1), 202 + 212 = (25 + h)2

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201

so that (25 + h)2 = 841,

or

25 + h = 29.

Equation (2) now gives dh 21 # 84 = 4 = ≈ 2.9 ft>sec 29 29 dt as the rate at which the weight is being raised when x = 21 ft.

Exercises  3.10 1. Area  Suppose that the radius r and area A = pr 2 of a circle are differentiable functions of t. Write an equation that relates dA>dt to dr>dt. 2. Surface area  Suppose that the radius r and surface area S = 4pr 2 of a sphere are differentiable functions of t. Write an equation that relates dS>dt to dr>dt.

increasing at the rate of 1 volt>sec while I is decreasing at the rate of 1>3 amp>sec. Let t denote time in seconds. + V− I

3. Assume that y = 5x and dx>dt = 2. Find dy>dt. 4. Assume that 2x + 3y = 12 and dy>dt = -2. Find dx>dt.

R

2

5. If y = x and dx>dt = 3, then what is dy>dt when x = -1? 6. If x = y3 - y and dy>dt = 5, then what is dx>dt when y = 2? 7. If x2 + y2 = 25 and dx>dt = -2, then what is dy>dt when x = 3 and y = - 4? 2 3

8. If x y = 4>27 and dy>dt = 1>2, then what is dx>dt when x = 2? 9. If L = 2x2 + y2, dx>dt = - 1, and dy>dt = 3, find dL>dt when x = 5 and y = 12. 10. If r + s2 + y3 = 12, dr>dt = 4, and ds>dt = -3, find dy>dt when r = 3 and s = 1. 11. If the original 24 m edge length x of a cube decreases at the rate of 5 m>min, when x = 3 m at what rate does the cube’s a. surface area change? b. volume change? 12. A cube’s surface area increases at the rate of 72 in2 >sec. At what rate is the cube’s volume changing when the edge length is x = 3 in?

13. Volume  The radius r and height h of a right circular cylinder are related to the cylinder’s volume V by the formula V = pr 2h. a. How is dV>dt related to dh>dt if r is constant? b. How is dV>dt related to dr>dt if h is constant? c. How is dV>dt related to dr>dt and dh>dt if neither r nor h is constant? 14. Volume  The radius r and height h of a right circular cone are related to the cone’s volume V by the equation V = (1>3)pr 2h. a. How is dV>dt related to dh>dt if r is constant? b. How is dV>dt related to dr>dt if h is constant? c. How is dV>dt related to dr>dt and dh>dt if neither r nor h is constant? 15. Changing voltage  The voltage V (volts), current I (amperes), and resistance R (ohms) of an electric circuit like the one shown here are related by the equation V = IR. Suppose that V is

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a. What is the value of dV>dt? b. What is the value of dI>dt? c. What equation relates dR>dt to dV>dt and dI>dt? d. Find the rate at which R is changing when V = 12 volts and I = 2 amps. Is R increasing, or decreasing? 16. Electrical power  The power P (watts) of an electric circuit is related to the circuit’s resistance R (ohms) and current I (amperes) by the equation P = RI 2. a. How are dP>dt, dR>dt, and dI>dt related if none of P, R, and I are constant? b. How is dR>dt related to dI>dt if P is constant? 17. Distance Let x and y be differentiable functions of t and let s = 2x2 + y2 be the distance between the points (x, 0) and (0, y) in the xy-plane. a. How is ds>dt related to dx>dt if y is constant? b. How is ds>dt related to dx>dt and dy>dt if neither x nor y is constant? c. How is dx>dt related to dy>dt if s is constant? 18. Diagonals If x, y, and z are lengths of the edges of a rectangular box, the common length of the box’s diagonals is s = 2x2 + y2 + z2. a. Assuming that x, y, and z are differentiable functions of t, how is ds>dt related to dx>dt, dy>dt, and dz>dt?

b. How is ds>dt related to dy>dt and dz>dt if x is constant? c. How are dx>dt, dy>dt, and dz>dt related if s is constant? 19. Area  The area A of a triangle with sides of lengths a and b enclosing an angle of measure u is A =

1 ab sin u. 2

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a. How is dA>dt related to du>dt if a and b are constant? b. How is dA>dt related to du>dt and da>dt if only b is constant? c. How is dA>dt related to du>dt, da>dt, and db>dt if none of a, b, and u are constant? 20. Heating a plate  When a circular plate of metal is heated in an oven, its radius increases at the rate of 0.01 cm > min. At what rate is the plate’s area increasing when the radius is 50 cm?

21. Changing dimensions in a rectangle  The length l of a rectangle is decreasing at the rate of 2 cm>sec while the width w is increasing at the rate of 2 cm>sec. When l = 12 cm and w = 5 cm, find the rates of change of (a) the area, (b) the perimeter, and (c) the lengths of the diagonals of the rectangle. Which of these quantities are decreasing, and which are increasing? 22. Changing dimensions in a rectangular box  Suppose that the edge lengths x, y, and z of a closed rectangular box are changing at the following rates: dy dz dx = 1 m>sec, = -2 m>sec, = 1 m>sec. dt dt dt Find the rates at which the box’s (a) volume, (b) surface area, and (c) diagonal length s = 2x2 + y2 + z2 are changing at the instant when x = 4, y = 3, and z = 2. 23. A sliding ladder  A 13-ft ladder is leaning against a house when its base starts to slide away (see accompanying figure). By the time the base is 12 ft from the house, the base is moving at the rate of 5 ft>sec. a. How fast is the top of the ladder sliding down the wall then? b. At what rate is the area of the triangle formed by the ladder, wall, and ground changing then? c. At what rate is the angle u between the ladder and the ground changing then? y

u x(t)

x

24. Commercial air traffic  Two commercial airplanes are flying at an altitude of 40,000 ft along straight-line courses that intersect at right angles. Plane A is approaching the intersection point at a speed of 442 knots (nautical miles per hour; a nautical mile is 2000 yd). Plane B is approaching the intersection at 481 knots. At what rate is the distance between the planes changing when A is 5 nautical miles from the intersection point and B is 12 nautical miles from the intersection point? 25. Flying a kite  A girl flies a kite at a height of 300 ft, the wind carrying the kite horizontally away from her at a rate of 25 ft>sec. How fast must she let out the string when the kite is 500 ft away from her? 26. Boring a cylinder  The mechanics at Lincoln Automotive are reboring a 6-in.-deep cylinder to fit a new piston. The machine they are using increases the cylinder’s radius one-thousandth of an inch every 3 min. How rapidly is the cylinder volume increasing when the bore (diameter) is 3.800 in.?

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28. A draining conical reservoir  Water is flowing at the rate of 50 m3 >min from a shallow concrete conical reservoir (vertex down) of base radius 45 m and height 6 m. a. How fast (centimeters per minute) is the water level falling when the water is 5 m deep?

b. How fast is the radius of the water’s surface changing then? Answer in centimeters per minute. 29. A draining hemispherical reservoir  Water is flowing at the rate of 6 m3 >min from a reservoir shaped like a hemispherical bowl of radius 13 m, shown here in profile. Answer the following questions, given that the volume of water in a hemispherical bowl of radius R is V = (p>3)y2(3R - y) when the water is y meters deep. Center of sphere 13 Water level r y

a. At what rate is the water level changing when the water is 8 m deep? b. What is the radius r of the water’s surface when the water is y m deep? c. At what rate is the radius r changing when the water is 8 m deep? 30. A growing raindrop  Suppose that a drop of mist is a perfect sphere and that, through condensation, the drop picks up moisture at a rate proportional to its surface area. Show that under these circumstances the drop’s radius increases at a constant rate.

y(t)

0

27. A growing sand pile  Sand falls from a conveyor belt at the rate of 10 m3 >min onto the top of a conical pile. The height of the pile is always three-eighths of the base diameter. How fast are the (a) height and (b) radius changing when the pile is 4 m high? Answer in centimeters per minute.

31. The radius of an inflating balloon  A spherical balloon is inflated with helium at the rate of 100p ft3 >min. How fast is the balloon’s radius increasing at the instant the radius is 5 ft? How fast is the surface area increasing? 32. Hauling in a dinghy  A dinghy is pulled toward a dock by a rope from the bow through a ring on the dock 6 ft above the bow. The rope is hauled in at the rate of 2 ft>sec. a. How fast is the boat approaching the dock when 10 ft of rope are out? b. At what rate is the angle u changing at this instant (see the figure)? Ring at edge of dock

u 6'

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33. A balloon and a bicycle  A balloon is rising vertically above a level, straight road at a constant rate of 1 ft>sec. Just when the balloon is 65 ft above the ground, a bicycle moving at a constant rate of 17 ft>sec passes under it. How fast is the distance s(t) between the bicycle and balloon increasing 3 sec later?

where Q is the number of milliliters of CO2 you exhale in a minute and D is the difference between the CO2 concentration (ml>L) in the blood pumped to the lungs and the CO2 concentration in the blood returning from the lungs. With Q = 233 ml>min and D = 97 - 56 = 41 ml>L,

y

y =

233 ml>min ≈ 5.68 L>min, 41 ml>L

fairly close to the 6 L>min that most people have at basal (resting) conditions. (Data courtesy of J. Kenneth Herd, M.D., Quillan College of Medicine, East Tennessee State University.)   Suppose that when Q = 233 and D = 41, we also know that D is decreasing at the rate of 2 units a minute but that Q remains unchanged. What is happening to the cardiac output?

y(t)

36. Moving along a parabola  A particle moves along the parabola y = x2 in the first quadrant in such a way that its x-coordinate (measured in meters) increases at a steady 10 m>sec. How fast is the angle of inclination u of the line joining the particle to the origin changing when x = 3 m?

s(t)

0

x(t)

x

34. Making coffee  Coffee is draining from a conical filter into a cylindrical coffeepot at the rate of 10 in3 >min. a. How fast is the level in the pot rising when the coffee in the cone is 5 in. deep? b. How fast is the level in the cone falling then?

37. Motion in the plane  The coordinates of a particle in the metric xy-plane are differentiable functions of time t with dx>dt = -1 m>sec and dy>dt = -5 m>sec. How fast is the particle’s distance from the origin changing as it passes through the point (5, 12)? 38. Videotaping a moving car  You are videotaping a race from a stand 132 ft from the track, following a car that is moving at 180 mi>h (264 ft>sec), as shown in the accompanying figure. How fast will your camera angle u be changing when the car is right in front of you? A half second later? Camera

6″ u 6″ How fast is this level falling?

132′

Car How fast is this level rising? 6″

35. Cardiac output  In the late 1860s, Adolf Fick, a professor of physiology in the Faculty of Medicine in Würzberg, Germany, developed one of the methods we use today for measuring how much blood your heart pumps in a minute. Your cardiac output as you read this sentence is probably about 7 L>min. At rest it is likely to be a bit under 6 L>min. If you are a trained marathon runner running a marathon, your cardiac output can be as high as 30 L>min.    Your cardiac output can be calculated with the formula y =

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Q , D

39. A moving shadow  A light shines from the top of a pole 50 ft high. A ball is dropped from the same height from a point 30 ft away from the light. (See accompanying figure.) How fast is the shadow of the ball moving along the ground 1>2 sec later? (Assume the ball falls a distance s = 16t 2 ft in t sec.) Light

Ball at time t = 0 1/2 sec later

50-ft pole

0

30

x(t)

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40. A building’s shadow  On a morning of a day when the sun will pass directly overhead, the shadow of an 80-ft building on level ground is 60 ft long. At the moment in question, the angle u the sun makes with the ground is increasing at the rate of 0.27°>min. At what rate is the shadow decreasing? (Remember to use radians. Express your answer in inches per minute, to the nearest tenth.)

b. At what rates are angles u1 and u2 (see the figure) changing at that time? c. The player slides into second base at the rate of 15 ft>sec. At what rates are angles u1 and u2 changing as the player touches base? Second base 90′ Third base

u1

u2

Player 30′

First base

80′ Home u

41. A melting ice layer  A spherical iron ball 8 in. in diameter is coated with a layer of ice of uniform thickness. If the ice melts at the rate of 10 in3 >min, how fast is the thickness of the ice decreasing when it is 2 in. thick? How fast is the outer surface area of ice decreasing? 42. Highway patrol  A highway patrol plane flies 3 mi above a level, straight road at a steady 120 mi>h. The pilot sees an oncoming car and with radar determines that at the instant the line-of-sight distance from plane to car is 5 mi, the line-of-sight distance is decreasing at the rate of 160 mi>h. Find the car’s speed along the highway. 43. Baseball players  A baseball diamond is a square 90 ft on a side. A player runs from first base to second at a rate of 16 ft>sec.

44. Ships  Two ships are steaming straight away from a point O along routes that make a 120° angle. Ship A moves at 14 knots (nautical miles per hour; a nautical mile is 2000 yd). Ship B moves at 21 knots. How fast are the ships moving apart when OA = 5 and OB = 3 nautical miles? 45. Clock’s moving hands  At what rate is the angle between a clock’s minute and hour hands changing at 4 o’clock in the afternoon? 46. Oil spill  An explosion at an oil rig located in gulf waters causes an elliptical oil slick to spread on the surface from the rig. The slick is a constant 9 in. thick. After several days, when the major axis of the slick is 2 mi long and the minor axis is 3/4 mi wide, it is determined that its length is increasing at the rate of 30 ft/hr, and its width is increasing at the rate of 10 ft/hr. At what rate (in cubic feet per hour) is oil flowing from the site of the rig at that time?

a. At what rate is the player’s distance from third base changing when the player is 30 ft from first base?

3.11  Linearization and Differentials Sometimes we can approximate complicated functions with simpler ones that give the accuracy we want for specific applications and are easier to work with. The approximating functions discussed in this section are called linearizations, and they are based on tangent lines. Other approximating functions, such as polynomials, are discussed in Chapter 9. We introduce new variables dx and dy, called differentials, and define them in a way that makes Leibniz’s notation for the derivative dy>dx a true ratio. We use dy to estimate error in measurement, which then provides for a precise proof of the Chain Rule (Section 3.6).

Linearization As you can see in Figure 3.50, the tangent to the curve y = x2 lies close to the curve near the point of tangency. For a brief interval to either side, the y-values along the tangent line give good approximations to the y-values on the curve. We observe this phenomenon by zooming in on the two graphs at the point of tangency or by looking at tables of values for the difference between ƒ(x) and its tangent line near the x-coordinate of the point of tangency. The phenomenon is true not just for parabolas; every differentiable curve behaves locally like its tangent line.

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4

2 y = x2

y = x2 y = 2x − 1

y = 2x − 1

(1, 1)

(1, 1) −1

3

0

0

y = x 2 and its tangent y = 2x − 1 at (1, 1). 1.2

2

0

Tangent and curve very close near (1, 1). 1.003

y = x2

y = x2 y = 2x − 1

(1, 1)

(1, 1)

y = 2x − 1 0.8

1.2

0.8

Tangent and curve very close throughout entire x-interval shown.

0.997 0.997

1.003

Tangent and curve closer still. Computer screen cannot distinguish tangent from curve on this x-interval.

Figure 3.50  The more we magnify the graph of a function near a point where the function is differentiable, the flatter the graph becomes and the more it resembles its tangent. y

y = f (x) Slope = f ′(a) (a, f (a))

In general, the tangent to y = ƒ(x) at a point x = a, where ƒ is differentiable (Figure 3.51), passes through the point (a, ƒ(a)), so its point-slope equation is y = ƒ(a) + ƒ′(a)(x - a).

y = L(x)

Thus, this tangent line is the graph of the linear function L(x) = ƒ(a) + ƒ′(a)(x - a).

0

a

Figure 3.51  The tangent to the curve y = ƒ(x) at x = a is the line L(x) = ƒ(a) + ƒ′(a)(x - a).

x

For as long as this line remains close to the graph of ƒ as we move off the point of tangency, L (x) gives a good approximation to ƒ(x).

Definitions  If ƒ is differentiable at x = a, then the approximating function L(x) = ƒ(a) + ƒ′(a)(x - a) is the linearization of ƒ at a. The approximation ƒ(x) ≈ L(x) of ƒ by L is the standard linear approximation of ƒ at a. The point x = a is the center of the approximation.

EXAMPLE 1   Find the linearization of ƒ(x) = 21 + x at x = 0 (Figure 3.52).

Solution  Since

ƒ′(x) =

1 (1 + x)-1>2, 2

we have ƒ(0) = 1 and ƒ′(0) = 1>2, giving the linearization L(x) = ƒ(a) + ƒ′(a)(x - a) = 1 +

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x 1 (x - 0) = 1 + . 2 2

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Chapter 3: Derivatives

y x y= 1+ 2

2

5 x + 4 4

0

1

2

y=1+ x 2

1.1

y = "1 + x

1

−1

y=

y = "1 + x

1.0

3

x

4

Figure 3.52  The graph of y = 21 + x and its linearizations at x = 0 and x = 3. Figure 3.53 shows a magnified view of the small window about 1 on the y-axis.

0.9 −0.1

0

0.1

0.2

Figure 3.53  Magnified view of the window in Figure 3.52.

See Figure 3.53. The following table shows how accurate the approximation 21 + x ≈ 1 + (x>2) from Example 1 is for some values of x near 0. As we move away from zero, we lose accuracy. For example, for x = 2, the linearization gives 2 as the approximation for 23, which is not even accurate to one decimal place.

Approximation 21.2 ≈ 1 + 21.05 ≈ 1 +

21.005 ≈ 1 +

0.2 2

True value

True value − approximation

= 1.10 1.095445

0.004555 6 10-2

0.05 = 1.025 1.024695 2

0.000305 6 10-3

0.005 = 1.00250 1.002497 2

0.000003 6 10-5

Do not be misled by the preceding calculations into thinking that whatever we do with a linearization is better done with a calculator. In practice, we would never use a linearization to find a particular square root. The utility of a linearization is its ability to replace a complicated formula by a simpler one over an entire interval of values. If we have to work with 21 + x for x close to 0 and can tolerate the small amount of error involved, we can work with 1 + (x>2) instead. Of course, we then need to know how much error there is. We further examine the estimation of error in Chapter 9. A linear approximation normally loses accuracy away from its center. As Figure 3.52 suggests, the approximation 21 + x ≈ 1 + (x>2) will probably be too crude to be useful near x = 3. There, we need the linearization at x = 3.

EXAMPLE 2   Find the linearization of ƒ(x) = 21 + x at x = 3. Solution  We evaluate the equation defining L(x) at a = 3. With ƒ(3) = 2,

ƒ′(3) =

we have

L(x) = 2 +

1 1 (1 + x)-1>2 ` = , 2 4 x=3

5 x 1 (x - 3) = + . 4 4 4

At x = 3.2, the linearization in Example 2 gives 21 + x = 21 + 3.2 ≈

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5 3.2 + = 1.250 + 0.800 = 2.050, 4 4

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207

which differs from the true value 24.2 ≈ 2.04939 by less than one one-thousandth. The linearization in Example 1 gives 21 + x = 21 + 3.2 ≈ 1 +

3.2 = 1 + 1.6 = 2.6, 2

a result that is off by more than 25%. y

EXAMPLE 3   Find the linearization of ƒ(x) = cos x at x = p>2 (Figure 3.54). Solution  Since ƒ(p>2) = cos (p>2) = 0, ƒ′(x) = -sin x, and ƒ′(p>2) = -sin (p>2) = -1, we find the linearization at a = p>2 to be

0

x

p 2

y = cos x

y = −x + p 2

Figure 3.54  The graph of ƒ(x) = cos x and its linearization at x = p>2. Near x = p>2, cos x ≈ -x + (p>2) (Example 3).

Approximations Near x = 0 21 + x ≈ 1 +

x 2

1 ≈ 1 + x 1 - x

x2 1 ≈ 1 + 2 21 - x2

L(x) = ƒ(a) + ƒ′(a)(x - a)

= 0 + (-1)ax -

= -x +

p . 2

p b 2

An important linear approximation for roots and powers is (1 + x)k ≈ 1 + kx

(x near 0; any number k)

(Exercise 15). This approximation, good for values of x sufficiently close to zero, has broad application. For example, when x is small,

21 + x ≈ 1 +

1 x 2

1 = (1 - x)-1 ≈ 1 + (-1)(-x) = 1 + x 1 - x 5 1 3 2 1 + 5x4 = (1 + 5x4)1>3 ≈ 1 + (5x4) = 1 + x4 3 3

1

21 - x2

k = 1>2

k = - 1; replace x by - x.

k = 1>3; replace x by 5x4.

1 1 = (1 - x2)-1>2 ≈ 1 + a- b (-x2) = 1 + x2 2 2

k = - 1>2; replace x by - x2 .

Differentials We sometimes use the Leibniz notation dy>dx to represent the derivative of y with respect to x. Contrary to its appearance, it is not a ratio. We now introduce two new variables dx and dy with the property that when their ratio exists, it is equal to the derivative.

DEFINITION Let y = ƒ(x) be a differentiable function. The differential dx is an independent variable. The differential dy is dy = ƒ′(x) dx.

Unlike the independent variable dx, the variable dy is always a dependent variable. It depends on both x and dx. If dx is given a specific value and x is a particular number in the domain of the function ƒ, then these values determine the numerical value of dy. Often the variable dx is chosen to be ∆x, the change in x.

EXAMPLE 4

(a) Find dy if y = x5 + 37x. (b) Find the value of dy when x = 1 and dx = 0.2.

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Solution (a) dy = (5x4 + 37) dx (b) Substituting x = 1 and dx = 0.2 in the expression for dy, we have dy = (5 # 14 + 37) 0.2 = 8.4.

The geometric meaning of differentials is shown in Figure 3.55. Let x = a and set dx = ∆x. The corresponding change in y = ƒ(x) is ∆y = ƒ(a + dx) - ƒ(a). The corresponding change in the tangent line L is

∆L = L(a + dx) - L(a) = ƒ(a) + ƒ′(a)3 (a + dx) - a4 - ƒ(a) (++++1+)+++++1*   ()* L (a + dx) L (a) = ƒ′(a) dx. y

y = f (x) (a + dx, f (a + dx)) Δy = f (a + dx) − f (a) ΔL = f ′(a)dx (a, f (a)) dx = Δx When dx is a small change in x, the corresponding change in the linearization is precisely dy.

Tangent line 0

a

x

a + dx

Figure 3.55  Geometrically, the differential dy is the change ∆L in the linearization of ƒ when x = a changes by an amount dx = ∆x.

That is, the change in the linearization of ƒ is precisely the value of the differential dy when x = a and dx = ∆x. Therefore, dy represents the amount the tangent line rises or falls when x changes by an amount dx = ∆x. If dx ≠ 0, then the quotient of the differential dy by the differential dx is equal to the derivative ƒ′(x) because dy ƒ′(x) dx dy , dx = = ƒ′(x) = . dx dx We sometimes write dƒ = ƒ′(x) dx in place of dy = ƒ′(x) dx, calling dƒ the differential of ƒ. For instance, if ƒ(x) = 3x2 - 6, then dƒ = d(3x2 - 6) = 6x dx. Every differentiation formula like d (u + y) du dy = + dx dx dx

or

d (sin u) du = cos u dx dx

or

d(sin u) = cos u du.

has a corresponding differential form like d(u + y) = du + dy

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209

EXAMPLE 5   We can use the Chain Rule and other differentiation rules to find differentials of functions. (a) d (tan 2x) = sec2 (2x) d (2x) = 2 sec2 2x dx (b) d a

(x + 1) dx - x d (x + 1) x dx + dx - x dx dx x b = = =  x + 1 (x + 1)2 (x + 1)2 (x + 1)2

Estimating with Differentials

Suppose we know the value of a differentiable function ƒ(x) at a point a and want to estimate how much this value will change if we move to a nearby point a + dx. If dx = ∆x is small, then we can see from Figure 3.55 that ∆y is approximately equal to the differential dy. Since ƒ(a + dx) = ƒ(a) + ∆y,   ∆x

= dx

the differential approximation gives ƒ(a + dx) ≈ ƒ(a) + dy when dx = ∆x. Thus the approximation ∆y ≈ dy can be used to estimate ƒ(a + dx) when ƒ(a) is known, dx is small, and dy = ƒ′(a) dx.

dr = 0.1

EXAMPLE 6   The radius r of a circle increases from a = 10 m to 10.1 m (Figure 3.56). Use dA to estimate the increase in the circle’s area A. Estimate the area of the enlarged circle and compare your estimate to the true area found by direct calculation. Solution Since A = pr 2, the estimated increase is

a = 10

dA = A′(a) dr = 2pa dr = 2p(10)(0.1) = 2p m2. Thus, since A(r + ∆r) ≈ A(r) + dA, we have

ΔA ≈ dA = 2 pa dr

Figure 3.56  When dr is small compared with a, the differential dA gives the estimate A(a + dr) = pa2 + dA (Example 6).

A(10 + 0.1) ≈ A(10) + 2p = p(10)2 + 2p = 102p. The area of a circle of radius 10.1 m is approximately 102p m2. The true area is A(10.1) = p(10.1)2 = 102.01p m2. The error in our estimate is 0.01p m2, which is the difference ∆A - dA.

EXAMPLE 7   Use differentials to estimate (a) 7.971>3   (b)  sin (p>6 + 0.01). Solution (a) The differential associated with the cube root function y = x1>3 is 1 dx. 3x2>3 We set a = 8, the closest number near 7.97 where we can easily compute ƒ(a) and ƒ′(a). To arrange that a + dx = 7.97, we choose dx = -0.03. Approximating with the differential gives dy =

ƒ(7.97) = ƒ(a + dx) ≈ ƒ(a) + dy = 81>3 + = 2 +

1 (-0.03) 3(8)2>3

1 (-0.03) = 1.9975 12

This gives an approximation to the true value of 7.971>3, which is 1.997497 to 6 decimals.

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(b) The differential associated with y = sin x is dy = cos x dx. To estimate sin (p>6 + 0.01), we set a = p>6 and dx = 0.01. Then ƒ(p>6 + 0.01) = ƒ(a + dx) ≈ ƒ(a) + dy

sin (a + dx) ≈ sin a + (cos a) dx

= sin =

p p + acos b (0.01) 6 6

23 1 + (0.01) ≈ 0.5087 2 2

For comparison, the true value of sin (p>6 + 0.01) to 6 decimals is 0.508635. The method in part (b) of Example 7 is used by some calculator and computer algorithms to give values of trigonometric functions. The algorithms store a large table of sine and cosine values between 0 and p>4. Values between these stored values are computed using differentials as in Example 7b. Values outside of 3 0, p>44 are computed from values in this interval using trigonometric identities.

Error in Differential Approximation Let ƒ(x) be differentiable at x = a and suppose that dx = ∆x is an increment of x. We have two ways to describe the change in ƒ as x changes from a to a + ∆x: The true change: The differential estimate:

∆ƒ = ƒ(a + ∆x) - ƒ(a) dƒ = ƒ′(a) ∆x.

How well does dƒ approximate ∆ƒ? We measure the approximation error by subtracting dƒ from ∆ƒ:

Approximation error = ∆ƒ - dƒ = ∆ƒ - ƒ′(a)∆x

= ƒ(a + ∆x) - ƒ(a) - ƒ′(a)∆x           (+++)+++* ∆ƒ

ƒ(a + ∆x) - ƒ(a) = a - ƒ′(a)b # ∆x ∆x           (++++1+)+++++1*

= P # ∆x.

Call this part P.

As ∆x S 0, the difference quotient ƒ(a + ∆x) - ƒ(a) ∆x approaches ƒ′(a) (remember the definition of ƒ′(a)), so the quantity in parentheses becomes a very small number (which is why we called it P). In fact, P S 0 as ∆x S 0. When ∆x is small, the approximation error P ∆x is smaller still.

∆ƒ = ƒ′(a)∆x + P ∆x ()* (+)+* ()*

true estimated error change change

Although we do not know the exact size of the error, it is the product P # ∆x of two small quantities that both approach zero as ∆x S 0. For many common functions, whenever ∆x is small, the error is still smaller.

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211

Change in y = ƒ(x) near x = a If y = ƒ(x) is differentiable at x = a and x changes from a to a + ∆x, the change ∆y in ƒ is given by ∆y = ƒ′(a) ∆x + P ∆x(1)

in which P S 0 as ∆x S 0.

In Example 6 we found that

∆A = p(10.1)2 - p(10)2 = (102.01 - 100)p = (2p + 0.01p) m2   ()* 6

dA

error

so the approximation error is ∆A - dA = P∆r = 0.01p and P = 0.01p> ∆r = 0.01p>0.1 = 0.1p m.

Proof of the Chain Rule Equation (1) enables us to prove the Chain Rule correctly. Our goal is to show that if ƒ(u) is a differentiable function of u and u = g(x) is a differentiable function of x, then the composite y = ƒ(g(x)) is a differentiable function of x. Since a function is differentiable if and only if it has a derivative at each point in its domain, we must show that whenever g is differentiable at x0 and ƒ is differentiable at g(x0), then the composite is differentiable at x0 and the derivative of the composite satisfies the equation dy 2 = ƒ′(g(x0)) # g′(x0). dx x = x0 Let ∆x be an increment in x and let ∆u and ∆y be the corresponding increments in u and y. Applying Equation (1) we have ∆u = g′(x0)∆x + P1 ∆x = (g′(x0) + P1)∆x, where P1 S 0 as ∆x S 0. Similarly, ∆y = ƒ′(u0)∆u + P2 ∆u = (ƒ′(u0) + P2)∆u, where P2 S 0 as ∆u S 0. Notice also that ∆u S 0 as ∆x S 0. Combining the equations for ∆u and ∆y gives ∆y = (ƒ′(u0) + P2)(g′(x0) + P1)∆x, so ∆y = ƒ′(u0)g′(x0) + P2 g′(x0) + ƒ′(u0)P1 + P2P1. ∆x Since P1 and P2 go to zero as ∆x goes to zero, the last three terms on the right vanish in the limit, leaving

dy ∆y 2 = lim = ƒ′(u0)g′(x0) = ƒ′(g(x0)) # g′(x0). dx x = x0 ∆x S 0 ∆x

Sensitivity to Change The equation df = ƒ′(x) dx tells how sensitive the output of ƒ is to a change in input at different values of x. The larger the value of ƒ′ at x, the greater the effect of a given change dx. As we move from a to a nearby point a + dx, we can describe the change in ƒ in three ways:

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Absolute change Relative change Percentage change

True Estimated ∆ƒ = ƒ(a + dx) - ƒ(a) ∆ƒ ƒ(a)

dƒ = ƒ′(a) dx dƒ ƒ(a)

∆ƒ * 100 ƒ(a)

dƒ * 100 ƒ(a)

EXAMPLE 8   You want to calculate the depth of a well from the equation s = 16t2 by timing how long it takes a heavy stone you drop to splash into the water below. How sensitive will your calculations be to a 0.1-sec error in measuring the time? Solution  The size of ds in the equation ds = 32t dt depends on how big t is. If t = 2 sec, the change caused by dt = 0.1 is about ds = 32(2)(0.1) = 6.4 ft. Three seconds later at t = 5 sec, the change caused by the same dt is ds = 32(5)(0.1) = 16 ft. For a fixed error in the time measurement, the error in using ds to estimate the depth is larger when it takes a longer time before the stone splashes into the water. That is, the estimate is more sensitive to the effect of the error for larger values of t.

Exercises  3.11 Finding Linearizations In Exercises 1–5, find the linearization L(x) of ƒ(x) at x = a.

1. ƒ(x) = x3 - 2x + 3, a = 2

14. ƒ(x) = sin-1 x, a = p>12 15. Show that the linearization of ƒ(x) = (1 + x)k at x = 0 is L(x) = 1 + kx. 16. Use the linear approximation (1 + x)k ≈ 1 + kx to find an approximation for the function ƒ(x) for values of x near zero. 2 a. ƒ(x) = (1 - x)6 b. ƒ(x) = 1 - x

2. ƒ(x) = 2x2 + 9, a = - 4 1 3. ƒ(x) = x + x , a = 1 3 4. ƒ(x) = 2x, a = - 8

5. ƒ(x) = tan x, a = p

6. Common linear approximations at x = 0  Find the linearizations of the following functions at x = 0. x

a. sin x  b.  cos x  c.  tan x  d.  e   e.  ln (1 + x)

c. ƒ(x) =

1 21 + x

d. ƒ(x) = 22 + x2

e. ƒ(x) = (4 + 3x)1>3 f. ƒ(x) =

3

B

a1 -

2 x b 2 + x

Linearization for Approximation In Exercises 7–14, find a linearization at a suitably chosen integer near a at which the given function and its derivative are easy to evaluate.

17. Faster than a calculator  Use the approximation (1 + x)k ≈ 1 + kx to estimate the following.

7. ƒ(x) = x2 + 2x, a = 0.1

18. Find the linearization of ƒ(x) = 2x + 1 + sin x at x = 0. How is it related to the individual linearizations of 2x + 1 and sin x at x = 0?

8. ƒ(x) = x-1, a = 0.9 9. ƒ(x) = 2x2 + 3x - 3, a = - 0.9 10. ƒ(x) = 1 + x, a = 8.1 3 11. ƒ(x) = 2 x, a = 8.5

12. ƒ(x) =

x , a = 1.3 x + 1

13. ƒ(x) = e-x, a = - 0.1

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3

21.009 a. (1.0002)50 b.

Derivatives in Differential Form In Exercises 19–38, find dy.

19. y = x3 - 3 2x 21. y =

2x 1 + x2

20. y = x 21 - x2 22. y =

2 1x 3(1 + 1x)

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3.11  Linearization and Differentials

23. 2y3>2 + xy - x = 0

24. xy2 - 4x3>2 - y = 0

25. y = sin (5 1x)

26. y = cos (x2)

29. y = 3 csc 1 1 - 2 2x 2

30. y = 2 cot a

33. y = ln (1 + x2)

34. y = ln a

3

31. y = e2x

b. Express the estimate as a percentage of the circle’s original area.

1 b 1x

x + 1 2x - 1

52. The diameter of a tree was 10 in. During the following year, the circumference increased 2 in. About how much did the tree’s diameter increase? The tree’s cross-sectional area? b

37. y = sec (e )

38. y = e

Approximation Error In Exercises 39–44, each function ƒ(x) changes value when x changes from x0 to x0 + dx. Find a. the change ∆ƒ = ƒ(x0 + dx) - ƒ(x0);

b. the value of the estimate dƒ = ƒ′(x0) dx; and c. the approximation error 0 ∆ƒ - dƒ 0 .

Δf = f (x 0 + dx) − f (x 0) df = f ′(x 0 ) dx

Tangent x0

x

x 0 + dx

39. ƒ(x) = x2 + 2x, x0 = 1, dx = 0.1 40. ƒ(x) = 2x2 + 4x - 3, x0 = - 1, dx = 0.1 41. ƒ(x) = x3 - x, x0 = 1, dx = 0.1

55. The radius r of a circle is measured with an error of at most 2%. What is the maximum corresponding percentage error in computing the circle’s 56. The edge x of a cube is measured with an error of at most 0.5%. What is the maximum corresponding percentage error in computing the cube’s 57. Tolerance  The height and radius of a right circular cylinder are equal, so the cylinder’s volume is V = ph3. The volume is to be calculated with an error of no more than 1% of the true value. Find approximately the greatest error that can be tolerated in the measurement of h, expressed as a percentage of h. 58. Tolerance a. About how accurately must the interior diameter of a 10-m-high cylindrical storage tank be measured to calculate the tank’s volume to within 1% of its true value?

42. ƒ(x) = x4, x0 = 1, dx = 0.1 43. ƒ(x) = x-1, x0 = 0.5, dx = 0.1 44. ƒ(x) = x3 - 2x + 3, x0 = 2, dx = 0.1 Differential Estimates of Change In Exercises 45–50, write a differential formula that estimates the given change in volume or surface area.

45. The change in the volume V = (4>3)pr 3 of a sphere when the radius changes from r0 to r0 + dr 46. The change in the volume V = x3 of a cube when the edge lengths change from x0 to x0 + dx 2

47. The change in the surface area S = 6x of a cube when the edge lengths change from x0 to x0 + dx 48. The change in the lateral surface area S = pr 2r 2 + h2 of a right circular cone when the radius changes from r0 to r0 + dr and the height does not change 49. The change in the volume V = pr 2h of a right circular cylinder when the radius changes from r0 to r0 + dr and the height does not change 50. The change in the lateral surface area S = 2prh of a right circular cylinder when the height changes from h0 to h0 + dh and the radius does not change

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54. Estimating height of a building  A surveyor, standing 30 ft from the base of a building, measures the angle of elevation to the top of the building to be 75°. How accurately must the angle be measured for the percentage error in estimating the height of the building to be less than 4%?

a. surface area? b. volume?

dx 0

30 in.

a. circumference? b. area?

y = f (x)

(x 0, f (x 0 ))

0.5 in.

6 in.

tan-1 2x2 + 1

-x

y

53. Estimating volume  Estimate the volume of material in a cylindrical shell with length 30 in., radius 6 in., and shell thickness 0.5 in.

1 36. y = cot-1 a 2 b + cos-1 2x x

2

35. y = tan-1 (ex ) -1

32. y = xe-x

Applications 51. The radius of a circle is increased from 2.00 to 2.02 m.

a. Estimate the resulting change in area.

28. y = sec (x2 - 1)

27. y = 4 tan (x >3)

213

b. About how accurately must the tank’s exterior diameter be measured to calculate the amount of paint it will take to paint the side of the tank to within 5% of the true amount? 59. The diameter of a sphere is measured as 100 { 1 cm and the volume is calculated from this measurement. Estimate the percentage error in the volume calculation. 60. Estimate the allowable percentage error in measuring the diameter D of a sphere if the volume is to be calculated correctly to within 3%. 61. The effect of flight maneuvers on the heart  The amount of work done by the heart’s main pumping chamber, the left ventricle, is given by the equation W = PV +

Vdy2 , 2g

where W is the work per unit time, P is the average blood pressure, V is the volume of blood pumped out during the unit of time, d (“delta”) is the weight density of the blood, y is the average velocity of the exiting blood, and g is the acceleration of gravity.

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When P, V, d, and y remain constant, W becomes a function of g, and the equation takes the simplified form b W = a + g (a, b constant). As a member of NASA’s medical team, you want to know how sensitive W is to apparent changes in g caused by flight maneuvers, and this depends on the initial value of g. As part of your investigation, you decide to compare the effect on W of a given change dg on the moon, where g = 5.2 ft>sec2, with the effect the same change dg would have on Earth, where g = 32 ft>sec2. Use the simplified equation above to find the ratio of dWmoon to dWEarth. 62. Drug concentration  The concentration C in milligrams per milliliter (mg>ml) of a certain drug in a person’s bloodstream t hrs after a pill is swallowed is modeled by 4t C (t) = 1 + - e-0.06t. 1 + t3 Estimate the change in concentration when t changes from 20 to 30 min. 63. Unclogging arteries  The formula V = kr 4, discovered by the physiologist Jean Poiseuille (1797–1869), allows us to predict how much the radius of a partially clogged artery has to be expanded in order to restore normal blood flow. The formula says that the volume V of blood flowing through the artery in a unit of time at a fixed pressure is a constant k times the radius of the artery to the fourth power. How will a 10% increase in r affect V   ? 64. Measuring acceleration of gravity  When the length L of a clock pendulum is held constant by controlling its temperature, the pendulum’s period T depends on the acceleration of gravity g. The period will therefore vary slightly as the clock is moved from place to place on the earth’s surface, depending on the change in g. By keeping track of ∆T, we can estimate the variation in g from the equation T = 2p(L>g)1>2 that relates T, g, and L. a. With L held constant and g as the independent variable, calculate dT and use it to answer parts (b) and (c). b. If g increases, will T increase or decrease? Will a pendulum clock speed up or slow down? Explain. c. A clock with a 100-cm pendulum is moved from a location where g = 980 cm>sec2 to a new location. This increases the period by dT = 0.001 sec. Find dg and estimate the value of g at the new location. 65. Quadratic approximations 2

a. Let Q(x) = b0 + b1(x - a) + b2(x - a) be a quadratic approximation to ƒ(x) at x = a with the properties: i) Q(a) = ƒ(a) ii) Q′(a) = ƒ′(a) iii) Q″(a) = ƒ″(a). Determine the coefficients b0, b1, and b2. b. Find the quadratic approximation to ƒ(x) = 1>(1 - x) at x = 0. T c. Graph ƒ(x) = 1>(1 - x) and its quadratic approximation at x = 0. Then zoom in on the two graphs at the point (0, 1). Comment on what you see. T d. Find the quadratic approximation to g(x) = 1>x at x = 1. Graph g and its quadratic approximation together. Comment on what you see.

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T e. Find the quadratic approximation to h(x) = 21 + x at x = 0. Graph h and its quadratic approximation together. Comment on what you see. f. What are the linearizations of ƒ, g, and h at the respective points in parts (b), (d), and (e)? 66. The linearization is the best linear approximation Suppose that y = ƒ(x) is differentiable at x = a and that g(x) = m(x - a) + c is a linear function in which m and c are constants. If the error E(x) = ƒ(x) - g(x) were small enough near x = a, we might think of using g as a linear approximation of ƒ instead of the linearization L(x) = ƒ(a) + ƒ′(a)(x - a). Show that if we impose on g the conditions 1. E(a) = 0

The approximation error is zero at x = a.

E(x) 2. lim x - a = 0 xSa

The error is negligible when compared with x - a.

then g(x) = ƒ(a) + ƒ′(a)(x - a). Thus, the linearization L(x) gives the only linear approximation whose error is both zero at x = a and negligible in comparison with x - a. The linearization, L(x): y = f(a) + f ′(a)(x − a)

Some other linear approximation, g(x): y = m(x − a) + c y = f (x)

(a, f (a)) x

a

67. The linearization of 2x a. Find the linearization of ƒ(x) = 2x at x = 0. Then round its coefficients to two decimal places. T b. Graph the linearization and function together for -3 … x … 3 and -1 … x … 1. 68. The linearization of log3 x a. Find the linearization of ƒ(x) = log3 x at x = 3. Then round its coefficients to two decimal places. T b. Graph the linearization and function together in the window 0 … x … 8 and 2 … x … 4. COMPUTER EXPLORATIONS In Exercises 69–74, use a CAS to estimate the magnitude of the error in using the linearization in place of the function over a specified interval I. Perform the following steps:

a. Plot the function ƒ over I. b. Find the linearization L of the function at the point a. c. Plot ƒ and L together on a single graph. d. Plot the absolute error  ƒ(x) - L(x)  over I and find its maximum value. e. From your graph in part (d), estimate as large a d 7 0 as you can, satisfying

0x - a0 6 d

1

0 ƒ(x) - L(x) 0 6 P

for P = 0.5, 0.1, and 0.01. Then check graphically to see if your d@estimate holds true.

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69. ƒ(x) = x3 + x2 - 2x, x - 1 , 70. ƒ(x) = 2 4x + 1

3- 1, 24, a = 1

3 1 c- , 1d, a = 4 2

71. ƒ(x) = x2>3(x - 2),

Chapter  3

3- 2, 34, a = 2

2. What role does the derivative play in defining slopes, tangents, and rates of change? 3. How can you sometimes graph the derivative of a function when all you have is a table of the function’s values? 4. What does it mean for a function to be differentiable on an open interval? On a closed interval? 5. How are derivatives and one-sided derivatives related? 6. Describe geometrically when a function typically does not have a derivative at a point. 7. How is a function’s differentiability at a point related to its continuity there, if at all? 8. What rules do you know for calculating derivatives? Give some examples. 9. Explain how the three formulas d n (x ) = nxn - 1 dx

d du (cu) = c dx dx dun du du d 1 2 c. (u + u2 + g + un) = + + + dx 1 dx dx g dx enable us to differentiate any polynomial. b.

10. What formula do we need, in addition to the three listed in Question 9, to differentiate rational functions? 11. What is a second derivative? A third derivative? How many derivatives do the functions you know have? Give examples. 12. What is the derivative of the exponential function ex ? How does the domain of the derivative compare with the domain of the function? 13. What is the relationship between a function’s average and instantaneous rates of change? Give an example. 14. How do derivatives arise in the study of motion? What can you learn about an object’s motion along a line by examining the derivatives of the object’s position function? Give examples. 15. How can derivatives arise in economics? 16. Give examples of still other applications of derivatives. 17. What do the limits limh S 0 ((sin h)>h) and limh S 0 ((cos h - 1)>h) have to do with the derivatives of the sine and cosine functions? What are the derivatives of these functions?

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30, 2p4, a = 2

30, 24, a = 1

74. ƒ(x) = 2x sin-1 x, 30, 14, a =

1 2

1. What is the derivative of a function ƒ? How is its domain related to the domain of ƒ? Give examples.

a.

72. ƒ(x) = 2x - sin x,

73. ƒ(x) = x2x,

215

18. Once you know the derivatives of sin x and cos x, how can you find the derivatives of tan x, cot x, sec x, and csc x? What are the derivatives of these functions? 19. At what points are the six basic trigonometric functions continuous? How do you know? 20. What is the rule for calculating the derivative of a composite of two differentiable functions? How is such a derivative evaluated? Give examples. 21. If u is a differentiable function of x, how do you find (d>dx)(un) if n is an integer? If n is a real number? Give examples. 22. What is implicit differentiation? When do you need it? Give examples. 23. What is the derivative of the natural logarithm function ln x? How does the domain of the derivative compare with the domain of the function? 24. What is the derivative of the exponential function ax, a 7 0 and a ≠ 1? What is the geometric significance of the limit of (ah - 1)>h as h S 0? What is the limit when a is the number e? 25. What is the derivative of loga x? Are there any restrictions on a? 26. What is logarithmic differentiation? Give an example. 27. How can you write any real power of x as a power of e? Are there any restrictions on x? How does this lead to the Power Rule for differentiating arbitrary real powers? 28. What is one way of expressing the special number e as a limit? What is an approximate numerical value of e correct to 7 decimal places? 29. What are the derivatives of the inverse trigonometric functions? How do the domains of the derivatives compare with the domains of the functions? 30. How do related rates problems arise? Give examples. 31. Outline a strategy for solving related rates problems. Illustrate with an example. 32. What is the linearization L (x) of a function ƒ(x) at a point x = a? What is required of ƒ at a for the linearization to exist? How are linearizations used? Give examples. 33. If x moves from a to a nearby value a + dx, how do you estimate the corresponding change in the value of a differentiable function ƒ(x)? How do you estimate the relative change? The percentage change? Give an example.

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Chapter  3

Practice Exercises

Derivatives of Functions Find the derivatives of the functions in Exercises 1–64. 3

2

5

1. y = x - 0.375x + 0.70x 5

2. y = 8 + 0.5x - 0.7x

57. y = ln sin-1 x 6

1 4. y = x + 25x p + 5

2

5

3. y = x - 5(x + 4)

6. y = (4x + 7)(2 - x)-2

5. y = (x + 3)2(x3 + 5x2) 7. y = (x3 + sin x + 3)4

8. y = a- 3 -

1s 9. r = 5 - 1s

5 10. r = 1s - 8

11. y = 2 cot2 x + csc2 x

12. y =

3 3

u sec u - b 3 6

3 1 cos2 x tan x

58. y = x sin-1 x + 21 - x2 59. y = x cot-1 x -

60. y = ( 1 - x2 ) tan-1 3x 61. y = t csc-1 t + 2t 2 - 1, t 7 1

62. y = 5 2t + 1 tan-1 1t

63. y = csc-1 (sec u), 0 6 u 6 p>2 64. y = 21 - x2 esin

15. r = (cos t + cot t)6

16. s = sec6 (8 - t 2 + 6t 3)

17. r = 23u tan u

18. r = 3u 2tan u

71. y2 =

21. y =

22. y = 5 2x cos 2x

23. y = x-3>2 csc (3x)3

24. y = 2x sec (x - 4)4

25. y = 6 tan x3 3

26. y = x3 cot 8x

2

2

27. y = x cos (4x ) 29. s = a

6t b t + 3

31. y = a 33. y =

30. s =

3

2 - x 3

x + x

3 cos u b sin u + 1

42. y = 25e25x 44. y = x3e-3>x

45. y = ln (tan2 u)

46. y = ln (cos2 u)

3

47. y = log3 (x >3)

48. y = log3 (8x + 5)

49. y = 9-x

50. y = 34x 52. y = 25t - 25

7.2

51. y = 8x

53. y = (x + 3)x + 3 55. y = cos-1(2x 21 - x2), 2x

2 - cos u 3 b 2 + sin u

40. y = (2 + sin2 2x)-1>2

1 5x 1 5x xe e 5 25

1

cos (x )

38. y = 30 (5x - 3)1>5 (5x - 3)-1>6

8 (9x3 + cos 2x)5>2

56. y = cos-1 a

5

-8 25(25t + 4)3

36. r = a

41. y = 16e-x>8 43. y =

2

34. y = 6x2 2x3 + 1x3

37. y = (3x + 2) 23x + 2

39. y =

-3

3 3 2x 32. y = a b 3 2x - 4

b

B x3

35. r = a

28. y = x

-3

2x

3

20. r = cos 1 u - 2u - 1 2

3 1 3 x sec x 3

b, x 7 1

54. y = 5 (ln x)x>5

1 12

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6 x 6 1

x

65. xy + 2x + 3y = 1 67. x + 4xy - 3y

19. r = tan 25u

-1

Implicit Differentiation In Exercises 65–78, find dy>dx by implicit differentiation.

6 14. r = cot2 a t b

13. r = tan4 (3 + 4t)

1 ln x 2

4>3

= 2x

66. x2 + xy + y2 - 5x = 2 68. 5x4>5 + 10y6>5 = 15

69. 1xy = 1

70. x2y2 = 1

73. ex + 2y = 1

74. y2 = 2e-1>x

75. ln (x>y) = 1

76. x sin-1 y = 1 + x2

x x + 1

tan-1 x

77. ye

72. y2 =

1 + x A1 - x

78. xy = 22

= 2

In Exercises 79 and 80, find dp>dq. 79. p3 + 4pq - 3q2 = 2

80. q = (5p2 + 2p)-3>2

In Exercises 81 and 82, find dr>ds. 81. r cos 2s + sin2 s = p 2

82. 2rs - r - s + s2 = - 3

2

83. Find d y>dx by implicit differentiation: 2 y2 = 1 - x a. x3 + y3 = 1 b. 84. a.  By differentiating dy>dx = x>y.

x2 - y2 = 1

implicitly, show that

b. Then show that d 2y>dx2 = - 1>y3. Numerical Values of Derivatives 85. Suppose that functions ƒ(x) and g(x) and their first derivatives have the following values at x = 0 and x = 1.

x 0 1

ƒ(x)

g(x)

ƒ′(x)

g′(x)

1 3

1 5

-3 1>2

1>2 -4

Find the first derivatives of the following combinations at the given value of x. ƒ(x)g2(x), x = 0 a. 6ƒ(x) - g(x), x = 1 b. ƒ(x) , x = 1 d. ƒ(g(x)), x = 0 c. g(x) + 1 e. g(ƒ(x)), x = 0 f. (x + ƒ(x))3>2, x = 1 g. ƒ(x + g(x)), x = 0

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Chapter 3  Practice Exercises

86. Suppose that the function ƒ(x) and its first derivative have the following values at x = 0 and x = 1. x

ƒ(x)

ƒ′(x)

0 1

9 - 3

-2 1>5

88. Find the value of ds>du at u = 2 if s = t 2 + 5t and t = (u2 + 2u)1>3. 89. Find the value of dw>ds at s = 0 if w = sin 1 e r = 3 sin (s + p>6). 2

1r

90. Find the value of dr>dt at t = 0 if r = (u + 7) u 2t + u = 1. 2

2

1>3

2

and and

91. If y + y = 2 cos x, find the value of d y>dx at the point (0, 1). 92. If x

+ y

2

= 4, find d y>dx at the point (8, 8).

Applying the Derivative Definition In Exercises 93 and 94, find the derivative using the definition.

93. ƒ(t) =

1 2t + 1

94. g(x) = 2x2 + 1 95. a. Graph the function ƒ(x) = e

x2, -1 … x 6 0 - x 2, 0 … x … 1.

b. Is ƒ continuous at x = 0? c. Is ƒ differentiable at x = 0? Give reasons for your answers. 96. a. Graph the function x, -1 … x 6 0 ƒ(x) = e tan x, 0 … x … p>4.

b. Is ƒ continuous at x = 0?

c. Is ƒ differentiable at x = 0? Give reasons for your answers. x, 0 … x … 1 2 - x, 1 6 x … 2.

b. Is ƒ continuous at x = 1? c. Is ƒ differentiable at x = 1? Give reasons for your answers.

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101. Horizontal tangents  Find the points on the curve y = 2x3 - 3x2 - 12x + 20 where the tangent is parallel to the x-axis. 102. Tangent intercepts  Find the x- and y-intercepts of the line that is tangent to the curve y = x3 at the point (- 2, - 8). 103. Tangents perpendicular or parallel to lines  Find the points on the curve y = 2x3 - 3x2 - 12x + 20 where the tangent is a. perpendicular to the line y = 1 - (x>24). b. parallel to the line y = 22 - 12x.

104. Intersecting tangents  Show that the tangents to the curve y = (p sin x)>x at x = p and x = - p intersect at right angles. 105. Normals parallel to a line  Find the points on the curve y = tan x, -p>2 6 x 6 p>2, where the normal is parallel to the line y = -x>2. Sketch the curve and normals together, labeling each with its equation.

106. Tangent and normal lines  Find equations for the tangent and normal to the curve y = 1 + cos x at the point (p>2, 1). Sketch the curve, tangent, and normal together, labeling each with its equation. 107. Tangent parabola  The parabola y = x2 + C is to be tangent to the line y = x. Find C. 108. Slope of tangent  Show that the tangent to the curve y = x3 at any point (a, a3) meets the curve again at a point where the slope is four times the slope at (a, a3). 109. Tangent curve  For what value of c is the curve y = c>(x + 1) tangent to the line through the points (0, 3) and (5, - 2)? 110. Normal to a circle  Show that the normal line at any point of the circle x2 + y2 = a2 passes through the origin. In Exercises 111–116, find equations for the lines that are tangent and normal to the curve at the given point. 111. x2 + 2y2 = 9, (1, 2)

97. a. Graph the function ƒ(x) = e

Slopes, Tangents, and Normals 99. Tangents with specified slope  Are there any points on the curve y = (x>2) + 1>(2x - 4) where the slope is - 3>2? If so, find them.

100. Tangents with specified slope  Are there any points on the curve y = x - e-x where the slope is 2? If so, find them.

87. Find the value of dy>dt at t = 0 if y = 3 sin 2x and x = t 2 + p.

2

sin 2x, x … 0 mx, x 7 0

ƒ(1 - 5 tan x), x = 0 c. ƒ1 2x 2, x = 1 d. ƒ(x) px , x = 0 f. 10 sin a b ƒ2(x), x = 1 e. 2 + cos x 2

1>3

ƒ(x) = e

b. differentiable at x = 0?

2ƒ(x), x = 0 a. 1x ƒ(x), x = 1 b.

1>3

98. For what value or values of the constant m, if any, is

a. continuous at x = 0?

Find the first derivatives of the following combinations at the given value of x.

3

217

112. ex + y2 = 2, (0, 1) 113. xy + 2x - 5y = 2, (3, 2) 114. (y - x)2 = 2x + 4, (6, 2) 115. x + 1xy = 6, (4, 1)

116. x3>2 + 2y3>2 = 17, (1, 4)

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Chapter 3: Derivatives

117. Find the slope of the curve x3y3 + y2 = x + y at the points (1, 1) and (1, -1).

125. lim

118. The graph shown suggests that the curve y = sin (x - sin x) might have horizontal tangents at the x-axis. Does it? Give reasons for your answer.

127.

y

lim

u S (p>2)-

uS0

129. lim −2p

0

−p

x

2p

p

A

120.

2

A

2

136. y = B

1 −1

0

x

1

0

1

2

x

−1 B

4 tan2 u + tan u + 1 tan2 u + 5

1 - 2 cot2 u 5 cot u - 7 cot u - 8

x sin x 2 - 2 cos x

tan (tan x) tan x

2(x2 + 1) 2cos 2x

135. y = a

3 1

sin (sin u) u

130. lim

uS0

1 - cos u u2

132. ƒ(x) =

tan (tan x) sin (sin x)

Logarithmic Differentiation In Exercises 133–138, use logarithmic differentiation to find the derivative of y with respect to the appropriate variable.

133. y =

y 4

uS0

2

131. g(x) =

Analyzing Graphs Each of the figures in Exercises 119 and 120 shows two graphs, the graph of a function y = ƒ(x) together with the graph of its derivative ƒ′(x). Which graph is which? How do you know? y

xS0

126. lim

Show how to extend the functions in Exercises 131 and 132 to be continuous at the origin.

−1

119.

sin r tan 2r

128. lim+

y = sin (x − sin x)

1

rS0

134. y =

(t + 1)(t - 1) 5 b , t 7 2 (t - 2)(t + 3)

2u2u 2u2 + 1

137. y = (sin u)2u

3x + 4 A 2x - 4

10

138. y = (ln x)1>(ln x)

Related Rates 139. Right circular cylinder  The total surface area S of a right circular cylinder is related to the base radius r and height h by the equation S = 2pr 2 + 2prh.

a. How is dS>dt related to dr>dt if h is constant?

−2

b. How is dS>dt related to dh>dt if r is constant?

121. Use the following information to graph the function y = ƒ(x) for - 1 … x … 6. i) The graph of ƒ is made of line segments joined end to end. ii) The graph starts at the point (- 1, 2). iii) The derivative of ƒ, where defined, agrees with the step function shown here. y

c. How is dS>dt related to dr>dt and dh>dt if neither r nor h is constant? d. How is dr>dt related to dh>dt if S is constant? 140. Cube’s changing edges  The volume of a cube is increasing at the rate of 1200 cm3 >min at the instant its edges are 20 cm long. At what rate are the lengths of the edges changing at that instant? 141. Resistors connected in parallel  If two resistors of R1 and R2 ohms are connected in parallel in an electric circuit to make an R-ohm resistor, the value of R can be found from the equation

y = f ′(x) 1 1

−1

2

3

4

5

6

1 1 1 + . = R R1 R2

x

−1 −2

+

122. Repeat Exercise 121, supposing that the graph starts at (-1, 0) instead of (- 1, 2). Trigonometric Limits Find the limits in Exercises 123–130.

123. lim

xS0

sin x 2x2 - x

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124. lim

xS0

3x - tan 7x 2x

R1

R2 R

If R1 is decreasing at the rate of 1 ohm > sec and R2 is increasing at the rate of 0.5 ohm > sec, at what rate is R changing when R1 = 75 ohms and R2 = 50 ohms?

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Chapter 3  Practice Exercises

142. Impedance in a series circuit  The impedance Z (ohms) in a series circuit is related to the resistance R (ohms) and reactance X (ohms) by the equation Z = 2R2 + X 2. If R is increasing at 3 ohms > sec and X is decreasing at 2 ohms > sec, at what rate is Z changing when R = 10 ohms and X = 20 ohms?

b. How many revolutions per minute is 0.6 rad>sec?

x u

143. Speed of moving particle  The coordinates of a particle moving in the metric xy-plane are differentiable functions of time t with dx>dt = 10 m>sec and dy>dt = 5 m>sec. How fast is the particle moving away from the origin as it passes through the point (3, -4)? 144. Motion of a particle  A particle moves along the curve y = x3>2 in the first quadrant in such a way that its distance from the origin increases at the rate of 11 units per second. Find dx>dt when x = 3.

145. Draining a tank  Water drains from the conical tank shown in the accompanying figure at the rate of 5 ft3 >min. a. What is the relation between the variables h and r in the figure?

b. How fast is the water level dropping when h = 6 ft?

219

A 1 km

148. Points moving on coordinate axes Points A and B move along the x- and y-axes, respectively, in such a way that the distance r (meters) along the perpendicular from the origin to the line AB remains constant. How fast is OA changing, and is it increasing, or decreasing, when OB = 2r and B is moving toward O at the rate of 0.3r m > sec? Linearization 149. Find the linearizations of

sec x at x = - p>4. a. tan x at x = - p>4 b.

4′

Graph the curves and linearizations together. 150. We can obtain a useful linear approximation of the function ƒ(x) = 1>(1 + tan x) at x = 0 by combining the approximations

r

1 ≈ 1 - x 1 + x

10′ h Exit rate: 5 ft3min

146. Rotating spool  As television cable is pulled from a large spool to be strung from the telephone poles along a street, it unwinds from the spool in layers of constant radius (see accompanying figure). If the truck pulling the cable moves at a steady 6 ft > sec (a touch over 4 mph), use the equation s = r u to find how fast (radians per second) the spool is turning when the layer of radius 1.2 ft is being unwound.

1.2′

and

tan x ≈ x

to get 1 ≈ 1 - x. 1 + tan x Show that this result is the standard linear approximation of 1>(1 + tan x) at x = 0. 151. Find the linearization of ƒ(x) = 21 + x + sin x - 0.5 at x = 0.

152. Find the linearization of ƒ(x) = 2>(1 - x) + 21 + x - 3.1 at x = 0. Differential Estimates of Change 153. Surface area of a cone  Write a formula that estimates the change that occurs in the lateral surface area of a right circular cone when the height changes from h0 to h0 + dh and the radius does not change.

h

r

147. Moving searchlight beam  The figure shows a boat 1 km offshore, sweeping the shore with a searchlight. The light turns at a constant rate, du>dt = - 0.6 rad/sec.

V = 1 pr2h 3 S = pr "r2 + h2 (Lateral surface area)

a.  How fast is the light moving along the shore when it reaches point A?

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Chapter 3: Derivatives

154. Controlling error a.  How accurately should you measure the edge of a cube to be reasonably sure of calculating the cube’s surface area with an error of no more than 2%?

measure the length a of its shadow, finding it to be 15 ft, give or take an inch. Calculate the height of the lamppost using the value a = 15 and estimate the possible error in the result.

b.  Suppose that the edge is measured with the accuracy required in part (a). About how accurately can the cube’s volume be calculated from the edge measurement? To find out, estimate the percentage error in the volume calculation that might result from using the edge measurement.

h

155. Compounding error  The circumference of the equator of a sphere is measured as 10 cm with a possible error of 0.4 cm. This measurement is used to calculate the radius. The radius is then used to calculate the surface area and volume of the sphere. Estimate the percentage errors in the calculated values of

6 ft 20 ft a

a. the radius.   b.  the surface area.   c.  the volume. 156. Finding height  To find the height of a lamppost (see accompanying figure), you stand a 6 ft pole 20 ft from the lamp and

Chapter  3 Additional and Advanced Exercises 1. An equation like sin2 u + cos2 u = 1 is called an identity because it holds for all values of u. An equation like sin u = 0.5 is not an identity because it holds only for selected values of u, not all. If you differentiate both sides of a trigonometric identity in u with respect to u, the resulting new equation will also be an identity.    Differentiate the following to show that the resulting equations hold for all u. a. sin 2u = 2 sin u cos u b. cos 2u = cos2 u - sin2 u 2. If the identity sin (x + a) = sin x cos a + cos x sin a is differentiated with respect to x, is the resulting equation also an identity? Does this principle apply to the equation x2 - 2x - 8 = 0? Explain. 3.  a. Find values for the constants a, b, and c that will make ƒ(x) = cos x and g(x) = a + bx + cx2 satisfy the conditions ƒ(0) = g(0), ƒ′(0) = g′(0), and ƒ″(0) = g″(0). b. Find values for b and c that will make ƒ(x) = sin (x + a) and g(x) = b sin x + c cos x satisfy the conditions ƒ(0) = g(0) and ƒ′(0) = g′(0).  c. For the determined values of a, b, and c, what happens for the third and fourth derivatives of ƒ and g in each of parts (a) and (b)? 4. An osculating circle  Find the values of h, k, and a that make the circle (x - h)2 + (y - k)2 = a2 tangent to the parabola

M03_THOM9580_03_SE_C03_129-222.indd 220

y = x2 + 1 at the point (1, 2) and that also make the second derivatives d 2y>dx2 have the same value on both curves there. Circles like this one that are tangent to a curve and have the same second derivative as the curve at the point of tangency are called osculating circles (from the Latin osculari, meaning “to kiss”). We encounter them again in Chapter 12. 5. Industrial production  a. Economists often use the expression “rate of growth” in relative rather than absolute terms. For example, let u = ƒ(t) be the number of people in the labor force at time t in a given industry. (We treat this function as though it were differentiable even though it is an integer-valued step function.)   Let y = g(t) be the average production per person in the labor force at time t. The total production is then y = uy. If the labor force is growing at the rate of 4% per year (du>dt = 0.04u) and the production per worker is growing at the rate of 5% per year (dy>dt = 0.05y), find the rate of growth of the total production, y. b. Suppose that the labor force in part (a) is decreasing at the rate of 2% per year while the production per person is increasing at the rate of 3% per year. Is the total production increasing, or is it decreasing, and at what rate? 6. Designing a gondola  The designer of a 30-ft-diameter spherical hot air balloon wants to suspend the gondola 8 ft below the bottom of the balloon with cables tangent to the surface of the balloon, as shown. Two of the cables are shown running from the top edges of the gondola to their points of tangency, (- 12, -9) and (12, - 9). How wide should the gondola be?

16/11/15 4:41 PM

12. Average and instantaneous velocity

y

a. Show that if the position x of a moving point is given by a quadratic function of t, x = At 2 + Bt + C, then the average velocity over any time interval 3t1, t2 4 is equal to the instantaneous velocity at the midpoint of the time interval.

x 2 + y 2 = 225

x

0

b. What is the geometric significance of the result in part (a)?

13. Find all values of the constants m and b for which the function 15 ft (12, −9)

(−12, −9) Suspension cables

8 ft Gondola Width NOT TO SCALE

7. Pisa by parachute  On August 5, 1988, Mike McCarthy of London jumped from the top of the Tower of Pisa. He then opened his parachute in what he said was a world record lowlevel parachute jump of 179 ft. Make a rough sketch to show the shape of the graph of his speed during the jump. (Source: Boston Globe, Aug. 6, 1988.) 8. Motion of a particle  The position at time t Ú 0 of a particle moving along a coordinate line is s = 10 cos (t + p>4).  a. What is the particle’s starting position (t = 0)? b. What are the points farthest to the left and right of the origin reached by the particle?

is

d. When does the particle first reach the origin? What are its velocity, speed, and acceleration then? 9. Shooting a paper clip  On Earth, you can easily shoot a paper clip 64 ft straight up into the air with a rubber band. In t sec after firing, the paper clip is s = 64t - 16t 2 ft above your hand.  a. How long does it take the paper clip to reach its maximum height? With what velocity does it leave your hand? b. On the moon, the same acceleration will send the paper clip to a height of s = 64t - 2.6t 2 ft in t sec. About how long will it take the paper clip to reach its maximum height, and how high will it go? 10. Velocities of two particles  At time t sec, the positions of two particles on a coordinate line are s1 = 3t 3 - 12t 2 + 18t + 5 m and s2 = -t 3 + 9t 2 - 12t m. When do the particles have the same velocities? 11. Velocity of a particle  A particle of constant mass m moves along the x-axis. Its velocity y and position x satisfy the equation 1 1 m (y2 - y0 2) = k (x0 2 - x2), 2 2 where k, y0, and x0 are constants. Show that whenever y ≠ 0,

M03_THOM9580_03_SE_C03_129-222.indd 221

y = e

sin x, x 6 p mx + b, x Ú p

a. continuous at x = p. b. differentiable at x = p. 14. Does the function ƒ(x) = •

1 - cos x , x ≠ 0 x 0,

x = 0

have a derivative at x = 0? Explain. 15.  a. For what values of a and b will ƒ(x) = e

ax, x 6 2 ax2 - bx + 3, x Ú 2

be differentiable for all values of x? b. Discuss the geometry of the resulting graph of ƒ. 16.  a. For what values of a and b will

c. Find the particle’s velocity and acceleration at the points in part (b).

m

221

g(x) = e

ax + b, x … -1 ax3 + x + 2b, x 7 - 1

be differentiable for all values of x? b. Discuss the geometry of the resulting graph of g. 17. Odd differentiable functions  Is there anything special about the derivative of an odd differentiable function of x? Give reasons for your answer. 18. Even differentiable functions  Is there anything special about the derivative of an even differentiable function of x? Give reasons for your answer. 19. Suppose that the functions ƒ and g are defined throughout an open interval containing the point x0, that ƒ is differentiable at x0, that ƒ(x0) = 0, and that g is continuous at x0. Show that the product ƒg is differentiable at x0. This process shows, for example, that although 0 x 0 is not differentiable at x = 0, the product x 0 x 0 is differentiable at x = 0.

20. (Continuation of Exercise 19.) Use the result of Exercise 19 to show that the following functions are differentiable at x = 0. 3  a. 0 x 0 sin x  b. x2>3 sin x  c. 2 x (1 - cos x)

d. h(x) = e

x2 sin (1>x), x ≠ 0 0, x = 0

dy = - kx. dt

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Chapter 3: Derivatives

21. Suppose that a function ƒ satisfies the following conditions for all real values of x and y: i) ƒ(x + y) = ƒ(x) # ƒ(y).

ii)  ƒ(x) = 1 + xg(x), where limx S 0 g(x) = 1. Show that the derivative ƒ′(x) exists at every value of x and that ƒ′(x) = ƒ(x). 22. Leibniz’s rule for higher-order derivatives of products Leibniz’s rule for higher-order derivatives of products of differentiable functions says that d 2(uy) du dy d 2y d 2u + u 2.  a. = 2y + 2 2 dx dx dx dx dx b.

d 3(uy) d 2u dy du d 2y d 3y d 3u + 3 = 3y + 3 2 + u 3. 3 2 dx dx dx dx dx dx dx

c.

d n(uy) d n - 1u dy d nu + g = n y + n n-1 n dx dx dx dx n(n - 1) g(n - k + 1) d n - ku d ky k! dxn - k dxk

+

d ny + g + u n. dx

M03_THOM9580_03_SE_C03_129-222.indd 222

The equations in parts (a) and (b) are special cases of the equation in part (c). Derive the equation in part (c) by mathematical induction, using m m m! m! a b + a b = + . k!(m - k)! (k + 1)!(m - k - 1)! k k + 1

23. The generalized product rule  Use mathematical induction to prove that if y = u1 u2 gun is a finite product of differentiable functions, then y is differentiable on their common domain and dun dy du1 du2 = u2 gun + u1 . un + g + u1 u2 gun - 1 g dx dx dx dx 24. The period of a clock pendulum  The period T of a clock pendulum (time for one full swing and back) is given by the formula T 2 = 4p2L>g, where T is measured in seconds, g = 32.2 ft>sec2, and L, the length of the pendulum, is measured in feet. Find approximately a. the length of a clock pendulum whose period is T = 1 sec. b. the change dT in T if the pendulum in part (a) is lengthened 0.01 ft. c. the amount the clock gains or loses in a day as a result of the period’s changing by the amount dT found in part (b).

16/11/15 4:42 PM

chapter

4

Applications of Derivatives

Overview  One of the most important applications of the derivative is its use as a tool for

finding the optimal (best) solutions to problems. For example, what are the dimensions of the strongest rectangular wooden beam that can be cut from a cylindrical log of given diameter? Based on production costs and sales revenue, how many items should a manufacturer produce to maximize profit? What is the branching angle at which blood vessels minimize the energy loss due to friction as blood flows through the branches? In this chapter we use derivatives to find extreme values of functions, to determine and analyze the shapes of graphs, and to solve equations numerically. We also introduce the idea of recovering a function from its derivative. The key to many of these applications is the Mean Value Theorem, which paves the way to integral calculus.

4.1 Extreme Values of Functions This section shows how to locate and identify extreme (maximum or minimum) values of a function from its derivative. Once we can do this, we can solve a variety of optimization problems (see Section 4.6). The domains of the functions we consider are intervals or unions of separate intervals. Definitions  Let ƒ be a function with domain D. Then ƒ has an absolute maximum value on D at a point c if y

ƒ(x) … ƒ(c)

1

and an absolute minimum value on D at c if

y = sin x

y = cos x

ƒ(x) Ú ƒ(c) 0

p 2

p 2

for all x in D for all x in D.

x

−1

Figure 4.1  Absolute extrema for the sine and cosine functions on 3- p>2, p>24. These values can depend on the domain of a function.

Maximum and minimum values are called extreme values of the function ƒ. Absolute maxima or minima are also referred to as global maxima or minima. For example, on the closed interval 3 -p>2, p>24 the function ƒ(x) = cos x takes on an absolute maximum value of 1 (once) and an absolute minimum value of 0 (twice). On the same interval, the function g(x) = sin x takes on a maximum value of 1 and a minimum value of -1 (Figure 4.1). Functions with the same defining rule or formula can have different extrema (maximum or minimum values), depending on the domain. We see this in the following example.

223

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Chapter 4: Applications of Derivatives

Example 1   The absolute extrema of the following functions on their domains can be seen in Figure 4.2. Each function has the same defining equation, y = x2, but the domains vary. Notice that a function might not have a maximum or minimum if the domain is unbounded or fails to contain an endpoint. Function rule

y = x2

Domain D

Absolute extrema on D

(a)  y = x2 (- q, q)

No absolute maximum Absolute minimum of 0 at x = 0

(b)  y = x2 3 0, 24

Absolute maximum of 4 at x = 2 Absolute minimum of 0 at x = 0

(c)  y = x2 (0, 24

Absolute maximum of 4 at x = 2 No absolute minimum

(d)  y = x2

No absolute extrema

(0, 2)

y = x2

y

D = (−∞, ∞)

x

2 (b) abs max and min

y = x2

y

D = (0, 2]

D = [0, 2]

2 (a) abs min only

y = x2

y

x

D = (0, 2)

2 (c) abs max only

y

x

2 (d) no max or min

x

Figure 4.2  Graphs for Example 1.

Historical Biography Daniel Bernoulli (1700–1789)

Some of the functions in Example 1 did not have a maximum or a minimum value. The following theorem asserts that a function which is continuous over (or on) a finite closed interval 3 a, b4 has an absolute maximum and an absolute minimum value on the interval. We look for these extreme values when we graph a function.

theorem 1—The Extreme Value Theorem  If ƒ is continuous on a closed interval 3 a, b4 , then ƒ attains both an absolute maximum value M and an absolute minimum value m in 3 a, b4 . That is, there are numbers x1 and x2 in 3 a, b4 with ƒ(x1) = m, ƒ(x2) = M, and m … ƒ(x) … M for every other x in 3 a, b4 . The proof of the Extreme Value Theorem requires a detailed knowledge of the real number system (see Appendix 7) and we will not give it here. Figure 4.3 illustrates possible locations for the absolute extrema of a continuous function on a closed interval 3 a, b4 . As we observed for the function y = cos x, it is possible that an absolute minimum (or absolute maximum) may occur at two or more different points of the interval. The requirements in Theorem 1 that the interval be closed and finite, and that the function be continuous, are key ingredients. Without them, the conclusion of the theorem need not hold. Example 1 shows that an absolute extreme value may not exist if the interval fails to be both closed and finite. The exponential function y = ex over (- q, q)

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4.1  Extreme Values of Functions

(x2, M) y = f (x)

y = f (x)

M

M x1

a

x2

b

0m0

m

x

b

a

x

Maximum and minimum at endpoints

(x1, m) Maximum and minimum at interior points

y = f (x) y = f (x)

M

m

m a

M

x2

b

x

Maximum at interior point, minimum at endpoint

a

x1

b

x

Minimum at interior point, maximum at endpoint

Figure 4.3  Some possibilities for a continuous function’s maximum and minimum on a closed interval 3a, b4. y

shows that neither extreme value need exist on an infinite interval. Figure 4.4 shows that the continuity requirement cannot be omitted.

No largest value

1

Local (Relative) Extreme Values

y=x 0≤ xx) on the interval (0, 14 . (We graphed this function in Figure 2.31.)

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226

Chapter 4: Applications of Derivatives Absolute maximum No greater value of f anywhere. Also a local maximum.

Local maximum No greater value of f nearby.

Local minimum No smaller value of f nearby.

y = f (x) Absolute minimum No smaller value of f anywhere. Also a local minimum.

Local minimum No smaller value of f nearby. a

c

e

d

b

x

Figure 4.5  How to identify types of maxima and minima for a function with domain a … x … b.

An absolute maximum is also a local maximum. Being the largest value overall, it is also the largest value in its immediate neighborhood. Hence, a list of all local maxima will automatically include the absolute maximum if there is one. Similarly, a list of all local minima will include the absolute minimum if there is one.

Finding Extrema The next theorem explains why we usually need to investigate only a few values to find a function’s extrema.

Local maximum value

y = f (x)

THEOREM 2—The First Derivative Theorem for Local Extreme Values  If ƒ has a local maximum or minimum value at an interior point c of its domain, and if ƒ′ is defined at c, then ƒ′(c) = 0. Secant slopes ≥ 0 (never negative)

x

Secant slopes ≤ 0 (never positive)

c

x

x

Figure 4.6  A curve with a local maximum value. The slope at c, simultaneously the limit of nonpositive numbers and nonnegative numbers, is zero.

Proof   To prove that ƒ′(c) is zero at a local extremum, we show first that ƒ′(c) cannot be positive and second that ƒ′(c) cannot be negative. The only number that is neither positive nor negative is zero, so that is what ƒ′(c) must be. To begin, suppose that ƒ has a local maximum value at x = c (Figure 4.6) so that ƒ(x) - ƒ(c) … 0 for all values of x near enough to c. Since c is an interior point of ƒ’s domain, ƒ′(c) is defined by the two-sided limit lim

xSc

ƒ(x) - ƒ(c) x - c .

This means that the right-hand and left-hand limits both exist at x = c and equal ƒ′(c). When we examine these limits separately, we find that

ƒ′(c) = lim+

ƒ(x) - ƒ(c) … 0.  Because (x x - c

- c) 7 0 and ƒ(x) … ƒ(c)

(1)

ƒ′(c) = lim-

ƒ(x) - ƒ(c) Ú 0.  Because (x x - c

- c) 6 0 and ƒ(x) … ƒ(c)

(2)

xSc

Similarly,

xSc

Together, Equations (1) and (2) imply ƒ′(c) = 0. This proves the theorem for local maximum values. To prove it for local minimum values, we simply use ƒ(x) Ú ƒ(c), which reverses the inequalities in Equations (1) and (2).

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4.1  Extreme Values of Functions

Theorem 2 says that a function’s first derivative is always zero at an interior point where the function has a local extreme value and the derivative is defined. If we recall that all the domains we consider are intervals or unions of separate intervals, the only places where a function ƒ can possibly have an extreme value (local or global) are

y y=

x3

1

−1

0

1

227

x

−1

1. interior points where ƒ′ = 0, 2. interior points where ƒ′ is undefined, 3. endpoints of the domain of ƒ.

At x = c and x = e in Fig. 4.5 At x = d in Fig. 4.5 At x = a and x = b in Fig. 4.5

The following definition helps us to summarize these results. (a)

Definition  An interior point of the domain of a function ƒ where ƒ′ is zero or undefined is a critical point of ƒ.

y 1 y = x13 −1

0

1

x

−1 (b)

Figure 4.7  Critical points without extreme values. (a) y′ = 3x2 is 0 at x = 0, but y = x3 has no extremum there. (b) y′ = (1>3)x-2>3 is undefined at x = 0, but y = x1>3 has no extremum there.

Thus the only domain points where a function can assume extreme values are critical points and endpoints. However, be careful not to misinterpret what is being said here. A function may have a critical point at x = c without having a local extreme value there. For instance, both of the functions y = x3 and y = x1>3 have critical points at the origin, but neither function has a local extreme value at the origin. Instead, each function has a point of inflection there (see Figure 4.7). We define and explore inflection points in Section 4.4. Most problems that ask for extreme values call for finding the absolute extrema of a continuous function on a closed and finite interval. Theorem 1 assures us that such values exist; Theorem 2 tells us that they are taken on only at critical points and endpoints. Often we can simply list these points and calculate the corresponding function values to find what the largest and smallest values are, and where they are located. Of course, if the interval is not closed or not finite (such as a 6 x 6 b or a 6 x 6 q), we have seen that absolute extrema need not exist. If an absolute maximum or minimum value does exist, it must occur at a critical point or at an included right- or left-hand endpoint of the interval.

How to Find the Absolute Extrema of a Continuous Function ƒ on a Finite Closed Interval 1. Evaluate ƒ at all critical points and endpoints. 2. Take the largest and smallest of these values.

Example 2  Find the absolute maximum and minimum values of ƒ(x) = x2 on 3 -2, 14 .

Solution  The function is differentiable over its entire domain, so the only critical point is where ƒ′(x) = 2x = 0, namely x = 0. We need to check the function’s values at x = 0 and at the endpoints x = -2 and x = 1: Critical point value: Endpoint values:

ƒ(0) = 0 ƒ(-2) = 4 ƒ(1) = 1.

The function has an absolute maximum value of 4 at x = -2 and an absolute minimum value of 0 at x = 0.

Example 3  Find the absolute maximum and minimum values of ƒ(x) = 10x (2 - ln x) on the interval 3 1, e2 4 .

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Chapter 4: Applications of Derivatives

Solution  Figure 4.8 suggests that ƒ has its absolute maximum value near x = 3 and its absolute minimum value of 0 at x = e2. Let’s verify this observation. We evaluate the function at the critical points and endpoints and take the largest and smallest of the resulting values. The first derivative is

y 30

(e, 10e)

25 20 15

(1, 20)

1 ƒ′(x) = 10(2 - ln x) - 10xa x b = 10(1 - ln x).

10 5

(e 2, 0) 1

0

2

3

4

5

6

7

x

8

Figure 4.8  The extreme values of ƒ(x) = 10x(2 - ln x) on 31, e2 4 occur at x = e and x = e2 (Example 3).

The only critical point in the domain 3 1, e2 4 is the point x = e, where ln x = 1. The values of ƒ at this one critical point and at the endpoints are Critical point value:

ƒ(e) = 10e

Endpoint values:

ƒ(1) = 10(2 - ln 1) = 20 ƒ(e2) = 10e2(2 - 2 ln e) = 0.

We can see from this list that the function’s absolute maximum value is 10e ≈ 27.2; it occurs at the critical interior point x = e. The absolute minimum value is 0 and occurs at the right endpoint x = e2.

Example 4  Find the absolute maximum and minimum values of ƒ(x) = x2>3 on the

interval 3 -2, 34 . y

Solution  We evaluate the function at the critical points and endpoints and take the largest and smallest of the resulting values. The first derivative

y = x 23, −2 ≤ x ≤ 3

−1

0

1 2 3 Absolute minimum; also a local minimum

2 -1>3 2 x = 3 3 3 2x

has no zeros but is undefined at the interior point x = 0. The values of ƒ at this one critical point and at the endpoints are Critical point value: ƒ(0) = 0

1 −2

ƒ′(x) =

Absolute maximum; also a local maximum

Local maximum 2

x

Endpoint values:

3 ƒ(-2) = (-2)2>3 = 2 4 3 2>3 ƒ(3) = (3) = 29.

3 We can see from this list that the function’s absolute maximum value is 2 9 ≈ 2.08, and it occurs at the right endpoint x = 3. The absolute minimum value is 0, and it occurs at the interior point x = 0 where the graph has a cusp (Figure 4.9).

Figure 4.9  The extreme values of ƒ(x) = x2>3 on 3- 2, 34 occur at x = 0 and x = 3 (Example 4).

Exercises  4.1 Finding Extrema from Graphs In Exercises 1–6, determine from the graph whether the function has any absolute extreme values on 3a, b4 . Then explain how your answer is consistent with .Theorem 1

1. y

2.

3. y y = f (x)

y = h(x)

y = f (x) 0

a

c1

c2

b

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x

y

y

y = h(x)

0

4.

0

a

c

b

a

c

b

x

0

a

c

b

x

x

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4.1  Extreme Values of Functions

5. y

6.

In Exercises 15–20, sketch the graph of each function and determine whether the function has any absolute extreme values on its domain. Explain how your answer is consistent with Theorem 1.

y

y = g(x)

229

y = g(x)

15. ƒ(x) = 0 x 0 , - 1 6 x 6 2

6 , -1 6 x 6 1 x2 + 2 - x, 0 … x 6 1 17. g(x) = e x - 1, 1 … x … 2 16. y =

0

a

c

b

x

0

a

c

x

b

In Exercises 7–10, find the absolute extreme values and where they occur. y

7.

y

8.

1

−2

0 … x … 4

19. y = 3 sin x, 0 6 x 6 2p

0

20. ƒ(x) = •

y 9. 10.

x

2

y

5

(1, 2)

2

x

2

−3

cos x,

0 6 x …

p 2

Absolute Extrema on Finite Closed Intervals In Exercises 21–40, find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.

21. ƒ(x) =

−1

2 x - 5, - 2 … x … 3 3

22. ƒ(x) = -x - 4, -4 … x … 1

x

2

2x,

x + 1, - 1 … x 6 0

−1

0

-1 … x 6 0

2 x

1

−1

18. h(x) = •

1 x,

23. ƒ(x) = x2 - 1, - 1 … x … 2 24. ƒ(x) = 4 - x3, - 2 … x … 1

In Exercises 11–14, match the table with a graph. 11. x

12. x

ƒ′(x)

ƒ′(x)

14. x

ƒ′(x)

a does not exist b   0 c -2

a

b c

(a)

3 27. h(x) = 2x, - 1 … x … 8

28. h(x) = - 3x2>3, - 1 … x … 1

29. g(x) = 24 - x2 , - 2 … x … 1

ƒ′(x)

30. g(x) = - 25 - x2 , - 25 … x … 0 p 5p 31. ƒ(u) = sin u, - … u … 2 6 p p 32. ƒ(u) = tan u, - … u … 3 4 p 2p 33. g(x) = csc x, … x … 3 3 p p 34. g(x) = sec x, - … x … 3 6 35. ƒ(t) = 2 - 0 t 0 , - 1 … t … 3

does not exist does not exist - 1.7

a b c

1 , 0.5 … x … 2 x2

1 26. F(x) = - x , -2 … x … -1

a   0 b   0 c -5

a 0 b 0 c 5 13. x

25. F(x) = -

a

b

c

36. ƒ(t) = 0 t - 5 0 , 4 … t … 7

(b)

37. g(x) = xe-x, -1 … x … 1

38. h(x) = ln (x + 1), 0 … x … 3 1 39. ƒ(x) = x + ln x, 0.5 … x … 4 2

a

b

c

(c)

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a

b

c

40. g(x) = e-x , - 2 … x … 1

(d)

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Chapter 4: Applications of Derivatives

In Exercises 41–44, find the function’s absolute maximum and minimum values and say where they are assumed. 41. ƒ(x) = x5>3, - 1 … x … 8 42. ƒ(x) = x5>3, - 1 … x … 8 43. g(u) = 9u 3>5, - 1 … u … 243 44. h(u) = 3u 2>3, - 27 … u … 8 Finding Critical Points In Exercises 45–52, determine all critical points for each function. 2

2

3

45. ƒ(x) = x - 10x + 6

46. ƒ(x) = 3x - 2x

47. ƒ(x) = x(4 - x)3

48. g(x) = (x - 1)(x - 9)3

2 49. y = x2 + x

50. ƒ(x) =

51. y = x2 - 32 2x

52. g(x) = 22x - x2

x2 x - 2

Finding Extreme Values In Exercises 53–68, find the extreme values (absolute and local) of the function over its natural domain, and where they occur.

53. y = 4x2 - 24x + 13

54. y = x3 - 2x + 4

55. y = 3x3 - 3x2 - 3x + 5

56. y = x3(x - 5)2

57. y = 2x2 - 1

58. y = x - 4 2x

1 3 21 - x2 x 61. y = 2 x + 1 59. y =

63. y = ex + e-x

60. y = 23 + 2x - x2 62. y =

x + 1 x2 + 2x + 2

64. y = ex - e-x 2

65. y = x ln x

66. y = x ln x

67. y = cos-1 (x2)

68. y = sin-1(ex)

Local Extrema and Critical Points In Exercises 69–76, find the critical points, domain endpoints, and extreme values (absolute and local) for each function.

69. y = x2>3(x + 2)

70. y = x2>3(x2 - 4)

71. y = x 24 - x2

72. y = x2 23 - x

4 - 2x, x … 1 73. y = e x + 1, x 7 1

75. y = e

3 - x, x 6 0 74. y = e 3 + 2x - x2, x Ú 0

-x2 - 2x + 4, x … 1 - x2 + 6x - 4, x 7 1

15 1 1 , x … 1 - x2 - x + 4 2 4 76. y = c x3 - 6x2 + 8x, x 7 1 In Exercises 77 and 78, give reasons for your answers. 77. Let ƒ(x) = (x - 2)2>3. a. Does ƒ′(2) exist? b. Show that the only local extreme value of ƒ occurs at x = 2.  c. Does the result in part (b) contradict the Extreme Value Theorem? d. Repeat parts (a) and (b) for ƒ(x) = (x - a)2>3, replacing 2 by a.

Theory and Examples 79. A minimum with no derivative  The function ƒ(x) = 0 x 0 has an absolute minimum value at x = 0 even though ƒ is not differentiable at x = 0. Is this consistent with Theorem 2? Give reasons for your answer.

80. Even functions  If an even function ƒ(x) has a local maximum value at x = c, can anything be said about the value of ƒ at x = -c? Give reasons for your answer. 81. Odd functions  If an odd function g(x) has a local minimum value at x = c, can anything be said about the value of g at x = -c? Give reasons for your answer. 82. No critical points or endpoints exist  We know how to find the extreme values of a continuous function ƒ(x) by investigating its values at critical points and endpoints. But what if there are no critical points or endpoints? What happens then? Do such functions really exist? Give reasons for your answers. 83. The function V(x) = x(10 - 2x)(16 - 2x),

0 6 x 6 5,

models the volume of a box. a. Find the extreme values of V. b. Interpret any values found in part (a) in terms of the volume of the box. 84. Cubic functions  Consider the cubic function ƒ(x) = ax3 + bx2 + cx + d. a. Show that ƒ can have 0, 1, or 2 critical points. Give examples and graphs to support your argument. b. How many local extreme values can ƒ have? 85. Maximum height of a vertically moving body  The height of a body moving vertically is given by 1 s = - gt 2 + y0 t + s0, 2

g 7 0,

with s in meters and t in seconds. Find the body’s maximum height. 86. Peak alternating current  Suppose that at any given time t (in seconds) the current i (in amperes) in an alternating current circuit is i = 2 cos t + 2 sin t. What is the peak current for this circuit (largest magnitude)? T Graph the functions in Exercises 87–90. Then find the extreme values of the function on the interval and say where they occur. 87. ƒ(x) = 0 x - 2 0 + 0 x + 3 0 , - 5 … x … 5

88. g(x) = 0 x - 1 0 - 0 x - 5 0 , - 2 … x … 7 89. h(x) = 0 x + 2 0 - 0 x - 3 0 , - q 6 x 6 q

90. k(x) = 0 x + 1 0 + 0 x - 3 0 , - q 6 x 6 q

Computer Explorations In Exercises 91–98, you will use a CAS to help find the absolute extrema of the given function over the specified closed interval. Perform the following steps.

a. Plot the function over the interval to see its general behavior there.

Does ƒ′(3) exist? a. Does ƒ′(0) exist? b.

b. Find the interior points where ƒ′ = 0. (In some exercises, you may have to use the numerical equation solver to approximate a solution.) You may want to plot ƒ′ as well.

Determine all extrema of ƒ. c. Does ƒ′(- 3) exist? d.

c. Find the interior points where ƒ′ does not exist.

78. Let ƒ(x) = 0 x3 - 9x 0 .

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4.2  The Mean Value Theorem

d. Evaluate the function at all points found in parts (b) and (c) and at the endpoints of the interval. e. Find the function’s absolute extreme values on the interval and identify where they occur. 91. ƒ(x) = x4 - 8x2 + 4x + 2, 92. ƒ(x) = - x4 + 4x3 - 4x + 1, 2>3

93. ƒ(x) = x (3 - x),

3- 2, 24

3- 20>25, 64>254 3- 3>4, 34

94. ƒ(x) = 2 + 2x - 3x2>3,

231

3- 1, 10>34

30, 2p4 1 96. ƒ(x) = x3>4 - sin x + , 30, 2p4 2 30, 54 97. ƒ(x) = px2e - 3x>2, 95. ƒ(x) = 2x + cos x,

98. ƒ(x) = ln (2x + x sin x),

31, 154

4.2 The Mean Value Theorem We know that constant functions have zero derivatives, but could there be a more complicated function whose derivative is always zero? If two functions have identical derivatives over an interval, how are the functions related? We answer these and other questions in this chapter by applying the Mean Value Theorem. First we introduce a special case, known as Rolle’s Theorem, which is used to prove the Mean Value Theorem.

y f ′(c) = 0

Rolle’s Theorem

y = f (x)

0

a

c

As suggested by its graph, if a differentiable function crosses a horizontal line at two different points, there is at least one point between them where the tangent to the graph is horizontal and the derivative is zero (Figure 4.10). We now state and prove this result.

x

b

(a)

THEOREM 3—Rolle’s Theorem  Suppose that y = ƒ(x) is continuous over the closed interval 3 a, b4 and differentiable at every point of its interior (a, b). If ƒ(a) = ƒ(b), then there is at least one number c in (a, b) at which ƒ′(c) = 0.

y f ′(c1 ) = 0

0

a

c1

f ′(c3 ) = 0 f ′(c2 ) = 0

c2

y = f (x)

c3

b

x

(b)

Figure 4.10  Rolle’s Theorem says that a differentiable curve has at least one horizontal tangent between any two points where it crosses a horizontal line. It may have just one (a), or it may have more (b).

Historical Biography Michel Rolle (1652–1719)

Proof   Being continuous, ƒ assumes absolute maximum and minimum values on 3 a, b4 by Theorem 1. These can occur only

1. at interior points where ƒ′ is zero, 2. at interior points where ƒ′ does not exist, 3. at endpoints of the function’s domain, in this case a and b.

By hypothesis, ƒ has a derivative at every interior point. That rules out possibility (2), leaving us with interior points where ƒ′ = 0 and with the two endpoints a and b. If either the maximum or the minimum occurs at a point c between a and b, then ƒ′(c) = 0 by Theorem 2 in Section 4.1, and we have found a point for Rolle’s Theorem. If both the absolute maximum and the absolute minimum occur at the endpoints, then because ƒ(a) = ƒ(b) it must be the case that ƒ is a constant function with ƒ(x) = ƒ(a) = ƒ(b) for every x∊ 3 a, b4 . Therefore ƒ′(x) = 0 and the point c can be taken anywhere in the interior (a, b). The hypotheses of Theorem 3 are essential. If they fail at even one point, the graph may not have a horizontal tangent (Figure 4.11). Rolle’s Theorem may be combined with the Intermediate Value Theorem to show when there is only one real solution of an equation ƒ(x) = 0, as we illustrate in the next example.

Example 1   Show that the equation x3 + 3x + 1 = 0 has exactly one real solution.

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Chapter 4: Applications of Derivatives y

y

y

y = f (x)

a

y = f (x)

b

(a) Discontinuous at an endpoint of [a, b]

y

x

a

x0 b

y = f(x)

x

(b) Discontinuous at an interior point of [a, b]

(1, 5)

a

x0

b

x

(c) Continuous on [a, b] but not differentiable at an interior point

Figure 4.11  There may be no horizontal tangent if the hypotheses of Rolle’s Theorem do not hold.

0

−1

Solution  We define the continuous function

y = x 3 + 3x + 1

1

1

ƒ(x) = x3 + 3x + 1.

x

(−1, −3)

Figure 4.12  The only real zero of the polynomial y = x3 + 3x + 1 is the one shown here where the curve crosses the x-axis between - 1 and 0 (Example 1).

Since ƒ(-1) = -3 and ƒ(0) = 1, the Intermediate Value Theorem tells us that the graph of ƒ crosses the x-axis somewhere in the open interval (-1, 0). (See Figure 4.12.) Now, if there were even two points x = a and x = b where ƒ(x) was zero, Rolle’s Theorem would guarantee the existence of a point x = c in between them where ƒ′ was zero. However, the derivative ƒ′(x) = 3x2 + 3 is never zero (because it is always positive). Therefore, ƒ has no more than one zero. Our main use of Rolle’s Theorem is in proving the Mean Value Theorem.

The Mean Value Theorem The Mean Value Theorem, which was first stated by Joseph-Louis Lagrange, is a slanted version of Rolle’s Theorem (Figure 4.13). The Mean Value Theorem guarantees that there is a point where the tangent line is parallel to the secant joining A and B. Tangent parallel to secant

y Slope f ′(c)

B

THEOREM 4—The Mean Value Theorem  Suppose y = ƒ(x) is continuous over a closed interval 3 a, b4 and differentiable on the interval’s interior (a, b). Then there is at least one point c in (a, b) at which

f (b) − f (a) Slope b−a A 0

a y = f(x)

c

b

x

Figure 4.13  Geometrically, the Mean Value Theorem says that somewhere between a and b the curve has at least one tangent parallel to the secant joining A and B.

ƒ(b) - ƒ(a) = ƒ′(c). (1) b - a

Proof  We picture the graph of ƒ and draw a line through the points A(a, ƒ(a)) and B(b, ƒ(b)). (See Figure 4.14.) The secant line is the graph of the function

g(x) = ƒ(a) +

ƒ(b) - ƒ(a) (x - a) (2) b - a

(point-slope equation). The vertical difference between the graphs of ƒ and g at x is h(x) = ƒ(x) - g(x) Historical Biography Joseph-Louis Lagrange (1736–1813)

= ƒ(x) - ƒ(a) -

ƒ(b) - ƒ(a) (x - a). (3) b - a

Figure 4.15 shows the graphs of ƒ, g, and h together.

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4.2  The Mean Value Theorem

y = f (x)

B(b, f (b))

233

B

y = f (x) h(x)

A(a, f (a))

A

y = g(x)

h(x) = f (x) − g(x)

a

y

0

x

1

Figure 4.16  The function ƒ(x) = 21 - x2 satisfies the hypotheses (and conclusion) of the Mean Value Theorem on 3- 1, 14 even though ƒ is not differentiable at - 1 and 1. y B(2, 4)

3

y = x2

2

1

1

x

2

Figure 4.17  As we find in Example 2, c = 1 is where the tangent is parallel to the secant line. s

Distance (ft)

400

s = f (t) (8, 352)

320 240 160 80 0

x

b

x

Figure 4.15  The secant AB is the graph of the function g(x). The function h(x) = ƒ(x) - g(x) gives the vertical distance between the graphs of ƒ and g at x.

ƒ(b) b ƒ(b) h′(c) = ƒ′(c) b ƒ(b) 0 = ƒ′(c) b ƒ(b) - ƒ(a) ƒ′(c) = , b - a h′(x) = ƒ′(x) -

ƒ(a) a ƒ(a) a ƒ(a) a

Derivative of Eq. (3) . . . . . . with x = c h′(c) = 0 Rearranged

which is what we set out to prove.

(1, 1)

A(0, 0)

a

The function h satisfies the hypotheses of Rolle’s Theorem on 3 a, b4 . It is continuous on 3 a, b4 and differentiable on (a, b) because both ƒ and g are. Also, h(a) = h(b) = 0 because the graphs of ƒ and g both pass through A and B. Therefore h′(c) = 0 at some point c∊(a, b). This is the point we want for Equation (1) in the theorem. To verify Equation (1), we differentiate both sides of Equation (3) with respect to x and then set x = c:

4

x

Figure 4.14  The graph of ƒ and the secant AB over the interval 3a, b4 .

y = "1 − x 2, −1 ≤ x ≤ 1 1

−1

b

At this point, the car’s speed was 30 mph. t

5 Time (sec)

Figure 4.18  Distance versus elapsed time for the car in Example 3.

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The hypotheses of the Mean Value Theorem do not require ƒ to be differentiable at either a or b. One-sided continuity at a and b is enough (Figure 4.16).

Example 2  The function ƒ(x) = x2 (Figure 4.17) is continuous for 0 … x … 2 and differentiable for 0 6 x 6 2. Since ƒ(0) = 0 and ƒ(2) = 4, the Mean Value Theorem says that at some point c in the interval, the derivative ƒ′(x) = 2x must have the value (4 - 0)>(2 - 0) = 2. In this case we can identify c by solving the equation 2c = 2 to get c = 1. However, it is not always easy to find c algebraically, even though we know it always exists. A Physical Interpretation We can think of the number (ƒ(b) - ƒ(a))>(b - a) as the average change in ƒ over 3 a, b4 and ƒ′(c) as an instantaneous change. Then the Mean Value Theorem says that at some interior point the instantaneous change must equal the average change over the entire interval.

Example 3  If a car accelerating from zero takes 8 sec to go 352 ft, its average velocity for the 8-sec interval is 352>8 = 44 ft>sec. The Mean Value Theorem says that at some point during the acceleration the speedometer must read exactly 30 mph (44 ft>sec) (Figure 4.18).

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Chapter 4: Applications of Derivatives

Mathematical Consequences At the beginning of the section, we asked what kind of function has a zero derivative over an interval. The first corollary of the Mean Value Theorem provides the answer that only constant functions have zero derivatives. Corollary 1  If ƒ′(x) = 0 at each point x of an open interval (a, b), then ƒ(x) = C for all x∊(a, b), where C is a constant.

Proof   We want to show that ƒ has a constant value on the interval (a, b). We do so by showing that if x1 and x2 are any two points in (a, b) with x1 6 x2 , then ƒ(x1) = ƒ(x2). Now ƒ satisfies the hypotheses of the Mean Value Theorem on 3 x1, x2 4 : It is differentiable at every point of 3 x1, x2 4 and hence continuous at every point as well. Therefore, ƒ(x2) - ƒ(x1) = ƒ′(c) x2 - x1

at some point c between x1 and x2. Since ƒ′ = 0 throughout (a, b), this equation implies successively that

ƒ(x2) - ƒ(x1) = 0, x2 - x1

ƒ(x2) - ƒ(x1) = 0,

and

ƒ(x1) = ƒ(x2).

At the beginning of this section, we also asked about the relationship between two functions that have identical derivatives over an interval. The next corollary tells us that their values on the interval have a constant difference. Corollary 2  If ƒ′(x) = g′(x) at each point x in an open interval (a, b), then there exists a constant C such that ƒ(x) = g(x) + C for all x∊(a, b). That is, ƒ - g is a constant function on (a, b). y

y = x2 + C

C=2

Proof  At each point x∊(a, b) the derivative of the difference function h = ƒ - g is

C=1

h′(x) = ƒ′(x) - g′(x) = 0.

C=0

Thus, h(x) = C on (a, b) by Corollary 1. That is, ƒ(x) - g(x) = C on (a, b), so ƒ(x) = g(x) + C.

C = −1 2

C = −2

1 0

x

Corollaries 1 and 2 are also true if the open interval (a, b) fails to be finite. That is, they remain true if the interval is (a, q), (- q, b), or (- q, q). Corollary 2 plays an important role when we discuss antiderivatives in Section 4.8. It tells us, for instance, that since the derivative of ƒ(x) = x2 on (- q, q) is 2x, any other function with derivative 2x on (- q, q) must have the formula x2 + C for some value of C (Figure 4.19).

−1 −2

Figure 4.19  From a geometric point of view, Corollary 2 of the Mean Value Theorem says that the graphs of functions with identical derivatives on an interval can differ only by a vertical shift there. The graphs of the functions with derivative 2x are the parabolas y = x2 + C, shown here for selected values of C.

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Example 4   Find the function ƒ(x) whose derivative is sin x and whose graph passes through the point (0, 2). Solution  Since the derivative of g(x) = -cos x is g′(x) = sin x, we see that ƒ and g have the same derivative. Corollary 2 then says that ƒ(x) = -cos x + C for some constant C. Since the graph of ƒ passes through the point (0, 2), the value of C is determined from the condition that ƒ(0) = 2: ƒ(0) = -cos (0) + C = 2,

so

C = 3.

The function is ƒ(x) = -cos x + 3.

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4.2  The Mean Value Theorem

235

Finding Velocity and Position from Acceleration We can use Corollary 2 to find the velocity and position functions of an object moving along a vertical line. Assume the object or body is falling freely from rest with acceleration 9.8 m>sec2. We assume the position s(t) of the body is measured positive downward from the rest position (so the vertical coordinate line points downward, in the direction of the motion, with the rest position at 0). We know that the velocity y(t) is some function whose derivative is 9.8. We also know that the derivative of g(t) = 9.8t is 9.8. By Corollary 2, y(t) = 9.8t + C for some constant C. Since the body falls from rest, y(0) = 0. Thus 9.8(0) + C = 0,

and

C = 0.

The velocity function must be y(t) = 9.8t. What about the position function s(t)? We know that s(t) is some function whose derivative is 9.8t. We also know that the derivative of ƒ(t) = 4.9t 2 is 9.8t. By Corollary 2, s(t) = 4.9t 2 + C for some constant C. Since s(0) = 0, 4.9(0)2 + C = 0,

and

C = 0.

2

The position function is s(t) = 4.9t until the body hits the ground. The ability to find functions from their rates of change is one of the very powerful tools of calculus. As we will see, it lies at the heart of the mathematical developments in Chapter 5.

Proofs of the Laws of Logarithms The algebraic properties of logarithms were stated in Section 1.6. We can prove those properties by applying Corollary 2 of the Mean Value Theorem to each of them. The steps in the proofs are similar to those used in solving problems involving logarithms.

Proof that ln bx = ln b + ln x    Recall that ln bx and ln x have the same derivative (Section 3.8). According to Corollary 2 of the Mean Value Theorem, this means that the functions then differ by a constant, ln bx = ln x + C (4)

for some C. Since this last equation holds for all positive values of x, it must hold for x = 1. Hence, ln (b # 1) = ln 1 + C ln b = 0 + C    ln 1 C = ln b.

= 0

By substituting in Equation (4), we conclude that ln bx = ln b + ln x.

Proof that ln x r = r ln x   We use the same-derivative argument again. For all positive values of x, d 1 d ln xr = r (xr) x dx dx =

1 r-1 rx xr

Chain Rule   Derivative Power Rule

d 1 = r#x = (r ln x). dx

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236

Chapter 4: Applications of Derivatives

Since ln xr and r ln x have the same derivative, ln xr = r ln x + C for some constant C. Taking x to be 1 identifies C as zero, and we’re done. You are asked to prove the Quotient Rule for logarithms, b ln a x b = ln b - ln x,

in Exercise 75. The Reciprocal Rule, ln (1>x) = -ln x, is a special case of the Quotient Rule, obtained by taking b = 1 and noting that ln 1 = 0.

Laws of Exponents The laws of exponents for the natural exponential ex are consequences of the algebraic properties of ln x. They follow from the inverse relationship between these functions.

Laws of Exponents for ex For all numbers x, x1, and x2, the natural exponential ex obeys the following laws: 1.  ex1 # ex2 = ex1 + x2 3.

2.  e-x =

1 ex

ex1 = ex1 - x2 4.  (ex1)x2 = ex1x2 = (ex2)x1 ex2

Proof of Law 1  Let y1 = ex1

y2 = ex2. (5)

and

Then x1 = ln y1 and x2 = ln y2  Take logs of both sides of Eqs. (5). x1 + x2 = ln y1 + ln y2 = ln y1 y2   Product Rule for logarithms x1 + x2 ln y1 y2 e = e   Exponentiate. = y1 y2   eln u = u x1 x2 = e e .  The proof of Law 4 is similar. Laws 2 and 3 follow from Law 1 (Exercises 77 and 78).

Exercises  4.2 Checking the Mean Value Theorem Find the value or values of c that satisfy Equation (1) in the conclusion of the Mean Value Theorem for the functions and intervals in Exercises 1–8. 2

1. ƒ(x) = x + 2x - 1, 2>3

2. ƒ(x) = x ,

30, 14

1 3. ƒ(x) = x + x ,

30, 14

1 c , 2d 2

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4. ƒ(x) = 2x - 1, -1

5. ƒ(x) = sin

x,

3-1, 14

6. ƒ(x) = ln (x - 1), 3

2

7. ƒ(x) = x - x ,

31, 34

32, 44

3-1, 24

x3, -2 … x … 0 8. g(x) = e 2 x, 0 6 x … 2

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237

4.2  The Mean Value Theorem

Which of the functions in Exercises 9–14 satisfy the hypotheses of the Mean Value Theorem on the given interval, and which do not? Give reasons for your answers. 9. ƒ(x) = x6>7, 10. ƒ(x) = x2>9,

3- 7, 84 30, 14

11. ƒ(x) = 2x(9 - x), sin x , 12. ƒ(x) = • x 0,

4 + 7, (- q, 0) x2

23. g(t) = 2t + 21 + t - 4, (0, q)

1 + 21 + t - 3.1, (-1, 1) 1 - t u 25. r(u) = u + sin2 a b - 8, (- q, q) 3

24. g(t) =

30, 94

-p … x 6 0

26. r(u) = 2u - cos2 u + 22, (- q, q)

x = 0

2

x - x, -5 … x … -3 13. ƒ(x) = e 2 2x - 3x - 15, - 3 6 x … - 1 2x - 3, 0 … x … 2 1 4. ƒ(x) = e 6x - x2 - 7, 2 6 x … 3

15. The function

ƒ(x) = e

x, 0 … x 6 1 0, x = 1

is zero at x = 0 and x = 1 and differentiable on (0, 1), but its derivative on (0, 1) is never zero. How can this be? Doesn’t Rolle’s Theorem say the derivative has to be zero somewhere in (0, 1)? Give reasons for your answer. 16. For what values of a, m, and b does the function 3, ƒ(x) = c - x2 + 3x + a, mx + b,

x = 0 0 6 x 6 1 1 … x … 2

satisfy the hypotheses of the Mean Value Theorem on the interval 30, 24 ? Roots (Zeros) 17. a.  Plot the zeros of each polynomial on a line together with the zeros of its first derivative.

22. ƒ(x) = x3 +

27. r(u) = sec u -

28. r(u) = tan u - cot u - u, (0, p>2) Finding Functions from Derivatives 29. Suppose that ƒ(-1) = 3 and that ƒ′(x) = 0 for all x. Must ƒ(x) = 3 for all x? Give reasons for your answer.

30. Suppose that ƒ(0) = 5 and that ƒ′(x) = 2 for all x. Must ƒ(x) = 2x + 5 for all x? Give reasons for your answer. 31. Suppose that ƒ′(x) = 2x for all x. Find ƒ(2) if a. ƒ(0) = 0  b.  ƒ(1) = 0  c.  ƒ(-2) = 3. 32. What can be said about functions whose derivatives are constant? Give reasons for your answer. In Exercises 33–38, find all possible functions with the given derivative. b.  y′ = x2

33. a.  y′ = x

b.  y′ = 3x + 9

1 x2

b.  y′ = 7 -

34. a.  y′ = 3x 35. a.  y′ = 36. a.  y′ =

ii) y = x2 + 8x + 15

c.  y′ = x3

2

1 2 2x

37. a.  y′ = sin 2t 38. a.  y′ = sec2 u

i) y = x2 - 4

1 + 5, (0, p>2) u3

2

b.  y′ =

3 x2

1 2x

c.  y′ = 4x3 - 3x2 + 9 c.  y′ = - 9 + c.  y′ = 4x -

5 x2 1

2x

t b.  y′ = cos 2

c.  y′ = sin 2t + cos

t 2

b.  y′ = 2u

c.  y′ = 2u - sec2 u

iii) y = x3 - 3x2 + 4 = (x + 1)(x - 2)2

In Exercises 39–42, find the function with the given derivative whose graph passes through the point P.

iv) y = x3 - 33x2 + 216x = x(x - 9)(x - 24)

39. ƒ′(x) = 2x - 3, P(3, 2)

b. Use Rolle’s Theorem to prove that between every two zeros of xn + an - 1xn - 1 + g + a1 x + a0 there lies a zero of

40. g′(x) =

nxn - 1 + (n - 1)an - 1xn - 2 + g + a1. 18. Suppose that ƒ″ is continuous on 3a, b4 and that ƒ has three zeros in the interval. Show that ƒ″ has at least one zero in (a, b). Generalize this result. 19. Show that if ƒ″ 7 0 throughout an interval 3a, b4 , then ƒ′ has at most one zero in 3a, b4 . What if ƒ″ 6 0 throughout 3a, b4 instead? 20. Show that a cubic polynomial can have at most three real zeros.

Show that the functions in Exercises 21–28 have exactly one zero in the given interval. 21. ƒ(x) = x4 + 3x + 1,

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3- 2, - 14

2 + 6x2, P(1, 1) x2

3 41. ƒ′(x) = e2x, P a0, b 2

42. r′(t) = sec t tan t - 1, P(0, 0) Finding Position from Velocity or Acceleration Exercises 43–46 give the velocity y = ds>dt and initial position of an object moving along a coordinate line. Find the object’s position at time t.

43. y = 9.8t + 2, s(0) = 14 44. y = 32t - 2, s(0.5) = 4 45. y = sin pt, s(0) = 0

2t 2 46. y = p cos p , s(p2) = 1

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Chapter 4: Applications of Derivatives

Exercises 47–50 give the acceleration a = d 2s>dt 2, initial velocity, and initial position of an object moving on a coordinate line. Find the object’s position at time t. 47. a = 18 e3t, y(0) = 20, s(0) = 7 48. a = 9.8, y(0) = - 16, s(0) = 0 49. a = - 9 sin 3t, y(0) = 3, s(0) = - 4 50. a =

9 3t cos p , y(0) = 0, s(0) = - 1 p2

Applications 51. Temperature change  It took 14 sec for a mercury thermometer to rise from - 19°C to 100°C when it was taken from a freezer and placed in boiling water. Show that somewhere along the way the mercury was rising at the rate of 8.5°C>sec.

52. A trucker handed in a ticket at a toll booth showing that in 2 hours she had covered 159 mi on a toll road with speed limit 65 mph. The trucker was cited for speeding. Why? 53. Classical accounts tell us that a 170-oar trireme (ancient Greek or Roman warship) once covered 184 sea miles in 24 hours. Explain why at some point during this feat the trireme’s speed exceeded 7.5 knots (sea or nautical miles per hour). 54. A marathoner ran the 26.2-mi New York City Marathon in 2.2 hours. Show that at least twice the marathoner was running at exactly 11 mph, assuming the initial and final speeds are zero. 55. Show that at some instant during a 2-hour automobile trip the car’s speedometer reading will equal the average speed for the trip. 56. Free fall on the moon  On our moon, the acceleration of gravity is 1.6 m>sec2. If a rock is dropped into a crevasse, how fast will it be going just before it hits bottom 30 sec later? Theory and Examples 57. The geometric mean of a and b The geometric mean of two positive numbers a and b is the number 2ab. Show that the value of c in the conclusion of the Mean Value Theorem for ƒ(x) = 1>x on an interval of positive numbers 3a, b4 is c = 2ab.

58. The arithmetic mean of a and b The arithmetic mean of two numbers a and b is the number (a + b)>2. Show that the value of c in the conclusion of the Mean Value Theorem for ƒ(x) = x2 on any interval 3a, b4 is c = (a + b)>2.

T 59. Graph the function

ƒ(x) = sin x sin (x + 2) - sin2 (x + 1). What does the graph do? Why does the function behave this way? Give reasons for your answers. 60. Rolle’s Theorem a. Construct a polynomial ƒ(x) that has zeros at x = -2, - 1, 0, 1, and 2. b. Graph ƒ and its derivative ƒ′ together. How is what you see related to Rolle’s Theorem? c. Do g(x) = sin x and its derivative g′ illustrate the same phenomenon as ƒ and ƒ′? 61. Unique solution  Assume that ƒ is continuous on 3a, b4 and differentiable on (a, b). Also assume that ƒ(a) and ƒ(b) have

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opposite signs and that ƒ′ ≠ 0 between a and b. Show that ƒ(x) = 0 exactly once between a and b. 62. Parallel tangents  Assume that ƒ and g are differentiable on 3a, b4 and that ƒ(a) = g(a) and ƒ(b) = g(b). Show that there is at least one point between a and b where the tangents to the graphs of ƒ and g are parallel or the same line. Illustrate with a sketch. 63. Suppose that ƒ′(x) … 1 for 1 … x … 4. Show that ƒ(4) ƒ(1) … 3. 64. Suppose that 0 6 ƒ′(x) 6 1>2 for all x-values. Show that ƒ(-1) 6 ƒ(1) 6 2 + ƒ(- 1). 65. Show that 0 cos x - 1 0 … 0 x 0 for all x-values. (Hint: Consider ƒ(t) = cos t on 30, x4 .)

66. Show that for any numbers a and b, the sine inequality 0 sin b sin a 0 … 0 b - a 0 is true.

67. If the graphs of two differentiable functions ƒ(x) and g(x) start at the same point in the plane and the functions have the same rate of change at every point, do the graphs have to be identical? Give reasons for your answer.

68. If 0 ƒ(w) - ƒ(x) 0 … 0 w - x 0 for all values w and x and ƒ is a differentiable function, show that -1 … ƒ′(x) … 1 for all x-values. 69. Assume that ƒ is differentiable on a … x … b and that ƒ(b) 6 ƒ(a). Show that ƒ′ is negative at some point between a and b.

70. Let ƒ be a function defined on an interval 3a, b4 . What conditions could you place on ƒ to guarantee that min ƒ′ …

ƒ(b) - ƒ(a) … max ƒ′, b - a

where min ƒ′ and max ƒ′ refer to the minimum and maximum values of ƒ′ on 3a, b4 ? Give reasons for your answers.

T 71. Use the inequalities in Exercise 70 to estimate ƒ(0.1) if ƒ′(x) = 1>(1 + x4 cos x) for 0 … x … 0.1 and ƒ(0) = 1. T 72. Use the inequalities in Exercise 70 to estimate ƒ(0.1) if ƒ′(x) = 1>(1 - x4) for 0 … x … 0.1 and ƒ(0) = 2.

73. Let ƒ be differentiable at every value of x and suppose that ƒ(1) = 1, that ƒ′ 6 0 on (- q, 1), and that ƒ′ 7 0 on (1, q). a. Show that ƒ(x) Ú 1 for all x. b. Must ƒ′(1) = 0? Explain. 74. Let ƒ(x) = px2 + qx + r be a quadratic function defined on a closed interval 3a, b4 . Show that there is exactly one point c in (a, b) at which ƒ satisfies the conclusion of the Mean Value Theorem.

75. Use the same-derivative argument, as was done to prove the Product and Power Rules for logarithms, to prove the Quotient Rule property. 76. Use the same-derivative argument to prove the identities a. tan-1 x + cot-1 x =

p p   b.  sec-1 x + csc-1 x = 2 2

77. Starting with the equation ex1ex2 = ex1 + x2, derived in the text, show that e-x = 1>ex for any real number x. Then show that ex1 >ex2 = ex1 - x2 for any numbers x1 and x2. 78. Show that (ex1)x2 = ex1 x2 = (ex2)x1 for any numbers x1 and x2.

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4.3  Monotonic Functions and the First Derivative Test

239

4.3 Monotonic Functions and the First Derivative Test In sketching the graph of a differentiable function, it is useful to know where it increases (rises from left to right) and where it decreases (falls from left to right) over an interval. This section gives a test to determine where it increases and where it decreases. We also show how to test the critical points of a function to identify whether local extreme values are present.

Increasing Functions and Decreasing Functions As another corollary to the Mean Value Theorem, we show that functions with positive derivatives are increasing functions and functions with negative derivatives are decreasing functions. A function that is increasing or decreasing on an interval is said to be monotonic on the interval. Corollary 3  Suppose that ƒ is continuous on 3 a, b4 and differentiable on (a, b). If ƒ′(x) 7 0 at each point x∊(a, b), then ƒ is increasing on 3 a, b4 . If ƒ′(x) 6 0 at each point x∊(a, b), then ƒ is decreasing on 3 a, b4 .

Proof  Let x1 and x2 be any two points in 3 a, b4 with x1 6 x2. The Mean Value Theorem applied to ƒ on 3 x1, x2 4 says that ƒ(x2) - ƒ(x1) = ƒ′(c)(x2 - x1)

for some c between x1 and x2. The sign of the right-hand side of this equation is the same as the sign of ƒ ′(c) because x2 - x1 is positive. Therefore, ƒ(x2) 7 ƒ(x1) if ƒ′ is positive on (a, b) and ƒ(x2) 6 ƒ(x1) if ƒ′ is negative on (a, b). Corollary 3 tells us that ƒ(x) = 2x is increasing on the interval 3 0, b4 for any b 7 0 because ƒ′(x) = 1> 2x is positive on (0, b). The derivative does not exist at x = 0, but Corollary 3 still applies. The corollary is valid for infinite as well as finite intervals, so ƒ(x) = 2x is increasing on 3 0, q). To find the intervals where a function ƒ is increasing or decreasing, we first find all of the critical points of ƒ. If a 6 b are two critical points for ƒ, and if the derivative ƒ′ is continuous but never zero on the interval (a, b), then by the Intermediate Value Theorem applied to ƒ′, the derivative must be everywhere positive on (a, b), or everywhere negative there. One way we can determine the sign of ƒ′ on (a, b) is simply by evaluating the derivative at a single point c in (a, b). If ƒ′(c) 7 0, then ƒ′(x) 7 0 for all x in (a, b) so ƒ is increasing on 3 a, b4 by Corollary 3; if ƒ′(c) 6 0, then ƒ is decreasing on 3 a, b4 . The next example illustrates how we use this procedure.

y y = x3 − 12x − 5 20 (−2, 11)

Example 1   Find the critical points of ƒ(x) = x3 - 12x - 5 and identify the open

10

−4 −3 −2 −1 0

intervals on which ƒ is increasing and on which ƒ is decreasing.

1

2

3

4

Solution  The function ƒ is everywhere continuous and differentiable. The first derivative ƒ′(x) = 3x2 - 12 = 3(x2 - 4) = 3(x + 2)(x - 2)

−10 −20

x

(2, −21)

Figure 4.20  The function ƒ(x) = x3 - 12x - 5 is monotonic on three separate intervals (Example 1).

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is zero at x = -2 and x = 2. These critical points subdivide the domain of ƒ to create nonoverlapping open intervals (- q, -2), (-2, 2), and (2, q) on which ƒ′ is either positive or negative. We determine the sign of ƒ′ by evaluating ƒ′ at a convenient point in each subinterval. The behavior of ƒ is determined by then applying Corollary 3 to each subinterval. The results are summarized in the following table, and the graph of ƒ is given in Figure 4.20.

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Interval

- q 6 x 6 -2

-2 6 x 6 2

2 6 x 6 q

ƒ′(-3) = 15

ƒ′(0) = -12

ƒ′(3) = 15

+

-

+

ƒ′ evaluated Sign of ƒ′

increasing

Behavior of ƒ

−3

−2

decreasing 0

−1

increasing 1

2

x

3

We used “strict” less-than inequalities to identify the intervals in the summary table for Example 1, since open intervals were specified. Corollary 3 says that we could use … inequalities as well. That is, the function ƒ in the example is increasing on - q 6 x … -2, decreasing on -2 … x … 2, and increasing on 2 … x 6 q. We do not talk about whether a function is increasing or decreasing at a single point.

Historical Biography Edmund Halley (1656–1742)

First Derivative Test for Local Extrema In Figure 4.21, at the points where ƒ has a minimum value, ƒ′ 6 0 immediately to the left and ƒ′ 7 0 immediately to the right. (If the point is an endpoint, there is only one side to consider.) Thus, the function is decreasing on the left of the minimum value and it is increasing on its right. Similarly, at the points where ƒ has a maximum value, ƒ′ 7 0 immediately to the left and ƒ′ 6 0 immediately to the right. Thus, the function is increasing on the left of the maximum value and decreasing on its right. In summary, at a local extreme point, the sign of ƒ′(x) changes.

Local max f′ = 0 No extremum f′ = 0 f′ > 0

Absolute max f ′ undefined y = f(x)

No extremum f′ = 0

f′ > 0

f′ < 0

f′ < 0 f′ < 0 Local min

Local min f′ = 0

f′ > 0 Absolute min a

c1

c2

c3

c4

c5

b

x

Figure 4.21  The critical points of a function locate where it is increasing and where it is decreasing. The first derivative changes sign at a critical point where a local extremum occurs.

These observations lead to a test for the presence and nature of local extreme values of differentiable functions. First Derivative Test for Local Extrema  Suppose that c is a critical point of a continuous function ƒ, and that ƒ is differentiable at every point in some interval containing c except possibly at c itself. Moving across this interval from left to right, 1.  if ƒ′ changes from negative to positive at c, then ƒ has a local minimum at c; 2.  if ƒ′ changes from positive to negative at c, then ƒ has a local maximum at c; 3.  if ƒ′ does not change sign at c (that is, ƒ′ is positive on both sides of c or negative on both sides), then ƒ has no local extremum at c.

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4.3  Monotonic Functions and the First Derivative Test

The test for local extrema at endpoints is similar, but there is only one side to consider in determining whether ƒ is increasing or decreasing, based on the sign of ƒ′.

Proof of the First Derivative Test   Part (1). Since the sign of ƒ′ changes from negative to positive at c, there are numbers a and b such that a 6 c 6 b, ƒ′ 6 0 on (a, c), and ƒ′ 7 0 on (c, b). If x∊(a, c), then ƒ(c) 6 ƒ(x) because ƒ′ 6 0 implies that ƒ is decreasing on 3 a, c4 . If x∊(c, b), then ƒ(c) 6 ƒ(x) because ƒ′ 7 0 implies that ƒ is increasing on 3 c, b4 . Therefore, ƒ(x) Ú ƒ(c) for every x∊(a, b). By definition, ƒ has a local minimum at c. Parts (2) and (3) are proved similarly.

Example 2   Find the critical points of ƒ(x) = x1>3(x - 4) = x4>3 - 4x1>3. Identify the open intervals on which ƒ is increasing and decreasing. Find the function’s local and absolute extreme values. Solution  The function ƒ is continuous at all x since it is the product of two continuous functions, x1>3 and (x - 4). The first derivative d 4>3 4 4 (x - 4x1>3) = x1>3 - x-2>3 3 3 dx 4(x - 1) 4 = x-2>3(x - 1) = 3 3x2>3 is zero at x = 1 and undefined at x = 0. There are no endpoints in the domain, so the critical points x = 0 and x = 1 are the only places where ƒ might have an extreme value. The critical points partition the x-axis into open intervals on which ƒ′ is either positive or negative. The sign pattern of ƒ′ reveals the behavior of ƒ between and at the critical points, as summarized in the following table. ƒ′(x) =

Interval Sign of ƒ′

y 4 y = x13(x − 4) 2

Behavior of ƒ

x 6 0 -

0 6 x 6 1 -

x 7 1 +

decreasing

decreasing

increasing

0

−1

1

x

2

1 −1 0 −1

1

2

3

4

x

−2 −3

(1, −3)

Figure 4.22  The function ƒ(x) = x1>3(x - 4) decreases when x 6 1 and increases when x 7 1 (Example 2).

Corollary 3 to the Mean Value Theorem implies that ƒ decreases on (- q, 0), decreases on (0, 1), and increases on (1, q). The First Derivative Test for Local Extrema tells us that ƒ does not have an extreme value at x = 0 (ƒ′ does not change sign) and that ƒ has a local minimum at x = 1 (ƒ′ changes from negative to positive). The value of the local minimum is ƒ(1) = 11>3(1 - 4) = -3. This is also an absolute minimum since ƒ is decreasing on (- q, 1) and increasing on (1, q). Figure 4.22 shows this value in relation to the function’s graph. Note that limx S 0 ƒ′(x) = - q, so the graph of ƒ has a vertical tangent at the origin. 

Example 3   Find the critical points of ƒ(x) = (x2 - 3)ex. Identify the open intervals on which ƒ is increasing and decreasing. Find the function’s local and absolute extreme values. Solution  The function ƒ is continuous and differentiable for all real numbers, so the critical points occur only at the zeros of ƒ′.

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Using the Derivative Product Rule, we find the derivative ƒ′(x) = (x2 - 3) #

d x d 2 e + (x - 3) # ex dx dx = (x2 - 3) # ex + (2x) # ex = (x2 + 2x - 3)ex.

Since ex is never zero, the first derivative is zero if and only if x2 + 2x - 3 = 0 (x + 3)(x - 1) = 0. The zeros x = -3 and x = 1 partition the x-axis into open intervals as follows.

y

y = (x 2 − 3)e x

4 3

Interval Sign of ƒ′

2 1 1

−5 −4 −3 −2 −1 −1

2

3

x

Behavior of ƒ

x 6 -3 +

-3 6 x 6 1 -

1 6 x +

increasing

decreasing

increasing

−4

−2

−3

−2

0

−1

1

2

x

3

−3 −4

We can see from the table that there is a local maximum (about 0.299) at x = -3 and a local minimum (about -5.437) at x = 1. The local minimum value is also an absolute minimum because ƒ(x) 7 0 for 0 x 0 7 23. There is no absolute maximum. The function increases on (- q, -3) and (1, q) and decreases on (-3, 1). Figure 4.23 shows the graph.

−5 −6

Figure 4.23  The graph of ƒ(x) = (x2 - 3)ex (Example 3).

Exercises  4.3 Analyzing Functions from Derivatives Answer the following questions about the functions whose derivatives are given in Exercises 1–14:

a. What are the critical points of ƒ? b. On what open intervals is ƒ increasing or decreasing? c. At what points, if any, does ƒ assume local maximum and minimum values? 1. ƒ′(x) = 2x(x + 3)

2. ƒ′(x) = (x - 1)(x + 2)

3. ƒ′(x) = (x - 1)2(x + 2)

4. ƒ′(x) = (x - 1)2(x + 2)2

Identifying Extrema In Exercises 15–44:

a. Find the open intervals on which the function is increasing and decreasing. b. Identify the function’s local and absolute extreme values, if any, saying where they occur. 15. 2

6. ƒ′(x) = (x - 7)(x + x2(x - 2) 7. ƒ′(x) = , x + 3 (x - 4)(x + 8. ƒ′(x) = (x + 3)(x 9. ƒ′(x) = 1 11. ƒ′(x) = x

-1>3

1)(x + 5)

(x + 2)

6

10. ƒ′(x) = 3 -

3

x

−3 −2 −1 −1

12. ƒ′(x) = x

-1>2

2x

2

3

1

2

3

x

−2

y

17.

1

18.

y

, x≠0

(x - 3)

13. ƒ′(x) = (sin x - 1)(2 cos x + 1), 0 … x … 2p 14. ƒ′(x) = (sin x + cos x)(sin x - cos x), 0 … x … 2p

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2

y = f (x)

1

−2

9) , x ≠ -3, 7 7)

4 , x ≠ 0 x2

1

−3 −2 −1 −1

x ≠ -3

2

y = f (x)

1

5. ƒ′(x) = (x + 5)e-x

y

16.

y

2 1 −3 −2 −1 −1 −2

y = f (x) 1

2

y = f (x) 3

x

2 1

−3 −2 −1 −1

x

−2

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4.3  Monotonic Functions and the First Derivative Test

19. g(t) = - t 2 - 3t + 3

20. g(t) = - 3t 2 + 9t + 5

3

2

22. h(x) = 2x - 18x

2

3

24. ƒ(u) = 6u - u 3

25. ƒ(r) = 3r + 16r

26. h(r) = (r + 7)3

23. ƒ(u) = 3u - 4u 3

4

63. ƒ(x) = csc2 x - 2 cot x, 0 6 x 6 p -p p 6 x 6 64. ƒ(x) = sec2 x - 2 tan x, 2 2

3

21. h(x) = - x + 2x

2

4

3

2

27. ƒ(x) = x - 8x + 16

28. g(x) = x - 4x + 4x

3 29. H(t) = t 4 - t 6 2

30. K(t) = 15t 3 - t 5

31. ƒ(x) = x - 6 2x - 1

32. g(x) = 4 2x - x2 + 3

33. g(x) = x 28 - x2

34. g(x) = x2 25 - x

37. ƒ(x) = x1>3(x + 8)

x3 3x + 1 38. g(x) = x2>3(x + 5)

39. h(x) = x1>3(x2 - 4)

40. k(x) = x2>3(x2 - 4)

41. ƒ(x) = e2x + e-x

42. ƒ(x) = e2x

35. ƒ(x) =

2

x - 3 , x ≠ 2 x - 2

36. ƒ(x) =

2

44. ƒ(x) = x2 ln x

43. ƒ(x) = x ln x

243

In Exercises 45–56: a. Identify the function’s local extreme values in the given domain, and say where they occur. b. Which of the extreme values, if any, are absolute? T c.  Support your findings with a graphing calculator or computer grapher. 45. ƒ(x) = 2x - x2, - q 6 x … 2 46. ƒ(x) = (x + 1)2, - q 6 x … 0 47. g(x) = x2 - 4x + 4, 1 … x 6 q 48. g(x) = - x2 - 6x - 9, - 4 … x 6 q 49. ƒ(t) = 12t - t 3, - 3 … t 6 q 50. ƒ(t) = t 3 - 3t 2, - q 6 t … 3 x3 - 2x2 + 4x, 0 … x 6 q 3 52. k(x) = x3 + 3x2 + 3x + 1, - q 6 x … 0 51. h(x) =

53. ƒ(x) = 225 - x2, - 5 … x … 5

54. ƒ(x) = 2x2 - 2x - 3, 3 … x 6 q

x - 2 , 0 … x 6 1 x2 - 1 x2 , -2 6 x … 1 56. g(x) = 4 - x2 55. g(x) =

Theory and Examples Show that the functions in Exercises 65 and 66 have local extreme values at the given values of u, and say which kind of local extreme the function has. u 65. h(u) = 3 cos , 0 … u … 2p, at u = 0 and u = 2p 2 u 66. h(u) = 5 sin , 0 … u … p, at u = 0 and u = p 2 67. Sketch the graph of a differentiable function y = ƒ(x) through the point (1, 1) if ƒ′(1) = 0 and

a. ƒ′(x) 7 0 for x 6 1 and ƒ′(x) 6 0 for x 7 1; b. ƒ′(x) 6 0 for x 6 1 and ƒ′(x) 7 0 for x 7 1; c. ƒ′(x) 7 0 for x ≠ 1; d. ƒ′(x) 6 0 for x ≠ 1. 68. Sketch the graph of a differentiable function y = ƒ(x) that has a. a local minimum at (1, 1) and a local maximum at (3, 3); b. a local maximum at (1, 1) and a local minimum at (3, 3); c. local maxima at (1, 1) and (3, 3); d. local minima at (1, 1) and (3, 3). 69. Sketch the graph of a continuous function y = g(x) such that a. g(2) = 2, 0 6 g′ 6 1 for x 6 2, g′(x) S 1- as x S 2-, - 1 6 g′ 6 0 for x 7 2, and g′(x) S -1+ as x S 2+; b. g(2) = 2, g′ 6 0 for x 6 2, g′(x) S - q as x S 2-, g′ 7 0 for x 7 2, and g′(x) S q as x S 2+. 70. Sketch the graph of a continuous function y = h(x) such that a. h(0) = 0, -2 … h(x) … 2 for all x, h′(x) S q as x S 0-, and h′(x) S q as x S 0+; b. h(0) = 0, -2 … h(x) … 0 for all x, h′(x) S q as x S 0-, and h′(x) S - q as x S 0+. 71. Discuss the extreme-value behavior of the function ƒ(x) = x sin (1>x), x ≠ 0. How many critical points does this function have? Where are they located on the x-axis? Does ƒ have an absolute minimum? An absolute maximum? (See Exercise 49 in Section 2.3.)

In Exercises 57–64:

72. Find the open intervals on which the function ƒ(x) = ax2 + bx + c, a ≠ 0, is increasing and decreasing. Describe the reasoning behind your answer.

a. Find the local extrema of each function on the given interval, and say where they occur.

73. Determine the values of constants a and b so that ƒ(x) = ax2 + bx has an absolute maximum at the point (1, 2).

T b. Graph the function and its derivative together. Comment on the behavior of ƒ in relation to the signs and values of ƒ′.

74. Determine the values of constants a, b, c, and d so that ƒ(x) = ax3 + bx2 + cx + d has a local maximum at the point (0, 0) and a local minimum at the point (1, -1).

57. ƒ(x) = sin 2x, 0 … x … p 58. ƒ(x) = sin x - cos x, 0 … x … 2p 59. ƒ(x) = 23 cos x + sin x, -p 60. ƒ(x) = -2x + tan x, 2 x x 61. ƒ(x) = - 2 sin , 0 … 2 2 62. ƒ(x) = - 2 cos x - cos2 x,

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0 … x … 2p p 6 x 6 2 x … 2p -p … x … p

75. Locate and identify the absolute extreme values of a. ln (cos x) on 3- p>4, p>34, b. cos (ln x) on 31>2, 24.

76.  a. Prove that ƒ(x) = x - ln x is increasing for x 7 1. b. Using part (a), show that ln x 6 x if x 7 1. 77. Find the absolute maximum and minimum values of ƒ(x) = ex - 2x on 30, 14 .

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78. Where does the periodic function ƒ(x) = 2esin (x>2) take on its extreme values and what are these values?

b. Use the result in part (a) to show that ex Ú 1 + x +

y

81. Show that increasing functions and decreasing functions are oneto-one. That is, show that for any x1 and x2 in I, x2 ≠ x1 implies ƒ(x2) ≠ ƒ(x1).

y = 2e sin (x2)

x

0

79. Find the absolute maximum value of ƒ(x) = x2 ln (1>x) and say where it is assumed.

Use the results of Exercise 81 to show that the functions in Exercises 82–86 have inverses over their domains. Find a formula for dƒ -1 >dx using Theorem 3, Section 3.8. 82. ƒ(x) = (1>3)x + (5>6) 3

84. ƒ(x) = 1 - 8x

80. a. Prove that ex Ú 1 + x if x Ú 0.

1 2 x. 2

83. ƒ(x) = 27x3

85. ƒ(x) = (1 - x)3

86. ƒ(x) = x5>3

4.4  Concavity and Curve Sketching y

CA

VE

UP

y = x3

CO N W

f ′ increases

0

x

CO NC AV E

DO

f ′ decreases

N

Figure 4.24  The graph of ƒ(x) = x3 is concave down on (- q, 0) and concave up on (0, q) (Example 1a).

We have seen how the first derivative tells us where a function is increasing, where it is decreasing, and whether a local maximum or local minimum occurs at a critical point. In this section we see that the second derivative gives us information about how the graph of a differentiable function bends or turns. With this knowledge about the first and second derivatives, coupled with our previous understanding of symmetry and asymptotic behavior studied in Sections 1.1 and 2.6, we can now draw an accurate graph of a function. By organizing all of these ideas into a coherent procedure, we give a method for sketching graphs and revealing visually the key features of functions. Identifying and knowing the locations of these features is of major importance in mathematics and its applications to science and engineering, especially in the graphical analysis and interpretation of data.

Concavity As you can see in Figure 4.24, the curve y = x3 rises as x increases, but the portions defined on the intervals (- q, 0) and (0, q) turn in different ways. As we approach the origin from the left along the curve, the curve turns to our right and falls below its tangents. The slopes of the tangents are decreasing on the interval (- q, 0). As we move away from the origin along the curve to the right, the curve turns to our left and rises above its tangents. The slopes of the tangents are increasing on the interval (0, q). This turning or bending behavior defines the concavity of the curve. Definition  The graph of a differentiable function y = ƒ(x) is (a) concave up on an open interval I if ƒ′ is increasing on I; (b) concave down on an open interval I if ƒ′ is decreasing on I. If y = ƒ(x) has a second derivative, we can apply Corollary 3 of the Mean Value Theorem to the first derivative function. We conclude that ƒ′ increases if ƒ″ 7 0 on I, and decreases if ƒ″ 6 0. The Second Derivative Test for Concavity Let y = ƒ(x) be twice-differentiable on an interval I. 1.  If ƒ″ 7 0 on I, the graph of ƒ over I is concave up. 2.  If ƒ″ 6 0 on I, the graph of ƒ over I is concave down.

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If y = ƒ(x) is twice-differentiable, we will use the notations ƒ″ and y″ interchangeably when denoting the second derivative.

y 4

y = x2

Example 1

3

P

−2

−1

1

CO NC AV E

U VE

y″ > 0

(a) The curve y = x3 (Figure 4.24) is concave down on (- q, 0) where y″ = 6x 6 0 and

UP

CA CON

2

245

0

1

concave up on (0, q) where y″ = 6x 7 0. (b) The curve y = x2 (Figure 4.25) is concave up on (- q, q) because its second derivative y″ = 2 is always positive.

y″ > 0 x

2

Figure 4.25  The graph of ƒ(x) = x2 is concave up on every interval (Example 1b).

Example 2   Determine the concavity of y = 3 + sin x on 3 0, 2p4 .

Solution  The first derivative of y = 3 + sin x is y′ = cos x, and the second derivative is y″ = -sin x. The graph of y = 3 + sin x is concave down on (0, p), where y″ = -sin x is negative. It is concave up on (p, 2p), where y″ = -sin x is positive (Figure 4.26).

Points of Inflection The curve y = 3 + sin x in Example 2 changes concavity at the point (p, 3). Since the first derivative y′ = cos x exists for all x, we see that the curve has a tangent line of slope -1 at the point (p, 3). This point is called a point of inflection of the curve. Notice from Figure 4.26 that the graph crosses its tangent line at this point and that the second derivative y″ = -sin x has value 0 when x = p. In general, we have the following definition.

y 4

y = 3 + sin x (p, 3)

3

Definition  A point (c, ƒ(c)) where the graph of a function has a tangent line and where the concavity changes is a point of inflection.

2 1 0 −1

2p

p

x

y ″ = −sin x

Figure 4.26  Using the sign of y″ to determine the concavity of y (Example 2).

At a point of inflection (c, ƒ(c)), either ƒ″(c) = 0 or ƒ″(c) fails to exist.

y 2

y=

x53

The next example illustrates a function having a point of inflection where the first derivative exists, but the second derivative fails to exist.

1 −1

0 −1

We observed that the second derivative of ƒ(x) = 3 + sin x is equal to zero at the inflection point (p, 3). Generally, if the second derivative exists at a point of inflection (c, ƒ(c)), then ƒ″(c) = 0. This follows immediately from the Intermediate Value Theorem whenever ƒ″ is continuous over an interval containing x = c because the second derivative changes sign moving across this interval. Even if the continuity assumption is dropped, it is still true that ƒ″(c) = 0, provided the second derivative exists (although a more advanced argument is required in this noncontinuous case). Since a tangent line must exist at the point of inflection, either the first derivative ƒ′(c) exists (is finite) or the graph has a vertical tangent at the point. At a vertical tangent neither the first nor second derivative exists. In summary, we conclude the following result.

1 Point of inflection

x

−2

Figure 4.27  The graph of ƒ(x) = x5>3 has a horizontal tangent at the origin where the concavity changes, although ƒ″ does not exist at x = 0 (Example 3).

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Example 3   The graph of ƒ(x) = x5>3 has a horizontal tangent at the origin because ƒ′(x) = (5>3)x2>3 = 0 when x = 0. However, the second derivative ƒ″(x) =

d 5 2>3 10 -1>3 a x b = x 9 dx 3

fails to exist at x = 0. Nevertheless, ƒ″(x) 6 0 for x 6 0 and ƒ″(x) 7 0 for x 7 0, so the second derivative changes sign at x = 0 and there is a point of inflection at the origin. The graph is shown in Figure 4.27.

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Here is an example showing that an inflection point need not occur even though both derivatives exist and ƒ″ = 0.

y y = x4 2

Example 4   The curve y = x4 has no inflection point at x = 0 (Figure 4.28). Even

1

though the second derivative y″ = 12x2 is zero there, it does not change sign.

y″ = 0 −1

1

0

x

As our final illustration, we show a situation in which a point of inflection occurs at a vertical tangent to the curve where neither the first nor the second derivative exists.

Figure 4.28  The graph of y = x4 has no inflection point at the origin, even though y″ = 0 there (Example 4).

Example 5   The graph of y = x1>3 has a point of inflection at the origin because the

second derivative is positive for x 6 0 and negative for x 7 0: y″ =

y Point of inflection

However, both y′ = x-2>3 >3 and y″ fail to exist at x = 0, and there is a vertical tangent there. See Figure 4.29.

y = x13 0

d 2 1>3 d 1 -2>3 2 1 x 2 = dx a x b = - x-5>3. 3 9 dx2

x

Figure 4.29  A point of inflection where y′ and y″ fail to exist (Example 5).

Caution  Example 4 in Section 4.1 (Figure 4.9) shows that the function ƒ(x) = x2>3 does not have a second derivative at x = 0 and does not have a point of inflection there (there is no change in concavity at x = 0). Combined with the behavior of the function in Example 5 above, we see that when the second derivative does not exist at x = c, an inflection point may or may not occur there. So we need to be careful about interpreting functional behavior whenever first or second derivatives fail to exist at a point. At such points the graph can have vertical tangents, corners, cusps, or various discontinuities. To study the motion of an object moving along a line as a function of time, we often are interested in knowing when the object’s acceleration, given by the second derivative, is positive or negative. The points of inflection on the graph of the object’s position function reveal where the acceleration changes sign.

Example 6   A particle is moving along a horizontal coordinate line (positive to the right) with position function s(t) = 2t 3 - 14t 2 + 22t - 5,

t Ú 0.

Find the velocity and acceleration, and describe the motion of the particle. Solution  The velocity is y(t) = s′(t) = 6t 2 - 28t + 22 = 2(t - 1)(3t - 11), and the acceleration is a(t) = y′(t) = s″(t) = 12t - 28 = 4(3t - 7). When the function s(t) is increasing, the particle is moving to the right; when s(t) is decreasing, the particle is moving to the left. Notice that the first derivative (y = s′) is zero at the critical points t = 1 and t = 11>3. Interval 0 6 t 6 1 1 6 t 6 11>3 11>3 6 t Sign of Y = s′ + - + Behavior of s increasing decreasing increasing Particle motion right left right The particle is moving to the right in the time intervals 3 0, 1) and (11>3, q), and moving to the left in (1, 11>3). It is momentarily stationary (at rest) at t = 1 and t = 11>3.

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4.4  Concavity and Curve Sketching

247

The acceleration a(t) = s″(t) = 4(3t - 7) is zero when t = 7>3. Interval Sign of a = s″ Graph of s

0 6 t 6 7>3 - concave down

7>3 6 t + concave up

The particle starts out moving to the right while slowing down, and then reverses and begins moving to the left at t = 1 under the influence of the leftward acceleration over the time interval 3 0, 7>3). The acceleration then changes direction at t = 7>3 but the particle continues moving leftward, while slowing down under the rightward acceleration. At t = 11>3 the particle reverses direction again: moving to the right in the same direction as the acceleration, so it is speeding up.

Second Derivative Test for Local Extrema Instead of looking for sign changes in ƒ′ at critical points, we can sometimes use the following test to determine the presence and nature of local extrema. THEOREM 5—Second Derivative Test for Local Extrema  Suppose ƒ″ is continuous on an open interval that contains x = c. 1. If ƒ′(c) = 0 and ƒ″(c) 6 0, then ƒ has a local maximum at x = c. 2. If ƒ′(c) = 0 and ƒ″(c) 7 0, then ƒ has a local minimum at x = c. 3. If ƒ′(c) = 0 and ƒ″(c) = 0, then the test fails. The function ƒ may have a local maximum, a local minimum, or neither.

f ′ = 0, f ″ < 0 1 local max

f ′ = 0, f ″ > 0 1 local min

Proof  Part (1). If ƒ″(c) 6 0, then ƒ″(x) 6 0 on some open interval I containing the point c, since ƒ″ is continuous. Therefore, ƒ′ is decreasing on I. Since ƒ′(c) = 0, the sign of ƒ′ changes from positive to negative at c so ƒ has a local maximum at c by the First Derivative Test. The proof of Part (2) is similar. For Part (3), consider the three functions y = x4, y = -x4, and y = x3. For each function, the first and second derivatives are zero at x = 0. Yet the function y = x4 has a local minimum there, y = -x4 has a local maximum, and y = x3 is increasing in any open interval containing x = 0 (having neither a maximum nor a minimum there). Thus the test fails. This test requires us to know ƒ″ only at c itself and not in an interval about c. This makes the test easy to apply. That’s the good news. The bad news is that the test is inconclusive if ƒ″ = 0 or if ƒ″ does not exist at x = c. When this happens, use the First Derivative Test for local extreme values. Together ƒ′ and ƒ″ tell us the shape of the function’s graph—that is, where the critical points are located and what happens at a critical point, where the function is increasing and where it is decreasing, and how the curve is turning or bending as defined by its concavity. We use this information to sketch a graph of the function that captures its key features.

Example 7   Sketch a graph of the function ƒ(x) = x4 - 4x3 + 10 using the following steps. (a) Identify where the extrema of ƒ occur. (b) Find the intervals on which ƒ is increasing and the intervals on which ƒ is decreasing.

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Chapter 4: Applications of Derivatives

(c) Find where the graph of ƒ is concave up and where it is concave down. (d) Sketch the general shape of the graph for ƒ. (e) Plot some specific points, such as local maximum and minimum points, points of

inflection, and intercepts. Then sketch the curve. Solution  The function ƒ is continuous since ƒ′(x) = 4x3 - 12x2 exists. The domain of ƒ is (- q, q), and the domain of ƒ′ is also (- q, q). Thus, the critical points of ƒ occur only at the zeros of ƒ′. Since ƒ′(x) = 4x3 - 12x2 = 4x2(x - 3), the first derivative is zero at x = 0 and x = 3. We use these critical points to define intervals where ƒ is increasing or decreasing. Interval x 6 0 0 6 x 6 3 3 6 x Sign of ƒ′ - - + Behavior of ƒ decreasing decreasing increasing (a) Using the First Derivative Test for local extrema and the table above, we see that there is no extremum at x = 0 and a local minimum at x = 3. (b) Using the table above, we see that ƒ is decreasing on (- q, 04 and 3 0, 34 , and increasing on 3 3, q). (c) ƒ″(x) = 12x2 - 24x = 12x(x - 2) is zero at x = 0 and x = 2. We use these points to define intervals where ƒ is concave up or concave down. x 6 0 + concave up

Interval Sign of ƒ″ Behavior of ƒ

0 6 x 6 2 - concave down

2 6 x + concave up

We see that ƒ is concave up on the intervals (- q, 0) and (2, q), and concave down on (0, 2). (d) Summarizing the information in the last two tables, we obtain the following. x * 0

y 20

0 * x * 2

2 * x * 3

3 * x

decreasing decreasing decreasing increasing concave up concave down concave up concave up

y = x 4 − 4x 3 + 10

15 Inflection 10 point 5 −1

0 −5 −10

(0, 10)

1 Inflection point

The general shape of the curve is shown in the accompanying figure.

2

3

4

(2, −6)

x

decr

decr

decr

incr

conc up

conc down

conc up

conc up

0

2

3

infl point

infl point

local min

General shape

−15 −20

(3, −17) Local minimum

Figure 4.30  The graph of ƒ(x) = x4 - 4x3 + 10 (Example 7).

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(e) Plot the curve’s intercepts (if possible) and the points where y′ and y″ are zero. Indicate any local extreme values and inflection points. Use the general shape as a guide to sketch the curve. (Plot additional points as needed.) Figure 4.30 shows the graph of ƒ.

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249

4.4  Concavity and Curve Sketching

The steps in Example 7 give a procedure for graphing the key features of a function. Asymptotes were defined and discussed in Section 2.6. We can find them for rational functions, and the methods in the next section give tools to help find them for more general functions.

Procedure for Graphing y = ƒ(x) 1. Identify the domain of ƒ and any symmetries the curve may have. 2. Find the derivatives y′ and y″. 3. Find the critical points of ƒ, if any, and identify the function’s behavior at each one. 4. Find where the curve is increasing and where it is decreasing. 5. Find the points of inflection, if any occur, and determine the concavity of the curve. 6. Identify any asymptotes that may exist. 7. Plot key points, such as the intercepts and the points found in Steps 3–5, and sketch the curve together with any asymptotes that exist.

Example 8   Sketch the graph of ƒ(x) =

(x + 1)2 . 1 + x2

Solution 1. The domain of ƒ is (- q, q) and there are no symmetries about either axis or the origin (Section 1.1). 2. Find ƒ′ and ƒ″. (x + 1)2 1 + x2 (1 + x2) # 2(x + 1) - (x + 1)2 # 2x ƒ′(x) = (1 + x2)2 2 2(1 - x ) = (1 + x2)2 ƒ(x) =

ƒ″(x) = =

x@intercept at x = - 1, = 1) at x = 0

y@intercept (y

Critical points: x

= - 1, x = 1

(1 + x2)2 # 2( -2x) - 2(1 - x2)3 2(1 + x2) # 2x4 (1 + x2)4 4x(x2 - 3) (1 + x2)3

After some algebra

3. Behavior at critical points. The critical points occur only at x = {1 where ƒ′(x) = 0 (Step 2) since ƒ′ exists everywhere over the domain of ƒ. At x = -1, ƒ″(-1) = 1 7 0, yielding a relative minimum by the Second Derivative Test. At x = 1, f″(1) = -1 6 0, yielding a relative maximum by the Second Derivative test. 4. Increasing and decreasing. We see that on the interval (- q, -1) the derivative ƒ′(x) 6 0, and the curve is decreasing. On the interval (-1, 1), ƒ′(x) 7 0 and the curve is increasing; it is decreasing on (1, q) where ƒ′(x) 6 0 again. 5. Inflection points. Notice that the denominator of the second derivative (Step 2) is always positive. The second derivative ƒ″ is zero when x = - 23, 0, and 23. The second derivative changes sign at each of these points: negative on 1 - q, - 23 2, positive on 1 - 23, 0 2, negative on 1 0, 23 2, and positive again on 1 23, q 2. Thus each point is a point of inflection. The curve is concave down on the interval 1 - q, - 23 2, concave up on 1 - 23, 0 2, concave down on 1 0, 23 2, and concave up again on 1 23, q 2.

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Chapter 4: Applications of Derivatives

y 2

(1, 2)

Point of inflection where x = " 3

6. Asymptotes. Expanding the numerator of ƒ(x) and then dividing both numerator and denominator by x2 gives (x + 1)2 x2 + 2x + 1 =   Expanding numerator 1 + x2 1 + x2 1 + (2>x) + (1>x2) .   Dividing by x2 = (1>x2) + 1

ƒ(x) = y=1

1 Horizontal asymptote −1 Point of inflection where x = − " 3

x

1

(x + 1)2 1 + x2

Figure 4.31  The graph of y = (Example 8).

We see that ƒ(x) S 1+ as x S q and that ƒ(x) S 1- as x S - q. Thus, the line y = 1 is a horizontal asymptote. Since ƒ decreases on (- q, -1) and then increases on (-1, 1), we know that ƒ(-1) = 0 is a local minimum. Although ƒ decreases on (1, q), it never crosses the horizontal asymptote y = 1 on that interval (it approaches the asymptote from above). So the graph never becomes negative, and ƒ(-1) = 0 is an absolute minimum as well. Likewise, ƒ(1) = 2 is an absolute maximum because the graph never crosses the asymptote y = 1 on the interval (- q, -1), approaching it from below. Therefore, there are no vertical asymptotes (the range of ƒ is 0 … y … 2). 7. The graph of ƒ is sketched in Figure 4.31. Notice how the graph is concave down as it approaches the horizontal asymptote y = 1 as x S - q, and concave up in its approach to y = 1 as x S q.

Example 9   Sketch the graph of ƒ(x) =

x2 + 4 . 2x

Solution 1. The domain of ƒ is all nonzero real numbers. There are no intercepts because neither x nor ƒ(x) can be zero. Since ƒ(-x) = -ƒ(x), we note that ƒ is an odd function, so the graph of ƒ is symmetric about the origin. 2. We calculate the derivatives of the function, but first rewrite it in order to simplify our computations: x2 + 4 x 2 = + x   Function simplified for differentiation 2x 2

ƒ(x) =

x2 - 4 1 2   Combine fractions to solve easily ƒ′(x) = - 2 = 2 x 2x2 4 ƒ″(x) = 3   Exists throughout the entire domain of ƒ x ƒ′(x) =

y 4 2 −4

−2 (−2, −2)

0

3. The critical points occur at x = {2 where ƒ′(x) = 0. Since ƒ″(-2) 6 0 and ƒ″(2) 7 0, we see from the Second Derivative Test that a relative maximum occurs at x = -2 with ƒ(-2) = -2, and a relative minimum occurs at x = 2 with ƒ(2) = 2. 4. On the interval (- q, -2) the derivative ƒ′ is positive because x2 - 4 7 0 so the graph is increasing; on the interval (-2, 0) the derivative is negative and the graph is decreasing. Similarly, the graph is decreasing on the interval (0, 2) and increasing on (2, q). 5. There are no points of inflection because ƒ″(x) 6 0 whenever x 6 0, ƒ″(x) 7 0 whenever x 7 0, and ƒ″ exists everywhere and is never zero throughout the domain of ƒ. The graph is concave down on the interval (- q, 0) and concave up on the interval (0, q). 6. From the rewritten formula for ƒ(x), we see that

2 y= x +4 2x

(2, 2) y= x 2 2

4

x

−2 −4

Figure 4.32  The graph of y = (Example 9).

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0.

x2 + 4 2x

lim a

xS0 +

x 2 + x b = + q and 2

lim a

xS0 -

x 2 + x b = - q, 2

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4.4  Concavity and Curve Sketching

251

so the y-axis is a vertical asymptote. Also, as x S q or as x S - q, the graph of ƒ(x) approaches the line y = x>2. Thus y = x>2 is an oblique asymptote. 7. The graph of ƒ is sketched in Figure 4.32.

y 5

Example 10   Sketch the graph of ƒ(x) = e2>x. y = e 2x

Solution  The domain of ƒ is (- q, 0)h(0, q) and there are no symmetries about either axis or the origin. The derivatives of ƒ are

4 3 Inflection 2 point 1 −2

−1

ƒ′(x) = e2>x a-

y=1 0

1

2

x

3

Figure 4.33  The graph of y = e has a point of inflection at (- 1, e-2). The line y = 1 is a horizontal asymptote and x = 0 is a vertical asymptote (Example 10). 2>x

and ƒ″(x) = -

2e2>x 2 b = - 2 2 x x

x2(2e2>x)(-2>x2) - 2e2>x(2x)

=

x4

4e2>x(1 + x) . x4

Both derivatives exist everywhere over the domain of ƒ. Moreover, since e2>x and x2 are both positive for all x ≠ 0, we see that ƒ′ 6 0 everywhere over the domain and the graph is everywhere decreasing. Examining the second derivative, we see that ƒ″(x) = 0 at x = -1. Since e2>x 7 0 and x4 7 0, we have ƒ″ 6 0 for x 6 -1 and ƒ″ 7 0 for x 7 -1, x ≠ 0. Therefore, the point (-1, e-2) is a point of inflection. The curve is concave down on the interval (- q, -1) and concave up over (-1, 0) h (0, q). From Example 7, Section 2.6, we see that limx S 0- ƒ(x) = 0. As x S 0+, we see that 2>x S q, so limx S 0+ ƒ(x) = q and the y-axis is a vertical asymptote. Also, as x S - q or x S q, 2>x S 0 and so limx S - q ƒ(x) = limx Sq ƒ(x) = e0 = 1. Therefore, y = 1 is a horizontal asymptote. There are no absolute extrema, since ƒ never takes on the value 0 and has no absolute maximum. The graph of ƒ is sketched in Figure 4.33.

Example 11   Sketch the graph of ƒ(x) = cos x -

22

2

x over 0 … x … 2p.

Solution  The derivatives of ƒ are ƒ′(x) = -sin x -

y 4 2

!2 y = cos x − x 2 π 2

5π 3π 7π 4 2 4 2π

0

x

−2 −4

inflection points

Figure 4.34  The graph of the function in (Example 11).

22

2

and   ƒ″(x) = -cos x.

Both derivatives exist everywhere over the interval (0, 2p). Within that open interval, the first derivative is zero when sin x = - 22>2, so the critical points are x = 5p>4 and x = 7p>4. Since ƒ″(5p>4) = -cos (5p>4) = 22>2 7 0, the function has a local minimum value of ƒ(5p>4) = -3.48 (evaluated with a calculator) by the Second Derivative Test. Also, ƒ″(7p>4) = -cos (7p>4) = 22>2 6 0, so the function has a local maximum value of ƒ(7p>4) = -3.18. Examining the second derivative, we find that ƒ″ = 0 when x = p>2 or x = 3p>2. We conclude that (p>2, ƒ(p>2) ≈ (p>2, -1.11) and (3p>2, ƒ(3p>2)) ≈ (3p>2, -3.33) are points of inflection because ƒ″ changes sign at these two points. Finally, we evaluate ƒ at the endpoints of the interval to find ƒ(2p) ≈ -3.44. Therefore, the values ƒ(0) = 1 and ƒ(5p>4) ≈ -3.48 are the absolute maximum and absolute minimum values of ƒ over the closed interval 3 0, 2p4 . The graph of ƒ is sketched in Figure 4.34.

Graphical Behavior of Functions from Derivatives As we saw in Examples 7–11, we can learn much about a twice-differentiable function y = ƒ(x) by examining its first derivative. We can find where the function’s graph rises and falls and where any local extrema are located. We can differentiate y′ to learn how the graph bends as it passes over the intervals of rise and fall. We can determine the shape of the function’s graph. Information we cannot get from the derivative is how to place the

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Chapter 4: Applications of Derivatives

graph in the xy-plane. But, as we discovered in Section 4.2, the only additional information we need to position the graph is the value of ƒ at one point. Information about the asymptotes is found using limits (Section 2.6). The following figure summarizes how the first derivative and second derivative affect the shape of a graph.

y = f (x)

y = f (x)

Differentiable 1 smooth, connected; graph may rise and fall

y′ > 0 1 rises from left to right; may be wavy

or

y = f (x)

y′ < 0 1 falls from left to right; may be wavy

or +

y″ > 0 1 concave up throughout; no waves; graph may rise or fall

+

or

y″ < 0 1 concave down throughout; no waves; graph may rise or fall

y″ changes sign at an inflection point

y′ = 0 and y″ < 0 at a point; graph has local maximum

y′ = 0 and y″ > 0 at a point; graph has local minimum

+

y′ changes sign 1 graph has local maximum or local minimum

Exercises  4.4 Analyzing Functions from Graphs Identify the inflection points and local maxima and minima of the functions graphed in Exercises 1–8. Identify the intervals on which the functions are concave up and concave down.

1. y = x 3 − x 2 − 2x + 1 3 y

2

2.

3

4 y = x − 2x2 + 4 4 y

5.

6. p p y = x + sin 2x, − 2p ≤ x ≤ 2p y = tan x − 4x, − < x < 3 2 3 2 y y

2p 3

0

2p 3

7. y = sin 0 x 0 , −2p ≤ x ≤ 2p

0

3.

y = 3 (x 2 − 1)23 4 y

0

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4.

x

x 0

y = 9 x13(x 2 − 7) 14 y

0

x

x

0

8. y = 2 cos x −

y

x

0

x

−p

3p " 2 x, −p ≤ x ≤ 2 y

0

3p 2

x

NOT TO SCALE

Graphing Functions In Exercises 9–58, identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. x

9. y = x2 - 4x + 3 3

11. y = x - 3x + 3

10. y = 6 - 2x - x2 12. y = x(6 - 2x)2

18/11/15 11:58 AM

13. y = - 2x3 + 6x2 - 3

14. y = 1 - 9x - 6x2 - x3

3

15. y = (x - 2) + 1 16. y = 1 - (x + 1)3 17. y = x4 - 2x2 = x2(x2 - 2) 18. y = - x4 + 6x2 - 4 = x2(6 - x2) - 4 19. y = 4x3 - x4 = x3(4 - x) 20. y = x4 + 2x3 = x3(x + 2) 21. y = x5 - 5x4 = x4(x - 5) 22. y = xa

65. y′ = (8x - 5x2)(4 - x)2 66. y′ = (x2 - 2x)(x - 5)2 p p 67. y′ = sec2 x, - 6 x 6 2 2 p p 68. y′ = tan x, - 6 x 6 2 2 u u 69. y′ = cot , 0 6 u 6 2p 70. y′ = csc2 , 0 6 u 6 2p 2 2 p p 71. y′ = tan2 u - 1, - 6 u 6 2 2 72. y′ = 1 - cot2 u, 0 6 u 6 p 73. y′ = cos t, 0 … t … 2p

4 x - 5b 2

74. y′ = sin t, 0 … t … 2p 75. y′ = (x + 1)-2>3

23. y = x + sin x, 0 … x … 2p

-2>3

24. y = x - sin x, 0 … x … 2p

77. y′ = x

25. y = 23x - 2 cos x, 0 … x … 2p -p p 4 6 x 6 26. y = x - tan x, 3 2 2

79. y′ = 2 0 x 0 = e

80. y′ = e

27. y = sin x cos x, 0 … x … p 28. y = cos x + 23 sin x, 0 … x … 2p

29. y = x1>5 31. y =

30. y = x2>5

x 2

2x + 1

32. y =

33. y = 2x - 3x2>3 35. y = x2>3 a

5 - xb 2

37. y = x 28 - x2 2

39. y = 216 - x x2 - 3 41. y = x - 2 8x 43. y = 2 x + 4 45. y = 0 x2 - 1 0

2- x, 2x,

51. y = ln (3 - x2) 53. y = e - 2e

34. y = 5x2>5 - 2x

- 3x

55. y = ln (cos x) 1 1 + e-x

y

82.

y = f ′(x) x P

P

y = f ′(x) x

0 y = f ″(x)

84.

y y = f ′(x)

54. y = xe-x

59. y′ = 2 + x - x2

60. y′ = x2 - x - 6

61. y′ = x(x - 3)2

62. y′ = x2(2 - x)

63. y′ = x(x2 - 12)

64. y′ = (x - 1)2(2x + 3)

y = f ″(x)

83. y

52. y = x (ln x)2 ln x 2x ex 58. y = 1 + ex

x

y = f ″(x)

x 6 0 x Ú 0

56. y =

y

P

3

Sketching the General Shape, Knowing y ′ Each of Exercises 59–80 gives the first derivative of a continuous function y = ƒ(x). Find y″ and then use Steps 2–4 of the graphing procedure on page 263 to sketch the general shape of the graph of ƒ.

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-2x, x … 0 2x, x 7 0

y = f ′(x)

38. y = (2 - x2)3>2 2 40. y = x2 + x

5 x4 + 5 46. y = 0 x2 - 2 x 0

78. y′ = x-4>5(x + 1)

(x - 1)

-x2, x … 0 x 2, x 7 0

81.

36. y = x2>3(x - 5)

ex 50. y = x

49. y = xe1>x

-x

2x + 1

76. y′ = (x - 2)-1>3

Sketching y from Graphs of y′ and y ″ Each of Exercises 81–84 shows the graphs of the first and second derivatives of a function y = ƒ(x). Copy the picture and add to it a sketch of the approximate graph of ƒ, given that the graph passes through the point P.

44. y =

48. y = 2 0 x - 4 0 x

21 - x2

42. y = 2x3 + 1

47. y = 2 0 x 0 = e

57. y =

253

4.4  Concavity and Curve Sketching

x

0 y = f ″(x) P

Graphing Rational Functions Graph the rational functions in Exercises 85–102 using all the steps in the graphing procedure on page 249. x2 - 49 2x2 + x - 1 85. y = 86. y = 2 2 x - 1 x + 5x - 14

x4 + 1 x2 1 89. y = 2 x - 1 87. y =

x2 - 4 2x x2 90. y = 2 x - 1 88. y =

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Chapter 4: Applications of Derivatives

93. y = 95. y = 97. y = 99. y = 101. y = 102. y =

Motion Along a Line  The graphs in Exercises 107 and 108 show the position s = ƒ(t) of an object moving up and down on a coordinate line. (a) When is the object moving away from the origin? Toward the origin? At approximately what times is the (b) velocity equal to zero? (c) Acceleration equal to zero? (d) When is the acceleration positive? Negative?

x2 - 4 x2 - 2 x2 - 4 x + 1 x2 - x + 1 x - 1 3 x + x - 2 x - x2 x - 1 x2(x - 2)

s = f (t)

5

0

Theory and Examples 103. The accompanying figure shows a portion of the graph of a twice-differentiable function y = ƒ(x). At each of the five labeled points, classify y′ and y″ as positive, negative, or zero. y

R

5

104. Sketch a smooth connected curve y = ƒ(x) with

ƒ(0) = 4,

ƒ′(x) 6 0 for

ƒ(2) = 0,

ƒ″(x) 6 0 for x 6 0,

0 x 0 7 2,

ƒ″(x) 7 0 for x 7 0.

0 x 0 6 2,

c = f (x)

105. Sketch the graph of a twice-differentiable function y = ƒ(x) with the following properties. Label coordinates where possible. y Derivatives 0, 0, 0, 0, 0, 0, 0,

y″ y″ y″ y″ y″ y″ y″

7 7 7 = 6 6 6

0 0 0 0 0 0 0

106. Sketch the graph of a twice-differentiable function y = ƒ(x) that passes through the points (- 2, 2), (- 1, 1), (0, 0), (1, 1), and (2, 2) and whose first two derivatives have the following sign patterns. + + y′: -2 0 2 + y″: -1 1

M04_THOM9580_03_SE_C04_223-296.indd 254

t

c

ƒ′(2) = ƒ′(- 2) = 0,

6 = 7 7 7 = 6

15

Cost

ƒ(- 2) = 8,

y′ x 6 2   2 1 y′ 2 6 x 6 4   y′ 4 4 y′ 4 6 x 6 6   y′ 6 7 y′ x 7 6   y′

10 Time (sec)

109. Marginal cost  The accompanying graph shows the hypothetical cost c = ƒ(x) of manufacturing x items. At approximately what production level does the marginal cost change from decreasing to increasing?

x

x

t

s = f (t)

0

T

0

15

s

108.

Q

ƒ′(x) 7 0 for

10 Time (sec)

S

y = f (x) P

s

107. Displacement

x2 - 2 92. y = x2 - 1 x2 94. y = x + 1 x2 - x + 1 96. y = x - 1 x3 - 3x2 + 3x - 1 98. y = x2 + x - 2 x 100. y = x2 - 1 8 (Agnesi>s witch) x2 + 4 4x (Newton>s serpentine) x2 + 4

91. y = -

Displacement

254

20 40 60 80 100 120 Thousands of units produced

x

110. The accompanying graph shows the monthly revenue of the Widget Corporation for the past 12 years. During approximately what time intervals was the marginal revenue increasing? Decreasing? y y = r(t)

0

5

10

t

111. Suppose the derivative of the function y = ƒ(x) is y′ = (x - 1)2(x - 2). At what points, if any, does the graph of ƒ have a local minimum, local maximum, or point of inflection? (Hint: Draw the sign pattern for y′.)

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4.5  Indeterminate Forms and L’Hôpital’s Rule

112. Suppose the derivative of the function y = ƒ(x) is

255

120. Suppose that the second derivative of the function y = ƒ(x) is

y′ = (x - 1)2(x - 2)(x - 4).

y″ = x2(x - 2)3(x + 3).

At what points, if any, does the graph of ƒ have a local minimum, local maximum, or point of inflection?

For what x-values does the graph of ƒ have an inflection point?

121. Find the values of constants a, b, and c so that the graph of y = ax3 + bx2 + cx has a local maximum at x = 3, local minimum at x = -1, and inflection point at (1, 11). 122. Find the values of constants a, b, and c so that the graph of y = (x2 + a)>(bx + c) has a local minimum at x = 3 and a local maximum at (-1, -2). Computer Explorations In Exercises 123–126, find the inflection points (if any) on the graph of the function and the coordinates of the points on the graph where the function has a local maximum or local minimum value. Then graph the function in a region large enough to show all these points simultaneously. Add to your picture the graphs of the function’s first and second derivatives. How are the values at which these graphs intersect the x-axis related to the graph of the function? In what other ways are the graphs of the derivatives related to the graph of the function?

123. y = x5 - 5x4 - 240

124. y = x3 - 12x2

125. y =

4 5 x + 16x2 - 25 5

126. y =

x3 x4 - 4x2 + 12x + 20 4 3

127. Graph ƒ(x) = 2x4 - 4x2 + 1 and its first two derivatives together. Comment on the behavior of ƒ in relation to the signs and values of ƒ′ and ƒ″. 128. Graph ƒ(x) = x cos x and its second derivative together for 0 … x … 2p. Comment on the behavior of the graph of ƒ in relation to the signs and values of ƒ″.

4.5 Indeterminate Forms and L’Hôpital’s Rule HISTORICAL BIOGRAPHY Guillaume François Antoine de l’Hôpital (1661–1704) Johann Bernoulli (1667–1748)

John (Johann) Bernoulli discovered a rule using derivatives to calculate limits of fractions whose numerators and denominators both approach zero or + q. The rule is known today as l’Hôpital’s Rule, after Guillaume de l’Hôpital. He was a French nobleman who wrote the first introductory differential calculus text, where the rule first appeared in print. Limits involving transcendental functions often require some use of the rule for their calculation.

Indeterminate Form 0 , 0

If we want to know how the function F(x) =

x - sin x x3

behaves near x = 0 (where it is undefined), we can examine the limit of F(x) as x S 0. We cannot apply the Quotient Rule for limits (Theorem 1 of Chapter 2) because the limit of the denominator is 0. Moreover, in this case, both the numerator and denominator approach 0, and 0>0 is undefined. Such limits may or may not exist in general, but the limit does exist for the function F(x) under discussion by applying l’Hôpital’s Rule, as we will see in Example 1d.

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If the continuous functions ƒ(x) and g (x) are both zero at x = a, then ƒ(x) x S a g(x) lim

cannot be found by substituting x = a. The substitution produces 0>0, a meaningless expression, which we cannot evaluate. We use 0>0 as a notation for an expression known as an indeterminate form. Other meaningless expressions often occur, such as q > q, q # 0, q - q, 00, and 1q, which cannot be evaluated in a consistent way; these are called indeterminate forms as well. Sometimes, but not always, limits that lead to indeterminate forms may be found by cancelation, rearrangement of terms, or other algebraic manipulations. This was our experience in Chapter 2. It took considerable analysis in Section 2.4 to find limx S 0 (sin x)>x. But we have had success with the limit ƒ′(a) = lim

xSa

ƒ(x) - ƒ(a) x - a ,

from which we calculate derivatives and which produces the indeterminant form 0>0 when we attempt to substitute x = a. L’Hôpital’s Rule enables us to draw on our success with derivatives to evaluate limits that otherwise lead to indeterminate forms.

THEOREM 6—L’Hôpital’s Rule  Suppose that ƒ(a) = g(a) = 0, that ƒ and g are differentiable on an open interval I containing a, and that g′(x) ≠ 0 on I if x ≠ a. Then lim

xSa

ƒ(x) ƒ′(x) = lim , g(x) x S a g′(x)

assuming that the limit on the right side of this equation exists.

We give a proof of Theorem 6 at the end of this section.

Caution To apply l’Hôpital’s Rule to ƒ>g, divide the derivative of ƒ by the derivative of g. Do not fall into the trap of taking the derivative of ƒ>g. The quotient to use is ƒ′>g′, not (ƒ>g)′.

Example 1   The following limits involve 0>0 indeterminate forms, so we apply l’Hôpital’s Rule. In some cases, it must be applied repeatedly. (a) lim

xS0

1 21 + x - 1 2 21 + x 1 (b) lim = lim = x 2 xS0 xS0 1 (c) lim

xS0

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3x - sin x 3 - cos x 3 - cos x 2 = lim = 2 = x 1 1 xS0 x=0

21 + x - 1 - x>2

x2

0 ; apply l’Hôpital’s Rule. 0

(1>2)(1 + x)-1>2 - 1>2 2x xS0

0 Still ; apply l’Hôpital’s Rule again. 0

-(1>4)(1 + x)-3>2 1 = - 8 2 xS0

0 Not ; limit is found. 0

= lim

= lim

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(d) lim

xS0

x - sin x x3

0 ; apply l’Hôpital’s Rule. 0

1 - cos x 3x2 sin x = lim x S 0 6x cos x 1 = lim = 6 xS0 6

0 Still ; apply l’Hôpital’s Rule again. 0

= lim

xS0

257

0 Still ; apply l’Hôpital’s Rule again. 0 0 Not ; limit is found. 0

Here is a summary of the procedure we followed in Example 1. Using L’Hôpital’s Rule  To find lim

xSa

ƒ(x) g(x)

by l’Hôpital’s Rule, we continue to differentiate ƒ and g, so long as we still get the form 0>0 at x = a. But as soon as one or the other of these derivatives is different from zero at x = a we stop differentiating. L’Hôpital’s Rule does not apply when either the numerator or denominator has a finite nonzero limit.

Example 2   Be careful to apply l’Hôpital’s Rule correctly: 1 - cos x    00   x + x2 sin x = lim   Not 00 x S 0 1 + 2x

lim

xS0

It is tempting to try to apply l’Hôpital’s Rule again, which would result in lim

xS0

cos x 1 = , 2 2

but this is not the correct limit. l’Hôpital’s Rule can be applied only to limits that give indeterminate forms, and limx S 0 (sin x)>(1 + 2x) does not give an indeterminate form. Instead, this limit is 0>1 = 0, and the correct answer for the original limit is 0. L’Hôpital’s Rule applies to one-sided limits as well.

Example 3   In this example the one-sided limits are different. (a) lim+ xS0

Recall that q and + q mean the same thing.

(b) limxS0

sin x x2

0 0

= lim+ xS0

cos x q = 2x

sin x x2

Positive for x 7 0 0 0

= limxS0

cos x = - q 2x

Indeterminate Forms

Negative for x 6 0 

H , H, H # 0, H − H

Sometimes when we try to evaluate a limit as x S a by substituting x = a we get an indeterminant form like q > q, q # 0, or q - q, instead of 0>0. We first consider the form q > q.

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More advanced treatments of calculus prove that l’Hôpital’s Rule applies to the indeterminate form q > q , as well as to 0>0. If ƒ(x) S { q and g(x) S { q as x S a, then lim

xSa

ƒ(x) ƒ′(x) = lim g(x) x S a g′(x)

provided the limit on the right exists. In the notation x S a, a may be either finite or infinite. Moreover, x S a may be replaced by the one-sided limits x S a+ or x S a-.

Example 4   Find the limits of these q > q forms: (a)

sec x x S p>2 1 + tan x lim

(b)  lim

x Sq

ln x 2 2x

(c)  lim

x Sq

ex . x2

Solution (a) The numerator and denominator are discontinuous at x = p>2, so we investigate the one-sided limits there. To apply l’Hôpital’s Rule, we can choose I to be any open interval with x = p>2 as an endpoint. sec x q lim    q from the left so we apply l’Hôpital’s Rule. x S (p>2)- 1 + tan x =

lim

x S (p>2)-

sec x tan x = sec2 x

lim sin x = 1

x S (p>2)-

The right-hand limit is 1 also, with (- q)>(- q) as the indeterminate form. Therefore, the two-sided limit is equal to 1. (b) lim

x Sq

(c) lim

x Sq

1>x 1>x ln x 1 = lim = lim = 0   1> 2x 2 2x x Sq 1> 2x x Sq 2x

2x 1 = x =

2x

ex ex ex q = lim = lim =  2 2x 2 Sq Sq x x x

Next we turn our attention to the indeterminate forms q # 0 and q - q. Sometimes these forms can be handled by using algebra to convert them to a 0>0 or q > q form. Here again we do not mean to suggest that q # 0 or q - q is a number. They are only notations for functional behaviors when considering limits. Here are examples of how we might work with these indeterminate forms.

Example 5   Find the limits of these q # 0 forms: 1

(a) lim ax sin x b

(b)  lim+ 2x ln x

x Sq

xS0

Solution

sin h 1 1 = 1 a. lim ax sin x b = lim+ a sin hb = lim+ h x Sq hS0 h hS0 b. lim+ 2x ln x = lim+ xS0

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xS0

ln x 1> 2x

1>x

= lim+

= lim+1 -2 2x 2 = 0

xS0

-1>2x3>2

q # 0; let h = 1>x. q # 0 converted to q > q l’Hôpital’s Rule applied

xS0

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259

Example 6   Find the limit of this q - q form: lim a

xS0

1 1 - xb. sin x

Solution  If x S 0+, then sin x S 0+ and

1 1 - x S q - q. sin x Similarly, if x S 0-, then sin x S 0- and 1 1 - x S - q - (- q) = - q + q. sin x Neither form reveals what happens in the limit. To find out, we first combine the fractions: 1 1 x - sin x - x = .  Common denominator is x sin x. sin x x sin x Then we apply l’Hôpital’s Rule to the result: lim a

xS0

x - sin x 1 1 - x b = lim sin x x S 0 x sin x

00

= lim

1 - cos x sin x + x cos x

= lim

sin x 0 = = 0. 2 cos x - x sin x 2

xS0

xS0

Still 00

Indeterminate Powers Limits that lead to the indeterminate forms 1q, 00, and q0 can sometimes be handled by first taking the logarithm of the function. We use l’Hôpital’s Rule to find the limit of the logarithm expression and then exponentiate the result to find the original function limit. This procedure is justified by the continuity of the exponential function and Theorem 10 in Section 2.5, and it is formulated as follows. (The formula is also valid for one-sided limits.) If limx S a ln ƒ(x) = L, then lim ƒ(x) = lim eln ƒ(x) = eL.

xSa

xSa

Here a may be either finite or infinite.

Example 7   Apply l’Hôpital’s Rule to show that limx S 0+ (1 + x)1>x = e. Solution  The limit leads to the indeterminate form 1q . We let ƒ(x) = (1 + x)1>x and find limx S 0+ ln ƒ(x). Since 1 ln ƒ(x) = ln (1 + x)1>x = x ln (1 + x), l’Hôpital’s Rule now applies to give lim ln ƒ(x) = lim+

x S 0+

xS0

ln (1 + x)    00 x

1 1 + x = lim+ 1 xS0 =

l’Hôpital’s Rule applied

1 = 1. 1

Therefore, lim+ (1 + x)1>x = lim+ ƒ(x) = lim+ eln ƒ(x) = e1 = e. xS0

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xS0

xS0

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Example 8  Find limx Sq x1>x. Solution  The limit leads to the indeterminate form q0 . We let ƒ(x) = x1>x and find limx Sq ln ƒ(x). Since ln x ln ƒ(x) = ln x1>x = x , l’Hôpital’s Rule gives ln x lim ln ƒ(x) = lim x x Sq

x Sq

1>x x Sq 1

= lim =

q

q   l’Hôpital’s Rule applied

0 = 0. 1

Therefore lim x1>x = lim ƒ(x) = lim eln ƒ(x) = e0 = 1. x Sq

x Sq

x Sq

Proof of L’Hôpital’s Rule y y = f ′(a)(x − a) f (x) y = g′(a)(x − a)

ƒ(x) = ƒ′(a)(x - a) + P1(x - a) and g(x) = g′(a)(x - a) + P2(x - a)

g(x) 0

a

Before we prove l’Hôpital’s Rule, we consider a special case to provide some geometric insight for its reasonableness. Consider the two functions ƒ(x) and g(x) having continuous derivatives and satisfying ƒ(a) = g(a) = 0, g′(a) ≠ 0. The graphs of ƒ(x) and g(x), together with their linearizations y = ƒ′(a)(x - a) and y = g′(a)(x - a), are shown in Figure 4.35. We know that near x = a, the linearizations provide good approximations to the functions. In fact,

x

where P1 S 0 and P2 S 0 as x S a. So, as Figure 4.35 suggests, lim

xSa

Figure 4.35  The two functions in l’Hôpital’s Rule, graphed with their linear approximations at x = a.

ƒ′(a)(x - a) + P1(x - a) ƒ(x) = lim g(x) x S a g′(a)(x - a) + P2(x - a) = lim

ƒ′(a) + P1 ƒ′(a) = g′(a) + P2 g′(a)

g′(a) ≠ 0

= lim

ƒ′(x) , g′(x)

Continuous derivatives

xSa

xSa

Historical Biography Augustin-Louis Cauchy (1789–1857)

as asserted by l’Hôpital’s Rule. We now proceed to a proof of the rule based on the more general assumptions stated in Theorem 6, which do not require that g′(a) ≠ 0 and that the two functions have continuous derivatives. The proof of l’Hôpital’s Rule is based on Cauchy’s Mean Value Theorem, an extension of the Mean Value Theorem that involves two functions instead of one. We prove Cauchy’s Theorem first and then show how it leads to l’Hôpital’s Rule. THEOREM 7—Cauchy’s Mean Value Theorem  Suppose functions ƒ and g are continuous on 3 a, b4 and differentiable throughout (a, b) and also suppose g′(x) ≠ 0 throughout (a, b). Then there exists a number c in (a, b) at which

When g(x) = x, Theorem 7 is the Mean Value Theorem.

ƒ′(c) ƒ(b) - ƒ(a) . = g′(c) g(b) - g(a)

Proof  We apply the Mean Value Theorem of Section 4.2 twice. First we use it to show that g(a) ≠ g(b). For if g(b) did equal g(a), then the Mean Value Theorem would give g′(c) =

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g(b) - g(a) = 0 b - a

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4.5  Indeterminate Forms and L’Hôpital’s Rule

261

for some c between a and b, which cannot happen because g′(x) ≠ 0 in (a, b). We next apply the Mean Value Theorem to the function F(x) = ƒ(x) - ƒ(a) -

ƒ(b) - ƒ(a) 3 g(x) - g(a) 4 . g(b) - g(a)

This function is continuous and differentiable where ƒ and g are, and F(b) = F(a) = 0. Therefore, there is a number c between a and b for which F′(c) = 0. When expressed in terms of ƒ and g, this equation becomes F′(c) = ƒ′(c) so that y

ƒ′(c) ƒ(b) - ƒ(a) . = g′(c) g(b) - g(a)

slope =

f ′(c) g′(c) B (g(b), f (b))

P

slope =

A

f (b) − f (a) g(b) − g(a)

(g(a), f (a))

0

Cauchy’s Mean Value Theorem has a geometric interpretation for a general winding curve C in the plane joining the two points A = (g(a), ƒ(a)) and B = (g(b), ƒ(b)). In Chapter 10 you will learn how the curve C can be formulated so that there is at least one point P on the curve for which the tangent to the curve at P is parallel to the secant line joining the points A and B. The slope of that tangent line turns out to be the quotient ƒ′>g′ evaluated at the number c in the interval (a, b), which is the left-hand side of the equation in Theorem 7. Because the slope of the secant line joining A and B is ƒ(b) - ƒ(a) , g(b) - g(a)

x

Figure 4.36  There is at least one point P on the curve C for which the slope of the tangent to the curve at P is the same as the slope of the secant line joining the points A(g(a), ƒ(a)) and B(g(b), ƒ(b)).

ƒ(b) - ƒ(a) 3 g′(c) 4 = 0 g(b) - g(a)

the equation in Cauchy’s Mean Value Theorem says that the slope of the tangent line equals the slope of the secant line. This geometric interpretation is shown in Figure 4.36. Notice from the figure that it is possible for more than one point on the curve C to have a tangent line that is parallel to the secant line joining A and B.

Proof of l’Hôpital’s Rule   We first establish the limit equation for the case x S a+. The method needs almost no change to apply to x S a-, and the combination of these two cases establishes the result. Suppose that x lies to the right of a. Then g′(x) ≠ 0, and we can apply Cauchy’s Mean Value Theorem to the closed interval from a to x. This step produces a number c between a and x such that ƒ′(c) ƒ(x) - ƒ(a) = . g′(c) g(x) - g(a) But ƒ(a) = g(a) = 0, so ƒ′(c) ƒ(x) = . g′(c) g(x) As x approaches a, c approaches a because it always lies between a and x. Therefore, lim

x S a+

ƒ(x) ƒ′(c) ƒ′(x) = lim = lim , g(x) c S a+ g′(c) x S a+ g′(x)

which establishes l’Hôpital’s Rule for the case where x approaches a from above. The case where x approaches a from below is proved by applying Cauchy’s Mean Value Theorem to the closed interval 3 x, a4 , x 6 a.

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Chapter 4: Applications of Derivatives

Exercises  4.5 Finding Limits in Two Ways In Exercises 1–6, use l’Hôpital’s Rule to evaluate the limit. Then evaluate the limit using a method studied in Chapter 2.

2x - 10 x S 5 7x 2 - 175

sin 5x x xS0

1. lim

2. lim

5x2 - 3x 3. lim x Sq 9x 2 + 1

x3 - 1 4. lim 3 x S 1 4x - x - 3

1 - cos 10x 5. lim xS0 100x2

3x2 + 7x 6. lim 3 x Sq 5x + 3x + 7

49. lim

uS0

x Sq

55. lim+ x xS0

-1>ln x

59. lim+ xx

8x2 15. lim x S 0 cos x - 1

sin x - x 16. lim xS0 x3

2u - p 17. lim u S p>2 cos (2p - u)

3u + p 18. lim u S -p>3 sin (u + (p>3))

61. lim a

1 - sin u 19. lim u S p>2 1 + cos 2u

x - 1 20. lim x S 1 ln x - sin px

25.

lim - ax -

x S (p>2)

p b sec x 2

3sin u - 1 u uS0

27. lim

x2x - 1

29. lim

26.

x S (p>2)

32. lim

33. lim+

ln (x2 + 2x) ln x

34. lim+

yS0

25y + 25 - 5

y

37. lim (ln 2x - ln (x + 1)) x Sq

39. lim+ xS0

p - xb tan x 2

(ln x)2 ln (sin x)

1 1 41. lim+ a b x - 1 ln x xS1

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36. lim

yS0

2x + 1

sec x 69. lim - tan x x S (p>2) 2x - 3x + 4x

xS0

1 x 60. lim+ a1 + x b xS0

62. lim a x Sq

x2 + 1 1>x b x + 2

64. lim+ x (ln x)2 xS0

66. lim+ sin x # ln x xS0

68. lim+ xS0

2x

2sin x

cot x 70. lim+ csc x xS0 72. lim

2x + 4x - 2x

x x S -q 5

2

ex x xe Sq x

x e-1>x 75. Which one is correct, and which one is wrong? Give reasons for your answers.

ln (ex - 1) ln x

2ay + a2 - a

y

a. lim

xS3

, a 7 0

xS0

3x + 1 1 b x sin x

42. lim+ (csc x - cot x + cos x) xS0

p - xb 2

29x + 1

73. lim

38. lim+ (ln x - ln sin x)

xS0

x Sq

x x Sq 3

log2 x (x + 3)

40. lim+ a

65. lim+ x tan a

71. lim

x Sq log 3

xS0

xS0

67. lim

3x - 1 - 1

ln (x + 1) log2 x

35. lim

a

63. lim+ x2 ln x

58. lim (ex + x)1>x

Theory and Applications L’Hôpital’s Rule does not help with the limits in Exercises 67–74. Try it—you just keep on cycling. Find the limits some other way.

x xS0 2

31. lim

xS0

-

x + 2 x b x - 1

xS0

(1>2)u - 1 u uS0

28. lim 30. lim

x xS0 2

x Sq

x Sq

t sin t - cos t

lim

xS q

1>(2 ln x)

xS0

ln (csc x) x S p>2 (x - (p>2))2 tS0 1

56. lim x1>ln x

x Sq

22. lim 24. lim

xSe

57. lim (1 + 2x)

x - 8x2 x Sq 12x 2 + 5x

t(1 - cos t) t - sin t

sin 3x - 3x + x2 sin x sin 2x

54. lim+ (ln x)1>(x - e)

53. lim (ln x)

3 sin 8x 14. lim 5x xS0

tS0

xS0

xS1

1>x

2 sin 8t 8 13. lim - 4t tS0

23. lim

50. lim

52. lim+ x1>(x - 1)

xS1

12. lim

x2 x S 0 ln (sec x)

(ex - 1)2 x S 0 x sin x

48. lim

u - sin u cos u tan u - u

11. lim

21. lim

x Sq

51. lim+ x1>(1 - x)

3t 3 + 3 10. lim 3 t S -1 4t - t + 3

5x3 - 2x x Sq 7x 3 + 3

46. lim x2e-x

Indeterminate Powers and Products Find the limits in Exercises 51–66.

x2 - 25 8. lim xS - 5 x + 5

t 3 - 3t + 198 9. lim 2 t S -6 t - t - 42

44. lim

x - sin x 47. lim x tan x xS0

Applying l’Hôpital’s Rule Use l’Hôpital’s rule to find the limits in Exercises 7–50.

7x - 49 7. lim 2 x S 7 6x - 294

eh - (1 + h) hS0 h2

cos u - 1 u S 0 eu - u - 1 et + t 2 45. lim t t Sq e - t 43. lim

74. lim+ xS0

x - 3 x - 3 0 1 1 =  b. lim 2 = lim = = 0 6 xS3 x - 3 x2 - 3 x S 3 2x 6

76. Which one is correct, and which one is wrong? Give reasons for your answers. a. lim

x2 - 2x 2x - 2 = lim - sin x x S 0 2x - cos x 2 2 = = 1 = lim 2 + 0 x S 0 2 + sin x

x S 0 x2

b. lim

x2 - 2x 2x - 2 -2 = = 2 = lim - sin x x S 0 2x - cos x 0 - 1

x S 0 x2

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77. Only one of these calculations is correct. Which one? Why are the others wrong? Give reasons for your answers. a. lim x ln x = 0 # (- q) = 0 xS0

a. Use l’Hôpital’s Rule to show that

b. lim+ x ln x = 0 # (- q) = - q

T b. Graph

xS0

c. lim+ x ln x = lim+

ln x -q = q = -1 (1>x)

d. lim+ x ln x = lim+

ln x (1>x)

xS0

xS0

xS0

xS0

= lim+

xS0

(1>x) (- 1>x2)

ƒ(x) = a1 +

= lim+ (- x) = 0 xS0

a. ƒ(x) = x,

g(x) = x2,

(a, b) = (- 2, 0)

b. ƒ(x) = x,

g(x) = x2,

(a, b) arbitrary

g(x) = x2,

(a, b) = (0, 3)

79. Continuous extension  Find a value of c that makes the function 9x - 3 sin 3x , x ≠ 0 5x3 ƒ(x) = c c, x = 0 continuous at x = 0. Explain why your value of c works. 80. For what values of a and b is lim a

xS0

T 81. H − H Form

tan 2x a sin bx + 2 + x b = 0? x3 x

a. Estimate the value of lim 1 x - 2x + x 2 2

x Sq

by graphing ƒ(x) = x - 2x2 + x over a suitably large interval of x-values. b. Now confirm your estimate by finding the limit with l’Hôpital’s Rule. As the first step, multiply ƒ(x) by the fraction 1 x + 2x2 + x 2 > 1 x + 2x2 + x 2 and simplify the new numerator.

82. Find lim

x Sq

1 2x

2

+ 1 - 2x 2.

T 83. 0 , 0 Form  Estimate the value of

2x2 - (3x + 1) 2x + 2 x - 1 xS1 lim

by graphing. Then confirm your estimate with l’Hôpital’s Rule. 84. This exercise explores the difference between the limit lim a1 +

x Sq

and the limit

1 x b x2

1 x lim a1 + x b = e. x Sq

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1 x 1 x b and g(x) = a1 + x b 2 x

together for x Ú 0. How does the behavior of ƒ compare with that of g? Estimate the value of limx Sq ƒ(x).

78. Find all values of c that satisfy the conclusion of Cauchy’s Mean Value Theorem for the given functions and interval.

c. ƒ(x) = x3 >3 - 4x,

1 x lim a1 + x b = e.

x Sq

+

c. Confirm your estimate of limx Sq ƒ(x) by calculating it with l’Hôpital’s Rule. 85. Show that r k lim a1 + b = er. k k Sq

86. Given that x 7 0, find the maximum value, if any, of a. x1>x 2

b. x1>x

n

c. x1>x (n a positive integer) n

d. Show that limx Sq x1>x = 1 for every positive integer n. 87. Use limits to find horizontal asymptotes for each function. 3x + e2x 1 a. y = x tan a x b     b. y = 2x + e3x 2

88. Find ƒ′(0) for ƒ(x) = e

e-1/x , 0,

x ≠ 0 x = 0.

T 89. The continuous extension of (sin x)x to 30, p4

a. Graph ƒ(x) = (sin x)x on the interval 0 … x … p. What value would you assign to ƒ to make it continuous at x = 0?

b. Verify your conclusion in part (a) by finding limx S 0+ ƒ(x) with l’Hôpital’s Rule. c. Returning to the graph, estimate the maximum value of ƒ on 30, p4 . About where is max ƒ taken on?

d. Sharpen your estimate in part (c) by graphing ƒ′ in the same window to see where its graph crosses the x-axis. To simplify your work, you might want to delete the exponential factor from the expression for ƒ′ and graph just the factor that has a zero.

T 90. The function (sin x)tan x (Continuation of Exercise 89.) a. Graph ƒ(x) = (sin x)tan x on the interval - 7 … x … 7. How do you account for the gaps in the graph? How wide are the gaps? b. Now graph ƒ on the interval 0 … x … p. The function is not defined at x = p>2, but the graph has no break at this point. What is going on? What value does the graph appear to give for ƒ at x = p>2? (Hint: Use l’Hôpital’s Rule to find lim ƒ as x S (p>2)- and x S (p>2)+.) c. Continuing with the graphs in part (b), find max ƒ and min ƒ as accurately as you can and estimate the values of x at which they are taken on.

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4.6  Applied Optimization What are the dimensions of a rectangle with fixed perimeter having maximum area? What are the dimensions for the least expensive cylindrical can of a given volume? How many items should be produced for the most profitable production run? Each of these questions asks for the best, or optimal, value of a given function. In this section we use derivatives to solve a variety of optimization problems in mathematics, physics, economics, and business. x

Solving Applied Optimization Problems 1. Read the problem. Read the problem until you understand it. What is given? What is the unknown quantity to be optimized? 2. Draw a picture. Label any part that may be important to the problem. 3. Introduce variables. List every relation in the picture and in the problem as an equation or algebraic expression, and identify the unknown variable. 4. Write an equation for the unknown quantity. If you can, express the unknown as a function of a single variable or in two equations in two unknowns. This may require considerable manipulation. 5. Test the critical points and endpoints in the domain of the unknown. Use what you know about the shape of the function’s graph. Use the first and second derivatives to identify and classify the function’s critical points.

12

x x

x

12 (a)

x

Example 1   An open-top box is to be made by cutting small congruent squares from the corners of a 12-in.-by-12-in. sheet of tin and bending up the sides. How large should the squares cut from the corners be to make the box hold as much as possible?

12 − 2x 12 12 − 2x

x

x

Solution  We start with a picture (Figure 4.37). In the figure, the corner squares are x in. on a side. The volume of the box is a function of this variable:

(b)

Figure 4.37  An open box made by cutting the corners from a square sheet of tin. What size corners maximize the box’s volume (Example 1)? Maximum y

Volume 0

min 2

6

= hlw

Since the sides of the sheet of tin are only 12 in. long, x … 6 and the domain of V is the interval 0 … x … 6. A graph of V (Figure 4.38) suggests a minimum value of 0 at x = 0 and x = 6 and a maximum near x = 2. To learn more, we examine the first derivative of V with respect to x: dV = 144 - 96x + 12x2 = 12(12 - 8x + x2) = 12(2 - x)(6 - x). dx

y = x(12 − 2x)2, 0≤x≤6

min

V(x) = x(12 - 2x)2 = 144x - 48x2 + 4x3.   V

x

NOT TO SCALE

Figure 4.38  The volume of the box in Figure 4.37 graphed as a function of x.

Of the two zeros, x = 2 and x = 6, only x = 2 lies in the interior of the function’s domain and makes the critical-point list. The values of V at this one critical point and two endpoints are Critical point value: V(2) = 128 Endpoint values:

V(0) = 0,

V(6) = 0.

The maximum volume is 128 in3. The cutout squares should be 2 in. on a side.

Example 2   You have been asked to design a one-liter can shaped like a right circular cylinder (Figure 4.39). What dimensions will use the least material? Solution  Volume of can: If r and h are measured in centimeters, then the volume of the can in cubic centimeters is pr 2h = 1000.   1 liter

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= 1000 cm3

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265

2 Surface area of can:  A = 2pr + 2prh ()* ()*

2r

h

circular cylindrical ends wall

How can we interpret the phrase “least material”? For a first approximation we can ignore the thickness of the material and the waste in manufacturing. Then we ask for dimensions r and h that make the total surface area as small as possible while satisfying the constraint pr 2h = 1000. To express the surface area as a function of one variable, we solve for one of the variables in pr 2h = 1000 and substitute that expression into the surface area formula. Solving for h is easier:

Figure 4.39  This one-liter can uses the least material when h = 2r (Example 2).

h =

1000 . pr 2

Thus, A = 2pr 2 + 2prh = 2pr 2 + 2pr a = 2pr 2 +

1000 b pr 2

2000 r .

Our goal is to find a value of r 7 0 that minimizes the value of A. Figure 4.40 suggests that such a value exists. A Tall and thin can Short and wide can —— , r > 0 A = 2pr2 + 2000 r

Tall and thin

min

0

3

Short and wide

r

500 p

Figure 4.40  The graph of A = 2pr 2 + 2000>r is concave up.

Notice from the graph that for small r (a tall, thin cylindrical container), the term 2000>r dominates (see Section 2.6) and A is large. For large r (a short, wide cylindrical container), the term 2pr 2 dominates and A again is large. Since A is differentiable on r 7 0, an interval with no endpoints, it can have a minimum value only where its first derivative is zero. dA 2000 = 4pr dr r2 2000 0 = 4pr r2 4pr 3 = 2000 r = 3 What happens at r = 2 500>p?

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Set dA>dr

= 0.

Multiply by r 2.

3 500 ≈ 5.42  Solve for r. A p

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The second derivative d 2A 4000 = 4p + 3 r dr 2 is positive throughout the domain of A. The graph is therefore everywhere concave up and 3 the value of A at r = 2 500>p is an absolute minimum. The corresponding value of h (after a little algebra) is h =

1000 500 = 2 3 p = 2r. 2 pr A

The one-liter can that uses the least material has height equal to twice the radius, here with r ≈ 5.42 cm and h ≈ 10.84 cm.

Examples from Mathematics and Physics

Example 3   A rectangle is to be inscribed in a semicircle of radius 2. What is the largest area the rectangle can have, and what are its dimensions?

y x 2 + y2 = 4 Qx, "4 − x 2 R

2 −2 −x

0

x 2

Figure 4.41  The rectangle inscribed in the semicircle in Example 3.

x

Solution Let 1 x, 24 - x2 2 be the coordinates of the corner of the rectangle obtained by placing the circle and rectangle in the coordinate plane (Figure 4.41). The length, height, and area of the rectangle can then be expressed in terms of the position x of the lower right-hand corner: Length: 2x,

Height: 24 - x2,

Area: 2x 24 - x2.

Notice that the values of x are to be found in the interval 0 … x … 2, where the selected corner of the rectangle lies. Our goal is to find the absolute maximum value of the function on the domain 3 0, 24 . The derivative

A(x) = 2x 24 - x2

dA -2x2 = + 2 24 - x2 dx 24 - x2

is not defined when x = 2 and is equal to zero when

-2x2 + 2 24 - x2 = 0 24 - x2 -2x2 + 2(4 - x2) = 0 8 - 4x2 = 0 x2 = 2 x = { 22.

Of the two zeros, x = 22 and x = - 22, only x = 22 lies in the interior of A’s domain and makes the critical-point list. The values of A at the endpoints and at this one critical point are Critical point value: A1 22 2 = 2 22 24 - 2 = 4 Endpoint values: A(0) = 0, A(2) = 0.

The area has a maximum value of 4 when the rectangle is 24 - x2 = 22 units high and 2x = 2 22 units long.

Historical Biography Willebrord Snell van Royen (1580–1626)

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Example 4   The speed of light depends on the medium through which it travels, and is generally slower in denser media. Fermat’s principle in optics states that light travels from one point to another along a path for which the time of travel is a minimum. Describe the path that a ray of light will

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4.6  Applied Optimization

follow in going from a point A in a medium where the speed of light is c1 to a point B in a second medium where its speed is c2.

y A a

u1

Angle of incidence u1 x

Medium 2

Solution  Since light traveling from A to B follows the quickest route, we look for a path that will minimize the travel time. We assume that A and B lie in the xy-plane and that the line separating the two media is the x-axis (Figure 4.42). In a uniform medium, where the speed of light remains constant, “shortest time” means “shortest path,” and the ray of light will follow a straight line. Thus the path from A to B will consist of a line segment from A to a boundary point P, followed by another line segment from P to B. Distance traveled equals rate times time, so

Medium 1 P

0

267

b

u2 d−x

d Angle of refraction

x

B

Figure 4.42  A light ray refracted (deflected from its path) as it passes from one medium to a denser medium (Example 4).

Time =

distance rate .

From Figure 4.42, the time required for light to travel from A to P is 2a2 + x2 AP t1 = c = . c1 1

From P to B, the time is 2b2 + (d - x)2 PB t2 = c = . c2 2

The time from A to B is the sum of these: t = t1 + t2 =

2a2 + x2

c1

+

2b2 + (d - x)2

c2

.

This equation expresses t as a differentiable function of x whose domain is 3 0, d4 . We want to find the absolute minimum value of t on this closed interval. We find the derivative dt x d - x = dx c1 2a2 + x2 c2 2b2 + (d - x)2

and observe that it is continuous. In terms of the angles u1 and u2 in Figure 4.42, dtdx negative

0

dtdx zero

− − − − − +++++++++ x0

sin u1 sin u2 dt = c - c . 1 2 dx

dtdx positive x d

Figure 4.43  The sign pattern of dt>dx in Example 4.

The function t has a negative derivative at x = 0 and a positive derivative at x = d. Since dt>dx is continuous over the interval 3 0, d4 , by the Intermediate Value Theorem for continuous functions (Section 2.5), there is a point x0 ∊ 3 0, d4 where dt>dx = 0 (Figure 4.43). There is only one such point because dt>dx is an increasing function of x (Exercise 62). At this unique point we then have sin u1 sin u2 c1 = c2 . This equation is Snell’s Law or the Law of Refraction, and is an important principle in the theory of optics. It describes the path the ray of light follows.

Examples from Economics Suppose that r(x) = the revenue from selling x items c(x) = the cost of producing the x items p(x) = r(x) - c(x) = the profit from producing and selling x items. Although x is usually an integer in many applications, we can learn about the behavior of these functions by defining them for all nonzero real numbers and by assuming they are differentiable functions. Economists use the terms marginal revenue,

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marginal cost, and marginal profit to name the derivatives r′(x), c′(x), and p′(x) of the revenue, cost, and profit functions. Let’s consider the relationship of the profit p to these derivatives. If r(x) and c(x) are differentiable for x in some interval of production possibilities, and if p(x) = r(x) - c(x) has a maximum value there, it occurs at a critical point of p(x) or at an endpoint of the interval. If it occurs at a critical point, then p′(x) = r′(x) c′(x) = 0 and we see that r′(x) = c′(x). In economic terms, this last equation means that At a production level yielding maximum profit, marginal revenue equals marginal cost (Figure 4.44).

y

Dollars

Cost c(x)

Revenue r(x) Break-even point B

Maximum profit, c′(x) = r ′(x)

Local maximum for loss (minimum profit), c′(x) = r ′(x) x Items produced

0

Figure 4.44  The graph of a typical cost function starts concave down and later turns concave up. It crosses the revenue curve at the break-even point B. To the left of B, the company operates at a loss. To the right, the company operates at a profit, with the maximum profit occurring where c′(x) = r′(x). Farther to the right, cost exceeds revenue (perhaps because of a combination of rising labor and material costs and market saturation) and production levels become unprofitable again.

Example 5  Suppose that r(x) = 9x and c(x) = x3 - 6x2 + 15x, where x represents millions of MP3 players produced. Is there a production level that maximizes profit? If so, what is it?

y c(x) = x 3 − 6x2 + 15x

Solution  Notice that r′(x) = 9 and c′(x) = 3x2 - 12x + 15. 3x2 - 12x + 15 = 9  Set c′(x) 3x2 - 12x + 6 = 0

r(x) = 9x

The two solutions of the quadratic equation are Maximum for profit

x1 =

Local maximum for loss 0 2 − "2

2

= r′(x).

2 + "2

NOT TO SCALE

Figure 4.45  The cost and revenue curves for Example 5.

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x

x2 =

12 - 272 = 2 - 22 ≈ 0.586 6

and

12 + 272 = 2 + 22 ≈ 3.414. 6

The possible production levels for maximum profit are x ≈ 0.586 million MP3 players or x ≈ 3.414 million. The second derivative of p(x) = r(x) - c(x) is p″(x) = -c″(x) since r″(x) is everywhere zero. Thus, p″(x) = 6(2 - x), which is negative at x = 2 + 22 and positive at x = 2 - 22. By the Second Derivative Test, a maximum profit occurs at about x = 3.414 (where revenue exceeds costs) and maximum loss occurs at about x = 0.586. The graphs of r(x) and c(x) are shown in Figure 4.45.

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269

Exercises  4.6 Mathematical Applications Whenever you are maximizing or minimizing a function of a single variable, we urge you to graph it over the domain that is appropriate to the problem you are solving. The graph will provide insight before you calculate and will furnish a visual context for understanding your answer.

1. Minimizing perimeter  What is the smallest perimeter possible for a rectangle whose area is 16 in2, and what are its dimensions? 2. Show that among all rectangles with an 8-m perimeter, the one with largest area is a square. 3. The figure shows a rectangle inscribed in an isosceles right triangle whose hypotenuse is 2 units long.

engineer, your job is to find dimensions for the base and height that will make the tank weigh as little as possible. a. What dimensions do you tell the shop to use? b. Briefly describe how you took weight into account. 10. Catching rainwater A 1125 ft3 open-top rectangular tank with a square base x ft on a side and y ft deep is to be built with its top flush with the ground to catch runoff water. The costs associated with the tank involve not only the material from which the tank is made but also an excavation charge proportional to the product xy. a. If the total cost is c = 5(x2 + 4xy) + 10xy,

a. Express the y-coordinate of P in terms of x. (Hint: Write an equation for the line AB.)

what values of x and y will minimize it?

b. Express the area of the rectangle in terms of x.

b. Give a possible scenario for the cost function in part (a).

c. What is the largest area the rectangle can have, and what are its dimensions? y

11. Designing a poster  You are designing a rectangular poster to contain 50 in2 of printing with a 4-in. margin at the top and bottom and a 2-in. margin at each side. What overall dimensions will minimize the amount of paper used? 12. Find the volume of the largest right circular cone that can be inscribed in a sphere of radius 3.

B

P(x, ?) 3 A −1

0

x

1

x

4. A rectangle has its base on the x-axis and its upper two vertices on the parabola y = 12 - x2. What is the largest area the rectangle can have, and what are its dimensions? 5. You are planning to make an open rectangular box from an 8-in.-by15-in. piece of cardboard by cutting congruent squares from the corners and folding up the sides. What are the dimensions of the box of largest volume you can make this way, and what is its volume? 6. You are planning to close off a corner of the first quadrant with a line segment 20 units long running from (a, 0) to (0, b). Show that the area of the triangle enclosed by the segment is largest when a = b. 7. The best fencing plan  A rectangular plot of farmland will be bounded on one side by a river and on the other three sides by a single-strand electric fence. With 800 m of wire at your disposal, what is the largest area you can enclose, and what are its dimensions? 8. The shortest fence A 216 m2 rectangular pea patch is to be enclosed by a fence and divided into two equal parts by another fence parallel to one of the sides. What dimensions for the outer rectangle will require the smallest total length of fence? How much fence will be needed? 9. Designing a tank  Your iron works has contracted to design and build a 500 ft3, square-based, open-top, rectangular steel holding tank for a paper company. The tank is to be made by welding thin stainless steel plates together along their edges. As the production

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3

y x

13. Two sides of a triangle have lengths a and b, and the angle between them is u. What value of u will maximize the triangle’s area? (Hint: A = (1>2)ab sin u.) 14. Designing a can  What are the dimensions of the lightest opentop right circular cylindrical can that will hold a volume of 1000 cm3? Compare the result here with the result in Example 2. 15. Designing a can  You are designing a 1000 cm3 right circular cylindrical can whose manufacture will take waste into account. There is no waste in cutting the aluminum for the side, but the top and bottom of radius r will be cut from squares that measure 2r units on a side. The total amount of aluminum used up by the can will therefore be A = 8r 2 + 2prh rather than the A = 2pr 2 + 2prh in Example 2. In Example 2, the ratio of h to r for the most economical can was 2 to 1. What is the ratio now? T 16. Designing a box with a lid  A piece of cardboard measures 10 in. by 15 in. Two equal squares are removed from the corners of a 10-in. side as shown in the figure. Two equal rectangles are removed from the other corners so that the tabs can be folded to form a rectangular box with lid.

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NOT TO SCALE

x

20. a. The U.S. Postal Service will accept a box for domestic shipment only if the sum of its length and girth (distance around) does not exceed 108 in. What dimensions will give a box with a square end the largest possible volume?

x

x

x

10″

Base

Lid

x

Girth = distance around here

x x

x 15″

a. Write a formula V(x) for the volume of the box. b. Find the domain of V for the problem situation and graph V over this domain. c. Use a graphical method to find the maximum volume and the value of x that gives it. d. Confirm your result in part (c) analytically. T 17. Designing a suitcase  A 24-in.-by-36-in. sheet of cardboard is folded in half to form a 24-in.-by-18-in. rectangle as shown in the accompanying figure. Then four congruent squares of side length x are cut from the corners of the folded rectangle. The sheet is unfolded, and the six tabs are folded up to form a box with sides and a lid.

Length

Square end

T b. Graph the volume of a 108-in. box (length plus girth equals 108 in.) as a function of its length and compare what you see with your answer in part (a). 21. (Continuation of Exercise 20.) a. Suppose that instead of having a box with square ends you have a box with square sides so that its dimensions are h by h by w and the girth is 2h + 2w. What dimensions will give the box its largest volume now?

a. Write a formula V(x) for the volume of the box. Girth

b. Find the domain of V for the problem situation and graph V over this domain. c. Use a graphical method to find the maximum volume and the value of x that gives it. d. Confirm your result in part (c) analytically.

h

e. Find a value of x that yields a volume of 1120 in3. f. Write a paragraph describing the issues that arise in part (b). x

w

x

x

h

x

24″

24″ x

x x

36″

x 18″

The sheet is then unfolded.

24″

T b. Graph the volume as a function of h and compare what you see with your answer in part (a). 22. A window is in the form of a rectangle surmounted by a semicircle. The rectangle is of clear glass, whereas the semicircle is of tinted glass that transmits only half as much light per unit area as clear glass does. The total perimeter is fixed. Find the proportions of the window that will admit the most light. Neglect the thickness of the frame.

Base

36″

18. A rectangle is to be inscribed under the arch of the curve y = 4 cos (0.5x) from x = - p to x = p. What are the dimensions of the rectangle with largest area, and what is the largest area? 19. Find the dimensions of a right circular cylinder of maximum volume that can be inscribed in a sphere of radius 10 cm. What is the maximum volume?

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23. A silo (base not included) is to be constructed in the form of a cylinder surmounted by a hemisphere. The cost of construction per square unit of surface area is twice as great for the hemisphere as it is for the cylindrical sidewall. Determine the dimensions to be used if the volume is fixed and the cost of construction is to be kept to a minimum. Neglect the thickness of the silo and waste in construction.

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24. The trough in the figure is to be made to the dimensions shown. Only the angle u can be varied. What value of u will maximize the trough’s volume?

1′ u

27. Constructing cones  A right triangle whose hypotenuse is 23 m long is revolved about one of its legs to generate a right circular cone. Find the radius, height, and volume of the cone of greatest volume that can be made this way.

h

u 1′ 1′

"3 r

20′

y x 28. Find the point on the line a + = 1 that is closest to the origin. b 25. Paper folding  A rectangular sheet of 8.5-in.-by-11-in. paper is placed on a flat surface. One of the corners is placed on the opposite longer edge, as shown in the figure, and held there as the paper is smoothed flat. The problem is to make the length of the crease as small as possible. Call the length L. Try it with paper. a. Show that L2 = 2x3 >(2x - 8.5). b. What value of x minimizes L2?

C

Crease

L

Q (originally at A)

x P

34. Determine the dimensions of the rectangle of largest area that can be inscribed in a semicircle of radius 3. (See accompanying figure.) 35. What value of a makes ƒ(x) = x2 + (a>x) have

x A

31. A wire b m long is cut into two pieces. One piece is bent into an equilateral triangle and the other is bent into a circle. If the sum of the areas enclosed by each part is a minimum, what is the length of each part? w

5

33. Determine the dimensions of the rectangle of largest area that can be inscribed in the right triangle shown in the accompanying figure.

R

"L2 − x 2

30. Find a positive number for which the sum of its reciprocal and four times its square is the smallest possible.

32. Answer Exercise 31 if one piece is bent into a square and the other into a circle.

c. What is the minimum value of L? D

29. Find a positive number for which the sum of it and its reciprocal is the smallest (least) possible.

4

h 3 w

h r=3

a. a local minimum at x = 2?

B

26. Constructing cylinders  Compare the answers to the following two construction problems. a. A rectangular sheet of perimeter 36 cm and dimensions x cm by y cm is to be rolled into a cylinder as shown in part (a) of the figure. What values of x and y give the largest volume? b. The same sheet is to be revolved about one of the sides of length y to sweep out the cylinder as shown in part (b) of the figure. What values of x and y give the largest volume?

b. a point of inflection at x = 1? 36. What values of a and b make ƒ(x) = x3 + ax2 + bx have a. a local maximum at x = -1 and a local minimum at x = 3? b. a local minimum at x = 4 and a point of inflection at x = 1? Physical Applications 37. Vertical motion  The height above ground of an object moving vertically is given by

s = - 16t 2 + 96t + 112, with s in feet and t in seconds. Find

y

a. the object’s velocity when t = 0; b. its maximum height and when it occurs;

x Circumference = x

x y

y

(a)

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(b)

c. its velocity when s = 0. 38. Quickest route  Jane is 2 mi offshore in a boat and wishes to reach a coastal village 6 mi down a straight shoreline from the point nearest the boat. She can row 2 mph and can walk 5 mph. Where should she land her boat to reach the village in the least amount of time?

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39. Shortest beam  The 8-ft wall shown here stands 27 ft from the building. Find the length of the shortest straight beam that will reach to the side of the building from the ground outside the wall.

Beam

Building

8′ wall 27′

40. Motion on a line  The positions of two particles on the s-axis are s1 = sin t and s2 = sin (t + p>3), with s1 and s2 in meters and t in seconds. a. At what time(s) in the interval 0 … t … 2p do the particles meet? b. What is the farthest apart that the particles ever get? c. When in the interval 0 … t … 2p is the distance between the particles changing the fastest?

T 44. Stiffness of a beam  The stiffness S of a rectangular beam is proportional to its width times the cube of its depth. a. Find the dimensions of the stiffest beam that can be cut from a 12-in.-diameter cylindrical log. b. Graph S as a function of the beam’s width w, assuming the proportionality constant to be k = 1. Reconcile what you see with your answer in part (a). c. On the same screen, graph S as a function of the beam’s depth d, again taking k = 1. Compare the graphs with one another and with your answer in part (a). What would be the effect of changing to some other value of k? Try it. 45. Frictionless cart  A small frictionless cart, attached to the wall by a spring, is pulled 10 cm from its rest position and released at time t = 0 to roll back and forth for 4 sec. Its position at time t is s = 10 cos pt. a. What is the cart’s maximum speed? When is the cart moving that fast? Where is it then? What is the magnitude of the acceleration then? b. Where is the cart when the magnitude of the acceleration is greatest? What is the cart’s speed then?

41. The intensity of illumination at any point from a light source is proportional to the square of the reciprocal of the distance between the point and the light source. Two lights, one having an intensity eight times that of the other, are 6 m apart. How far from the stronger light is the total illumination least? 42. Projectile motion The range R of a projectile fired from the origin over horizontal ground is the distance from the origin to the point of impact. If the projectile is fired with an initial velocity y0 at an angle a with the horizontal, then in Chapter 12 we find that y02 R = g sin 2a, where g is the downward acceleration due to gravity. Find the angle a for which the range R is the largest possible.

0

10

s

46. Two masses hanging side by side from springs have positions s1 = 2 sin t and s2 = sin 2t, respectively. a. At what times in the interval 0 6 t do the masses pass each other? (Hint: sin 2t = 2 sin t cos t.) b. When in the interval 0 … t … 2p is the vertical distance between the masses the greatest? What is this distance? (Hint: cos 2t = 2 cos2 t - 1.)

T 43. Strength of a beam  The strength S of a rectangular wooden beam is proportional to its width times the square of its depth. (See the accompanying figure.) a. Find the dimensions of the strongest beam that can be cut from a 12-in.-diameter cylindrical log. b. Graph S as a function of the beam’s width w, assuming the proportionality constant to be k = 1. Reconcile what you see with your answer in part (a). c. On the same screen, graph S as a function of the beam’s depth d, again taking k = 1. Compare the graphs with one another and with your answer in part (a). What would be the effect of changing to some other value of k? Try it.

12″ d

m1

s1 0 s2

m2

s

47. Distance between two ships  At noon, ship A was 12 nautical miles due north of ship B. Ship A was sailing south at 12 knots (nautical miles per hour; a nautical mile is 2000 yd) and continued to do so all day. Ship B was sailing east at 8 knots and continued to do so all day. a. Start counting time with t = 0 at noon and express the distance s between the ships as a function of t.

w

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b. How rapidly was the distance between the ships changing at noon? One hour later?

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c. The visibility that day was 5 nautical miles. Did the ships ever sight each other? T d. Graph s and ds>dt together as functions of t for -1 … t … 3, using different colors if possible. Compare the graphs and reconcile what you see with your answers in parts (b) and (c). e. The graph of ds>dt looks as if it might have a horizontal asymptote in the first quadrant. This in turn suggests that ds>dt approaches a limiting value as t S q. What is this value? What is its relation to the ships’ individual speeds? 48. Fermat’s principle in optics  Light from a source A is reflected by a plane mirror to a receiver at point B, as shown in the accompanying figure. Show that for the light to obey Fermat’s principle, the angle of incidence must equal the angle of reflection, both measured from the line normal to the reflecting surface. (This result can also be derived without calculus. There is a purely geometric argument, which you may prefer.) Normal Light receiver Light source A

Angle of incidence u1

Angle of reflection u2

B

Plane mirror

49. Tin pest  When metallic tin is kept below 13.2°C, it slowly becomes brittle and crumbles to a gray powder. Tin objects eventually crumble to this gray powder spontaneously if kept in a cold climate for years. The Europeans who saw tin organ pipes in their churches crumble away years ago called the change tin pest because it seemed to be contagious, and indeed it was, for the gray powder is a catalyst for its own formation. A catalyst for a chemical reaction is a substance that controls the rate of reaction without undergoing any permanent change in itself. An autocatalytic reaction is one whose product is a catalyst for its own formation. Such a reaction may proceed slowly at first if the amount of catalyst present is small and slowly again at the end, when most of the original substance is used up. But in between, when both the substance and its catalyst product are abundant, the reaction proceeds at a faster pace. In some cases, it is reasonable to assume that the rate y = dx>dt of the reaction is proportional both to the amount of the original substance present and to the amount of product. That is, y may be considered to be a function of x alone, and y = kx(a - x) = kax - kx2, where x = the amount of product a = the amount of substance at the beginning k = a positive constant. At what value of x does the rate y have a maximum? What is the maximum value of y? 50. Airplane landing path  An airplane is flying at altitude H when it begins its descent to an airport runway that is at horizontal ground distance L from the airplane, as shown in the figure. Assume that the landing path of the airplane is the graph of a cubic polynomial function y = ax3 + bx2 + cx + d, where y(- L) = H and y(0) = 0.

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a. What is dy>dx at x = 0? b. What is dy>dx at x = - L? c. Use the values for dy>dx at x = 0 and x = - L together with y(0) = 0 and y(- L) = H to show that x 3 x 2 y(x) = H c 2a b + 3a b d . L L

y

Landing path

H = Cruising altitude

Airport L

x

Business and Economics 51. It costs you c dollars each to manufacture and distribute backpacks. If the backpacks sell at x dollars each, the number sold is given by a n = x - c + b(100 - x), where a and b are positive constants. What selling price will bring a maximum profit?

52. You operate a tour service that offers the following rates:  \$200 per person if 50 people (the minimum number to book the tour) go on the tour.  For each additional person, up to a maximum of 80 people total, the rate per person is reduced by \$2. It costs \$6000 (a fixed cost) plus \$32 per person to conduct the tour. How many people does it take to maximize your profit? 53. Wilson lot size formula  One of the formulas for inventory management says that the average weekly cost of ordering, paying for, and holding merchandise is hq km , A(q) = q + cm + 2 where q is the quantity you order when things run low (shoes, radios, brooms, or whatever the item might be), k is the cost of placing an order (the same, no matter how often you order), c is the cost of one item (a constant), m is the number of items sold each week (a constant), and h is the weekly holding cost per item (a constant that takes into account things such as space, utilities, insurance, and security). a. Your job, as the inventory manager for your store, is to find the quantity that will minimize A(q). What is it? (The formula you get for the answer is called the Wilson lot size formula.) b. Shipping costs sometimes depend on order size. When they do, it is more realistic to replace k by k + bq, the sum of k and a constant multiple of q. What is the most economical quantity to order now? 54. Production level  Prove that the production level (if any) at which average cost is smallest is a level at which the average cost equals marginal cost. 55. Show that if r(x) = 6x and c(x) = x3 - 6x2 + 15x are your revenue and cost functions, then the best you can do is break even (have revenue equal cost).

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56. Production level  Suppose that c(x) = x3 - 20x2 + 20,000x is the cost of manufacturing x items. Find a production level that will minimize the average cost of making x items. 57. You are to construct an open rectangular box with a square base and a volume of 48 ft3. If material for the bottom costs \$6>ft2 and material for the sides costs \$4>ft2, what dimensions will result in the least expensive box? What is the minimum cost? 58. The 800-room Mega Motel chain is filled to capacity when the room charge is \$50 per night. For each \$10 increase in room charge, 40 fewer rooms are filled each night. What charge per room will result in the maximum revenue per night? Biology 59. Sensitivity to medicine  (Continuation of Exercise 72, Section 3.3.) Find the amount of medicine to which the body is most sensitive by finding the value of M that maximizes the derivative dR>dM, where

and C is a constant.

b. Show that g(x) =

d - x 2b2 + (d - x)2

is a decreasing function of x. c. Show that

dt x d - x = dx c1 2a2 + x2 c2 2b2 + (d - x)2

is an increasing function of x.

63. Let ƒ(x) and g(x) be the differentiable functions graphed here. Point c is the point where the vertical distance between the curves is the greatest. Is there anything special about the tangents to the two curves at c? Give reasons for your answer.

y = f (x)

C M R = M2 a - b 2 3

y = g(x)

60. How we cough a. When we cough, the trachea (windpipe) contracts to increase the velocity of the air going out. This raises the questions of how much it should contract to maximize the velocity and whether it really contracts that much when we cough. Under reasonable assumptions about the elasticity of the tracheal wall and about how the air near the wall is slowed by friction, the average flow velocity y can be modeled by the equation y = c(r0 - r)r 2 cm>sec,

r0 … r … r0 , 2

where r0 is the rest radius of the trachea in centimeters and c is a positive constant whose value depends in part on the length of the trachea. Show that y is greatest when r = (2>3)r0; that is, when the trachea is about 33% contracted. The remarkable fact is that X-ray photographs confirm that the trachea contracts about this much during a cough. T b. Take r0 to be 0.5 and c to be 1 and graph y over the interval 0 … r … 0.5. Compare what you see with the claim that y is at a maximum when r = (2>3)r0.

a

c

a. Explain why you need to consider values of x only in the interval 30, 2p4. b. Is ƒ ever negative? Explain.

65. a.  The function y = cot x - 22 csc x has an absolute maximum value on the interval 0 6 x 6 p. Find it. T b. Graph the function and compare what you see with your answer in part (a). 66. a.  The function y = tan x + 3 cot x has an absolute minimum value on the interval 0 6 x 6 p>2. Find it. T b. Graph the function and compare what you see with your answer in part (a). 67. a.   How close does the curve y = 2x come to the point (3>2, 0)? (Hint: If you minimize the square of the distance, you can avoid square roots.) T b. Graph the distance function D(x) and y = 2x together and reconcile what you see with your answer in part (a). y

(x, " x)

(a2 + 1)(b2 + 1)(c2 + 1)(d 2 + 1) Ú 16. abcd 62. The derivative dt>dx in Example 4 0

ƒ(x) =

x 2a2 + x2

is an increasing function of x.

x

64. You have been asked to determine whether the function ƒ(x) = 3 + 4 cos x + cos 2x is ever negative.

Theory and Examples 61. An inequality for positive integers  Show that if a, b, c, and d are positive integers, then

a. Show that

b

3 a , 0b 2

y = "x

x

68. a.  How close does the semicircle y = 216 - x2 come to the point 1 1, 23 2?

T b. Graph the distance function and y = 216 - x2 together and reconcile what you see with your answer in part (a).

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4.7  Newton’s Method

4.7 Newton’s Method In this section we study a numerical method, called Newton’s method or the Newton– Raphson method, which is a technique to approximate the solution to an equation ƒ(x) = 0. Essentially it uses tangent lines of the graph of y = ƒ(x) near the points where ƒ is zero to estimate the solution. (A value of x where ƒ is zero is a root of the function ƒ and a solution of the equation ƒ(x) = 0.)

Procedure for Newton’s Method y y = f (x) (x0, f (x0))

(x1, f (x1)) (x2, f (x2 )) Root sought 0

x3 Fourth

x2 Third

x1 Second

x

x0 First

The goal of Newton’s method for estimating a solution of an equation ƒ(x) = 0 is to produce a sequence of approximations that approach the solution. We pick the first number x0 of the sequence. Then, under favorable circumstances, the method does the rest by moving step by step toward a point where the graph of ƒ crosses the x-axis (Figure 4.46). At each step the method approximates a zero of ƒ with a zero of one of its linearizations. Here is how it works. The initial estimate, x0, may be found by graphing or just plain guessing. The method then uses the tangent to the curve y = ƒ(x) at (x0, ƒ(x0)) to approximate the curve, calling the point x1 where the tangent meets the x-axis (Figure 4.46). The number x1 is usually a better approximation to the solution than is x0. The point x2 where the tangent to the curve at (x1, ƒ(x1)) crosses the x-axis is the next approximation in the sequence. We continue on, using each approximation to generate the next, until we are close enough to the root to stop. We can derive a formula for generating the successive approximations in the following way. Given the approximation xn, the point-slope equation for the tangent to the curve at (xn, ƒ(xn)) is y = ƒ(xn) + ƒ′(xn)(x - xn).

APPROXIMATIONS

We can find where it crosses the x-axis by setting y = 0 (Figure 4.47):

Figure 4.46  Newton’s method starts with an initial guess x0 and (under favorable circumstances) improves the guess one step at a time.

0 = ƒ(xn) + ƒ′(xn)(x - xn) ƒ(xn) = x - xn ƒ′(xn)

y

x = xn -

y = f (x)

Point: (xn, f (xn )) Slope: f ′(xn ) Tangent line equation: y − f (xn ) = f ′(xn )(x − xn ) (xn, f (xn))

Tangent line (graph of linearization of f at xn )

xn

≠ 0

Newton’s Method 1. Guess a first approximation to a solution of the equation ƒ(x) = 0. A graph of y = ƒ(x) may help. 2. Use the first approximation to get a second, the second to get a third, and so on, using the formula

x

f (xn ) xn+1 = xn − f '(xn )

Figure 4.47  The geometry of the successive steps of Newton’s method. From xn we go up to the curve and follow the tangent line down to find xn + 1.

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If ƒ′(xn)

This value of x is the next approximation xn + 1. Here is a summary of Newton’s method.

Root sought 0

ƒ(xn) ƒ′(xn)

xn + 1 = xn -

ƒ(xn) , ƒ′(xn)

if ƒ′(xn) ≠ 0.(1)

Applying Newton’s Method Applications of Newton’s method generally involve many numerical computations, making them well suited for computers or calculators. Nevertheless, even when the calculations

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are done by hand (which may be very tedious), they give a powerful way to find solutions of equations. In our first example, we find decimal approximations to 22 by estimating the positive root of the equation ƒ(x) = x2 - 2 = 0.

Example 1   Find the positive root of the equation ƒ(x) = x2 - 2 = 0. Solution  With ƒ(x) = x2 - 2 and ƒ′(x) = 2x, Equation (1) becomes xn 2 - 2 2xn xn 1 = xn + x n 2 xn 1 = + x . n 2

xn + 1 = xn -

The equation xn + 1 =

xn 1 + x n 2

enables us to go from each approximation to the next with just a few keystrokes. With the starting value x0 = 1, we get the results in the first column of the following table. (To five decimal places, 22 = 1.41421.)   Error x0 x1 x2 x3

y = x3 − x − 1

15

Example 2  Find the x-coordinate of the point where the curve y = x3 - x crosses

10

the horizontal line y = 1.

5

−1

0

1 -0.41421 1 1.5 0.08579 1 1.41667 0.00246 3 1.41422 0.00001 5

Newton’s method is the method used by most software applications to calculate roots because it converges so fast (more about this later). If the arithmetic in the table in Example 1 had been carried to 13 decimal places instead of 5, then going one step further would have given 22 correctly to more than 10 decimal places.

y 20

= = = =

Number of correct digits

1

2

3

x

Figure 4.48  The graph of ƒ(x) = x3 - x - 1 crosses the x-axis once; this is the root we want to find (Example 2).

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Solution  The curve crosses the line when x3 - x = 1 or x3 - x - 1 = 0. When does ƒ(x) = x3 - x - 1 equal zero? Since ƒ(1) = -1 and ƒ(2) = 5, we know by the Intermediate Value Theorem there is a root in the interval (1, 2) (Figure 4.48). We apply Newton’s method to ƒ with the starting value x0 = 1. The results are displayed in Table 4.1 and Figure 4.49. At n = 5, we come to the result x6 = x5 = 1.3247 17957. When xn + 1 = xn, Equation (1) shows that ƒ(xn) = 0. We have found a solution of ƒ(x) = 0 to nine decimals.

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4.7  Newton’s Method

y = x3 − x − 1

Table 4.1  The result of applying Newton’s method to ƒ(x) = x 3 - x - 1

(1.5, 0.875)

with x0 = 1

Root sought

1

n

x1

x2

x0

xn

ƒ(xn)

xn+1 = xn −

ƒ′(xn)

x

1.5 1.3478 (1, −1)

Figure 4.49  The first three x-values in Table 4.1 (four decimal places).

-1 2 0 1 1 1.5   0.875 5.75 2 1.3478 26087   0.1006 82173 4.4499 05482 3 1.3252 00399   0.0020 58362 4.2684 68292 4 1.3247 18174   0.0000 00924 4.2646 34722 5 1.3247 17957 -1.8672E@13 4.2646 32999

ƒ(xn) ƒ′(xn)

1.5 1.3478 26087 1.3252 00399 1.3247 18174 1.3247 17957 1.3247 17957

In Figure 4.50 we have indicated that the process in Example 2 might have started at the point B0(3, 23) on the curve, with x0 = 3. Point B0 is quite far from the x-axis, but the tangent at B0 crosses the x-axis at about (2.12, 0), so x1 is still an improvement over x0. If we use Equation (1) repeatedly as before, with ƒ(x) = x3 - x - 1 and ƒ′(x) = 3x2 - 1, we obtain the nine-place solution x7 = x6 = 1.3247 17957 in seven steps.

y 25 B0(3, 23) 20 y = x3 − x − 1

Convergence of the Approximations

15

10 B1(2.12, 6.35) 5 −1" 3 −1

Root sought 1" 3

0

x2 x1 1

1.6 2.12

x0 3

x

Figure 4.50  Any starting value x0 to the right of x = 1> 23 will lead to the root in Example 2.

ƒ(x) = e

y y = f (x)

r 0

In Chapter 9 we define precisely the idea of convergence for the approximations xn in Newton’s method. Intuitively, we mean that as the number n of approximations increases without bound, the values xn get arbitrarily close to the desired root r. (This notion is similar to the idea of the limit of a function g(t) as t approaches infinity, as defined in Section 2.6.) In practice, Newton’s method usually gives convergence with impressive speed, but this is not guaranteed. One way to test convergence is to begin by graphing the function to estimate a good starting value for x0. You can test that you are getting closer to a zero of the function by evaluating 0 ƒ(xn) 0 , and check that the approximations are converging by evaluating 0 xn - xn + 1 0 . Newton’s method does not always converge. For instance, if

x0

x1

x

Figure 4.51  Newton’s method fails to converge. You go from x0 to x1 and back to x0, never getting any closer to r.

- 2r - x,

2x - r,

x 6 r x Ú r,

the graph will be like the one in Figure 4.51. If we begin with x0 = r - h, we get x1 = r + h, and successive approximations go back and forth between these two values. No amount of iteration brings us closer to the root than our first guess. If Newton’s method does converge, it converges to a root. Be careful, however. There are situations in which the method appears to converge but no root is there. Fortunately, such situations are rare. When Newton’s method converges to a root, it may not be the root you have in mind. Figure 4.52 shows two ways this can happen. y = f (x) Starting point Root sought

x0

Root found

y = f (x) x1

x

x1 Root found

x2

x

x0 Root sought

Starting point

Figure 4.52  If you start too far away, Newton’s method may miss the root you want.

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Exercises  4.7 Root Finding 1. Use Newton’s method to estimate the solutions of the equation x2 + x - 1 = 0. Start with x0 = -1 for the left-hand solution and with x0 = 1 for the solution on the right. Then, in each case, find x2.

2. Use Newton’s method to estimate the one real solution of x3 + 3x + 1 = 0. Start with x0 = 0 and then find x2.

T 14. Real solutions of a quartic  Use Newton’s method to find the two real solutions of the equation x4 - 2x3 - x2 - 2x + 2 = 0.  ow many solutions does the equation sin 3x = 0.99 - x2 T 15. a. H have? b. Use Newton’s method to find them. 16. Intersection of curves a. Does cos 3x ever equal x? Give reasons for your answer.

3. Use Newton’s method to estimate the two zeros of the function ƒ(x) = x4 + x - 3. Start with x0 = -1 for the left-hand zero and with x0 = 1 for the zero on the right. Then, in each case, find x2.

17. Find the four real zeros of the function ƒ(x) = 2x4 - 4x2 + 1.

4. Use Newton’s method to estimate the two zeros of the function ƒ(x) = 2x - x2 + 1. Start with x0 = 0 for the left-hand zero and with x0 = 2 for the zero on the right. Then, in each case, find x2.

T 18. Estimating pi Estimate p to as many decimal places as your calculator will display by using Newton’s method to solve the equation tan x = 0 with x0 = 3.

5. Use Newton’s method to find the positive fourth root of 2 by solving the equation x4 - 2 = 0. Start with x0 = 1 and find x2.

19. Intersection of curves  At what value(s) of x does cos x = 2x?

6. Use Newton’s method to find the negative fourth root of 2 by solving the equation x4 - 2 = 0. Start with x0 = -1 and find x2.

21. The graphs of y = x2(x + 1) and y = 1>x (x 7 0) intersect at one point x = r. Use Newton’s method to estimate the value of r to four decimal places.

7. Guessing a root  Suppose that your first guess is lucky, in the sense that x0 is a root of ƒ(x) = 0. Assuming that ƒ′(x0) is defined and not 0, what happens to x1 and later approximations?

b. Use Newton’s method to find where.

20. Intersection of curves  At what value(s) of x does cos x = -x?

y

8. Estimating pi  You plan to estimate p>2 to five decimal places by using Newton’s method to solve the equation cos x = 0. Does it matter what your starting value is? Give reasons for your answer.

3

Theory and Examples 9. Oscillation  Show that if h 7 0, applying Newton’s method to

1

ƒ(x) = e

x Ú 0 2- x, x 6 0

2

2x,

leads to x1 = - h if x0 = h and to x1 = h if x0 = -h. Draw a picture that shows what is going on. 10. Approximations that get worse and worse  Apply Newton’s method to ƒ(x) = x1>3 with x0 = 1 and calculate x1, x2, x3, and x4. Find a formula for 0 xn 0 . What happens to 0 xn 0 as n S q? Draw a picture that shows what is going on.

11. Explain why the following four statements ask for the same information: i)  Find the roots of ƒ(x) = x3 - 3x - 1. ii)  Find the x-coordinates of the intersections of the curve y = x3 with the line y = 3x + 1. iii)  Find the x-coordinates of the points where the curve y = x3 - 3x crosses the horizontal line y = 1.

T 13. Intersecting curves  The curve y = tan x crosses the line y = 2x between x = 0 and x = p>2. Use Newton’s method to find where.

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ar, 1rb

0

−1

y = 1x

1

2

x

22. The graphs of y = 2x and y = 3 - x2 intersect at one point x = r. Use Newton’s method to estimate the value of r to four decimal places. 2

23. Intersection of curves  At what value(s) of x does e-x = x2 - x + 1? 24. Intersection of curves  At what value(s) of x does ln (1 - x2) = x - 1? 25. Use the Intermediate Value Theorem from Section 2.5 to show that ƒ(x) = x3 + 2x - 4 has a root between x = 1 and x = 2. Then find the root to five decimal places. 26. Factoring a quartic  Find the approximate values of r1 through r4 in the factorization 8x4 - 14x3 - 9x2 + 11x - 1 = 8(x - r1)(x - r2)(x - r3)(x - r4).

iv)  F  ind the values of x where the derivative of g(x) = (1>4)x4 - (3>2)x2 - x + 5 equals zero. 12. Locating a planet  To calculate a planet’s space coordinates, we have to solve equations like x = 1 + 0.5 sin x. Graphing the function ƒ(x) = x - 1 - 0.5 sin x suggests that the function has a root near x = 1.5. Use one application of Newton’s method to improve this estimate. That is, start with x0 = 1.5 and find x1. (The value of the root is 1.49870 to five decimal places.) Remember to use radians.

y = x 2(x + 1)

y

y = 8x 4 − 14x 3 − 9x 2 + 11x − 1

2 −1

−2 −4 −6

1

2

x

−8 −10 −12

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T 27. Converging to different zeros  Use Newton’s method to find the zeros of ƒ(x) = 4x4 - 4x2 using the given starting values. a. x0 = - 2 and x0 = -0.8, lying in 1 - q, - 22>2 2 b. x0 = - 0.5 and x0 = 0.25, lying in 1 - 221>7, 221>7 2 c. x0 = 0.8 and x0 = 2, lying in 1 22>2, q 2

T 29. Curves that are nearly flat at the root  Some curves are so flat that, in practice, Newton’s method stops too far from the root to give a useful estimate. Try Newton’s method on ƒ(x) = (x - 1)40 with a starting value of x0 = 2 to see how close your machine comes to the root x = 1. See the accompanying graph. y

d. x0 = - 221>7 and x0 = 221>7

28. The sonobuoy problem  In submarine location problems, it is often necessary to find a submarine’s closest point of approach (CPA) to a sonobuoy (sound detector) in the water. Suppose that the submarine travels on the parabolic path y = x2 and that the buoy is located at the point (2, - 1>2). a. Show that the value of x that minimizes the distance between the submarine and the buoy is a solution of the equation x = 1>(x2 + 1). b. Solve the equation x = 1>(x2 + 1) with Newton’s method.

y = (x − 1) 40 Slope = −40

Slope = 40

1

(2, 1)

y Nearly flat

y = x2 Submarine track in two dimensions

1 CPA 0

1

x

2

1 Sonobuoy a2, − 2b

0

1

2

x

30. The accompanying figure shows a circle of radius r with a chord of length 2 and an arc s of length 3. Use Newton’s method to solve for r and u (radians) to four decimal places. Assume 0 6 u 6 p. s=3

r u

2

r

4.8 Antiderivatives We have studied how to find the derivative of a function and how to use it to solve a wide range of problems. However, many other problems require that we recover a function from its known derivative (from its known rate of change). For instance, the laws of physics tell us the acceleration of an object falling from an initial height, and we can use this to compute its velocity and its height at any time. More generally, starting with a function ƒ, we want to find a function F whose derivative is ƒ. If such a function F exists, it is called an antiderivative of ƒ. We will see in the next chapter that antiderivatives are the link connecting the two major elements of calculus: derivatives and definite integrals.

Finding Antiderivatives Definition  A function F is an antiderivative of ƒ on an interval I if F′(x) = ƒ(x) for all x in I.

The process of recovering a function F(x) from its derivative ƒ(x) is called antidifferentiation. We use capital letters such as F to represent an antiderivative of a function ƒ, G to represent an antiderivative of g, and so forth.

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Example 1  Find an antiderivative for each of the following functions. 1

(a) ƒ(x) = 2x   (b)  g(x) = cos x   (c)  h(x) = x + 2e2x

Solution  We need to think backward here: What function do we know has a derivative equal to the given function? (a) F(x) = x2    (b)  G(x) = sin x   (c)  H(x) = ln 0 x 0 + e2x

Each answer can be checked by differentiating. The derivative of F(x) = x2 is 2x. The derivative of G(x) = sin x is cos x, and the derivative of H(x) = ln 0 x 0 + e2x is (1>x) + 2e2x.

The function F(x) = x2 is not the only function whose derivative is 2x. The function x + 1 has the same derivative. So does x2 + C for any constant C. Are there others? Corollary 2 of the Mean Value Theorem in Section 4.2 gives the answer: Any two antiderivatives of a function differ by a constant. So the functions x2 + C, where C is an arbitrary constant, form all the antiderivatives of ƒ(x) = 2x. More generally, we have the following result. 2

Theorem 8  If F is an antiderivative of ƒ on an interval I, then the most general antiderivative of ƒ on I is F(x) + C where C is an arbitrary constant.

y

y = x3 + C 2 1

C=2 C=1 C=0 C = −1 C = −2

Example 2   Find an antiderivative of ƒ(x) = 3x2 that satisfies F(1) = -1. Solution  Since the derivative of x3 is 3x2 , the general antiderivative

x

0 −1

Thus the most general antiderivative of ƒ on I is a family of functions F(x) + C whose graphs are vertical translations of one another. We can select a particular antiderivative from this family by assigning a specific value to C. Here is an example showing how such an assignment might be made.

(1, −1)

−2

F(x) = x3 + C gives all the antiderivatives of ƒ(x). The condition F(1) = -1 determines a specific value for C. Substituting x = 1 into F(x) = x3 + C gives F(1) = (1)3 + C = 1 + C. Since F(1) = -1, solving 1 + C = -1 for C gives C = -2. So

Figure 4.53  The curves y = x3 + C fill the coordinate plane without overlapping. In Example 2, we identify the curve y = x3 - 2 as the one that passes through the given point (1, -1).

F(x) = x3 - 2 is the antiderivative satisfying F(1) = -1. Notice that this assignment for C selects the particular curve from the family of curves y = x3 + C that passes through the point (1, -1) in the plane (Figure 4.53). By working backward from assorted differentiation rules, we can derive formulas and rules for antiderivatives. In each case there is an arbitrary constant C in the general expression representing all antiderivatives of a given function. Table 4.2 gives antiderivative formulas for a number of important functions.

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4.8  Antiderivatives

Table 4.2  Antiderivative formulas, k a nonzero constant

Function

General antiderivative

Function

General antiderivative

1. xn

1 xn + 1 + C, n ≠ -1 n + 1

8. ekx

1 kx e + C k

2. sin kx

1 - cos kx + C k

1   9. x

3. cos kx

1 sin kx + C k

10.

ln 0 x 0 + C, x ≠ 0

4. sec2 kx

1 tan kx + C k

11.

5. csc2 kx

1 - cot kx + C k

6. sec kx tan kx

1 sec kx + C k

7. csc kx cot kx

1 - csc kx + C k

12.

1 21 - k 2x2

1 -1 sin kx + C k

1 1 + k 2x2

1 tan-1 kx + C k

1 x 2k x - 1

sec-1 kx + C, kx 7 1

2 2

13. akx

a

1 b akx + C, a 7 0, a ≠ 1 k ln a

The rules in Table 4.2 are easily verified by differentiating the general antiderivative formula to obtain the function to its left. For example, the derivative of (tan kx)>k + C is sec2 kx, whatever the value of the constants C or k ≠ 0, and this establishes Formula 4 for the most general antiderivative of sec2 kx.

Example 3   Find the general antiderivative of each of the following functions. 1

(a)  ƒ(x) = x5

(b)  g(x) =

x (d)  i(x) = cos 2

(e)  j(x) = e-3x

2x

(c)  h(x) = sin 2x (f)  k(x) = 2x

Solution  In each case, we can use one of the formulas listed in Table 4.2. x6 + C 6 ( b) g(x) = x-1>2, so

Formula 1 with n = 5

x1>2 + C = 2 2x + C 1>2 -cos 2x (c) H(x) = + C 2 sin (x>2) x ( d) I(x) = + C = 2 sin + C 2 1>2

Formula 1 with n = - 1>2

1 (e) J(x) = - e-3x + C 3

Formula 8 with k = - 3

(a) F(x) =

G(x) =

(f) K(x) = a

1 b 2x + C ln 2

Formula 2 with k = 2 Formula 3 with k = 1>2

Formula 13 with a = 2, k = 1



Other derivative rules also lead to corresponding antiderivative rules. We can add and subtract antiderivatives and multiply them by constants. The formulas in Table 4.3 are easily proved by differentiating the antiderivatives and verifying that the result agrees with the original function. Formula 2 is the special case k = -1 in Formula 1.

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Table 4.3  Antiderivative linearity rules

Function

General antiderivative

1. Constant Multiple Rule: 2. Negative Rule: 3. Sum or Difference Rule:

kƒ(x) -ƒ(x) ƒ(x) { g(x)

kF(x) + C, k a constant -F(x) + C F(x) { G(x) + C

Example 4   Find the general antiderivative of ƒ(x) =

3 2x

+ sin 2x.

Solution  We have that ƒ(x) = 3g(x) + h(x) for the functions g and h in Example 3. Since G(x) = 2 2x is an antiderivative of g(x) from Example 3b, it follows from the Constant Multiple Rule for antiderivatives that 3G(x) = 3 # 2 2x = 6 2x is an antiderivative of 3g(x) = 3> 2x. Likewise, from Example 3c we know that H(x) = (-1>2) cos 2x is an antiderivative of h(x) = sin 2x. From the Sum Rule for antiderivatives, we then get that F(x) = 3G(x) + H(x) + C 1 = 6 2x - cos 2x + C 2 is the general antiderivative formula for ƒ(x), where C is an arbitrary constant.

Initial Value Problems and Differential Equations Antiderivatives play several important roles in mathematics and its applications. Methods and techniques for finding them are a major part of calculus, and we take up that study in Chapter 8. Finding an antiderivative for a function ƒ(x) is the same problem as finding a function y(x) that satisfies the equation dy = ƒ(x). dx This is called a differential equation, since it is an equation involving an unknown function y that is being differentiated. To solve it, we need a function y(x) that satisfies the equation. This function is found by taking the antiderivative of ƒ(x). We can fix the arbitrary constant arising in the antidifferentiation process by specifying an initial condition y(x0) = y0. This condition means the function y(x) has the value y0 when x = x0. The combination of a differential equation and an initial condition is called an initial value problem. Such problems play important roles in all branches of science. The most general antiderivative F(x) + C (such as x3 + C in Example 2) of the function ƒ(x) gives the general solution y = F(x) + C of the differential equation dy>dx = ƒ(x). The general solution gives all the solutions of the equation (there are infinitely many, one for each value of C). We solve the differential equation by finding its general solution. We then solve the initial value problem by finding the particular solution that satisfies the initial condition y(x0) = y0. In Example 2, the function y = x3 - 2 is the particular solution of the differential equation dy>dx = 3x2 satisfying the initial condition y(1) = -1.

Antiderivatives and Motion We have seen that the derivative of the position function of an object gives its velocity, and the derivative of its velocity function gives its acceleration. If we know an object’s acceleration, then by finding an antiderivative we can recover the velocity, and from an antiderivative

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4.8  Antiderivatives

s y(0) = 12

283

of the velocity we can recover its position function. This procedure was used as an application of Corollary 2 in Section 4.2. Now that we have a terminology and conceptual framework in terms of antiderivatives, we revisit the problem from the point of view of differential equations.

Example 5   A hot-air balloon ascending at the rate of 12 ft>sec is at a height 80 ft above the ground when a package is dropped. How long does it take the package to reach the ground? dy = −32 dt s(t)

Solution  Let y(t) denote the velocity of the package at time t, and let s(t) denote its height above the ground. The acceleration of gravity near the surface of the earth is 32 ft>sec2. Assuming no other forces act on the dropped package, we have Negative because gravity acts in the dy = -32.     direction of decreasing s dt

0

ground

Figure 4.54  A package dropped from a rising hot-air balloon (Example 5).

This leads to the following initial value problem (Figure 4.54):

Differential equation:

Initial condition:

dy = -32 dt y(0) = 12.   Balloon initially rising

This is our mathematical model for the package’s motion. We solve the initial value problem to obtain the velocity of the package. 1. Solve the differential equation: The general formula for an antiderivative of -32 is y = -32t + C.

Having found the general solution of the differential equation, we use the initial condition to find the particular solution that solves our problem. 2. Evaluate C: 12 = -32(0) + C  Initial condition y(0) C = 12.

= 12

The solution of the initial value problem is y = -32t + 12.

Since velocity is the derivative of height, and the height of the package is 80 ft at time t = 0 when it is dropped, we now have a second initial value problem: Differential equation: Initial condition:

ds = -32t + 12  Set y dt s(0) = 80.

= ds>dt in the previous equation.

We solve this initial value problem to find the height as a function of t. 1. Solve the differential equation: Finding the general antiderivative of -32t + 12 gives s = -16t 2 + 12t + C. 2. Evaluate C: 80 = -16(0)2 + 12(0) + C  Initial condition s(0) C = 80.

= 80

The package’s height above ground at time t is s = -16t 2 + 12t + 80.

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Chapter 4: Applications of Derivatives

Use the solution: To find how long it takes the package to reach the ground, we set s equal to 0 and solve for t: -16t 2 + 12t + 80 = 0 -4t 2 + 3t + 20 = 0 -3 { 2329 Quadratic formula -8 t ≈ -1.89, t ≈ 2.64. t =

The package hits the ground about 2.64 sec after it is dropped from the balloon. (The negative root has no physical meaning.)

Indefinite Integrals A special symbol is used to denote the collection of all antiderivatives of a function ƒ.

Definition  The collection of all antiderivatives of ƒ is called the indefinite integral of ƒ with respect to x, and is denoted by L

ƒ(x) dx.

The symbol 1 is an integral sign. The function ƒ is the integrand of the integral, and x is the variable of integration.

After the integral sign in the notation we just defined, the integrand function is always followed by a differential to indicate the variable of integration. We will have more to say about why this is important in Chapter 5. Using this notation, we restate the solutions of Example 1, as follows: L

2x dx = x2 + C,

L

cos x dx = sin x + C,

L This notation is related to the main application of antiderivatives, which will be explored in Chapter 5. Antiderivatives play a key role in computing limits of certain infinite sums, an unexpected and wonderfully useful role that is described in a central result of Chapter 5, called the Fundamental Theorem of Calculus. 1 a x + 2e2x b dx = ln 0 x 0 + e2x + C.

Example 6  Evaluate L

(x2 - 2x + 5) dx.

Solution  If we recognize that (x3 >3) - x2 + 5x is an antiderivative of x2 - 2x + 5, we can evaluate the integral as antiderivative

\$++%++& x3 (x2 - 2x + 5) dx = - x2 + 5x + " C. 3 L arbitrary constant

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4.8  Antiderivatives

If we do not recognize the antiderivative right away, we can generate it term-by-term with the Sum, Difference, and Constant Multiple Rules: L

(x2 - 2x + 5) dx =

L

x2 dx -

=

L

x2 dx - 2 x dx + 5 1 dx L L

= a =

L

2x dx +

L

5 dx

x3 x2 + C1 b - 2a + C2 b + 5(x + C3) 3 2

x3 + C1 - x2 - 2C2 + 5x + 5C3. 3

This formula is more complicated than it needs to be. If we combine C1, -2C2, and 5C3 into a single arbitrary constant C = C1 - 2C2 + 5C3, the formula simplifies to x3 - x2 + 5x + C 3 and still gives all the possible antiderivatives there are. For this reason, we recommend that you go right to the final form even if you elect to integrate term-by-term. Write L

(x2 - 2x + 5) dx = =

L

x2 dx -

L

2x dx +

L

5 dx

x3 - x2 + 5x + C. 3

Find the simplest antiderivative you can for each part and add the arbitrary constant of integration at the end.

Exercises  4.8 Finding Antiderivatives In Exercises 1–24, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.

12. a.

1 3x

b.

2 5x

c. 1 +

4 1 3x x2

13. a. -p sin px

b. 3 sin x

c. sin px - 3 sin 3x

2. a. 6x b. x7 c. x7 - 6x + 8

14. a. p cos px

c. cos

3. a. - 3x-4

15. a. sec2 x

p px cos 2 2 x 2 b. sec2 3 3 3 3x b. - csc2 2 2

17. a. csc x cot x

b. -csc 5x cot 5x

c. -p csc

18. a. sec x tan x

b. 4 sec 3x tan 3x

1

19. a. e3x

b. e-x

c. ex>2

2x

20. a. e-2x

b. e4x>3

c. e-x>5

21. a. 3x

b. 2-x

1 c. - x-4>3 3

22. a. x 23

b. xp

x2 c. x2 - 2x + 1 1. a. 2x b. b. x-4 -3

c. x-4 + 2x + 3

2 9. a. x-1>3 3

x b. + x2 2 5 b. 2 x 1 b. 3 2x 1 b. 2 2x 1 b. 3 3 2x 1 b. x-2>3 3

1 10. a. x-1>2 2

1 b. - x-3>2 2

3 c. - x-5>2 2

23. a.

1 11. a. x

7 b. x

5 c. 1 - x

1 x 24. a. x - a b 2

4. a. 2x-3 1 x2 2 6. a. - 3 x 3 7. a. 2x 2 5. a.

4 3 8. a. 2 x 3

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c. -x-3 + x - 1 5 x2 1 c. x3 - 3 x c. 2 -

c. 2x + 3

c. 2x +

1

3

2x

16. a. csc2 x

2 21 - x2

b.

b.

1 2(x2 + 1)

b. x2 + 2x

px + p cos x 2 3x c. -sec2 2 c. 1 - 8 csc2 2x px px cot 2 2 px px c. sec tan 2 2

5 x c. a b 3

c. x 22 - 1 c.

1 1 + 4x2

c. px - x-1

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Finding Indefinite Integrals In Exercises 25–70, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

25.

L

27.

L

29.

L

31. 33.

(x + 1) dx a3t 2 +

t b dt 2

(2x3 - 5x + 7) dx

1 1 a 2 - x2 - b dx 3 L x L

x-1>3 dx

26. 28. 30.

L

(5 - 6x) dx

cot2 x dx L (Hint: 1 + cot2 x = csc2 x) 67.

t2 + 4t 3 b dt L 2

L

(1 + tan2 u) du 66. (2 + tan2 u) du L L (Hint: 1 + tan2 u = sec2 u) 65.

a

(1 - x2 - 3x5) dx

69.

L

68.

cos u (tan u + sec u) du 70.

34.

L

x-5>4 dx

71.

L

(7x - 2)3 dx =

36.

L

2x

72.

L

(3x + 5)-2 dx = -

73.

L

sec2 (5x - 1) dx =

(7x - 2)4 + C 28 (3x + 5)-1 + C 3

37.

L

39.

L

2x(1 - x-3) dx

40.

L

x-3(x + 1) dx

74.

L

41.

L

t 2t + 2t dt t2

42.

L

4 + 2t dt t3

75.

1 1 + C dx = 2 x + 1 L (x + 1)

43.

L

(- 2 cos t) dt

44.

L

(- 5 sin t) dt

76.

x 1 + C dx = 2 x + 1 L (x + 1)

45.

L

7 sin

u du 3

46.

L

3 cos 5u du

77.

47.

L

(- 3 csc2 x) dx

48.

L

49.

L

csc u cot u du 2

50.

2 sec u tan u du L5

79.

51.

L

(e3x + 5e-x) dx

52.

L

(2ex - 3e-2x) dx

80.

53.

L

(e-x + 4x) dx

54.

L

(1.3)x dx

55.

L

(4 sec x tan x - 2 sec2 x) dx

56.

1 (csc2 x - csc x cot x) dx L2

57.

L

+ 2x 2 dx

a8y -

2

y

1>4

b dy

(sin 2x - csc2 x) dx

1 + cos 4t 59. dt 2 L 61.

L

63.

L

1 ax -

5 b dx x2 + 1

3x 23 dx

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csc u du csc u - sin u L

Checking Antiderivative Formulas Verify the formulas in Exercises 71–82 by differentiation.

L

1 2x

(1 - cot2 x) dx

1 2 a - 3 + 2xb dx L 5 x

32.

35.

3

L

38.

58.

a

2

+

2 2x

b dx

1 1 b dy L 7 y5>4

L

a

a-

sec2 x b dx 3

64.

(2 cos 2x - 3 sin 3x) dx

2 1 a - 1>4 b dy y L 21 - y2 L

x - 1 x - 1 b dx = - 3 cot a b + C 3 3

1 dx = ln 0 x + 1 0 + C, x ≠ -1 Lx + 1 L

xex dx = xex - ex + C

x dx 1 = a tan-1 a a b + C 2 2 La + x

x dx = sin-1 a a b + C L 2a2 - x2

81.

tan-1 x tan-1 x 1 dx = ln x - ln (1 + x2) + C x 2 2 L x

82.

L

(sin-1 x)2 dx = x(sin-1 x)2 - 2x + 2 21 - x2 sin-1 x + C

83. Right, or wrong? Say which for each formula and give a brief reason for each answer.

1 - cos 6t 60. dt 2 L 62.

78.

csc2 a

1 tan (5x - 1) + C 5

a.

L

x sin x dx =

b.

L

x sin x dx = - x cos x + C

c.

L

x sin x dx = - x cos x + sin x + C

x2 sin x + C 2

x 22 - 1 dx

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4.8  Antiderivatives

84. Right, or wrong? Say which for each formula and give a brief reason for each answer. a. b.

L

tan u sec2 u du =

sec3 u + C 3

L

tan u sec2 u du =

1 2 tan u + C 2

y

1 tan u sec2 u du = sec2 u + C 2 L 85. Right, or wrong? Say which for each formula and give a brief reason for each answer. (2x + 1)3 a. + C (2x + 1)2 dx = 3 L b.

L

3(2x + 1)2 dx = (2x + 1)3 + C

c.

L

6(2x + 1)2 dx = (2x + 1)3 + C

86. Right, or wrong? Say which for each formula and give a brief reason for each answer. L

22x + 1 dx = 2x2 + x + C

b.

L

22x + 1 dx = 2x2 + x + C

L 87. Right, or wrong? Give a brief reason why. 22x + 1 dx =

c.

1 1 22x + 1 23 + C 3

- 15(x + 3)2 x + 3 3 dx = a b + C 4 x - 2 L (x - 2)

88. Right, or wrong? Give a brief reason why.

x cos (x2) - sin (x2) sin (x2) dx = + C x x2 L Initial Value Problems 89. Which of the following graphs shows the solution of the initial value problem

dy = 2x, y = 4 when x = 1? dx y

(1, 4)

3

(1, 4)

4

2

3 2

3

1

1

1

−1 0

1

x

(a)

−1 0 (b)

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0

x

−1 0

x

(a)

(−1, 1) 0

x

(b)

0

x

(c)

Give reasons for your answer. Solve the initial value problems in Exercises 91–112. dy = 2x - 7, y(2) = 0 91. dx dy = 10 - x, y(0) = - 1 92. dx dy 1 = + x, x 7 0; y(2) = 1 93. dx x2 dy = 9x2 - 4x + 5, y(-1) = 0 94. dx dy = 3x-2>3, y(-1) = -5 95. dx dy 1 = , y(4) = 0 96. dx 2 2x

97.

ds = 1 + cos t, s(0) = 4 dt

98.

ds = cos t + sin t, s(p) = 1 dt

99.

dr = -p sin pu, r(0) = 0 du

100.

dr = cos pu, r(0) = 1 du

101.

dy 1 = sec t tan t, y(0) = 1 2 dt

102.

p dy = 8t + csc2 t, y a b = -7 2 dt

105.

d 2y = 2 - 6x; y′(0) = 4, y(0) = 1 dx2

106.

d 2y = 0; y′(0) = 2, y(0) = 0 dx2

107.

d 2r 2 dr 2 = 3; = 1, r(1) = 1 dt t = 1 dt 2 t

108.

d 2s 3t ds 2 = ; = 3, s(4) = 4 8 dt t = 4 dt 2

109.

d 3y = 6; y″(0) = -8, y′(0) = 0, y(0) = 5 dx3

(1, 4)

2

1

y

dy 3 = , t 7 1, y(2) = 0 dt t 2t 2 - 1 dy 8 = + sec2 t, y(0) = 1 104. dt 1 + t2

y

4

y

(−1, 1) (−1, 1)

103.

y

4

90. Which of the following graphs shows the solution of the initial value problem dy = -x, y = 1 when x = - 1? dx

c.

a.

287

1

x

(c)

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Chapter 4: Applications of Derivatives

b. Suppose that the position s of a body moving along a coordinate line is a differentiable function of time t. Is it true that once you know an antiderivative of the velocity function ds>dt you can find the body’s displacement from t = a to t = b even if you do not know the body’s exact position at either of those times? Give reasons for your answer.

d3 u 1 = 0; u″(0) = - 2, u′(0) = - , u(0) = 22 2 dt 3 (4) 111. y = - sin t + cos t ; y‴(0) = 7, y″(0) = y′(0) = - 1, y(0) = 0 110.

112. y(4) = - cos x + 8 sin 2x ; y‴(0) = 0, y″(0) = y′(0) = 1, y(0) = 3 113. Find the curve y = ƒ(x) in the xy-plane that passes through the point (9, 4) and whose slope at each point is 3 2x. 114. a.  Find a curve y = ƒ(x) with the following properties: d 2y i)  2 = 6x dx ii) Its graph passes through the point (0, 1) and has a horizontal tangent there. b. How many curves like this are there? How do you know?

120. Liftoff from Earth  A rocket lifts off the surface of Earth with a constant acceleration of 20 m>sec2. How fast will the rocket be going 1 min later? 121. Stopping a car in time  You are driving along a highway at a steady 60 mph (88 ft>sec) when you see an accident ahead and slam on the brakes. What constant deceleration is required to stop your car in 242 ft? To find out, carry out the following steps.

1. Solve the initial value problem

Solution (Integral) Curves Exercises 115–118 show solution curves of differential equations. In each exercise, find an equation for the curve through the labeled point.

Differential equation:

115.

Initial conditions:

116. dy = 1 − 4 x13 3 dx

y

y dy = x − 1 dx

1

involve k.)

2

x

3. Find the value of k that makes s = 242 for the value of t you

1 (−1, 1)

found in Step 2.

1

0

−1

122. Stopping a motorcycle  The State of Illinois Cycle Rider Safety Program requires motorcycle riders to be able to brake from 30 mph (44 ft>sec) to 0 in 45 ft. What constant deceleration does it take to do that?

x

2

−1 −1

117.

118. dy = sin x − cos x dx y

y

123. Motion along a coordinate line  A particle moves on a coordinate line with acceleration a = d 2s>dt 2 = 15 2t - 1 3> 2t 2, subject to the conditions that ds>dt = 4 and s = 0 when t = 1. Find

dy = 1 + psin px dx 2" x

6

a. the velocity y = ds>dt in terms of t

1 0

2

x

(−p, −1)

b. the position s in terms of t.

4

2

0

(1, 2)

1

2

3

x

−2

Applications 119. Finding displacement from an antiderivative of velocity

a. Suppose that the velocity of a body moving along the s-axis is ds = y = 9.8t - 3. dt

ds = 88 and s = 0 when t = 0. dt

2. Find the value of t that makes ds>dt = 0. (The answer will

(1, 0.5) 0

(k constant)

Measuring time and distance from when the brakes are applied

2 1

d 2s = -k dt 2

i) Find the body’s displacement over the time interval from t = 1 to t = 3 given that s = 5 when t = 0. ii) Find the body’s displacement from t = 1 to t = 3 given that s = - 2 when t = 0. iii) Now find the body’s displacement from t = 1 to t = 3 given that s = s0 when t = 0.

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T 124. The hammer and the feather When Apollo 15 astronaut David Scott dropped a hammer and a feather on the moon to demonstrate that in a vacuum all bodies fall with the same (constant) acceleration, he dropped them from about 4 ft above the ground. The television footage of the event shows the hammer and the feather falling more slowly than on Earth, where, in a vacuum, they would have taken only half a second to fall the 4 ft. How long did it take the hammer and feather to fall 4 ft on the moon? To find out, solve the following initial value problem for s as a function of t. Then find the value of t that makes s equal to 0. Differential equation: Initial conditions:

d 2s = -5.2 ft>sec2 dt 2 ds = 0 and s = 4 when t = 0 dt

125. Motion with constant acceleration  The standard equation for the position s of a body moving with a constant acceleration a along a coordinate line is a s = t 2 + y0 t + s0 ,(1) 2

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Chapter 4  Questions to Guide Your Review

where y0 and s0 are the body’s velocity and position at time t = 0. Derive this equation by solving the initial value problem

Find:

d 2s = a Differential equation: dt 2 Initial conditions:

ds = y0 and s = s0 when t = 0. dt

126. Free fall near the surface of a planet  For free fall near the surface of a planet where the acceleration due to gravity has a constant magnitude of g length@units>sec2, Equation (1) in Exercise 125 takes the form

a.

L

ƒ(x) dx

b.

L

c.

L

[-ƒ(x)] dx

d.

L

e.

L

3ƒ(x) + g(x) 4 dx

f.

L

g(x) dx 3- g(x) 4 dx 3ƒ(x) - g(x) 4 dx

128. Uniqueness of solutions  If differentiable functions y = F(x) and y = g(x) both solve the initial value problem

1 s = - gt 2 + y0 t + s0 ,(2) 2

289

dy = ƒ(x), dx

y(x0) = y0,

where s is the body’s height above the surface. The equation has a minus sign because the acceleration acts downward, in the direction of decreasing s. The velocity y0 is positive if the object is rising at time t = 0 and negative if the object is falling. Instead of using the result of Exercise 125, you can derive Equation (2) directly by solving an appropriate initial value problem. What initial value problem? Solve it to be sure you have the right one, explaining the solution steps as you go along.

on an interval I, must F(x) = G(x) for every x in I? Give reasons for your answer.

127. Suppose that

1 130. y′ = x + x, y(1) = - 1

ƒ(x) =

d d 1 1 - 2x 2 and g(x) = dx (x + 2). dx

Chapter  4

Computer Explorations Use a CAS to solve the initial value problems in Exercises 129–132. Plot the solution curves.

129. y′ = cos2 x + sin x, y(p) = 1

131. y′ =

1 24 - x2

, y(0) = 2

2 132. y″ = x + 2x, y(1) = 0, y′(1) = 0

1. What can be said about the extreme values of a function that is continuous on a closed interval?

11. What do the derivatives of a function tell you about the shape of its graph?

2. What does it mean for a function to have a local extreme value on its domain? An absolute extreme value? How are local and absolute extreme values related, if at all? Give examples.

12. What is a cusp? Give examples.

3. How do you find the absolute extrema of a continuous function on a closed interval? Give examples.

14. Outline a general strategy for solving max-min problems. Give examples.

4. What are the hypotheses and conclusion of Rolle’s Theorem? Are the hypotheses really necessary? Explain.

15. Describe l’Hôpital’s Rule. How do you know when to use the rule and when to stop? Give an example.

5. What are the hypotheses and conclusion of the Mean Value Theorem? What physical interpretations might the theorem have?

16. How can you sometimes handle limits that lead to indeterminate forms q > q, q # 0, q - q , 1q, 00, and qq ? Give examples.

6. State the Mean Value Theorem’s three corollaries.

13. List the steps you would take to graph a rational function. Illustrate with an example.

7. What is the First Derivative Test for Local Extreme Values? Give examples of how it is applied.

17. Describe Newton’s method for solving equations. Give an example. What is the theory behind the method? What are some of the things to watch out for when you use the method?

8. How do you test a twice-differentiable function to determine where its graph is concave up or concave down? Give examples.

18. Can a function have more than one antiderivative? If so, how are the antiderivatives related? Explain.

9. What is an inflection point? Give an example. What physical significance do inflection points sometimes have?

19. What is an indefinite integral? How do you evaluate one? What general formulas do you know for finding indefinite integrals?

10. What is the Second Derivative Test for Local Extreme Values? Give examples of how it is applied.

20. What is an initial value problem? How do you solve one? Give an example.

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Chapter 4: Applications of Derivatives

Chapter  4

Practice Exercises

Extreme Values 1. Does ƒ(x) = x3 + 2x + tan x have any local maximum or minimum values? Give reasons for your answer.

2. Does g(x) = csc x + 2 cot x have any local maximum values? Give reasons for your answer. 3. Does ƒ(x) = (7 + x)(11 - 3x)1>3 have an absolute minimum value? An absolute maximum? If so, find them or give reasons why they fail to exist. List all critical points of ƒ. 4. Find values of a and b such that the function ƒ(x) =

ax + b x2 - 1

c. Zoom in on the graph to find a viewing window that shows 3 the presence of the extreme values at x = 2 5 and x = 23.

The moral here is that without calculus the existence of two of the three extreme values would probably have gone unnoticed. On any normal graph of the function, the values would lie close enough together to fall within the dimensions of a single pixel on the screen. (Source: Uses of Technology in the Mathematics Curriculum, by Benny Evans and Jerry Johnson, Oklahoma State University, published in 1990 under a grant from the National Science Foundation, USE-8950044.)

T 14. (Continuation of Exercise 13.)

has a local extreme value of 1 at x = 3. Is this extreme value a local maximum, or a local minimum? Give reasons for your answer.

a. Graph ƒ(x) = (x8 >8) - (2>5)x5 - 5x - (5>x2) + 11 over the interval -2 … x … 2. Where does the graph appear to have local extreme values or points of inflection?

5. Does g(x) = ex - x have an absolute minimum value? An absolute maximum? If so, find them or give reasons why they fail to exist. List all critical points of g.

7 b. Show that ƒ has a local maximum value at x = 25 ≈ 1.2585 3 and a local minimum value at x = 22 ≈ 1.2599.

6. Does ƒ(x) = 2ex >(1 + x2) have an absolute minimum value? An absolute maximum? If so, find them or give reasons why they fail to exist. List all critical points of ƒ. In Exercises 7 and 8, find the absolute maximum and absolute minimum values of ƒ over the interval. 7. ƒ(x) = x - 2 ln x, 1 … x … 3 8. ƒ(x) = (4>x) + ln x2, 1 … x … 4 9. The greatest integer function ƒ(x) = : x ; , defined for all values of x, assumes a local maximum value of 0 at each point of 30, 1). Could any of these local maximum values also be local minimum values of ƒ? Give reasons for your answer. 10. a.  Give an example of a differentiable function ƒ whose first derivative is zero at some point c even though ƒ has neither a local maximum nor a local minimum at c. b. How is this consistent with Theorem 2 in Section 4.1? Give reasons for your answer. 11. The function y = 1>x does not take on either a maximum or a minimum on the interval 0 6 x 6 1 even though the function is continuous on this interval. Does this contradict the Extreme Value Theorem for continuous functions? Why?

c. Zoom in to find a viewing window that shows the presence of 7 3 the extreme values at x = 2 5 and x = 2 2.

The Mean Value Theorem 15. a.  Show that g(t) = sin2 t - 3t decreases on every interval in its domain.

b. How many solutions does the equation sin2 t - 3t = 5 have? Give reasons for your answer. 16. a.  Show that y = tan u increases on every open interval in its domain. b. If the conclusion in part (a) is really correct, how do you explain the fact that tan p = 0 is less than tan (p>4) = 1? 17. a.  S  how that the equation x4 + 2x2 - 2 = 0 has exactly one solution on 30, 14 . T b. Find the solution to as many decimal places as you can.

18. a.  S  how that ƒ(x) = x>(x + 1) increases on every open interval in its domain. b. Show that ƒ(x) = x3 + 2x has no local maximum or minimum values.

12. What are the maximum and minimum values of the function y = 0 x 0 on the interval - 1 … x 6 1? Notice that the interval is not closed. Is this consistent with the Extreme Value Theorem for continuous functions? Why?

19. Water in a reservoir  As a result of a heavy rain, the volume of water in a reservoir increased by 1400 acre-ft in 24 hours. Show that at some instant during that period the reservoir’s volume was increasing at a rate in excess of 225,000 gal>min. (An acre-foot is 43,560 ft3, the volume that would cover 1 acre to the depth of 1 ft. A cubic foot holds 7.48 gal.)

T 13. A graph that is large enough to show a function’s global behavior may fail to reveal important local features. The graph of ƒ(x) = (x8 >8) - (x6 >2) - x5 + 5x3 is a case in point.

20. The formula F(x) = 3x + C gives a different function for each value of C. All of these functions, however, have the same derivative with respect to x, namely F′(x) = 3. Are these the only differentiable functions whose derivative is 3? Could there be any others? Give reasons for your answers.

a. Graph ƒ over the interval - 2.5 … x … 2.5. Where does the graph appear to have local extreme values or points of inflection?

b. Now factor ƒ′(x) and show that ƒ has a local maximum at 3 x =2 5 ≈ 1.70998 and local minima at x = { 23 ≈ {1.73205.

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21. Show that d x d 1 a b = ab x + 1 dx x + 1 dx

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291

Chapter 4  Practice Exercises

Graphs and Graphing Graph the curves in Exercises 27– 42.

even though x 1 ≠ . x + 1 x + 1

27. y = x2 - (x3 >6)

28. y = x3 - 3x2 + 3

22. Calculate the first derivatives of ƒ(x) = x2 >(x2 + 1) and g(x) = - 1>(x2 + 1). What can you conclude about the graphs of these functions?

31. y = x3(8 - x)

32. y = x2(2x2 - 9)

33. y = x - 3x2>3

34. y = x1>3(x - 4)

Analyzing Graphs In Exercises 23 and 24, use the graph to answer the questions.

35. y = x 23 - x

37. y = (x - 3)2 ex

36. y = x 24 - x2

38. y = xe-x

23. Identify any global extreme values of ƒ and the values of x at which they occur.

39. y = ln (x2 - 4x + 3)

40. y = ln (sin x)

1 41. y = sin-1 a x b

1 42. y = tan-1 a x b

43. y′ = 16 - x2

44. y′ = x2 - x - 6

45. y′ = 6x(x + 1)(x - 2)

46. y′ = x2(6 - 4x)

y

1 a2, 2b

x

0

2

29. y = -x + 6x - 9x + 3 30. y = (1>8)(x3 + 3x2 - 9x - 27)

2

Each of Exercises 43– 48 gives the first derivative of a function y = ƒ(x). (a) At what points, if any, does the graph of ƒ have a local maximum, local minimum, or inflection point? (b) Sketch the general shape of the graph.

y = f (x) (1, 1)

3

4

24. Estimate the open intervals on which the function y = ƒ(x) is

2

48. y′ = 4x2 - x4

47. y′ = x - 2x

a. increasing.

In Exercises 49–52, graph each function. Then use the function’s first derivative to explain what you see.

b. decreasing.

49. y = x2>3 + (x - 1)1>3

c. Use the given graph of ƒ′ to indicate where any local extreme values of the function occur, and whether each extreme is a relative maximum or minimum.

51. y = x

54. y =

2x x + 5

55. y =

x2 + 1 x

56. y =

x2 - x + 1 x

57. y =

x3 + 2 2x

58. y =

−1

59. y =

x2 - 4 x2 - 3

−2

Using L’Hôpital’s Rule Use l’Hôpital’s Rule to find the limits in Exercises 61–72. x3 - 5x + 2 x5 - 1 61. lim 62. lim 7 x - 2 xS1 x - 1 xS2 sin x sin x 63. lim x 64. lim xS0 x S 0 2x + tan x tan mx 1 - x 65. lim 66. lim tan nx px x S 1 cos ( 2 ) xS0

(−3, 1) x

Each of the graphs in Exercises 25 and 26 is the graph of the position function s = ƒ(t) of an object moving on a coordinate line (t represents time). At approximately what times (if any) is each object’s (a) velocity equal to zero? (b) Acceleration equal to zero? During approximately what time intervals does the object move (c) forward? (d) Backward? 25. s s = f (t) 9

12 14

t

s

s = f (t)

x4 - 1 x2 x2 60. y = 2 x - 4

67. lim sec 9x cos 5x

68. lim+ 2x sec x

69. lim (sec x - tan x)

70. lim a

x S p>2

x S p>2

26.

52. y = x2>3 - (x - 1)1>3

Sketch the graphs of the rational functions in Exercises 53–60.

(2, 3)

6

50. y = x2>3 + (x - 1)2>3

x + 1 x - 3

y = f ′(x)

3

+ (x - 1)

1>3

53. y =

y

0

1>3

xS0

xS0

71. lim 1 2x4 - x2 + 5 - 2x4 + x2 2

1 1 - 3b x x5

x Sq

0

2

4

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6

8

t

72. lim a x Sq

x4 x4 - 3 b x - 1 x + 1 3

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Chapter 4: Applications of Derivatives

Find the limits in Exercises 73–84. 15x - 1 5u - 1 73. lim 74. lim x u xS0 uS0 75. lim

xS0

tan x

5

- 1 ex - 1

76. lim

xS0

-tan x

5

- 1 ex - 1

7 - 7ex 78. lim a b xex S x 0

tan 3x - 2x 77. lim a b x S 0 3x - sin2 x t - ln (1 + 2t) 7 9. lim+ tS0 t2 e4x 1 81. lim+ x - x S t 0

82. lim+ e-1>y ln y

b kx 83. lim a1 + x b x Sq

5 10 84. lim a6 - x - 3 b x Sq x

x2 - 4 x S 2 2x + 2 - 23x - 2

80. lim

yS0

Optimization 85. The sum of two nonnegative numbers is 36. Find the numbers if

Your profit on a grade A tire is twice your profit on a grade B tire. What is the most profitable number of each kind to make? 92. Particle motion  The positions of two particles on the s-axis are s1 = cos t and s2 = cos (t + p>4). a. What is the farthest apart the particles ever get? b. When do the particles collide? T 93. Open-top box  An open-top rectangular box is constructed from a 10-in.-by-16-in. piece of cardboard by cutting squares of equal side length from the corners and folding up the sides. Find analytically the dimensions of the box of largest volume and the maximum volume. Support your answers graphically. 94. The ladder problem  What is the approximate length (in feet) of the longest ladder you can carry horizontally around the corner of the corridor shown here? Round your answer down to the nearest foot. y

a. the difference of their square roots is to be as large as possible. b. the sum of their square roots is to be as large as possible. 86. The sum of two nonnegative numbers is 20. Find the numbers a. if the product of one number and the square root of the other is to be as large as possible.

6

b. if one number plus the square root of the other is to be as large as possible.

0

87. An isosceles triangle has its vertex at the origin and its base parallel to the x-axis with the vertices above the axis on the curve y = 27 - x2. Find the largest area the triangle can have. 88. A customer has asked you to design an open-top rectangular stainless steel vat. It is to have a square base and a volume of 32 ft3, to be welded from quarter-inch plate, and to weigh no more than necessary. What dimensions do you recommend? 89. Find the height and radius of the largest right circular cylinder that can be put in a sphere of radius 23.

90. The figure here shows two right circular cones, one upside down inside the other. The two bases are parallel, and the vertex of the smaller cone lies at the center of the larger cone’s base. What values of r and h will give the smaller cone the largest possible volume?

12′

96. Let ƒ(x) = x4 - x3. Show that the equation ƒ(x) = 75 has a solution in the interval 33, 44 and use Newton’s method to find it.

Finding Indefinite Integrals Find the indefinite integrals (most general antiderivatives) in Exercises 97–120. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

97.

L

99.

L

101.

dr (r 3)3 L

102.

103.

L

9u 2 2u 3 - 1 du

104.

105.

L

x4(1 - x5)-1>5 dx

106.

107.

L

sec2

h

91. Manufacturing tires  Your company can manufacture x hundred grade A tires and y hundred grade B tires a day, where 0 … x … 4 and 40 - 10x . y = 5 - x

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x

8

Newton’s Method 95. Let ƒ(x) = 3x - x3. Show that the equation ƒ(x) = -4 has a solution in the interval 32, 34 and use Newton’s method to find it.

r

6′

(8, 6)

(x5 + 4x2 + 9) dx at 2t +

5 b dt t3

s ds 10

109. csc 23u cot 23u du L

98. 100.

L

a6t 5 -

t3 - tb dt 4

5 4 a - 3 b dt L t 2t t

5 dr L 1 r + 23 22

x dx L 25 + x2 L

(5 - x)5>7 dx

108.

L

csc2 es ds

110.

L

sec

t t tan dt 5 5

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Chapter 4  Practice Exercises

x 1 + cos 2x b 111. cos dx  a Hint: cos2 x = 2 6 L 2

2

on the curve y = e-x . What dimensions give the rectangle its largest area, and what is that area? y

x 112. sec dx 2 L 2

113. 114. 115. 116. 117. 119.

L

1

5 a x + x3 b dx

9 4 a 3 + 2 b dx x + 9 L x

L

x

118.

dx

120.

L

p+x

8

dx L 225 - x2

3 d 2r = 15 2t + ; r′(1) = 8, r (1) = 0 dt 2 2t d 3r 124. 3 = - cos t; r″(0) = r′(0) = 0, r (0) = -1 dt

123.

Applications and Examples

125. Can the integrations in (a) and (b) both be correct? Explain. dx = sin-1 x + C L 21 - x2

b.

dx dx = - = -cos-1 x + C L 21 - x2 L 21 - x2 dx dx = - = -cos-1 x + C L 21 - x2 L 21 - x2

dx - du = 2 L 21 - x L 21 - (- u)2

x = -u dx = - du

=

= cos-1 u + C

= cos-1 (- x) + C

0.1 0

x

1

In Exercises 129 and 130, find the absolute maximum and minimum values of each function on the given interval. 129. y = x ln 2x - x,

c

1 e , d 2e 2

130. y = 10x(2 - ln x), (0, e2 4

In Exercises 131 and 132, find the absolute maxima and minima of the functions and say where they are assumed. 4

131. ƒ(x) = ex> 2x

+1

2

132. g(x) = e23 - 2x - x

T 133. Graph the following functions and use what you see to locate and estimate the extreme values, identify the coordinates of the inflection points, and identify the intervals on which the graphs are concave up and concave down. Then confirm your estimates by working with the functions’ derivatives. 2

a. y = (ln x)> 2x  b.  y = e-x   c.  y = (1 + x)e-x

T 135. Graph ƒ(x) = (sin x)sin x over 30, 3p4. Explain what you see.

136. A round underwater transmission cable consists of a core of copper wires surrounded by nonconducting insulation. If x denotes the ratio of the radius of the core to the thickness of the insulation, it is known that the speed of the transmission signal is given by the equation y = x2 ln (1>x). If the radius of the core is 1 cm, what insulation thickness h will allow the greatest transmission speed? Insulation

- du 2 1 - u2 L

ln x x2

T 134. Graph ƒ(x) = x ln x. Does the function appear to have an absolute minimum value? Confirm your answer with calculus.

126. Can the integrations in (a) and (b) both be correct? Explain. a.

y=

0.2

dx

Initial Value Problems Solve the initial value problems in Exercises 121–124. dy x2 + 1 121. = , y(1) = -1 dx x2 dy 1 2 = ax + x b , y(1) = 1 122. dx

b.

x

y

5 dx L 4x 2x2 - 9

a.

2

128. The rectangle shown here has one side on the positive y-axis, one side on the positive x-axis, and its upper right-hand vertex on the curve y = (ln x)>x2. What dimensions give the rectangle its largest area, and what is that area?

(ax - xa) dx 4+p

y = e−x

0

5 a e2x - e-2x b dx L 2 L

293

x= r h

Core

h r

u = -x

127. The rectangle shown here has one side on the positive y-axis, one side on the positive x-axis, and its upper right-hand vertex

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Chapter 4: Applications of Derivatives

Chapter  4

Functions and Derivatives 1. What can you say about a function whose maximum and minimum values on an interval are equal? Give reasons for your answer.

2. Is it true that a discontinuous function cannot have both an absolute maximum and an absolute minimum value on a closed interval? Give reasons for your answer. 3. Can you conclude anything about the extreme values of a continuous function on an open interval? On a half-open interval? Give reasons for your answer. 4. Local extrema  Use the sign pattern for the derivative dƒ = 6(x - 1)(x - 2)2(x - 3)3(x - 4)4 dx to identify the points where ƒ has local maximum and minimum values. 5. Local extrema a. Suppose that the first derivative of y = ƒ(x) is y′ = 6(x + 1)(x - 2)2. At what points, if any, does the graph of ƒ have a local maximum, local minimum, or point of inflection? b. Suppose that the first derivative of y = ƒ(x) is y′ = 6x(x + 1)(x - 2). At what points, if any, does the graph of ƒ have a local maximum, local minimum, or point of inflection? 6. If ƒ′(x) … 2 for all x, what is the most the values of ƒ can increase on 30, 64 ? Give reasons for your answer.

7. Bounding a function  Suppose that ƒ is continuous on 3a, b4 and that c is an interior point of the interval. Show that if ƒ′(x) … 0 on 3a, c) and ƒ′(x) Ú 0 on (c, b4 , then ƒ(x) is never less than ƒ(c) on 3a, b4 . 8. An inequality

a. Show that - 1>2 … x>(1 + x2) … 1>2 for every value of x.

b. Suppose that ƒ is a function whose derivative is ƒ′(x) = x>(1 + x2). Use the result in part (a) to show that

0 ƒ(b) - ƒ(a) 0 … 1 0 b - a 0 2

for any a and b.

9. The derivative of ƒ(x) = x2 is zero at x = 0, but ƒ is not a constant function. Doesn’t this contradict the corollary of the Mean Value Theorem that says that functions with zero derivatives are constant? Give reasons for your answer. 10. Extrema and inflection points Let h = ƒg be the product of two differentiable functions of x.

In either case, if the answer is yes, give a proof. If the answer is no, give a counterexample. 11. Finding a function   Use the following information to find the values of a, b, and c in the formula ƒ(x) = (x + a)> (bx2 + cx + 2). i)  The values of a, b, and c are either 0 or 1. ii)  The graph of ƒ passes through the point (- 1, 0). iii)  The line y = 1 is an asymptote of the graph of ƒ. 12. Horizontal tangent  For what value or values of the constant k will the curve y = x3 + kx2 + 3x - 4 have exactly one horizontal tangent? Optimization 13. Largest inscribed triangle Points A and B lie at the ends of a diameter of a unit circle and point C lies on the circumference. Is it true that the area of triangle ABC is largest when the triangle is isosceles? How do you know?

14. Proving the second derivative test  The Second Derivative Test for Local Maxima and Minima (Section 4.4) says: a. ƒ has a local maximum value at x = c if ƒ′(c) = 0 and ƒ″(c) 6 0 b. ƒ has a local minimum value at x = c if ƒ′(c) = 0 and ƒ″(c) 7 0. To prove statement (a), let P = (1>2) 0 ƒ″(c) 0 . Then use the fact that ƒ″(c) = lim

hS0

ƒ′(c + h) - ƒ′(c) ƒ′(c + h) = lim h h hS0

to conclude that for some d 7 0, 0 6 0h0 6 d

1

ƒ′(c + h) 6 ƒ″(c) + P 6 0. h

Thus, ƒ′(c + h) is positive for -d 6 h 6 0 and negative for 0 6 h 6 d. Prove statement (b) in a similar way. 15. Hole in a water tank  You want to bore a hole in the side of the tank shown here at a height that will make the stream of water coming out hit the ground as far from the tank as possible. If you drill the hole near the top, where the pressure is low, the water will exit slowly but spend a relatively long time in the air. If you drill the hole near the bottom, the water will exit at a higher velocity but have only a short time to fall. Where is the best place, if any, for the hole? (Hint: How long will it take an exiting droplet of water to fall from height y to the ground?) Tank kept full, top open

y

h y

Exit velocity = " 64(h − y)

a. If ƒ and g are positive, with local maxima at x = a, and if ƒ′ and g′ change sign at a, does h have a local maximum at a? b. If the graphs of ƒ and g have inflection points at x = a, does the graph of h have an inflection point at a?

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Ground

0

x

Range

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295

16. Kicking a field goal  An American football player wants to kick a field goal with the ball being on a right hash mark. Assume that the goal posts are b feet apart and that the hash mark line is a distance a 7 0 feet from the right goal post. (See the accompanying figure.) Find the distance h from the goal post line that gives the kicker his largest angle b. Assume that the football field is flat. Goal posts b

Goal post line

a

h

20. L’Hôpital’s Rule does not help with the following limits. Find them some other way. a. lim

x Sq

2x + 5 2x + 5

2x b. lim x Sq x + 7 2x

Theory and Examples 21. Suppose that it costs a company y = a + bx dollars to produce x units per week. It can sell x units per week at a price of P = c - ex dollars per unit. Each of a, b, c, and e represents a positive constant. (a) What production level maximizes the profit? (b) What is the corresponding price? (c) What is the weekly profit at this level of production? (d) At what price should each item be sold to maximize profits if the government imposes a tax of t dollars per item sold? Comment on the difference between this price and the price before the tax.

22. Estimating reciprocals without division  You can estimate the value of the reciprocal of a number a without ever dividing by a if you apply Newton’s method to the function ƒ(x) = (1>x) - a. For example, if a = 3, the function involved is ƒ(x) = (1>x) - 3.

b u

Football

17. A max-min problem with a variable answer  Sometimes the solution of a max-min problem depends on the proportions of the shapes involved. As a case in point, suppose that a right circular cylinder of radius r and height h is inscribed in a right circular cone of radius R and height H, as shown here. Find the value of r (in terms of R and H) that maximizes the total surface area of the cylinder (including top and bottom). As you will see, the solution depends on whether H … 2R or H 7 2R.

a. Graph y = (1>x) - 3. Where does the graph cross the x-axis? b. Show that the recursion formula in this case is xn + 1 = xn(2 - 3xn), so there is no need for division. q

23. To find x = 2a, we apply Newton’s method to ƒ(x) = xq - a. Here we assume that a is a positive real number and q is a positive integer. Show that x1 is a “weighted average” of x0 and a>x0q - 1, and find the coefficients m0, m1 such that x1 = m0 x0 + m1 a

r H

h

a b, x0 q - 1

m0 7 0, m1 7 0, m0 + m1 = 1.

What conclusion would you reach if x0 and a>x0 q - 1 were equal? What would be the value of x1 in that case? 24. The family of straight lines y = ax + b (a, b arbitrary constants) can be characterized by the relation y″ = 0. Find a similar relation satisfied by the family of all circles (x - h)2 + (y - h)2 = r 2,

R

18. Minimizing a parameter  Find the smallest value of the positive constant m that will make mx - 1 + (1>x) greater than or equal to zero for all positive values of x. Limits 19. Evaluate the following limits.

a. lim

xS0

2 sin 5x b. lim sin 5x cot 3x 3x xS0

c. lim x csc2 22x d. lim (sec x - tan x) xS0

x S p>2

x - sin x sin x2 e. lim x - tan x f. lim S S x x 0 x 0 sin x

sec x - 1 x3 - 8 h. lim 2 2 xS0 xS2 x - 4 x

g. lim

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where h and r are arbitrary constants. (Hint: Eliminate h and r from the set of three equations including the given one and two obtained by successive differentiation.) 25. Free fall in the fourteenth century  In the middle of the fourteenth century, Albert of Saxony (1316–1390) proposed a model of free fall that assumed that the velocity of a falling body was proportional to the distance fallen. It seemed reasonable to think that a body that had fallen 20 ft might be moving twice as fast as a body that had fallen 10 ft. And besides, none of the instruments in use at the time were accurate enough to prove otherwise. Today we can see just how far off Albert of Saxony’s model was by solving the initial value problem implicit in his model. Solve the problem and compare your solution graphically with the equation s = 16t 2. You will see that it describes a motion that starts too slowly at first and then becomes too fast too soon to be realistic. T 26. Group blood testing  During World War II it was necessary to administer blood tests to large numbers of recruits. There are two standard ways to administer a blood test to N people. In method 1,

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each person is tested separately. In method 2, the blood samples of x people are pooled and tested as one large sample. If the test is negative, this one test is enough for all x people. If the test is positive, then each of the x people is tested separately, requiring a total of x + 1 tests. Using the second method and some probability theory it can be shown that, on the average, the total number of tests y will be 1 y = N a1 - qx + x b.

With q = 0.99 and N = 1000, find the integer value of x that minimizes y. Also find the integer value of x that maximizes y. (This second result is not important to the real-life situation.) The group testing method was used in World War II with a savings of 80% over the individual testing method, but not with the given value of q. 27. Assume that the brakes of an automobile produce a constant deceleration of k ft>sec2. (a) Determine what k must be to bring an automobile traveling 60 mi>hr (88 ft>sec) to rest in a distance of 100 ft from the point where the brakes are applied. (b) With the same k, how far would a car traveling 30 mi>hr go before being brought to a stop?

b. Show that equality holds in Schwarz’s inequality only if there exists a real number x that makes ai x equal - bi for every value of i from 1 to n. 35. The best branching angles for blood vessels and pipes When a smaller pipe branches off from a larger one in a flow system, we may want it to run off at an angle that is best from some energysaving point of view. We might require, for instance, that energy loss due to friction be minimized along the section AOB shown in the accompanying figure. In this diagram, B is a given point to be reached by the smaller pipe, A is a point in the larger pipe upstream from B, and O is the point where the branching occurs. A law due to Poiseuille states that the loss of energy due to friction in nonturbulent flow is proportional to the length of the path and inversely proportional to the fourth power of the radius. Thus, the loss along AO is (kd1)>R4 and along OB is (kd2)>r 4, where k is a constant, d1 is the length of AO, d2 is the length of OB, R is the radius of the larger pipe, and r is the radius of the smaller pipe. The angle u is to be chosen to minimize the sum of these two losses: L = k

28. Let ƒ(x), g(x) be two continuously differentiable functions satisfying the relationships ƒ′(x) = g(x) and ƒ″(x) = - ƒ(x). Let h(x) = ƒ2(x) + g2(x). If h(0) = 5, find h(10).

B d2

29. Can there be a curve satisfying the following conditions? d 2y>dx2 is everywhere equal to zero and, when x = 0, y = 0 and dy>dx = 1. Give a reason for your answer.

32. A particle moves with acceleration a = 2t - 1 1> 2t 2. Assuming that the velocity y = 4>3 and the position s = -4>15 when t = 0, find a. the velocity y in terms of t. b. the position s in terms of t.

d1

34. Schwarz’s inequality a. In Exercise 33, let ƒ(x) = (a1 x + b1)2 + (a2 x + b2)2 + g + (an x + bn)2, and deduce Schwarz’s inequality: (a1 b1 + a2 b2 + g + an bn)2

O a

C d 2 cos u

In our model, we assume that AC = a and BC = b are fixed. Thus we have the relations d1 + d2 cos u = a d2 sin u = b, so that d2 = b csc u, d1 = a - d2 cos u = a - b cot u. We can express the total loss L as a function of u:

2

33. Given ƒ(x) = ax + 2bx + c with a 7 0. By considering the minimum, prove that ƒ(x) Ú 0 for all real x if and only if b2 - ac … 0.

b = d 2 sin u u

A

30. Find the equation for the curve in the xy-plane that passes through the point (1, - 1) if its slope at x is always 3x2 + 2. 31. A particle moves along the x-axis. Its acceleration is a = -t 2. At t = 0, the particle is at the origin. In the course of its motion, it reaches the point x = b, where b 7 0, but no point beyond b. Determine its velocity at t = 0.

d1 d2 + k 4. 4 R r

L = ka

b csc u a - b cot u + b. R4 r4

a. Show that the critical value of u for which dL>du equals zero is uc = cos-1

r4 . R4

b. If the ratio of the pipe radii is r>R = 5>6, estimate to the nearest degree the optimal branching angle given in part (a).

… 1 a1 2 + a2 2 + g + an 2 21 b1 2 + b2 2 + g + bn 2 2.

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chapter

5

Integrals

Overview  The definite integral is the key tool in calculus for defining and calculating

many important quantities, such as areas, volumes, and averages. As with the derivative, the definite integral also arises as a limit, this time of increasingly fine approximations to the quantity of interest. The idea behind the integral is that we can effectively compute such quantities by breaking them into small pieces, and then summing the contributions from each piece. As the number of terms contributing to the sum approaches infinity and we take the limit of these sums in a way described in Section 5.3, the result is a definite integral. By considering the rate of change of the area under a graph, we prove that defi­ nite integrals are connected to anti­derivatives, a connection that gives one of the most important relationships in calculus.

5.1  Area and Estimating with Finite Sums The basis for formulating definite integrals is the construction of appropriate approxima­ tions by finite sums. In this section we consider three examples of this construction proc­ ess: finding the area under a graph, the distance traveled by a moving object, and the average value of a function. Although we need to define precisely what we mean by the area of a general region in the plane, or the average value of a function over a closed inter­ val, we do have intuitive ideas of what these notions mean. So in this section we begin our approach to integration by approximating these quantities with finite sums. We also con­ sider what happens when we take more and more terms in the summation process. In sub­ sequent sections we look at taking the limit of these sums as the number of terms goes to infinity, which then leads to precise definitions of the quantities being approximated here.

y

1 y = 1 − x2

0.5 R

0

Area 0.5

1

x

Figure 5.1  The area of the region R cannot be found by a simple formula.

Suppose we want to find the area of the shaded region R that lies above the x-axis, below the graph of y = 1 - x2, and between the vertical lines x = 0 and x = 1 (Figure 5.1). Unfortunately, there is no simple geometric formula for calculating the areas of general shapes having curved boundaries like the region R. How, then, can we find the area of R? While we do not yet have a method for determining the exact area of R, we can approximate it in a simple way. Figure 5.2a shows two rectangles that together contain the region R. Each rectangle has width 1>2 and they have heights 1 and 3>4, moving from left to right. The height of each rectangle is the maximum value of the function ƒ in each sub­ interval. Because the function ƒ is decreasing, the height is its value at the left endpoint of the subinterval of 3 0, 14 forming the base of the rectangle. The total area of the two rect­ angles approximates the area A of the region R, A ≈ 1#

1 3#1 7 + = = 0.875. 2 4 2 8

297

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Chapter 5: Integrals

y

1

y y = 1 − x2

(0, 1)

1

Q2 , 4R

Q2 , 4R

1 3

1 3

0.5

y = 1 − x2

(0, 1) Q1 , 15R 4 16

Q4 , 16R 3

0.5

R

7

R

0

0.5

x

1

0

0.25

0.5

0.75

1

x

(b)

(a)

Figure 5.2  (a) We get an upper estimate of the area of R by using two rectangles containing R. (b) Four rectangles give a better upper estimate. Both estimates overshoot the true value for the area by the amount shaded in light red.

This estimate is larger than the true area A since the two rectangles contain R. We say that 0.875 is an upper sum because it is obtained by taking the height of each rectangle as the maximum (uppermost) value of ƒ(x) for a point x in the base interval of the rectangle. In Figure 5.2b, we improve our estimate by using four thinner rectangles, each of width 1>4, which taken together contain the region R. These four rectangles give the approximation A ≈ 1#

7 # 1 25 1 15 # 1 3 # 1 + + + = = 0.78125, 4 16 4 4 4 16 4 32 which is still greater than A since the four rectangles contain R. Suppose instead we use four rectangles contained inside the region R to estimate the area, as in Figure 5.3a. Each rectangle has width 1>4 as before, but the rectangles are shorter and lie entirely beneath the graph of ƒ. The function ƒ(x) = 1 - x2 is decreasing on 3 0, 14 , so the height of each of these rectangles is given by the value of ƒ at the right endpoint of the subinterval forming its base. The fourth rectangle has zero height and therefore contributes no area. Summing these rectangles with heights equal to the minimum value of ƒ(x) for a point x in each base subinterval gives a lower sum approximation to the area, A ≈

15 # 1 3 # 1 7 #1 1 17 + + + 0# = = 0.53125. 4 32 16 4 4 4 16 4

y 1

y 15 Q14 , 16 R

y=1−

x2

1

63 Q18 , 64 R 55 Q38 , 64 R

y = 1 − x2

1 3 Q2 , 4R 39 Q58 , 64 R 7 Q34 , 16 R

0.5

0.5 15 Q78 , 64 R

0

0.25

0.5 (a)

0.75

1

x

0

0.25 0.5 0.75 1 0.125 0.375 0.625 0.875

x

(b)

Figure 5.3  (a) Rectangles contained in R give an estimate for the area that under­ shoots the true value by the amount shaded in light blue. (b) The midpoint rule uses rect­ angles whose height is the value of y = ƒ(x) at the midpoints of their bases. The estimate appears closer to the true value of the area because the light red overshoot areas roughly balance the light blue undershoot areas.

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5.1  Area and Estimating with Finite Sums

299

This estimate is smaller than the area A since the rectangles all lie inside of the region R. The true value of A lies somewhere between these lower and upper sums:

y 1

0.53125 6 A 6 0.78125.

y = 1 − x2

1

0

x

(a)

By considering both lower and upper sum approximations, we get not only estimates for the area, but also a bound on the size of the possible error in these estimates, since the true value of the area lies somewhere between them. Here the error cannot be greater than the difference 0.78125 - 0.53125 = 0.25. Yet another estimate can be obtained by using rectangles whose heights are the values of ƒ at the midpoints of their bases (Figure 5.3b). This method of estimation is called the midpoint rule for approximating the area. The midpoint rule gives an estimate that is between a lower sum and an upper sum, but it is not quite so clear whether it overestimates or underestimates the true area. With four rectangles of width 1>4 as before, the midpoint rule estimates the area of R to be A ≈

y 1

63 # 1 55 # 1 39 # 1 15 # 1 172 # 1 + + + = = 0.671875. 64 4 64 4 64 4 64 4 64 4

In each of our computed sums, the interval 3 a, b4  over which the function ƒ is defined was subdivided into n subintervals of equal width (also called length) ∆x = (b - a)>n, and ƒ was evaluated at a point in each subinterval: c1 in the first subinterval, c2 in the sec­ ond subinterval, and so on. The finite sums then all take the form

y = 1 − x2

ƒ(c1) ∆x + ƒ(c2) ∆x + ƒ(c3) ∆x + g + ƒ(cn) ∆x.

1

0

x

(b)

Figure 5.4  (a) A lower sum using 16 rectangles of equal width ∆x = 1>16. (b) An upper sum using 16 rectangles.

By taking more and more rectangles, with each rectangle thinner than before, it appears that these finite sums give better and better approximations to the true area of the region R. Figure 5.4a shows a lower sum approximation for the area of R using 16 rectangles of equal width. The sum of their areas is 0.634765625, which appears close to the true area, but is still smaller since the rectangles lie inside R. Figure 5.4b shows an upper sum approximation using 16 rectangles of equal width. The sum of their areas is 0.697265625, which is somewhat larger than the true area because the rectangles taken together contain R. The midpoint rule for 16 rectangles gives a total area approximation of 0.6669921875, but it is not immediately clear whether this estimate is larger or smaller than the true area.

Example 1   Table 5.1 shows the values of upper and lower sum approximations to the area of R, using up to 1000 rectangles. In Section 5.2 we will see how to get an exact value of the areas of regions such as R by taking a limit as the base width of each rectangle goes to zero and the number of rectangles goes to infinity. With the techniques developed there, we will be able to show that the area of R is exactly 2>3.

Table 5.1  Finite approximations for the area of R

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Number of subintervals

Lower sum

Midpoint sum

Upper sum

2 4 16 50 100 1000

0.375 0.53125 0.634765625 0.6566 0.66165 0.6661665

0.6875 0.671875 0.6669921875 0.6667 0.666675 0.66666675

0.875 0.78125 0.697265625 0.6766 0.67165 0.6671665

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Distance Traveled Suppose we know the velocity function y(t) of a car moving down a highway, without changing direction, and want to know how far it traveled between times t = a and t = b. The position function s(t) of the car has derivative y(t). If we can find an antiderivative F(t) of y(t) then we can find the car’s position function s(t) by setting s(t) = F(t) + C. The distance traveled can then be found by calculating the change in position, s(b) - s(a) = F(b) - F(a). If the velocity function is known only by the readings at various times of a speedometer on the car, then we have no formula from which to obtain an antiderivative function for velocity. So what do we do in this situation? When we don’t know an antiderivative for the velocity function y(t), we can approxi­ mate the distance traveled with finite sums in a way similar to our estimates for area dis­ cussed before. We subdivide the interval 3 a, b4 into short time intervals on each of which the velocity is considered to be fairly constant. Then we approximate the distance traveled on each time subinterval with the usual distance formula distance = velocity * time and add the results across 3 a, b4 . Suppose the subdivided interval looks like Δt a

t1

Δt

Δt

t2

b

t3

t (sec)

with the subintervals all of equal length ∆t. Pick a number t1 in the first interval. If ∆t is so small that the velocity barely changes over a short time interval of duration ∆t, then the distance traveled in the first time interval is about y(t1) ∆t. If t2 is a number in the second interval, the distance traveled in the second time interval is about y(t2) ∆t. The sum of the distances traveled over all the time intervals is D ≈ y(t1) ∆t + y(t2) ∆t + g + y(tn) ∆t, where n is the total number of subintervals.

Example 2   The velocity function of a projectile fired straight into the air is ƒ(t) = 160 - 9.8t m>sec. Use the summation technique just described to estimate how far the projectile rises during the first 3 sec. How close do the sums come to the exact value of 435.9 m? (You will learn how to compute the exact value easily in Section 5.4.) Solution  We explore the results for different numbers of intervals and different choices of evaluation points. Notice that ƒ(t) is decreasing, so choosing left endpoints gives an upper sum estimate; choosing right endpoints gives a lower sum estimate. (a)  Three subintervals of length 1, withƒ evaluated at left endpoints giving an upper sum:

t1

t2

t3

0

1

2

3

t

Δt

With ƒ evaluated at t = 0, 1, and 2, we have D ≈ ƒ(t1) ∆t + ƒ(t2) ∆t + ƒ(t3) ∆t = 3 160 - 9.8(0)4 (1) + 3 160 - 9.8(1)4 (1) + 3 160 - 9.8(2)4 (1) = 450.6.

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(b) Three subintervals of length 1, with ƒ evaluated at right endpoints giving a lower sum:

0

t1

t2

t3

1

2

3

t

Δt

With ƒ evaluated at t = 1, 2, and 3, we have D ≈ ƒ(t1) ∆t + ƒ(t2) ∆t + ƒ(t3) ∆t = 3 160 - 9.8(1) 4 (1) + 3 160 - 9.8(2) 4 (1) + 3 160 - 9.8(3)4 (1) = 421.2. (c) With six subintervals of length 1>2, we get

t1

t 2 t 3 t 4 t 5 t6

0

1

2

3

t

Δt

t1 t 2 t 3 t 4 t 5 t 6 0

1

2

3

t

Δt

These estimates give an upper sum using left endpoints: D ≈ 443.25; and a lower sum using right endpoints: D ≈ 428.55. These six-interval estimates are somewhat closer than the three-interval estimates. The results improve as the subintervals get shorter.

As we can see in Table 5.2, the left-endpoint upper sums approach the true value 435.9 from above, whereas the right-endpoint lower sums approach it from below. The true value lies between these upper and lower sums. The magnitude of the error in the closest entries is 0.23, a small percentage of the true value. Error magnitude = 0 true value - calculated value 0 = 0 435.9 - 435.67 0 = 0.23.

Error percentage =

0.23 ≈ 0.05,. 435.9

It would be reasonable to conclude from the table’s last entries that the projectile rose about 436 m during its first 3 sec of flight.

Displacement Versus Distance Traveled If an object with position function s(t) moves along a coordinate line without chang­ ing direction, we can calculate the total distance it travels from t = a to t = b by summing the distance traveled over small intervals, as in Example 2. If the object Table 5.2  Travel-distance estimates

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Number of Length of each subintervals subinterval

Upper Lower sum sum

3 6 12 24 48 96 192

450.6 421.2 443.25 428.55 439.58 432.23 437.74 434.06 436.82 434.98 436.36 435.44 436.13 435.67

1 1>2 1>4 1>8 1>16 1>32 1>64

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Chapter 5: Integrals

reverses direction one or more times during the trip, then we need to use the object’s speed 0 y(t) 0 , which is the absolute value of its velocity function, y(t), to find the total distance traveled. Using the velocity itself, as in Example 2, gives instead an estimate to the object’s displacement, s(b) - s(a), the difference between its initial and final positions. To see why using the velocity function in the summation process gives an estimate to the displacement, partition the time interval 3 a, b4 into small enough equal subintervals ∆t so that the object’s velocity does not change very much from time tk - 1 to tk. Then y(tk) gives a good approximation of the velocity throughout the interval. Accordingly, the change in the object’s position coordinate, which is its displacement during the time interval, is about y(tk) ∆t. The change is positive if y(tk) is positive and negative if y(tk) is negative. In either case, the distance traveled by the object during the subinterval is about

s

Height (ft)

256

0 y(tk) 0 ∆t.

s(5)

400 (+)

(−)

s(2)

s(8)

The total distance traveled over the time interval is approximately the sum

144

0 y(t1) 0 ∆t + 0 y(t2) 0 ∆t + g + 0 y(tn) 0 ∆t.

We revisit these ideas in Section 5.4.

s=0

Example 3   In Example 4 in Section 3.4, we analyzed the motion of a heavy rock

s(0)

Figure 5.5  The rock in Example 3. The height s = 256 ft is reached at t = 2 and t = 8 sec. The rock falls 144 ft from its maximum height when t = 8.

Table 5.3  Velocity function t

Y(t)

t

Y(t)

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

160 144 128 112 96 80 64 48 32

4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0

16 0 -16 -32 -48 -64 -80 -96

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blown straight up by a dynamite blast. In that example, we found the velocity of the rock at any time during its motion to be y(t) = 160 - 32t ft>sec. The rock was 256 ft above the ground 2 sec after the explosion, continued upward to reach a maximum height of 400 ft at 5 sec after the explosion, and then fell back down to reach the height of 256 ft again at t = 8 sec after the explosion. (See Figure 5.5.) If we follow a procedure like that presented in Example 2, and use the velocity func­ tion y(t) in the summation process over the time interval 3 0, 84 , we will obtain an esti­ mate to the rock’s 256 ft height above the ground at t = 8. The positive upward motion (which yields a positive distance change of 144 ft from the height of 256 ft to the maxi­ mum height) is canceled by the negative downward motion (giving a negative change of 144 ft from the maximum height down to 256 ft again), so the displacement or height above the ground is estimated from the velocity function. On the other hand, if the absolute value 0 y(t) 0 is used in the summation process, we will obtain an estimate to the total distance the rock has traveled: the maximum height reached of 400 ft plus the additional distance of 144 ft it has fallen back down from that maximum when it again reaches the height of 256 ft at t = 8 sec. That is, using the abso­ lute value of the velocity function in the summation process over the time interval 3 0, 84 , we obtain an estimate to 544 ft, the total distance up and down that the rock has traveled in 8 sec. There is no cancelation of distance changes due to sign changes in the velocity func­ tion, so we estimate distance traveled rather than displacement when we use the absolute value of the velocity function (that is, the speed of the rock). As an illustration of our discussion, we subdivide the interval 3 0, 84 into sixteen sub­ intervals of length ∆t = 1>2 and take the right endpoint of each subinterval in our calcu­ lations. Table 5.3 shows the values of the velocity function at these endpoints. Using y(t) in the summation process, we estimate the displacement at t = 8: (144 + 128 + 112 + 96 + 80 + 64 + 48 + 32 + 16 1 + 0 - 16 - 32 - 48 - 64 - 80 - 96) # = 192 2 Error magnitude = 256 - 192 = 64

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303

Using 0 y(t) 0 in the summation process, we estimate the total distance traveled over the time interval 3 0, 84 : (144 + 128 + 112 + 96 + 80 + 64 + 48 + 32 + 16 1 + 0 + 16 + 32 + 48 + 64 + 80 + 96) # = 528 2 Error magnitude = 544 - 528 = 16

If we take more and more subintervals of 3 0, 84 in our calculations, the estimates to the heights 256 ft and 544 ft improve, approaching them as shown in Table 5.4. Table 5.4  Travel estimates for a rock blown straight up during the time interval 3 0, 8 4

Number of Length of each Total subintervals subinterval Displacement distance 1>2 1>4 1>8 1>16 1>32 1>64

16  32  64 128 256 512

192.0 528.0 224.0 536.0 240.0 540.0 248.0 542.0 252.0 543.0 254.0 543.5

Average Value of a Nonnegative Continuous Function The average value of a collection of n numbers x1, x2, c, xn is obtained by adding them together and dividing by n. But what is the average value of a continuous functionƒon an interval 3 a, b4?  Such a function can assume infinitely many values. For example, the tem­ perature at a certain location in a town is a continuous function that goes up and down each day. What does it mean to say that the average temperature in the town over the course of a day is 73 degrees? When a function is constant, this question is easy to answer. A function with constant value c on an interval 3 a, b4 has average value c. When c is positive, its graph over 3 a, b4 gives a rectangle of height c. The average value of the function can then be interpreted geo­ metrically as the area of this rectangle divided by its width b - a (Figure 5.6a). What if we want to find the average value of a nonconstant function, such as the func­ tion g in Figure 5.6b? We can think of this graph as a snapshot of the height of some water

y

y

0

y = g(x)

y=c

c

a

(a)

b

c

x

0

a

(b)

b

x

Figure 5.6  (a) The average value of ƒ(x) = c on 3a, b4 is the area of the rectangle divided by b - a. (b) The average value of g(x) on 3a, b4 is the area beneath its graph divided by b - a.

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that is sloshing around in a tank between enclosing walls at x = a and x = b. As the water moves, its height over each point changes, but its average height remains the same. To get the average height of the water, we let it settle down until it is level and its height is constant. The resulting height c equals the area under the graph of g divided by b - a. We are led to define the average value of a nonnegative function on an interval 3 a, b4 to be the area under its graph divided by b - a. For this definition to be valid, we need a precise understanding of what is meant by the area under a graph. This will be obtained in Section 5.3, but for now we look at an example.

y f(x) = sin x 1

0

p 2

p

x

Figure 5.7  Approximating the area under ƒ(x) = sin x between 0 and p to compute the average value of sin x over 30, p4, using eight rectangles (Example 4).

Table 5.5  Average value of sin x on 0 … x … p

Number of Upper sum subintervals estimate    8 0.75342   16 0.69707   32 0.65212   50 0.64657  100 0.64161 1000 0.63712

Example 4   Estimate the average value of the function ƒ(x) = sin x on the interval 3 0, p4 .

Solution  Looking at the graph of sin x between 0 and p in Figure 5.7, we can see that its average height is somewhere between 0 and 1. To find the average, we need to calculate the area A under the graph and then divide this area by the length of the interval, p - 0 = p. We do not have a simple way to determine the area, so we approximate it with finite sums. To get an upper sum approximation, we add the areas of eight rectangles of equal width p>8 that together contain the region beneath the graph of y = sin x and above the x-axis on 3 0, p4 . We choose the heights of the rectangles to be the largest value of sin x on each subinterval. Over a particular subinterval, this largest value may occur at the left endpoint, the right endpoint, or somewhere between them. We evaluate sin x at this point to get the height of the rectangle for an upper sum. The sum of the rectangular areas then estimates the total area (Figure 5.7): p p 3p p p 5p 3p 7p p + sin + sin + sin + sin + sin + sin + sin b # 8 8 8 8 8 4 2 2 4 p p = (6.02) # ≈ 2.364. ≈ (.38 + .71 + .92 + 1 + 1 + .92 + .71 + .38) # 8 8

A ≈ asin

To estimate the average value of sin x on 3 0, p4 we divide the estimated area by the length p of the interval and obtain the approximation 2.364>p ≈ 0.753. Since we used an upper sum to approximate the area, this estimate is greater than the actual average value of sin x over 3 0, p4 . If we use more and more rectangles, with each rectangle getting thinner and thinner, we get closer and closer to the true average value as shown in Table 5.5. Using the techniques covered in Section 5.3, we will show that the true average value is 2>p ≈ 0.63662. As before, we could just as well have used rectangles lying under the graph of y = sin x and calculated a lower sum approximation, or we could have used the midpoint rule. In Section 5.3 we will see that in each case, the approximations are close to the true area if all the rectangles are sufficiently thin.

Summary The area under the graph of a positive function, the distance traveled by a moving object that doesn’t change direction, and the average value of a nonnegative function over an interval can all be approximated by finite sums constructed in a certain way. First we sub­ divide the interval into subintervals, treating some function ƒ as if it were constant over each particular subinterval. Then we multiply the width of each subinterval by the value of ƒ at some point within it, and add these products together. If the interval 3 a, b4 is subdi­ vided into n subintervals of equal widths ∆x = (b - a)>n, and if ƒ(ck) is the value ofƒat the chosen point ck in the kth subinterval, this process gives a finite sum of the form ƒ(c1) ∆x + ƒ(c2) ∆x + ƒ(c3) ∆x + g + ƒ(cn) ∆x.

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305

The choices for the ck could maximize or minimize the value of ƒ in the kth subinterval, or give some value in between. The true value lies somewhere between the approximations given by upper sums and lower sums. The finite sum approximations we looked at improved as we took more subintervals of thinner width.

Exercises  5.1 Area In Exercises 1–4, use finite approximations to estimate the area under the graph of the function using

a. a lower sum with two rectangles of equal width.

b. a lower sum with four rectangles of equal width.

c. an upper sum with two rectangles of equal width. d. an upper sum with four rectangles of equal width. 1. ƒ(x) = x2 between x = 0 and x = 1. 2. ƒ(x) = x3 between x = 0 and x = 1.

Time Velocity Time Velocity (min) (m , sec) (min) (m , sec)  0  5 10 15 20 25 30

1 1.2 1.7 2.0 1.8 1.6 1.4

35 40 45 50 55 60

1.2 1.0 1.8 1.5 1.2 0

3. ƒ(x) = 1>x between x = 1 and x = 5. 4. ƒ(x) = 4 - x2 between x = - 2 and x = 2. Using rectangles each of whose height is given by the value of the function at the midpoint of the rectangle’s base (the midpoint rule), estimate the area under the graphs of the following functions, using first two and then four rectangles. 5. ƒ(x) = x2 between x = 0 and x = 1. 6. ƒ(x) = x3 between x = 0 and x = 1. 7. ƒ(x) = 1>x between x = 1 and x = 5.

11. Length of a road  You and a companion are about to drive a twisty stretch of dirt road in a car whose speedometer works but whose odometer (mileage counter) is broken. To find out how long this particular stretch of road is, you record the car’s velocity at 10-sec intervals, with the results shown in the accompanying table. Estimate the length of the road using a. left-endpoint values. b. right-endpoint values.

8. ƒ(x) = 4 - x2 between x = - 2 and x = 2. Distance 9. Distance traveled  The accompanying table shows the velocity of a model train engine moving along a track for 10 sec. Estimate the distance traveled by the engine using 10 subintervals of length 1 with

a. left-endpoint values. b. right-endpoint values. Time Velocity Time Velocity (sec) (in. , sec) (sec) (in. , sec) 0  0  6 11 1 12  7  6 2 22  8  2 3 10  9  6 4  5 10  0 5 13     10. Distance traveled upstream  You are sitting on the bank of a tidal river watching the incoming tide carry a bottle upstream. You record the velocity of the flow every 5 minutes for an hour, with the results shown in the accompanying table. About how far upstream did the bottle travel during that hour? Find an estimate using 12 subintervals of length 5 with a. left-endpoint values. b. right-endpoint values.

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Velocity Velocity Time (converted to ft , sec) Time (converted to ft , sec) (sec) (30 mi , h = 44 ft , sec) (sec) (30 mi , h = 44 ft , sec)  0  0  70 10 44  80 20 15  90 30 35 100 40 30 110 50 44 120 60 35

15 22 35 44 30 35

12. Distance from velocity data  The accompanying table gives data for the velocity of a vintage sports car accelerating from 0 to 142 mi > h in 36 sec (10 thousandths of an hour). Time Velocity Time Velocity (h) (mi , h) (h) (mi , h) 0.0   0 0.006 116 0.001   40 0.007 125 0.002   62 0.008 132 0.003   82 0.009 137 0.004   96 0.010 142 0.005 108

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Chapter 5: Integrals

mihr

18. ƒ(t) = 1 - acos

160

140

pt 4 b  on  30, 44 4

y

4 y = 1 − acos ptb 4

1

120 100 80

0

1

2

3

4

t

60

Examples of Estimations 19. Water pollution  Oil is leaking out of a tanker damaged at sea. The damage to the tanker is worsening as evidenced by the increased leakage each hour, recorded in the following table.

40 20 0

hours

0.002 0.004 0.006 0.008 0.01

a. Use rectangles to estimate how far the car traveled during the 36 sec it took to reach 142 mi > h. b. Roughly how many seconds did it take the car to reach the halfway point? About how fast was the car going then?

13. Free fall with air resistance  An object is dropped straight down from a helicopter. The object falls faster and faster but its accelera­ tion (rate of change of its velocity) decreases over time because of air resistance. The acceleration is measured in ft>sec2 and recorded every second after the drop for 5 sec, as shown: t

0

1

2

3

4

5

a

32.00

19.41

11.77

7.14

4.33

2.63

a. Find an upper estimate for the speed when t = 5. b. Find a lower estimate for the speed when t = 5. c. Find an upper estimate for the distance fallen when t = 3. 14. Distance traveled by a projectile    An object is shot straight upward from sea level with an initial velocity of 400 ft > sec.

a. Assuming that gravity is the only force acting on the object, give an upper estimate for its velocity after 5 sec have elapsed. Use g = 32 ft>sec2 for the gravitational acceleration. b. Find a lower estimate for the height attained after 5 sec. Average Value of a Function

In Exercises 15–18, use a finite sum to estimate the average value of ƒ on the given interval by partitioning the interval into four subintervals of equal length and evaluating ƒ at the subinterval midpoints. 15. ƒ(x) = x3  on  30, 24

1.5

y = 1 + sin 2 pt 2

0.5

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2

3

4

Leakage (gal , h)

50

70

97

136

190

Time (h) Leakage (gal , h)

1

5

6

7

8

265

369

516

720

a. Give an upper and a lower estimate of the total quantity of oil that has escaped after 5 hours.

b.  Repeat part (a) for the quantity of oil that has escaped after 8 hours.

c.  The tanker continues to leak 720 gal > h after the first 8 hours. If the tanker originally contained 25,000 gal of oil, approxi­ mately how many more hours will elapse in the worst case before all the oil has spilled? In the best case?

20. Air pollution  A power plant generates electricity by burning oil. Pollutants produced as a result of the burning process are removed by scrubbers in the smokestacks. Over time, the scrubbers become less efficient and eventually they must be replaced when the amount of pollution released exceeds government standards. Measurements are taken at the end of each month determining the rate at which pollutants are released into the atmosphere, recorded as follows. Month Pollutant release rate (tons > day)

Jan

Feb

Mar

Apr

May

Jun

0.20

0.25

0.27

0.34

0.45

0.52

Jul

Aug

Sep

Oct

Nov

Dec

0.63

0.70

0.81

0.85

0.89

0.95

a. Assuming a 30-day month and that new scrubbers allow only 0.05 ton > day to be released, give an upper estimate of the total tonnage of pollutants released by the end of June. What is a lower estimate?

1

0

1

Pollutant release rate (tons > day)

17. ƒ(t) = (1>2) + sin2 pt on  30, 24 y

0

Month

16. ƒ(x) = 1>x on  31, 94

Time (h)

2

t

b. In the best case, approximately when will a total of 125 tons of pollutants have been released into the atmosphere?

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5.2  Sigma Notation and Limits of Finite Sums

21. Inscribe a regular n-sided polygon inside a circle of radius 1 and compute the area of the polygon for the following values of n:

COMPUTER EXPLORATIONS In Exercises 23–26, use a CAS to perform the following steps.

a. 4 (square)   b.  8 (octagon)   c.  16

a. Plot the functions over the given interval.

d. Compare the areas in parts (a), (b), and (c) with the area of the circle.

b. Subdivide the interval into n = 100, 200, and 1000 subinter­ vals of equal length and evaluate the function at the midpoint of each subinterval.

22. (Continuation of Exercise 21.)

c. Compute the average value of the function values generated in part (b).

a. Inscribe a regular n-sided polygon inside a circle of radius 1 and compute the area of one of the n congruent triangles formed by drawing radii to the vertices of the polygon.

d. Solve the equation ƒ(x) = (average value) for x using the aver­ age value calculated in part (c) for the n = 1000 partitioning.

b. Compute the limit of the area of the inscribed polygon as n S q.

23. ƒ(x) = sin x on  30, p4

c. Repeat the computations in parts (a) and (b) for a circle of radius r.

24. ƒ(x) = sin2 x on  30, p4

p p 1 1 25. ƒ(x) = x sin x  on  c , p d 26. ƒ(x) = x sin2 x  on  c , p d 4 4

5.2  Sigma Notation and Limits of Finite Sums In estimating with finite sums in Section 5.1, we encountered sums with many terms (up to 1000 in Table 5.1, for instance). In this section we introduce a more convenient notation for sums with a large number of terms. After describing the notation and stating several of its properties, we look at what happens to a finite sum approximation as the number of terms approaches infinity.

Finite Sums and Sigma Notation Sigma notation enables us to write a sum with many terms in the compact form a ak = a1 + a2 + a3 + g + an - 1 + an . n

k=1

The Greek letter Σ (capital sigma, corresponding to our letter S), stands for “sum.” The index of summation k tells us where the sum begins (at the number below the Σ symbol) and where it ends (at the number above Σ). Any letter can be used to denote the index, but the letters i, j, and k are customary.

The index k ends at k = n.

n The summation symbol (Greek letter sigma)

ak k=1

a k is a formula for the kth term. The index k starts at k = 1.

Thus we can write

12 + 22 + 32 + 42 + 52 + 62 + 72 + 82 + 92 + 102 + 112 = a k 2,

and

11

ƒ(1) + ƒ(2) + ƒ(3) + g + ƒ(100) = a ƒ(i).

k=1

100

i=1

The lower limit of summation does not have to be 1; it can be any integer.

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Chapter 5: Integrals

Example 1   A sum in sigma notation

The sum written out, one term for each value of k

The value of the sum

ak

1 + 2 + 3 + 4 + 5

15

k a (-1) k

(- 1)1(1) + (- 1)2(2) + (- 1)3(3)

-1 + 2 - 3 = - 2

2 k ak + 1 k=1

1 2 + 1 + 1 2 + 1

1 2 7 + = 2 3 6

52 42 + 4 - 1 5 - 1

16 25 139 + = 3 4 12

5

k=1 3

k=1

5 k2 ak - 1 k=4

Example 2   Express the sum 1 + 3 + 5 + 7 + 9 in sigma notation. Solution  The formula generating the terms changes with the lower limit of summation, but the terms generated remain the same. It is often simplest to start with k = 0 or k = 1, but we can start with any integer. Starting with k = 0:

1 + 3 + 5 + 7 + 9 = a (2k + 1)

Starting with k = 1:

1 + 3 + 5 + 7 + 9 = a (2k - 1)

Starting with k = 2:

1 + 3 + 5 + 7 + 9 = a (2k - 3)

Starting with k = -3:

1 + 3 + 5 + 7 + 9 = a (2k + 7)

4

k=0 5

k=1 6

k=2 1

k = -3

When we have a sum such as 2 a (k + k ) 3

k=1

we can rearrange its terms, 2 2 2 2 a (k + k ) = (1 + 1 ) + (2 + 2 ) + (3 + 3 ) 3

k=1

= (1 + 2 + 3) + (12 + 22 + 32)    Regroup terms. = a k + a k 2. 3

3

k=1

k=1

This illustrates a general rule for finite sums: a (ak + bk) = a ak + a bk . n

n

n

k=1

k=1

k=1

Four such rules are given below. A proof that they are valid can be obtained using mathe­ matical induction (see Appendix 2).

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309

Algebra Rules for Finite Sums

a (ak + bk) = a ak + a bk n

n

n

k=1

k=1

k=1

n

n

n

k=1

k=1

1. Sum Rule:

a (ak - bk) = a ak - a bk

2. Difference Rule:

k=1

# a cak = c a ak   (Any number c) n

n

k=1

k=1

3. Constant Multiple Rule:

# a c = n c    (c is any constant value.) n

4. Constant Value Rule:

k=1

Example 3   We demonstrate the use of the algebra rules.

(a) a (3k - k 2) = 3 a k - a k 2           Multiple Rule n

k=1 n

n

k=1

k=1

n

n

k=1

k=1

Difference Rule and Constant

(b) a (-ak) = a (-1) ak = -1 a ak = - a ak   Constant Multiple Rule

#

#

n

n

k=1

k=1

(c) a (k + 4) = a k + a 4              Sum Rule 3

k=1

3

3

k=1

k=1

= (1 + 2 + 3) + (3 # 4)       Constant Value Rule = 6 + 12 = 18

(d) a n = n n = 1               (1>n is constant) n

k=1

HISTORICAL BIOGRAPHY Carl Friedrich Gauss (1777–1855)

1

#1

Constant Value Rule

Over the years people have discovered a variety of formulas for the values of finite sums. The most famous of these are the formula for the sum of the first n integers (Gauss is said to have discovered it at age 8) and the formulas for the sums of the squares and cubes of the first n integers.

Example 4   Show that the sum of the first n integers is ak = n

k=1

n(n + 1) . 2

Solution  The formula tells us that the sum of the first 4 integers is (4)(5) = 10. 2 Addition verifies this prediction: 1 + 2 + 3 + 4 = 10. To prove the formula in general, we write out the terms in the sum twice, once forward and once backward. 1 n

+ +

2 (n - 1)

+ +

3 (n - 2)

+ +

g g

+ +

n 1

If we add the two terms in the first column we get 1 + n = n + 1. Similarly, if we add the two terms in the second column we get 2 + (n - 1) = n + 1. The two terms in any column sum to n + 1. When we add the n columns together we get n terms, each equal to n + 1, for a total of n(n + 1). Since this is twice the desired quantity, the sum of the first n integers is (n)(n + 1)>2.

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Formulas for the sums of the squares and cubes of the first n integers are proved using mathematical induction (see Appendix 2). We state them here. 2 ak = n

The first n squares:

k=1

3 ak = a n

The first n cubes:

n(n + 1)(2n + 1) 6

k=1

n(n + 1) 2 b 2

Limits of Finite Sums The finite sum approximations we considered in Section 5.1 became more accurate as the number of terms increased and the subinterval widths (lengths) narrowed. The next exam­ ple shows how to calculate a limiting value as the widths of the subintervals go to zero and their number grows to infinity.

Example 5   Find the limiting value of lower sum approximations to the area of the region R below the graph of y = 1 - x2 and above the interval 3 0, 14 on the x-axis using equal-width rectangles whose widths approach zero and whose number approaches infin­ ity. (See Figure 5.4a.) Solution  We compute a lower sum approximation using n rectangles of equal width ∆x = (1 - 0)>n, and then we see what happens as n S q. We start by subdividing 3 0, 14 into n equal width subintervals n - 1 n 1 1 2 c 0, n d , c n , n d , . . . , c n , n d .

Each subinterval has width 1>n. The function 1 - x2 is decreasing on 3 0, 14 , and its small­ est value in a subinterval occurs at the subinterval’s right endpoint. So a lower sum is con­ structed with rectangles whose height over the subinterval 3 (k - 1)>n, k>n4 is ƒ(k>n) = 1 - (k>n)2, giving the sum k n 1 1 2 1 1 1 c ƒa n b d a n b + c ƒa n b d a n b + g + c ƒa n b d a n b + g + c ƒa n b d a n b .

We write this in sigma notation and simplify,

n n k 1 k 2 1 a ƒa n b a n b = a a1 - a n b b a n b

k=1

k=1

n 1 k2 = a an - 3 b n k=1

n n k2 1 = a n - a 3 k=1 k=1 n

1 n 1 = n # n - 3 a k2 n k=1 1 (n)(n + 1)(2n + 1) = 1 - a 3b 6 n = 1 -

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2n3 + 3n2 + n . 6n3

Difference Rule

Constant Value and Constant Multiple Rules

Sum of the First n Squares

Numerator expanded

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311

We have obtained an expression for the lower sum that holds for any n. Taking the limit of this expression as n S q, we see that the lower sums converge as the number of subintervals increases and the subinterval widths approach zero: lim a1 -

nS q

2n3 + 3n2 + n 2 2 b = 1 - = . 3 3 6 6n

The lower sum approximations converge to 2>3. A similar calculation shows that the upper n sum approximations also converge to 2>3. Any finite sum approximation g k = 1 ƒ(ck)(1>n) also converges to the same value, 2>3. This is because it is possible to show that any finite sum approximation is trapped between the lower and upper sum approximations. For this reason we are led to define the area of the region R as this limiting value. In Section 5.3 we study the limits of such finite approximations in a general setting.

Riemann Sums The theory of limits of finite approximations was made precise by the German mathemati­ cian Bernhard Riemann. We now introduce the notion of a Riemann sum, which underlies the theory of the definite integral studied in the next section. We begin with an arbitrary bounded function ƒ defined on a closed interval 3 a, b4 . Like the function pictured in Figure 5.8, ƒ may have negative as well as positive values. We subdivide the interval 3 a, b4 into subintervals, not necessarily of equal widths (or lengths), and form sums in the same way as for the finite approximations in Section 5.1. To do so, we choose n - 1 points 5x1, x2, x3, c, xn - 1 6 between a and b satisfying

HISTORICAL BIOGRAPHY Georg Friedrich Bernhard Riemann (1826–1866)

a 6 x1 6 x2 6 g 6 xn - 1 6 b.

y

To make the notation consistent, we denote a by x0 and b by xn , so that a = x0 6 x1 6 x2 6 g 6 xn - 1 6 xn = b.

y = f (x)

The set 0 a

b

Figure 5.8  A typical continuous function y = ƒ(x) over a closed interval 3 a, b4 .

P = 5x0, x1, x2, c, xn - 1, xn 6

x

is called a partition of 3 a, b4 . The partition P divides 3 a, b4 into n closed subintervals

3 x0, x1 4 , 3 x1, x2 4 , c, 3 xn - 1, xn 4 .

The first of these subintervals is 3 x0, x1 4 , the second is 3 x1, x2 4 , and the kth subinterval of P is 3 xk - 1, xk 4 , for k an integer between 1 and n.

HISTORICAL BIOGRAPHY Richard Dedekind (1831–1916)

kth subinterval

x0 = a

x1

x2

...

x k−1

x

xk

...

x n−1

xn = b

The width of the first subinterval 3 x0, x1 4 is denoted ∆x1, the width of the second 3 x1, x2 4 is denoted ∆x2, and the width of the kth subinterval is ∆xk = xk - xk - 1. If all n subintervals have equal width, then the common width ∆x is equal to (b - a)>n.

Δ x1 x0 = a

Δ x2 x1

Δ xn

Δ xk x2

...

x k−1

xk

...

x xn−1

xn = b

In each subinterval we select some point. The point chosen in the kth subinterval 3 xk - 1, xk 4 is called ck. Then on each subinterval we stand a vertical rectangle that stretches from the x-axis to touch the curve at (ck, ƒ(ck)). These rectangles can be above or below the x-axis, depending on whether ƒ(ck) is positive or negative, or on the x-axis if ƒ(ck) = 0 (Figure 5.9).

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Chapter 5: Integrals y

y = f(x)

(cn, f (cn ))

(ck, f (ck )) kth rectangle

c1 0 x0 = a

c2 x1

x2

cn

ck xk

x k−1

x n−1

x

xn = b

(c1, f (c1))

(c 2, f (c 2))

Figure 5.9  The rectangles approximate the region between the graph of the func­ tion y = ƒ(x) and the x-axis. Figure 5.8 has been enlarged to enhance the partition of 3a, b4 and selection of points ck that produce the rectangles.

On each subinterval we form the product ƒ(ck) # ∆xk. This product is positive, negative, or zero, depending on the sign of ƒ(ck). When ƒ(ck) 7 0, the product ƒ(ck) # ∆xk is the area of a rectangle with height ƒ(ck) and width ∆xk . When ƒ(ck) 6 0, the product ƒ(ck) # ∆xk is a negative number, the negative of the area of a rectangle of width ∆xk that drops from the x-axis to the negative number ƒ(ck). Finally we sum all these products to get

y

SP = a ƒ(ck) ∆xk . n

y = f(x)

k=1

0 a

b

x

(a)

The sum SP is called a Riemann sum for ƒ on the interval 3a, b4. There are many such sums, depending on the partition P we choose, and the choices of the points ck in the sub­ intervals. For instance, we could choose n subintervals all having equal width ∆x = (b - a)>n to partition 3 a, b4 , and then choose the point ck to be the right-hand endpoint of each subinterval when forming the Riemann sum (as we did in Example 5). This choice leads to the Riemann sum formula Sn = a ƒaa + k

y

n

k=1

y = f(x)

0 a

b

x

(b)

Figure 5.10  The curve of Figure 5.9 with rectangles from finer partitions of 3a, b4 . Finer partitions create collections of rectangles with thinner bases that ap­ proximate the region between the graph of ƒ and the x-axis with increasing accuracy.

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(b - a) # b - a n b a n b.

Similar formulas can be obtained if instead we choose ck to be the left-hand endpoint, or the midpoint, of each subinterval. In the cases in which the subintervals all have equal width ∆x = (b - a)>n, we can make them thinner by simply increasing their number n. When a partition has subintervals of varying widths, we can ensure they are all thin by controlling the width of a widest (long­est) subinterval. We define the norm of a partition P, written } P } , to be the largest of all the subinterval widths. If } P } is a small number, then all of the subintervals in the partition P have a small width. Let’s look at an example of these ideas.

Example 6   The set P = 50, 0.2, 0.6, 1, 1.5, 26 is a partition of 3 0, 24 . There are five subintervals of P: 3 0, 0.24 , 3 0.2, 0.64 , 3 0.6, 14 , 3 1, 1.54 , and 3 1.5, 24 :

Δ x1

0

Δ x2

0.2

Δ x4

Δ x3

0.6

1

Δ x5

1.5

2

x

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5.2  Sigma Notation and Limits of Finite Sums

The lengths of the subintervals are   ∆ x1 = 0.2, ∆x2 = 0.4, ∆x3 = 0.4, ∆x4 = 0.5, and ∆x5 = 0.5. The longest subinterval length is 0.5, so the norm of the partition is } P } = 0.5. In this example, there are two subintervals of this length. Any Riemann sum associated with a partition of a closed interval 3 a, b4 defines rect­ angles that approximate the region between the graph of a continuous function ƒ and the x-axis. Partitions with norm approaching zero lead to collections of rectangles that approx­ imate this region with increasing accuracy, as suggested by Figure 5.10. We will see in the next section that if the function ƒ is continuous over the closed interval 3 a, b4 , then no matter how we choose the partition P and the points ck in its subintervals to construct a Riemann sum, a single limiting value is approached as the subinterval widths, controlled by the norm of the partition, approach zero.

Exercises  5.2 Sigma Notation Write the sums in Exercises 1–6 without sigma notation. Then evalu­ ate them.

180k 1. a k=1 k + 8

2. a

2

3

k=1

3. a cos kp 5. a (- 1) k=1

k+1

k

k=1

7. Which of the following express 1 + 3 + 9 + 27 + 81 in sigma notation?

k k+1 a. a 3k - 1 b.  a 3 c.  a3 5

4

3

k=1

k=0

k = -1

8. Which of the following express 1 - 3 + 9 - 27 + 81 - 243 in sigma notation? k k k+1 k+2 a. a (- 3)k - 1 b.  a (- 1) 3 c.  a (-1) 3 6

5

3

k=1

k=0

k = -2

9. Which formula is not equivalent to the other two? 4 (- 1)k - 1 2 (- 1)k 1 (-1)k a. a b.  a k + 1 c.  a k + 2 k 1 k=2 k=0 k = -1 10. Which formula is not equivalent to the other two? 2 2 a. a (k - 1)2 b.  a (k + 1) c.  ak 4

-1

3

k=1

k = -1

k = -3

Express the sums in Exercises 11–16 in sigma notation. The form of your answer will depend on your choice of the lower limit of summation. 11. 1 + 2 + 3 + 4 + 5 + 6 13.

7 7 7 7 + + + 2 4 8 16

1 1 1 1 15. 1 - + - + 2 3 4 5

12. 1 + 16 + 81 + 256 + 625 14. 8 + 16 + 24 + 32 + 40 1 2 3 4 5 16. - + - + 5 5 5 5 5

Values of Finite Sums

17. Suppose that a ak = 9 and a bk = 24. Find the values of n

n

k=1

k=1

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k=1

n

k=1

6. a (- 1) cos kp 4

n

18. Suppose that a ak = 0 and a bk = 1. Find the values of

k=2

p sin k

n

n

6

k=1

d. a (ak - bk) e.  a (bk - 5ak) k=1

4. a cos kp

4

3

k - 1 k

n n b n k a. a 7ak b.  a 24 c.  a (ak + bk) k=1 k=1 k=1

k=1

a. a 8ak b.  a 250bk n

n

k=1

k=1

n

n

k=1

k=1

c. a (ak + 1) d.  a (bk - 1) Evaluate the sums in Exercises 19–32.

2 3 19. a. a k b.  a k c.  ak 14

14

14

k=1

k=1

k=1

13

13

13

k=1

k=1

k=1

2 3 20. a. a k b.  a k c.  ak

21. a (-5k)

5 pk 22.  a k = 1 15

23. a (3 + n2)

24.  a (k 2 - 5)

25. a k(7k + 2)

26.  a k(2k + 1)

7

k=1 8

6

n=1

k=1

5

7

k=1

k=1

k 27. a + a a kb k = 1 225 k=1 5

3

5

3

7 2 7 k3 28.  a a kb - a k=1 k=1 4

29. a. a 3 b.  a 7 c.  a 10 7

500

k=1

k=1

264

k=3

2 30. a. a k b.  a k c.  a k(k - 1) 36

17

71

k=9

k=3

k = 18

31. a. a 4 b.  a c c.  a (k - 1) n

n

n

k=1

k=1

k=1

n c k 1 32. a. a a n + 2nb b.  a n c.  a n2 n

n

k=1

k=1

k=1

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Chapter 5: Integrals

Riemann Sums In Exercises 33–36, graph each function ƒ(x) over the given interval. Partition the interval into four subintervals of equal length. Then add to your sketch the rectangles associated with the Riemann sum Σ 4k = 1ƒ(ck) ∆xk , given that ck is the (a) left-hand endpoint, (b) righthand endpoint, (c) midpoint of the kth subinterval. (Make a separate sketch for each set of rectangles.)

33. ƒ(x) = x2 - 1, 34. ƒ(x) = -x2, 35. ƒ(x) = sin x,

30, 24

39. ƒ(x) = 1 - x2 over the interval 30, 14 . 40. ƒ(x) = 2x over the interval 30, 34 .

41. ƒ(x) = x2 + 1 over the interval 30, 34 .

30, 14

42. ƒ(x) = 3x2 over the interval 30, 14 .

3- p, p4

36. ƒ(x) = sin x + 1,

Limits of Riemann Sums For the functions in Exercises 39–46, find a formula for the Riemann sum obtained by dividing the interval 3a, b4 into n equal subintervals and using the right-hand endpoint for each ck . Then take a limit of these sums as n S q to calculate the area under the curve over 3a, b4 .

3- p, p4

37. Find the norm of the partition P = 50, 1.2, 1.5, 2.3, 2.6, 36.

38. Find the norm of the partition P = 5- 2, -1.6, -0.5, 0, 0.8, 16.

43. ƒ(x) = x + x2 over the interval 30, 14 .

44. ƒ(x) = 3x + 2x2 over the interval 30, 14 .

45. ƒ(x) = 2x3 over the interval 30, 14 .

46. ƒ(x) = x2 - x3 over the interval 3-1, 04 .

5.3  The Definite Integral

In Section 5.2 we investigated the limit of a finite sum for a function defined over a closed interval 3a, b4 using n subintervals of equal width (or length), (b - a)>n. In this section we consider the limit of more general Riemann sums as the norm of the partitions of 3a, b4  approaches zero. For general Riemann sums, the subintervals of the partitions need not have equal widths. The limiting process then leads to the definition of the definite integral of a function over a closed interval 3a, b4 .

Definition of the Definite Integral

The definition of the definite integral is based on the idea that for certain functions, as the norm of the partitions of 3a, b4 approaches zero, the values of the corresponding Riemann sums approach a limiting value J. What we mean by this limit is that a Riemann sum will be close to the number J provided that the norm of its partition is sufficiently small (so that all of its subintervals have thin enough widths). We introduce the symbol P as a small positive number that specifies how close to J the Riemann sum must be, and the symbol d as a second small positive number that specifies how small the norm of a partition must be in order for convergence to happen. We now define this limit precisely. DEFINITION  Let ƒ(x) be a function defined on a closed interval 3a, b4 . We say that a number J is the definite integral of ƒ over 3a, b4 and that J is the limit of the Riemann sums g nk = 1 ƒ(ck) ∆xk if the following condition is satisfied: Given any number P 7 0 there is a corresponding number d 7 0 such that for every partition P = 5x0, x1, c , xn 6  of  3a, b4  with } P } 6 d and any choice of ck in 3 xk - 1, xk 4 , we have 2

2 a ƒ(ck) ∆xk - J 6 P. n

k=1

The definition involves a limiting process in which the norm of the partition goes to zero. We have many choices for a partition P with norm going to zero, and many choices of points ck for each partition. The definite integral exists when we always get the same limit J, no matter what choices are made. When the limit exists we write it as the definite integral J = lim a ƒ(ck) ∆xk . 00 P 00 S 0 n

k=1

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315

The limit of any Riemann sum is always taken as the norm of the partitions approaches zero and the number of subintervals goes to infinity. Leibniz introduced a notation for the definite integral that captures its construction as n a limit of Riemann sums. He envisioned the finite sums g k = 1 ƒ(ck) ∆xk becoming an infi­ nite sum of function values ƒ(x) multiplied by “infinitesimal” subinterval widths dx. The sum symbol g is replaced in the limit by the integral symbol 1 , whose origin is in the letter “S.” The function values ƒ(ck) are replaced by a continuous selection of function values ƒ(x). The subinterval widths ∆xk become the differential dx. It is as if we are sum­ ming all products of the form ƒ(x) # dx as x goes from a to b. While this notation captures the process of constructing an integral, it is Riemann’s definition that gives a precise meaning to the definite integral. The symbol for the number J in the definition of the definite integral is b

ƒ(x) dx, La which is read as “the integral from a to b of ƒ of x dee x” or sometimes as “the integral from a to b of ƒ of x with respect to x.” The component parts in the integral symbol also have names:

Upper limit of integration Integral sign

L a

b

The function is the integrand. x is the variable of integration.

f(x) dx

Lower limit of integration Integral of f from a to b

When you find the value of the integral, you have evaluated the integral.

When the condition in the definition is satisfied, we say that the Riemann sums of ƒ on b 3 a, b4 converge to the definite integral J = 1a ƒ(x) dx and that ƒ is integrable over 3 a, b4 . In the cases where the subintervals all have equal width ∆x = (b - a)>n, we can form each Riemann sum as Sn = a ƒ(ck) ∆xk = a ƒ(ck)a n

n

k=1

k=1

b - a n b ,   ∆xk

= ∆x = (b - a)>n for all k

where ck is chosen in the kth subinterval. When the limit of these Riemann sums as n S q exists and is equal to J, then J is the definite integral ofƒover 3 a, b4 , so J =

La

b

ƒ(x) dx = lim a ƒ(ck)a n Sq n

k=1

b - a n b    } P }

S 0 means n S q.

If we pick the point ck at the right endpoint of the kth subinterval, so ck = a + k ∆x = a + k(b - a)>n, then the formula for the definite integral becomes

La

b

ƒ(x) dx = lim a ƒaa + k nS q n

k=1

(b - a) b - a n b a n b (1)

Equation (1) gives one explicit formula that can be used to compute definite integrals. Other choices of partitions and locations of points ck result in the same value for the defi­ nite integral when we take the limit as n S q provided that the norm of the partition approaches zero.

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Chapter 5: Integrals

The value of the definite integral of a function over any particular interval depends on the function, not on the letter we choose to represent its independent variable. If we decide to use t or u instead of x, we simply write the integral as b

b

b

ƒ(x) dx. La La La No matter how we write the integral, it is still the same number that is defined as a limit of Riemann sums. Since it does not matter what letter we use, the variable of integration is called a dummy variable representing the real numbers in the closed interval 3 a, b4. ƒ(t) dt

or

ƒ(u) du

Integrable and Nonintegrable Functions

Not every function defined over the closed interval 3 a, b4 is integrable there, even if the function is bounded. That is, the Riemann sums for some functions may not converge to the same limiting value, or to any value at all. A full development of exactly which func­ tions defined over 3 a, b4  are integrable requires advanced mathematical analysis, but for­ tunately most functions that commonly occur in applications are integrable. In particular, every continuous function over 3 a, b4 is integrable over this interval, and so is every func­ tion having no more than a finite number of jump discontinuities on 3 a, b4 . (See Figures 1.9 and 1.10. The latter functions are called piecewise-continuous functions, and they are defined in Additional Exercises 11–18 at the end of this chapter.) The following theorem, which is proved in more advanced courses, establishes these results.

THEOREM 1—Integrability of Continuous Functions  If a function ƒ is con­ tinuous over the interval 3 a, b4 , or if ƒ has at most finitely many jump disconti­ b nuities there, then the definite integral 1a ƒ(x) dx exists and ƒ is integrable over 3 a, b4 . The idea behind Theorem 1 for continuous functions is given in Exercises 86 and 87. Briefly, when ƒ is continuous we can choose each ck so that ƒ(ck) gives the maximum value of ƒ on the subinterval 3 xk - 1, xk 4 , resulting in an upper sum. Likewise, we can choose ck to give the minimum value of ƒ on 3 xk - 1, xk 4 to obtain a lower sum. The upper and lower sums can be shown to converge to the same limiting value as the norm of the partition P tends to zero. Moreover, every Riemann sum is trapped between the values of the upper and lower sums, so every Riemann sum converges to the same limit as well. Therefore, the number J in the definition of the definite integral exists, and the continuous function ƒ is integrable over 3 a, b4 . For integrability to fail, a function needs to be sufficiently discontinuous that the region between its graph and the x-axis cannot be approximated well by increasingly thin rectangles. Our first example shows a function that is not integrable over a closed interval.

Example 1   The function ƒ(x) = e

1, 0,

if x is rational if x is irrational

has no Riemann integral over 3 0, 14 . Underlying this is the fact that between any two numbers there is both a rational number and an irrational number. Thus the function jumps up and down too erratically over 3 0, 14 to allow the region beneath its graph and above the x-axis to be approximated by rectangles, no matter how thin they are. We show, in fact, that upper sum approximations and lower sum approximations converge to different limit­ ing values.

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317

If we pick a partition P of 3 0, 14 and choose ck to be the point giving the maximum value for ƒ on 3 xk - 1, xk 4 then the corresponding Riemann sum is U = a ƒ(ck) ∆xk = a (1) ∆xk = 1, n

n

k=1

k=1

since each subinterval 3 xk - 1, xk 4 contains a rational number where ƒ(ck) = 1. Note that n the lengths of the intervals in the partition sum to 1, g k = 1 ∆xk = 1. So each such Rie­ mann sum equals 1, and a limit of Riemann sums using these choices equals 1. On the other hand, if we pick ck to be the point giving the minimum value for ƒ on 3 xk - 1, xk 4 , then the Riemann sum is L = a ƒ(ck) ∆xk = a (0) ∆xk = 0, n

n

k=1

k=1

since