Cover
Title
Statement
Contents
Preface
To the Student
Calculators, Computers, and Other Graphing Devices
Diagnostic Tests
A Preview of Calculus
Ch 1: Functions and Models
1.1: Four Ways to Represent a Function
1.2: Mathematical Models: a Catalog of Essential Functions
1.3: New Functions from Old Functions
1.4: Exponential Functions
1.5: Inverse Functions and Logarithms
Ch 1: Review
Ch 1: Principles Ofproblem Solving
Ch 2: Limits and Derivatives
2.1: The Tangent and Velocity Problems
2.2: The Limit of a Function
2.3: Calculating Limits Using the Limit Laws
2.4: The Precise Definition of a Limit
2.5: Continuity
2.6: Limits at Infinity; Horizontal Asymptotes
2.7: Derivatives and Rates of Change
2.8: The Derivative as a Function
Ch 2: Review
Ch 2: Problems Plus
Ch 3: Differentiation Rules
3.1: Derivatives of Polynomials and Exponential Functions
3.2: The Product and Quotient Rules
3.3: Derivatives of Trigonometric Functions
3.4: The Chain Rule
3.5: Implicit Differentiation
3.6: Derivatives of Logarithmic Functions
3.7: Rates of Change in the Natural and Social Sciences
3.8: Exponential Growth and Decay
3.9: Related Rates
3.10: Linear Approximations and Differentials
3.11: Hyperbolic Functions
Ch 3: Review
Ch 3: Problems Plus
Ch 4: Applications of Differentiation
4.1: Maximum and Minimum Values
4.2: The Mean Value Theorem
4.3: How Derivatives Affect the Shape of a Graph
4.4: Indeterminate Forms and L’Hospital’s Rule
4.5: Summary of Curve Sketching
4.6: Graphing with Calculus and Calculators
4.7: Optimization Problems
4.8: Newton’s Method
4.9: Antiderivatives
Ch 4: Review
Ch 4: Problems Plus
Ch 5: Integrals
5.1: Areas and Distances
5.2: The Definite Integral
5.3: The Fundamental Theorem of Calculus
5.4: Indefinite Integrals and the Net Change Theorem
5.5: The Substitution Rule
Ch 5: Review
Ch 5: Problems Plus
Ch 6: Applications of Integration
6.1: Areas Between Curves
6.2: Volumes
6.3: Volumes by Cylindrical Shells
6.4: Work
6.5: Average Value of a Function
Ch 6: Review
Ch 6: Problems Plus
Ch 7: Techniques of Integration
7.1: Integration by Parts
7.2: Trigonometric Integrals
7.3: Trigonometric Substitution
7.4: Integration of Rational Functions by Partial Fractions
7.5: Strategy for Integration
7.6: Integration Using Tables and Computer Algebra Systems
7.7: Approximate Integration
7.8: Improper Integrals
Ch 7: Review
Ch 7: Problems Plus
Ch 8: Further Applications of Integration
8.1: Arc Length
8.2: Area of a Surface of Revolution
8.3: Applications to Physics and Engineering
8.4: Applications to Economics and Biology
8.5: Probability
Ch 8: Review
Ch 8: Problems Plus
Ch 9: Differential Equations
9.1: Modeling with Differential Equations
9.2: Direction Fields and Euler’s Method
9.3: Separable Equations
9.4: Models for Population Growth
9.5: Linear Equations
9.6: Predator-Prey Systems
Ch 9: Review
Ch 9: Problems Plus
Ch 10: Parametric Equations and Polar Coordinates
10.1: Curves Defined by Parametric Equations
10.2: Calculus with Parametric Curves
10.3: Polar Coordinates
10.4: Areas and Lengths in Polar Coordinates
10.5: Conic Sections
10.6: Conic Sections in Polar Coordinates
Ch 10: Review
Ch 10: Problems Plus
Ch 11: Infinite Sequences and Series
11.1: Sequences
11.2: Series
11.3: The Integral Test and Estimates of Sums
11.4: The Comparison Tests
11.5: Alternating Series
11.6: Absolute Convergence and the Ratio and Root Tests
11.7: Strategy for Testing Series
11.8: Power Series
11.9: Representations of Functions as Power Series
11.10: Taylor and Maclaurin Series
11.11: Applications of Taylor Polynomials
Ch 11: Review
Ch 11: Problems Plus
Ch 12: Vectors and the Geometry of Space
12.1: Three-Dimensional Coordinate Systems
12.2: Vectors
12.3: The Dot Product
12.4: The Cross Product
12.5: Equations of Lines and Planes
Ch 12: Review
Ch 12: Problems Plus
Ch 13: Vector Functions
13.1: Vector Functions and Space Curves
13.2: Derivatives and Integrals of Vector Functions
13.3: Arc Length and Curvature
13.4: Motion in Space: Velocity and Acceleration
Ch 13: Review
Ch 13: Problems Plus
Ch 14: Partial Derivatives
14.1: Functions of Several Variables
14.2: Limits and Continuity
14.3: Partial Derivatives
14.4: Tangent Planes and Linear Approximations
14.5: The Chain Rule
14.6: Directional Derivatives and the Gradient Vector
14.7: Maximum and Minimum Values
14.8: Lagrange Multipliers
Ch 14: Review
Ch 14: Problems Plus
Ch 15: Multiple Integrals
15.1: Double Integrals over Rectangles
15.2: Double Integrals over General Regions
15.3: Double Integrals in Polar Coordinates
15.4: Applications of Double Integrals
15.5: Surface Area
15.6: Triple Integrals
15.7: Triple Integrals in Cylindrical Coordinates
15.8: Triple Integrals in Spherical Coordinates
15.9: Change of Variables in Multiple Integrals
Ch 15: Review
Ch 15: Problems Plus
Ch 16: Vector Calculus
16.1: Vector Fields
16.2: Line Integrals
16.3: The Fundamental Theorem for Line Integrals
16.4: Green’s Theorem
16.5: Curl and Divergence
16.6: Parametric Surfaces and Their Areas
16.7: Surface Integrals
16.8: Stokes’ Theorem
16.9: The Divergence Theorem
16.10: Summary
Ch 16: Review
Ch 16: Problems Plus
Ch 17: Second-Order Differential Equations
17.1: Second-Order Linear Equations
17.2: Nonhomogeneous Linear Equations
17.3: Applications of Second-order Differential Equations
17.4: Series Solutions
Ch 17: Review
Appendixes
A: Numbers, Inequalities, and Absolute Values
B: Coordinate Geometry and Lines
C: Graphs of Second-Degree Equations
D: Trigonometry
E: Sigma Notation
F: Proofs of Theorems
G: the Logarithm Defined as an Integral
H: Complex Numbers
Index

##### Citation preview

CALCULUS Early TranscEndEnTals EighTh EdiTion mETric vErsion

JamEs sTEwarT M C MASTER UNIVERSITY AND UNIVERSITY OF TORONTO

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This is an electronic version of the print textbook. Due to electronic rights restrictions, some third party content may be suppressed. Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. The publisher reserves the right to remove content from this title at any time if subsequent rights restrictions require it. For valuable information on pricing, previous editions, changes to current editions, and alternate formats, please visit www.cengage.com/highered to search by ISBN#, author, title, or keyword for materials in your areas of interest. Important Notice: Media content referenced within the product description or the product text may not be available in the eBook version.

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Contents PrEfacE

xi

To ThE sTudEnT

xxiii

calculaTors, comPuTErs, and oThEr graPhing dEvicEs diagnosTic TEsTs

xxiv

xxvi

a Preview of calculus 1

1 1.1 1.2 1.3 1.4 1.5

Four Ways to Represent a Function 10 Mathematical Models: A Catalog of Essential Functions 23 New Functions from Old Functions 36 Exponential Functions 45 Inverse Functions and Logarithms 55 Review 68

Principles of Problem Solving 71

2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8

The Tangent and Velocity Problems 78 The Limit of a Function 83 Calculating Limits Using the Limit Laws 95 The Precise Definition of a Limit 104 Continuity 114 Limits at Infinity; Horizontal Asymptotes 126 Derivatives and Rates of Change 140 Writing Project • Early Methods for Finding Tangents 152 The Derivative as a Function 152 Review 165

Problems Plus 169

iii Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

iv

Contents

3 3.1

Derivatives of Polynomials and Exponential Functions 172 Applied Project • Building a Better Roller Coaster 182 3.2 The Product and Quotient Rules 183 3.3 Derivatives of Trigonometric Functions 190 3.4 The Chain Rule 197 Applied Project • Where Should a Pilot Start Descent? 208 3.5 Implicit Differentiation 208 Laboratory Project • Families of Implicit Curves 217 3.6 Derivatives of Logarithmic Functions 218 3.7 Rates of Change in the Natural and Social Sciences 224 3.8 Exponential Growth and Decay 237 Applied Project • Controlling Red Blood Cell Loss During Surgery 244 3.9 Related Rates 245 3.10 Linear Approximations and Differentials 251 Laboratory Project • Taylor Polynomials 258 3.11 Hyperbolic Functions 259 Review 266

Problems Plus 270

4 4.1 4.2 4.3 4.4 4.5 4.6 4.7

4.8 4.9

Maximum and Minimum Values 276 Applied Project • The Calculus of Rainbows 285 The Mean Value Theorem 287 How Derivatives Affect the Shape of a Graph 293 Indeterminate Forms and l’Hospital’s Rule 304 Writing Project • The Origins of l’Hospital’s Rule 314 Summary of Curve Sketching 315 Graphing with Calculus and Calculators 323 Optimization Problems 330 Applied Project • The Shape of a Can 343 Applied Project • Planes and Birds: Minimizing Energy 344 Newton’s Method 345 Antiderivatives 350 Review 358

Problems Plus 363

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

Contents

5 5.1 5.2 5.3 5.4 5.5

Areas and Distances 366 The Definite Integral 378 Discovery Project • Area Functions 391 The Fundamental Theorem of Calculus 392 Indefinite Integrals and the Net Change Theorem 402 Writing Project • Newton, Leibniz, and the Invention of Calculus 411 The Substitution Rule 412 Review 421

Problems Plus 425

6 6.1 6.2 6.3 6.4 6.5

Areas Between Curves 428 Applied Project • The Gini Index 436 Volumes 438 Volumes by Cylindrical Shells 449 Work 455 Average Value of a Function 461 Applied Project • Calculus and Baseball 464 Applied Project • Where to Sit at the Movies 465 Review 466

Problems Plus 468

7 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8

Integration by Parts 472 Trigonometric Integrals 479 Trigonometric Substitution 486 Integration of Rational Functions by Partial Fractions 493 Strategy for Integration 503 Integration Using Tables and Computer Algebra Systems 508 Discovery Project • Patterns in Integrals 513 Approximate Integration 514 Improper Integrals 527 Review 537

Problems Plus 540

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v

vi

Contents

8 8.1 8.2 8.3 8.4 8.5

Arc Length 544 Discovery Project • Arc Length Contest 550 Area of a Surface of Revolution 551 Discovery Project • Rotating on a Slant 557 Applications to Physics and Engineering 558 Discovery Project • Complementary Coffee Cups 568 Applications to Economics and Biology 569 Probability 573 Review 581

Problems Plus 583

9 9.1 9.2 9.3

9.4 9.5 9.6

Modeling with Differential Equations 586 Direction Fields and Euler’s Method 591 Separable Equations 599 Applied Project • How Fast Does a Tank Drain? 608 Applied Project • Which Is Faster, Going Up or Coming Down? 609 Models for Population Growth 610 Linear Equations 620 Predator-Prey Systems 627 Review 634

Problems Plus 637

10 10.1 10.2 10.3 10.4

Curves Defined by Parametric Equations 640 Laboratory Project • Running Circles Around Circles 648 Calculus with Parametric Curves 649 Laboratory Project • Bézier Curves 657 Polar Coordinates 658 Laboratory Project • Families of Polar Curves 668 Areas and Lengths in Polar Coordinates 669

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Contents

10.5 10.6

Conic Sections 674 Conic Sections in Polar Coordinates 682 Review 689

Problems Plus 692

11 11.1

Sequences 694 Laboratory Project • Logistic Sequences 707 11.2 Series 707 11.3 The Integral Test and Estimates of Sums 719 11.4 The Comparison Tests 727 11.5 Alternating Series 732 11.6 Absolute Convergence and the Ratio and Root Tests 737 11.7 Strategy for Testing Series 744 11.8 Power Series 746 11.9 Representations of Functions as Power Series 752 11.10 Taylor and Maclaurin Series 759 Laboratory Project • An Elusive Limit 773 Writing Project • How Newton Discovered the Binomial Series 773 11.11 Applications of Taylor Polynomials 774 Applied Project • Radiation from the Stars 783 Review 784

Problems Plus 787

12 12.1 12.2 12.3 12.4 12.5 12.6

Three-Dimensional Coordinate Systems 792 Vectors 798 The Dot Product 807 The Cross Product 814 Discovery Project • The Geometry of a Tetrahedron 823 Equations of Lines and Planes 823 Laboratory Project • Putting 3D in Perspective 833 Cylinders and Quadric Surfaces 834 Review 841

Problems Plus 844

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vii

viii

Contents

13 13.1 13.2 13.3 13.4

Vector Functions and Space Curves 848 Derivatives and Integrals of Vector Functions 855 Arc Length and Curvature 861 Motion in Space: Velocity and Acceleration 870 Applied Project • Kepler’s Laws 880 Review 881

Problems Plus 884

14 14.1 14.2 14.3 14.4 14.5 14.6 14.7

14.8

Functions of Several Variables 888 Limits and Continuity 903 Partial Derivatives 911 Tangent Planes and Linear Approximations 927 Applied Project • The Speedo LZR Racer 936 The Chain Rule 937 Directional Derivatives and the Gradient Vector 946 Maximum and Minimum Values 959 Applied Project • Designing a Dumpster 970 Discovery Project • Quadratic Approximations and Critical Points 970 Lagrange Multipliers 971 Applied Project • Rocket Science 979 Applied Project • Hydro-Turbine Optimization 980 Review 981

Problems Plus 985

15 15.1 15.2 15.3 15.4 15.5

Double Integrals over Rectangles 988 Double Integrals over General Regions 1001 Double Integrals in Polar Coordinates 1010 Applications of Double Integrals 1016 Surface Area 1026

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Contents

15.6 Triple Integrals 1029 Discovery Project • Volumes of Hyperspheres 1040

15.7 Triple Integrals in Cylindrical Coordinates 1040 Discovery Project • The Intersection of Three Cylinders 1044 15.8 Triple Integrals in Spherical Coordinates 1045 Applied Project • Roller Derby 1052

15.9 Change of Variables in Multiple Integrals 1052 Review 1061

Problems Plus 1065

16 16.1 16.2 16.3 16.4 16.5 16.6 16.7 16.8

Vector Fields 1068 Line Integrals 1075 The Fundamental Theorem for Line Integrals 1087 Green’s Theorem 1096 Curl and Divergence 1103 Parametric Surfaces and Their Areas 1111 Surface Integrals 1122 Stokes’ Theorem 1134 Writing Project • Three Men and Two Theorems 1140 16.9 The Divergence Theorem 1141 16.10 Summary 1147 Review 1148

Problems Plus 1151

17 17.1 17.2 17.3 17.4

Second-Order Linear Equations 1154 Nonhomogeneous Linear Equations 1160 Applications of Second-Order Differential Equations 1168 Series Solutions 1176 Review 1181

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

ix

x

Contents

A B C D E F G H I

Numbers, Inequalities, and Absolute Values A2 Coordinate Geometry and Lines A10 Graphs of Second-Degree Equations A16 Trigonometry A24 Sigma Notation A34 Proofs of Theorems A39 The Logarithm Defined as an Integral A50 Complex Numbers A57 Answers to Odd-Numbered Exercises A65

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Preface This International Metric Version differs from the regular version of Calculus: Early Transcendentals, Eighth Edition, in several ways: The units used in almost all of the examples and exercises have been changed from US Customary units to metric units. There are a small number of exceptions: In some engineering applications (principally in Section 8.3) it may be useful for some engineers to be familiar with US units. And I wanted to retain a few exercises (for example, those involving baseball) where it would be inappropriate to use metric units. I’ve changed the examples and exercises involving real-world data to be more international in nature, so that the vast majority of them now come from countries other than the United States. For example, there are now exercises and examples concerning Hong kong postal rates; Canadian public debt; unemployment rates in Australia; hours of daylight in Ankara, Turkey; isothermals in China; percentage of the population in rural Argentina; populations of Malaysia, Indonesia, Mexico, and India; and power consumption in Ontario, among many others. In addition to changing exercises so that the units are metric and the data have a more international flavor, a number of other exercises have been changed as well, the result being that about 10% of the exercises are different from those in the regular version.

The art of teaching, Mark Van Doren said, is the art of assisting discovery. I have tried to write a book that assists students in discovering calculus—both for its practical power and its surprising beauty. In this edition, as in the first seven editions, I aim to convey to the student a sense of the utility of calculus and develop technical competence, but I also strive to give some appreciation for the intrinsic beauty of the subject. Newton undoubtedly experienced a sense of triumph when he made his great discoveries. I want students to share some of that excitement. The emphasis is on understanding concepts. I think that nearly everybody agrees that this should be the primary goal of calculus instruction. In fact, the impetus for the current calculus reform movement came from the Tulane Conference in 1986, which formulated as their first recommendation: Focus on conceptual understanding. I have tried to implement this goal through the Rule of Three: “Topics should be presented geometrically, numerically, and algebraically.” Visualization, numerical and graphical experimentation, and other approaches have changed how we teach conceptual reasoning in fundamental ways. More recently, the Rule of Three has been expanded to become the Rule of Four by emphasizing the verbal, or descriptive, point of view as well. In writing the eighth edition my premise has been that it is possible to achieve conceptual understanding and still retain the best traditions of traditional calculus. The book contains elements of reform, but within the context of a traditional curriculum. xi Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

xii

Preface

I have written several other calculus textbooks that might be preferable for some instructors. Most of them also come in single variable and multivariable versions. ●

Calculus, Eighth Edition, International Metric Version, is similar to the present textbook except that the exponential, logarithmic, and inverse trigonometric functions are covered in the second semester. Essential Calculus, Second Edition, International Edition, is a much briefer book (840 pages), though it contains almost all of the topics in Calculus, Eighth Edition, International Metric Version. The relative brevity is achieved through briefer exposition of some topics and putting some features on the website. Essential Calculus: Early Transcendentals, Second Edition, International Edition, resembles Essential Calculus, International Edition, but the exponential, logarithmic, and inverse trigonometric functions are covered in Chapter 3. Calculus: Concepts and Contexts, Fourth Edition, Metric International Edition, emphasizes conceptual understanding even more strongly than this book. The coverage of topics is not encyclopedic and the material on transcendental functions and on parametric equations is woven throughout the book instead of being treated in separate chapters. Calculus: Early Vectors introduces vectors and vector functions in the first semester and integrates them throughout the book. It is suitable for students taking engineering and physics courses concurrently with calculus. Brief Applied Calculus, International Edition, is intended for students in business, the social sciences, and the life sciences. Biocalculus: Calculus for the Life Sciences is intended to show students in the life sciences how calculus relates to biology. Biocalculus: Calculus, Probability, and Statistics for the Life Sciences contains all the content of Biocalculus: Calculus for the Life Sciences as well as three additional chapters covering probability and statistics.

The changes have resulted from talking with my colleagues and students at the University of Toronto and from reading journals, as well as suggestions from users and reviewers. Here are some of the many improvements that I’ve incorporated into this edition: ●

The data in examples and exercises have been updated to be more timely.

New examples have been added (see Examples 6.1.5, 11.2.5, and 14.3.3, for instance). And the solutions to some of the existing examples have been amplified. Three new projects have been added: The project Controlling Red Blood Cell Loss During Surgery (page 244) describes the ANH procedure, in which blood is extracted from the patient before an operation and is replaced by saline solution. This dilutes the patient’s blood so that fewer red blood cells are lost during bleeding and the extracted blood is returned to the patient after surgery. The project Planes and Birds: Minimizing Energy (page 344) asks how birds can minimize power and energy by flapping their wings versus gliding. In the project The Speedo LZR Racer (page 936) it is explained that this suit reduces drag in the water and, as

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

Preface

xiii

a result, many swimming records were broken. Students are asked why a small decrease in drag can have a big effect on performance. I have streamlined Chapter 15 (Multiple Integrals) by combining the first two sections so that iterated integrals are treated earlier. More than 20% of the exercises in each chapter are new. Here are some of my favorites: 2.7.61, 2.8.36–38, 3.1.79–80, 3.11.54, 4.1.69, 4.3.34, 4.3.66, 4.4.80, 4.7.39, 4.7.67, 5.1.19–20, 5.2.67–68, 5.4.70, 6.1.51, 8.1.39, 12.5.81, 12.6.29–30, 14.6.65–66. In addition, there are some good new Problems Plus. (See Problems 12–14 on page 272, Problem 13 on page 363, Problems 16–17 on page 426, and Problem 8 on page 986.)

conceptual Exercises The most important way to foster conceptual understanding is through the problems that we assign. To that end I have devised various types of problems. Some exercise sets begin with requests to explain the meanings of the basic concepts of the section. (See, for instance, the first few exercises in Sections 2.2, 2.5, 11.2, 14.2, and 14.3.) Similarly, all the review sections begin with a Concept Check and a True-False Quiz. Other exercises test conceptual understanding through graphs or tables (see Exercises 2.7.17, 2.8.35–38, 2.8.47–52, 9.1.11–13, 10.1.24–27, 11.10.2, 13.2.1–2, 13.3.33–39, 14.1.1–2, 14.1.32–38, 14.1.41–44, 14.3.3–10, 14.6.1–2, 14.7.3–4, 15.1.6–8, 16.1.11–18, 16.2.17–18, and 16.3.1–2). Another type of exercise uses verbal description to test conceptual understanding (see Exercises 2.5.10, 2.8.66, 4.3.69–70, and 7.8.67). I particularly value problems that combine and compare graphical, numerical, and algebraic approaches (see Exercises 2.6.45–46, 3.7.27, and 9.4.4).

graded Exercise sets Each exercise set is carefully graded, progressing from basic conceptual exercises and skill-development problems to more challenging problems involving applications and proofs.

real-world data My assistants and I spent a great deal of time looking in libraries, contacting companies and government agencies, and searching the Internet for interesting real-world data to introduce, motivate, and illustrate the concepts of calculus. As a result, many of the examples and exercises deal with functions defined by such numerical data or graphs. See, for instance, Figure 1 in Section 1.1 (seismograms from the Northridge earthquake), Exercise 2.8.35 (unemployment rates), Exercise 5.1.16 (velocity of the space shuttle Endeavour), and Figure 4 in Section 5.4 (San Francisco power consumption). Functions of two variables are illustrated by a table of values of the wind-chill index as a function of air temperature and wind speed (Example 14.1.2). Partial derivatives are introduced in Section 14.3 by examining a column in a table of values of the heat index (perceived air temperature) as a function of the actual temperature and the relative humidity. This example is pursued further in connection with linear approximations (Example 14.4.3). Directional derivatives are introduced in Section 14.6 by using a temperature contour map to estimate the rate of change of temperature at Reno in the direction of Las Vegas. Double integrals are used to estimate the average snowfall in Colorado on December Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

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Preface

20–21, 2006 (Example 15.1.9). Vector fields are introduced in Section 16.1 by depictions of actual velocity vector fields showing San Francisco Bay wind patterns.

Projects One way of involving students and making them active learners is to have them work (perhaps in groups) on extended projects that give a feeling of substantial accomplishment when completed. I have included four kinds of projects: Applied Projects involve applications that are designed to appeal to the imagination of students. The project after Section 9.3 asks whether a ball thrown upward takes longer to reach its maximum height or to fall back to its original height. (The answer might surprise you.) The project after Section 14.8 uses Lagrange multipliers to determine the masses of the three stages of a rocket so as to minimize the total mass while enabling the rocket to reach a desired velocity. Laboratory Projects involve technology; the one following Section 10.2 shows how to use Bézier curves to design shapes that represent letters for a laser printer. Writing Projects ask students to compare present-day methods with those of the founders of calculus—Fermat’s method for finding tangents, for instance. Suggested references are supplied. Discovery Projects anticipate results to be discussed later or encourage discovery through pattern recognition (see the one following Section 7.6). Others explore aspects of geometry: tetrahedra (after Section 12.4), hyperspheres (after Section 15.6), and intersections of three cylinders (after Section 15.7). Additional projects can be found in the Instructor’s Guide (see, for instance, Group Exercise 5.1: Position from Samples).

Problem solving Students usually have difficulties with problems for which there is no single well-defined procedure for obtaining the answer. I think nobody has improved very much on George Polya’s four-stage problem-solving strategy and, accordingly, I have included a version of his problem-solving principles following Chapter 1. They are applied, both explicitly and implicitly, throughout the book. After the other chapters I have placed sections called Problems Plus, which feature examples of how to tackle challenging calculus problems. In selecting the varied problems for these sections I kept in mind the following advice from David Hilbert: “A mathematical problem should be difficult in order to entice us, yet not inaccessible lest it mock our efforts.” When I put these challenging problems on assignments and tests I grade them in a different way. Here I reward a student significantly for ideas toward a solution and for recognizing which problem-solving principles are relevant.

Technology The availability of technology makes it not less important but more important to clearly understand the concepts that underlie the images on the screen. But, when properly used, graphing calculators and computers are powerful tools for discovering and understanding those concepts. This textbook can be used either with or without technology and I use two special symbols to indicate clearly when a particular type of machine is required. The icon ; indicates an exercise that definitely requires the use of such technology, but that is not to say that it can’t be used on the other exercises as well. The symbol CAS is reserved for problems in which the full resources of a computer algebra system (like Maple, Mathematica, or the TI-89) are required. But technology doesn’t make pencil and paper obsolete. Hand calculation and sketches are often preferable to technology for illustrating and reinforcing some concepts. Both instructors and students need to develop the ability to decide where the hand or the machine is appropriate.

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

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xv

Tools for Enriching calculus TEC is a companion to the text and is intended to enrich and complement its contents. (It is now accessible in the eBook via CourseMate and Enhanced WebAssign. Selected Visuals and Modules are available at www.stewartcalculus.com.) Developed by Harvey keynes, Dan Clegg, Hubert Hohn, and myself, TEC uses a discovery and exploratory approach. In sections of the book where technology is particularly appropriate, marginal icons direct students to TEC Modules that provide a laboratory environment in which they can explore the topic in different ways and at different levels. Visuals are animations of figures in text; Modules are more elaborate activities and include exercises. Instructors can choose to become involved at several different levels, ranging from simply encouraging students to use the Visuals and Modules for independent exploration, to assigning specific exercises from those included with each Module, or to creating additional exercises, labs, and projects that make use of the Visuals and Modules. TEC also includes Homework Hints for representative exercises (usually odd-numbered) in every section of the text, indicated by printing the exercise number in red. These hints are usually presented in the form of questions and try to imitate an effective teaching assistant by functioning as a silent tutor. They are constructed so as not to reveal any more of the actual solution than is minimally necessary to make further progress.

Enhanced webassign Technology is having an impact on the way homework is assigned to students, particularly in large classes. The use of online homework is growing and its appeal depends on ease of use, grading precision, and reliability. With the Eighth Edition we have been working with the calculus community and WebAssign to develop an online homework system. Up to 70% of the exercises in each section are assignable as online homework, including free response, multiple choice, and multi-part formats. The system also includes Active Examples, in which students are guided in step-bystep tutorials through text examples, with links to the textbook and to video solutions.

website Visit CengageBrain.com or stewartcalculus.com for these additional materials: ●

Homework Hints

Algebra Review

Lies My Calculator and Computer Told Me

History of Mathematics, with links to the better historical websites

Additional Topics (complete with exercise sets): Fourier Series, Formulas for the Remainder Term in Taylor Series, Rotation of Axes

Archived Problems (Drill exercises that appeared in previous editions, together with their solutions)

Challenge Problems (some from the Problems Plus sections from prior editions)

Links, for particular topics, to outside Web resources

Selected Visuals and Modules from Tools for Enriching Calculus (TEC)

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

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Preface

diagnostic Tests

The book begins with four diagnostic tests, in Basic Algebra, Analytic Geometry, Functions, and Trigonometry.

a Preview of calculus

This is an overview of the subject and includes a list of questions to motivate the study of calculus.

1 functions and models

From the beginning, multiple representations of functions are stressed: verbal, numerical, visual, and algebraic. A discussion of mathematical models leads to a review of the standard functions, including exponential and logarithmic functions, from these four points of view.

2 limits and derivatives

The material on limits is motivated by a prior discussion of the tangent and velocity problems. Limits are treated from descriptive, graphical, numerical, and algebraic points of view. Section 2.4, on the precise definition of a limit, is an optional section. Sections 2.7 and 2.8 deal with derivatives (especially with functions defined graphically and numerically) before the differentiation rules are covered in Chapter 3. Here the examples and exercises explore the meanings of derivatives in various contexts. Higher derivatives are introduced in Section 2.8.

3 differentiation rules

All the basic functions, including exponential, logarithmic, and inverse trigonometric functions, are differentiated here. When derivatives are computed in applied situations, students are asked to explain their meanings. Exponential growth and decay are now covered in this chapter.

4 applications of differentiation

The basic facts concerning extreme values and shapes of curves are deduced from the Mean Value Theorem. Graphing with technology emphasizes the interaction between calculus and calculators and the analysis of families of curves. Some substantial optimization problems are provided, including an explanation of why you need to raise your head 42° to see the top of a rainbow.

5 integrals

The area problem and the distance problem serve to motivate the definite integral, with sigma notation introduced as needed. (Full coverage of sigma notation is provided in Appendix E.) Emphasis is placed on explaining the meanings of integrals in various contexts and on estimating their values from graphs and tables.

6 applications of integration

Here I present the applications of integration—area, volume, work, average value—that can reasonably be done without specialized techniques of integration. General methods are emphasized. The goal is for students to be able to divide a quantity into small pieces, estimate with Riemann sums, and recognize the limit as an integral.

7 Techniques of integration

All the standard methods are covered but, of course, the real challenge is to be able to recognize which technique is best used in a given situation. Accordingly, in Section 7.5, I present a strategy for integration. The use of computer algebra systems is discussed in Section 7.6.

8 further applications of integration

Here are the applications of integration—arc length and surface area—for which it is useful to have available all the techniques of integration, as well as applications to biology, economics, and physics (hydrostatic force and centers of mass). I have also included a section on probability. There are more applications here than can realistically be covered in a given course. Instructors should select applications suitable for their students and for which they themselves have enthusiasm.

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

Preface

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9 differential Equations

Modeling is the theme that unifies this introductory treatment of differential equations. Direction fields and Euler’s method are studied before separable and linear equations are solved explicitly, so that qualitative, numerical, and analytic approaches are given equal consideration. These methods are applied to the exponential, logistic, and other models for population growth. The first four or five sections of this chapter serve as a good introduction to first-order differential equations. An optional final section uses predatorprey models to illustrate systems of differential equations.

10 Parametric Equations and Polar coordinates

This chapter introduces parametric and polar curves and applies the methods of calculus to them. Parametric curves are well suited to laboratory projects; the two presented here involve families of curves and Bézier curves. A brief treatment of conic sections in polar coordinates prepares the way for Kepler’s Laws in Chapter 13.

11 infinite sequences and series

The convergence tests have intuitive justifications (see page 719) as well as formal proofs. Numerical estimates of sums of series are based on which test was used to prove convergence. The emphasis is on Taylor series and polynomials and their applications to physics. Error estimates include those from graphing devices.

12 vectors and the geometry of space

The material on three-dimensional analytic geometry and vectors is divided into two chapters. Chapter 12 deals with vectors, the dot and cross products, lines, planes, and surfaces.

13 vector functions

This chapter covers vector-valued functions, their derivatives and integrals, the length and curvature of space curves, and velocity and acceleration along space curves, culminating in Kepler’s laws.

14 Partial derivatives

Functions of two or more variables are studied from verbal, numerical, visual, and algebraic points of view. In particular, I introduce partial derivatives by looking at a specific column in a table of values of the heat index (perceived air temperature) as a function of the actual temperature and the relative humidity.

15 multiple integrals

Contour maps and the Midpoint Rule are used to estimate the average snowfall and average temperature in given regions. Double and triple integrals are used to compute probabilities, surface areas, and (in projects) volumes of hyperspheres and volumes of intersections of three cylinders. Cylindrical and spherical coordinates are introduced in the context of evaluating triple integrals.

16 vector calculus

Vector fields are introduced through pictures of velocity fields showing San Francisco Bay wind patterns. The similarities among the Fundamental Theorem for line integrals, Green’s Theorem, Stokes’ Theorem, and the Divergence Theorem are emphasized.

17 second-order differential Equations

Since first-order differential equations are covered in Chapter 9, this final chapter deals with second-order linear differential equations, their application to vibrating springs and electric circuits, and series solutions.

Calculus, Early Transcendentals, Eighth Edition, International Metric Version, is supported by a complete set of ancillaries developed under my direction. Each piece has been designed to enhance student understanding and to facilitate creative instruction. The tables on pages xxi–xxii describe each of these ancillaries.

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

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Preface

The preparation of this and previous editions has involved much time spent reading the reasoned (but sometimes contradictory) advice from a large number of astute reviewers. I greatly appreciate the time they spent to understand my motivation for the approach taken. I have learned something from each of them.

Eighth Edition reviewers Jay Abramson, Arizona State University Adam Bowers, University of California San Diego Neena Chopra, The Pennsylvania State University Edward Dobson, Mississippi State University Isaac Goldbring, University of Illinois at Chicago Lea Jenkins, Clemson University Rebecca Wahl, Butler University

Technology reviewers Maria Andersen, Muskegon Community College Eric Aurand, Eastfield College Joy Becker, University of Wisconsin–Stout Przemyslaw Bogacki, Old Dominion University Amy Elizabeth Bowman, University of Alabama in Huntsville Monica Brown, University of Missouri–St. Louis Roxanne Byrne, University of Colorado at Denver and Health Sciences Center Teri Christiansen, University of Missouri–Columbia Bobby Dale Daniel, Lamar University Jennifer Daniel, Lamar University Andras Domokos, California State University, Sacramento Timothy Flaherty, Carnegie Mellon University Lee Gibson, University of Louisville Jane Golden, Hillsborough Community College Semion Gutman, University of Oklahoma Diane Hoffoss, University of San Diego Lorraine Hughes, Mississippi State University Jay Jahangiri, Kent State University John Jernigan, Community College of Philadelphia

Brian karasek, South Mountain Community College Jason kozinski, University of Florida Carole krueger, The University of Texas at Arlington ken kubota, University of Kentucky John Mitchell, Clark College Donald Paul, Tulsa Community College Chad Pierson, University of Minnesota, Duluth Lanita Presson, University of Alabama in Huntsville karin Reinhold, State University of New York at Albany Thomas Riedel, University of Louisville Christopher Schroeder, Morehead State University Angela Sharp, University of Minnesota, Duluth Patricia Shaw, Mississippi State University Carl Spitznagel, John Carroll University Mohammad Tabanjeh, Virginia State University Capt. koichi Takagi, United States Naval Academy Lorna TenEyck, Chemeketa Community College Roger Werbylo, Pima Community College David Williams, Clayton State University Zhuan Ye, Northern Illinois University

Previous Edition reviewers B. D. Aggarwala, University of Calgary John Alberghini, Manchester Community College Michael Albert, Carnegie-Mellon University Daniel Anderson, University of Iowa Amy Austin, Texas A&M University Donna J. Bailey, Northeast Missouri State University Wayne Barber, Chemeketa Community College Marilyn Belkin, Villanova University Neil Berger, University of Illinois, Chicago David Berman, University of New Orleans Anthony J. Bevelacqua, University of North Dakota Richard Biggs, University of Western Ontario Robert Blumenthal, Oglethorpe University

Martina Bode, Northwestern University Barbara Bohannon, Hofstra University Jay Bourland, Colorado State University Philip L. Bowers, Florida State University Amy Elizabeth Bowman, University of Alabama in Huntsville Stephen W. Brady, Wichita State University Michael Breen, Tennessee Technological University Robert N. Bryan, University of Western Ontario David Buchthal, University of Akron Jenna Carpenter, Louisiana Tech University Jorge Cassio, Miami-Dade Community College Jack Ceder, University of California, Santa Barbara Scott Chapman, Trinity University

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

Preface

Zhen-Qing Chen, University of Washington—Seattle James Choike, Oklahoma State University Barbara Cortzen, DePaul University Carl Cowen, Purdue University Philip S. Crooke, Vanderbilt University Charles N. Curtis, Missouri Southern State College Daniel Cyphert, Armstrong State College Robert Dahlin M. Hilary Davies, University of Alaska Anchorage Gregory J. Davis, University of Wisconsin–Green Bay Elias Deeba, University of Houston–Downtown Daniel DiMaria, Suffolk Community College Seymour Ditor, University of Western Ontario Greg Dresden, Washington and Lee University Daniel Drucker, Wayne State University kenn Dunn, Dalhousie University Dennis Dunninger, Michigan State University Bruce Edwards, University of Florida David Ellis, San Francisco State University John Ellison, Grove City College Martin Erickson, Truman State University Garret Etgen, University of Houston Theodore G. Faticoni, Fordham University Laurene V. Fausett, Georgia Southern University Norman Feldman, Sonoma State University Le Baron O. Ferguson, University of California—Riverside Newman Fisher, San Francisco State University José D. Flores, The University of South Dakota William Francis, Michigan Technological University James T. Franklin, Valencia Community College, East Stanley Friedlander, Bronx Community College Patrick Gallagher, Columbia University–New York Paul Garrett, University of Minnesota–Minneapolis Frederick Gass, Miami University of Ohio Bruce Gilligan, University of Regina Matthias k. Gobbert, University of Maryland, Baltimore County Gerald Goff, Oklahoma State University Stuart Goldenberg, California Polytechnic State University John A. Graham, Buckingham Browne & Nichols School Richard Grassl, University of New Mexico Michael Gregory, University of North Dakota Charles Groetsch, University of Cincinnati Paul Triantafilos Hadavas, Armstrong Atlantic State University Salim M. Haïdar, Grand Valley State University D. W. Hall, Michigan State University Robert L. Hall, University of Wisconsin–Milwaukee Howard B. Hamilton, California State University, Sacramento Darel Hardy, Colorado State University Shari Harris, John Wood Community College Gary W. Harrison, College of Charleston Melvin Hausner, New York University/Courant Institute Curtis Herink, Mercer University Russell Herman, University of North Carolina at Wilmington Allen Hesse, Rochester Community College Randall R. Holmes, Auburn University James F. Hurley, University of Connecticut

xix

Amer Iqbal, University of Washington—Seattle Matthew A. Isom, Arizona State University Gerald Janusz, University of Illinois at Urbana-Champaign John H. Jenkins, Embry-Riddle Aeronautical University, Prescott Campus Clement Jeske, University of Wisconsin, Platteville Carl Jockusch, University of Illinois at Urbana-Champaign Jan E. H. Johansson, University of Vermont Jerry Johnson, Oklahoma State University Zsuzsanna M. kadas, St. Michael’s College Nets katz, Indiana University Bloomington Matt kaufman Matthias kawski, Arizona State University Frederick W. keene, Pasadena City College Robert L. kelley, University of Miami Akhtar khan, Rochester Institute of Technology Marianne korten, Kansas State University Virgil kowalik, Texas A&I University kevin kreider, University of Akron Leonard krop, DePaul University Mark krusemeyer, Carleton College John C. Lawlor, University of Vermont Christopher C. Leary, State University of New York at Geneseo David Leeming, University of Victoria Sam Lesseig, Northeast Missouri State University Phil Locke, University of Maine Joyce Longman, Villanova University Joan McCarter, Arizona State University Phil McCartney, Northern Kentucky University Igor Malyshev, San Jose State University Larry Mansfield, Queens College Mary Martin, Colgate University Nathaniel F. G. Martin, University of Virginia Gerald Y. Matsumoto, American River College James Mckinney, California State Polytechnic University, Pomona Tom Metzger, University of Pittsburgh Richard Millspaugh, University of North Dakota Lon H. Mitchell, Virginia Commonwealth University Michael Montaño, Riverside Community College Teri Jo Murphy, University of Oklahoma Martin Nakashima, California State Polytechnic University, Pomona Ho kuen Ng, San Jose State University Richard Nowakowski, Dalhousie University Hussain S. Nur, California State University, Fresno Norma Ortiz-Robinson, Virginia Commonwealth University Wayne N. Palmer, Utica College Vincent Panico, University of the Pacific F. J. Papp, University of Michigan–Dearborn Mike Penna, Indiana University–Purdue University Indianapolis Mark Pinsky, Northwestern University Lothar Redlin, The Pennsylvania State University Joel W. Robbin, University of Wisconsin–Madison Lila Roberts, Georgia College and State University E. Arthur Robinson, Jr., The George Washington University Richard Rockwell, Pacific Union College

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

xx

Preface

Rob Root, Lafayette College Richard Ruedemann, Arizona State University David Ryeburn, Simon Fraser University Richard St. Andre, Central Michigan University Ricardo Salinas, San Antonio College Robert Schmidt, South Dakota State University Eric Schreiner, Western Michigan University Mihr J. Shah, Kent State University–Trumbull Qin Sheng, Baylor University Theodore Shifrin, University of Georgia Wayne Skrapek, University of Saskatchewan Larry Small, Los Angeles Pierce College Teresa Morgan Smith, Blinn College William Smith, University of North Carolina Donald W. Solomon, University of Wisconsin–Milwaukee Edward Spitznagel, Washington University Joseph Stampfli, Indiana University kristin Stoley, Blinn College M. B. Tavakoli, Chaffey College

Magdalena Toda, Texas Tech University Ruth Trygstad, Salt Lake Community College Paul Xavier Uhlig, St. Mary’s University, San Antonio Stan Ver Nooy, University of Oregon Andrei Verona, California State University–Los Angeles klaus Volpert, Villanova University Russell C. Walker, Carnegie Mellon University William L. Walton, McCallie School Peiyong Wang, Wayne State University Jack Weiner, University of Guelph Alan Weinstein, University of California, Berkeley Theodore W. Wilcox, Rochester Institute of Technology Steven Willard, University of Alberta Robert Wilson, University of Wisconsin–Madison Jerome Wolbert, University of Michigan–Ann Arbor Dennis H. Wortman, University of Massachusetts, Boston Mary Wright, Southern Illinois University–Carbondale Paul M. Wright, Austin Community College Xian Wu, University of South Carolina

Instructor’s Guide by Douglas Shaw ISBN 978-1-305-39371-4 Each section of the text is discussed from several viewpoints. The Instructor’s Guide contains suggested time to allot, points to stress, text discussion topics, core materials for lecture, workshop/discussion suggestions, group work exercises in a form suitable for handout, and suggested homework assignments.

Complete Solutions Manual Single Variable Early Transcendentals ISBN 978-1-305-27262-0 Multivariable ISBN 978-1-305-38699-0 Includes worked-out solutions to all exercises in the text.

Printed Test Bank By William Steven Harmon ISBN 978-1-305-38722-5 Contains text-specific multiple-choice and free response test items.

TEC TOOLS FOR ENRICHING™ CALCULUS By James Stewart, Harvey keynes, Dan Clegg, and developer Hubert Hohn Tools for Enriching Calculus (TEC) functions as both a powerful tool for instructors and as a tutorial environment in which students can explore and review selected topics. The Flash simulation modules in TEC include instructions, written and audio explanations of the concepts, and exercises. TEC is accessible in the eBook via CourseMate and Enhanced WebAssign. Selected Visuals and Modules are available at www.stewartcalculus.com.

Enhanced WebAssign® www.webassign.net Printed Access Code: ISBN 978-1-285-85826-5 Instant Access Code ISBN: 978-1-285-85825-8 Exclusively from Cengage Learning, Enhanced WebAssign offers an extensive online program for Stewart’s Calculus to encourage the practice that is so critical for concept mastery. The meticulously crafted pedagogy and exercises in our proven texts become even more effective in Enhanced WebAssign, supplemented by multimedia tutorial support and immediate feedback as students complete their assignments. Key features include: n

Thousands of homework problems that match your textbook’s end-of-section exercises

n

Opportunities for students to review prerequisite skills and content both at the start of the course and at the beginning of each section

n

n

A customizable Cengage YouBook with highlighting, notetaking, and search features, as well as links to multimedia resources

n

Personal Study Plans (based on diagnostic quizzing) that identify chapter topics that students will need to master

n

A WebAssign Answer Evaluator that recognizes and accepts equivalent mathematical responses in the same way an instructor grades

n

A Show My Work feature that gives instructors the option of seeing students’ detailed solutions

n

Visualizing Calculus Animations, Lecture Videos, and more

Cengage Learning Testing Powered by Cognero (login.cengage.com) This flexible online system allows you to author, edit, and manage test bank content from multiple Cengage Learning solutions; create multiple test versions in an instant; and deliver tests from your LMS, your classroom, or wherever you want.

Stewart Website www.stewartcalculus.com Contents: Homework Hints n Algebra Review n Additional Topics n Drill exercises n Challenge Problems n Web Links n History of Mathematics n Tools for Enriching Calculus (TEC)

■ Electronic items ■ Printed items

(Table continues on page xxii)

xxi Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

Cengage Customizable YouBook YouBook is an eBook that is both interactive and customizable. Containing all the content from Stewart’s Calculus, YouBook features a text edit tool that allows instructors to modify the textbook narrative as needed. With YouBook, instructors can quickly reorder entire sections and chapters or hide any content they don’t teach to create an eBook that perfectly matches their syllabus. Instructors can further customize the text by adding instructor-created or YouTube video links. Additional media assets include animated figures, video clips, highlighting and note-taking features, and more. YouBook is available within Enhanced WebAssign. CourseMate CourseMate is a perfect self-study tool for students, and requires no set up from instructors. CourseMate brings course concepts to life with interactive learning, study, and exam preparation tools that support the printed textbook. CourseMate for Stewart’s Calculus includes an interactive eBook, Tools for Enriching Calculus, videos, quizzes, flashcards, and more. For instructors, CourseMate includes Engagement Tracker, a first-of-its-kind tool that monitors student engagement. CengageBrain.com To access additional course materials, please visit www.cengagebrain.com. At the CengageBrain.com home page, search for the ISBN of your title (from the back cover of your book) using the search box at the top of the page. This will take you to the product page where these resources can be found.

Student Solutions Manual Single Variable Early Transcendentals ISBN 978-1-305-27263-7 Multivariable ISBN 978-1-305-38698-3 Provides completely worked-out solutions to all oddnumbered exercises in the text, giving students a chance to check their answer and ensure they took the correct steps

to arrive at the answer. The Student Solutions Manual can be ordered or accessed online as an eBook at www.cengagebrain.com by searching the ISBN. Study Guide Single Variable Early Transcendentals By Richard St. Andre ISBN 978-1-305-27914-8 Multivariable By Richard St. Andre ISBN 978-1-305-27184-5 For each section of the text, the Study Guide provides students with a brief introduction, a short list of concepts to master, and summary and focus questions with explained answers. The Study Guide also contains self-tests with exam-style questions. The Study Guide can be ordered or accessed online as an eBook at www.cengagebrain.com by searching the ISBN. A Companion to Calculus By Dennis Ebersole, Doris Schattschneider, Alicia Sevilla, and kay Somers ISBN 978-0-495-01124-8 Written to improve algebra and problem-solving skills of students taking a calculus course, every chapter in this companion is keyed to a calculus topic, providing conceptual background and specific algebra techniques needed to understand and solve calculus problems related to that topic. It is designed for calculus courses that integrate the review of precalculus concepts or for individual use. Order a copy of the text or access the eBook online at www.cengagebrain.com by searching the ISBN. Linear Algebra for Calculus by konrad J. Heuvers, William P. Francis, John H. kuisti, Deborah F. Lockhart, Daniel S. Moak, and Gene M. Ortner ISBN 978-0-534-25248-9 This comprehensive book, designed to supplement the calculus course, provides an introduction to and review of the basic ideas of linear algebra. Order a copy of the text or access the eBook online at www.cengagebrain.com by searching the ISBN.

■ Electronic items ■ Printed items

xxii Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

which the full resources of a computer algebra system (like Maple, Mathematica, or the TI-89) are required. You will also encounter the symbol |, which warns you against committing an error. I have placed this symbol in the margin in situations where I have observed that a large proportion of my students tend to make the same mistake. Tools for Enriching Calculus, which is a companion to this text, is referred to by means of the symbol TEC and can be accessed in the eBook via Enhanced WebAssign and CourseMate (selected Visuals and Modules are available at www.stewartcalculus.com). It directs you to modules in which you can explore aspects of calculus for which the computer is particularly useful. You will notice that some exercise numbers are printed in red: 5. This indicates that Homework Hints are available for the exercise. These hints can be found on stewartcalculus.com as well as Enhanced WebAssign and CourseMate. The homework hints ask you questions that allow you to make progress toward a solution without actually giving you the answer. You need to pursue each hint in an active manner with pencil and paper to work out the details. If a particular hint doesn’t enable you to solve the problem, you can click to reveal the next hint. I recommend that you keep this book for reference purposes after you finish the course. Because you will likely forget some of the specific details of calculus, the book will serve as a useful reminder when you need to use calculus in subsequent courses. And, because this book contains more material than can be covered in any one course, it can also serve as a valuable resource for a working scientist or engineer. Calculus is an exciting subject, justly considered to be one of the greatest achievements of the human intellect. I hope you will discover that it is not only useful but also intrinsically beautiful. JAMES STEWART

xxiii Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

Calculators, Computers, and Other Graphing Devices

xxiv

You can also use computer software such as Graphing Calculator by Pacific Tech (www.pacifict.com) to perform many of these functions, as well as apps for phones and tablets, like Quick Graph (Colombiamug) or Math-Studio (Pomegranate Apps). Similar functionality is available using a web interface at WolframAlpha.com.

Advances in technology continue to bring a wider variety of tools for doing mathematics. Handheld calculators are becoming more powerful, as are software programs and Internet resources. In addition, many mathematical applications have been released for smartphones and tablets such as the iPad. Some exercises in this text are marked with a graphing icon ; , which indicates that the use of some technology is required. Often this means that we intend for a graphing device to be used in drawing the graph of a function or equation. You might also need technology to find the zeros of a graph or the points of intersection of two graphs. In some cases we will use a calculating device to solve an equation or evaluate a definite integral numerically. Many scientific and graphing calculators have these features built in, such as the Texas Instruments TI-84 or TI-Nspire CX. Similar calculators are made by Hewlett Packard, Casio, and Sharp.

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

The CAS icon is reserved for problems in which the full resources of a computer algebra system (CAS) are required. A CAS is capable of doing mathematics (like solving equations, computing derivatives or integrals) symbolically rather than just numerically. Examples of well-established computer algebra systems are the computer software packages Maple and Mathematica. The WolframAlpha website uses the Mathematica engine to provide CAS functionality via the Web. Many handheld graphing calculators have CAS capabilities, such as the TI-89 and TI-Nspire CX CAS from Texas Instruments. Some tablet and smartphone apps also provide these capabilities, such as the previously mentioned MathStudio.

In general, when we use the term “calculator” in this book, we mean the use of any of the resources we have mentioned.

xxv Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

Diagnostic Tests Success in calculus depends to a large extent on knowledge of the mathematics that precedes calculus: algebra, analytic geometry, functions, and trigonometry. The following tests are intended to diagnose weaknesses that you might have in these areas. After taking each test you can check your answers against the given answers and, if necessary, refresh your skills by referring to the review materials that are provided.

A 1. Evaluate each expression without using a calculator. (a) s23d4 (b) 234 (c) 324 (d)

5 23 5 21

(e)

SD 2 3

22

(f) 16 23y4

2. Simplify each expression. Write your answer without negative exponents. (a) s200 2 s32 (b) s3a 3b 3 ds4ab 2 d 2 (c)

S

3x 3y2 y 3 x 2 y21y2

D

22

3. Expand and simplify. (a) 3sx 1 6d 1 4s2x 2 5d

(b) sx 1 3ds4x 2 5d

(c) ssa 1 sb dssa 2 sb d

(d) s2x 1 3d2

(e) sx 1 2d3 4. Factor each expression. (a) 4x 2 2 25 (c) x 3 2 3x 2 2 4x 1 12 (e) 3x 3y2 2 9x 1y2 1 6x 21y2

(b) 2x 2 1 5x 2 12 (d) x 4 1 27x (f) x 3 y 2 4xy

5. Simplify the rational expression. (a)

x 2 1 3x 1 2 x2 2 x 2 2

(c)

x2 x11 2 x 24 x12 2

2x 2 2 x 2 1 x13 ? x2 2 9 2x 1 1 y x 2 x y (d) 1 1 2 y x (b)

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Diagnostic Tests

6. Rationalize the expression and simplify. s10 s5 2 2

(a)

(b)

7. Rewrite by completing the square. (a) x 2 1 x 1 1

s4 1 h 2 2 h

(b) 2x 2 2 12x 1 11

8. Solve the equation. (Find only the real solutions.)

(c) x 2 2 x 2 12 − 0

2x 2x 2 1 − x11 x (d) 2x 2 1 4x 1 1 − 0

(e) x 4 2 3x 2 1 2 − 0

(f) 3 x 2 4 − 10

(a) x 1 5 − 14 2 12 x

(g) 2xs4 2 xd

21y2

(b)

|

|

2 3 s4 2 x − 0

9. Solve each inequality. Write your answer using interval notation. (a) 24 , 5 2 3x < 17 (b) x 2 , 2x 1 8 (c) xsx 2 1dsx 1 2d . 0 (d) x 2 4 , 3

|

|

2x 2 3 (e) x 2 2 1

(e) x 2 1 y 2 , 4

(f) 9x 2 1 16y 2 − 144

ANSWERS TO DIAGNOSTIC TEST B: ANALYTIC GEOMETRY 1. (a) y − 23x 1 1 (c) x − 2

(b) y − 25 (d) y −

1 2x

26

5. (a)

(b)

y

(c)

y

3

1

2

2. sx 1 1d2 1 s y 2 4d2 − 52

1

y=1- 2 x

0

3. Center s3, 25d, radius 5 4. (a) (b) (c) (d) (e) (f)

y

x

_1

234 4x 1 3y 1 16 − 0; x-intercept 24, y-intercept 2 16 3 s21, 24d 20 3x 2 4y − 13 sx 1 1d2 1 s y 1 4d2 − 100

(d)

_4

4x

0

0

2

x

_2

y

(e)

y 2

(f ) ≈+¥=4

y 3

0 _1

1

x

0

2

x

0

4 x

y=≈-1

If you had difficulty with these problems, you may wish to consult the review of analytic geometry in Appendixes B and C.

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Diagnostic Tests

xxix

C y

1. The graph of a function f is given at the left. (a) State the value of f s21d. (b) Estimate the value of f s2d. (c) For what values of x is f sxd − 2? (d) Estimate the values of x such that f sxd − 0. (e) State the domain and range of f.

1 0

1

x

2. If f sxd − x 3, evaluate the difference quotient

f s2 1 hd 2 f s2d and simplify your answer. h

3. Find the domain of the function.

FIGURE FOR PROBLEM 1

(a) f sxd −

2x 1 1 x 1x22

(b) tsxd −

2

3 x s x 11

(c) hsxd − s4 2 x 1 sx 2 2 1

2

4. How are graphs of the functions obtained from the graph of f ? (a) y − 2f sxd (b) y − 2 f sxd 2 1 (c) y − f sx 2 3d 1 2 5. Without using a calculator, make a rough sketch of the graph. (a) y − x 3 (b) y − sx 1 1d3 (c) y − sx 2 2d3 1 3 2 (d) y − 4 2 x (e) y − sx (f) y − 2 sx (g) y − 22 x (h) y − 1 1 x21 6. Let f sxd −

H

1 2 x 2 if x < 0 2x 1 1 if x . 0

(a) Evaluate f s22d and f s1d.

(b) Sketch the graph of f.

7. If f sxd − x 2 1 2x 2 1 and tsxd − 2x 2 3, find each of the following functions. (a) f 8 t (b) t 8 f (c) t 8 t 8 t

ANSWERS TO DIAGNOSTIC TEST C: FUNCTIONS 1. (a) 22 (c) 23, 1 (e) f23, 3g, f22, 3g

(b) 2.8 (d) 22.5, 0.3

5. (a)

0

4. (a) Reflect about the x-axis (b) Stretch vertically by a factor of 2, then shift 1 unit downward (c) Shift 3 units to the right and 2 units upward

(d)

(g)

y (2, 3)

1 1

x

2

(h)

x 0

(f)

y

0

x

y

0

_1

(e)

y 4

0

(c)

y

1

2. 12 1 6h 1 h 2 3. (a) s2`, 22d ø s22, 1d ø s1, `d (b) s2`, `d (c) s2`, 21g ø f1, 4g

(b)

y

1

x

1

x

x

y

0

1

y 1

0 _1

1

x

0

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x

xxx

Diagnostic Tests

6. (a) 23, 3

(b)

7. (a) s f 8 tdsxd − 4x 2 2 8x 1 2

y

(b) s t 8 f dsxd − 2x 2 1 4x 2 5

1 _1

(c) s t 8 t 8 tdsxd − 8x 2 21

x

0

If you had difficulty with these problems, you should look at sections 1.1–1.3 of this book.

D 1. Convert from degrees to radians. (a) 3008 (b) 2188 2. Convert from radians to degrees. (a) 5y6 (b) 2 3. Find the length of an arc of a circle with radius 12 cm if the arc subtends a central angle of 308. 4. Find the exact values. (a) tansy3d (b) sins7y6d

(c) secs5y3d

5. Express the lengths a and b in the figure in terms of . 24

6. If sin x − 13 and sec y − 54, where x and y lie between 0 and y2, evaluate sinsx 1 yd.

a ¨

7. Prove the identities. b

FIGURE FOR PROBLEM 5

(a) tan  sin  1 cos  − sec

(b)

2 tan x − sin 2x 1 1 tan 2x

8. Find all values of x such that sin 2x − sin x and 0 < x < 2. 9. Sketch the graph of the function y − 1 1 sin 2x without using a calculator.

ANSWERS TO DIAGNOSTIC TEST D: TRIGONOMETRY

s4 1 6 s2 d

1. (a) 5y3

(b) 2y10

6.

2. (a) 1508

(b) 3608y < 114.68

8. 0, y3, , 5y3, 2

3. 2 cm

1 15

9.

4. (a) s3

(b) 221

5. (a) 24 sin

(b) 24 cos

y 2

(c) 2 _π

0

π

x

If you had difficulty with these problems, you should look at Appendix D of this book.

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A Preview of Calculus By the time you finish this course, you will be able to calculate the length of the curve used to design the Gateway Arch in St. Louis, determine where a pilot should start descent for a smooth landing, compute the force on a baseball bat when it strikes the ball, and measure the amount of light sensed by the human eye as the pupil changes size.

CALCULUS IS FUNDAMENTALLY DIFFERENT FROM the mathematics that you have studied previously: calculus is less static and more dynamic. It is concerned with change and motion; it deals with quantities that approach other quantities. For that reason it may be useful to have an overview of the subject before beginning its intensive study. Here we give a glimpse of some of the main ideas of calculus by showing how the concept of a limit arises when we attempt to solve a variety of problems.

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2

A PREVIEW OF CALCULUS

The Area Problem

The origins of calculus go back at least 2500 years to the ancient Greeks, who found areas using the “method of exhaustion.” They knew how to find the area A of any polygon by dividing it into triangles as in Figure 1 and adding the areas of these triangles. It is a much more difficult problem to find the area of a curved figure. The Greek method of exhaustion was to inscribe polygons in the figure and circumscribe polygons about the figure and then let the number of sides of the polygons increase. Figure 2 illustrates this process for the special case of a circle with inscribed regular polygons.

A∞

A™ A¢

A=A¡+A™+A£+A¢+A∞

FIGURE 1

A∞

A¡™

FIGURE 2

Let An be the area of the inscribed polygon with n sides. As n increases, it appears that An becomes closer and closer to the area of the circle. We say that the area of the circle is the limit of the areas of the inscribed polygons, and we write TEC In the Preview Visual, you can see how areas of inscribed and circumscribed polygons approximate the area of a circle.

y

A − lim An nl`

The Greeks themselves did not use limits explicitly. However, by indirect reasoning, Eudoxus (fifth century bc) used exhaustion to prove the familiar formula for the area of a circle: A − r 2. We will use a similar idea in Chapter 5 to find areas of regions of the type shown in Figure 3. We will approximate the desired area A by areas of rectangles (as in Figure 4), let the width of the rectangles decrease, and then calculate A as the limit of these sums of areas of rectangles. y

y

(1, 1)

y

(1, 1)

(1, 1)

(1, 1)

y=≈ A 0

1

x

0

1 4

1 2

3 4

1

x

0

1

x

0

1 n

1

x

FIGURE 3

The area problem is the central problem in the branch of calculus called integral calculus. The techniques that we will develop in Chapter 5 for finding areas will also enable us to compute the volume of a solid, the length of a curve, the force of water against a dam, the mass and center of gravity of a rod, and the work done in pumping water out of a tank.

The Tangent Problem Consider the problem of trying to find an equation of the tangent line t to a curve with equation y − f sxd at a given point P. (We will give a precise definition of a tangent line in

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A PREVIEW OF CALCULUS

3

Chapter 2. For now you can think of it as a line that touches the curve at P as in Figure 5.) Since we know that the point P lies on the tangent line, we can find the equation of t if we know its slope m. The problem is that we need two points to compute the slope and we know only one point, P, on t. To get around the problem we first find an approximation to m by taking a nearby point Q on the curve and computing the slope mPQ of the secant line PQ. From Figure 6 we see that

y

t y=ƒ P

0

x

FIGURE 5

1

mPQ −

f sxd 2 f sad x2a

Now imagine that Q moves along the curve toward P as in Figure 7. You can see that the secant line rotates and approaches the tangent line as its limiting position. This means that the slope mPQ of the secant line becomes closer and closer to the slope m of the tangent line. We write

The tangent line at P y

t

m − lim mPQ Q lP

Q { x, ƒ} ƒ-f(a)

P { a, f(a)}

and we say that m is the limit of mPQ as Q approaches P along the curve. Because x approaches a as Q approaches P, we could also use Equation 1 to write

x-a

a

0

x

x

FIGURE 6 The secant line at PQ y

t Q P

0

FIGURE 7 Secant lines approaching the tangent line

x

2

f sxd 2 f sad x2a

m − lim

xla

Specific examples of this procedure will be given in Chapter 2. The tangent problem has given rise to the branch of calculus called differential calculus, which was not invented until more than 2000 years after integral calculus. The main ideas behind differential calculus are due to the French mathematician Pierre Fermat (1601–1665) and were developed by the English mathematicians John Wallis (1616–1703), Isaac Barrow (1630–1677), and Isaac Newton (1642–1727) and the German mathematician Gottfried Leibniz (1646–1716). The two branches of calculus and their chief problems, the area problem and the tangent problem, appear to be very different, but it turns out that there is a very close connection between them. The tangent problem and the area problem are inverse problems in a sense that will be described in Chapter 5.

Velocity When we look at the speedometer of a car and read that the car is traveling at 48 kmyh, what does that information indicate to us? We know that if the velocity remains constant, then after an hour we will have traveled 48 km. But if the velocity of the car varies, what does it mean to say that the velocity at a given instant is 48 kmyh? In order to analyze this question, let’s examine the motion of a car that travels along a straight road and assume that we can measure the distance traveled by the car (in meters) at l-second intervals as in the following chart: t − Time elapsed ssd

0

1

2

3

4

5

d − Distance smd

0

2

9

24

42

71

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4

A PREVIEW OF CALCULUS

As a first step toward finding the velocity after 2 seconds have elapsed, we find the average velocity during the time interval 2 < t < 4: change in position time elapsed

average velocity −

42 2 9 422

− 16.5 mys Similarly, the average velocity in the time interval 2 < t < 3 is average velocity −

24 2 9 − 15 mys 322

We have the feeling that the velocity at the instant t − 2 can’t be much different from the average velocity during a short time interval starting at t − 2. So let’s imagine that the distance traveled has been measured at 0.l-second time intervals as in the following chart: t

2.0

2.1

2.2

2.3

2.4

2.5

d

9.00

10.02

11.16

12.45

13.96

15.80

Then we can compute, for instance, the average velocity over the time interval f2, 2.5g: 15.80 2 9.00 − 13.6 mys 2.5 2 2

average velocity −

The results of such calculations are shown in the following chart: Time interval

f2, 3g

f2, 2.5g

f2, 2.4g

f2, 2.3g

f2, 2.2g

f2, 2.1g

Average velocity smysd

15.0

13.6

12.4

11.5

10.8

10.2

The average velocities over successively smaller intervals appear to be getting closer to a number near 10, and so we expect that the velocity at exactly t − 2 is about 10 mys. In Chapter 2 we will define the instantaneous velocity of a moving object as the limiting value of the average velocities over smaller and smaller time intervals. In Figure 8 we show a graphical representation of the motion of the car by plotting the distance traveled as a function of time. If we write d − f std, then f std is the number of meters traveled after t seconds. The average velocity in the time interval f2, tg is

d

Q { t, f(t)}

average velocity −

which is the same as the slope of the secant line PQ in Figure 8. The velocity v when t − 2 is the limiting value of this average velocity as t approaches 2; that is,

20 10 0

change in position f std 2 f s2d − time elapsed t22

P { 2, f(2)} 1

2

FIGURE 8

3

4

5

t

v − lim

tl2

f std 2 f s2d t22

and we recognize from Equation 2 that this is the same as the slope of the tangent line to the curve at P.

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5

A PREVIEW OF CALCULUS

Thus, when we solve the tangent problem in differential calculus, we are also solving problems concerning velocities. The same techniques also enable us to solve problems involving rates of change in all of the natural and social sciences.

The Limit of a Sequence In the fifth century bc the Greek philosopher Zeno of Elea posed four problems, now known as Zeno’s paradoxes, that were intended to challenge some of the ideas concerning space and time that were held in his day. Zeno’s second paradox concerns a race between the Greek hero Achilles and a tortoise that has been given a head start. Zeno argued, as follows, that Achilles could never pass the tortoise: Suppose that Achilles starts at position a 1 and the tortoise starts at position t1. (See Figure 9.) When Achilles reaches the point a 2 − t1, the tortoise is farther ahead at position t2. When Achilles reaches a 3 − t2, the tortoise is at t3. This process continues indefinitely and so it appears that the tortoise will always be ahead! But this defies common sense. a¡

a™

a∞

...

t™

...

Achilles tortoise

FIGURE 9

One way of explaining this paradox is with the idea of a sequence. The successive positions of Achilles sa 1, a 2 , a 3 , . . .d or the successive positions of the tortoise st1, t2 , t3 , . . .d form what is known as a sequence. In general, a sequence ha nj is a set of numbers written in a definite order. For instance, the sequence

h1, 12 , 13 , 14 , 15 , . . . j can be described by giving the following formula for the nth term: an − a¢ a£

a™

0

1 n

We can visualize this sequence by plotting its terms on a number line as in Figure 10(a) or by drawing its graph as in Figure 10(b). Observe from either picture that the terms of the sequence a n − 1yn are becoming closer and closer to 0 as n increases. In fact, we can find terms as small as we please by making n large enough. We say that the limit of the sequence is 0, and we indicate this by writing

a¡ 1

(a) 1

lim

nl`

1 2 3 4 5 6 7 8

( b)

FIGURE 10

n

1 −0 n

In general, the notation lim a n − L

nl`

is used if the terms a n approach the number L as n becomes large. This means that the numbers a n can be made as close as we like to the number L by taking n sufficiently large.

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6

A PREVIEW OF CALCULUS

The concept of the limit of a sequence occurs whenever we use the decimal representation of a real number. For instance, if a 1 − 3.1 a 2 − 3.14 a 3 − 3.141 a 4 − 3.1415 a 5 − 3.14159 a 6 − 3.141592 a 7 − 3.1415926 f then

lim a n −

nl`

The terms in this sequence are rational approximations to . Let’s return to Zeno’s paradox. The successive positions of Achilles and the tortoise form sequences ha nj and htn j, where a n , tn for all n. It can be shown that both sequences have the same limit: lim a n − p − lim tn

nl`

nl`

It is precisely at this point p that Achilles overtakes the tortoise.

The Sum of a Series Another of Zeno’s paradoxes, as passed on to us by Aristotle, is the following: “A man standing in a room cannot walk to the wall. In order to do so, he would first have to go half the distance, then half the remaining distance, and then again half of what still remains. This process can always be continued and can never be ended.” (See Figure 11.)

1 2

FIGURE 11

1 4

1 8

1 16

Of course, we know that the man can actually reach the wall, so this suggests that perhaps the total distance can be expressed as the sum of infinitely many smaller distances as follows: 3

1−

1 1 1 1 1 1 1 1 1 ∙∙∙ 1 n 1 ∙∙∙ 2 4 8 16 2

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A PREVIEW OF CALCULUS

7

Zeno was arguing that it doesn’t make sense to add infinitely many numbers together. But there are other situations in which we implicitly use infinite sums. For instance, in decimal notation, the symbol 0.3 − 0.3333 . . . means 3 3 3 3 1 1 1 1 ∙∙∙ 10 100 1000 10,000 and so, in some sense, it must be true that 3 3 3 3 1 1 1 1 1 ∙∙∙ − 10 100 1000 10,000 3 More generally, if dn denotes the nth digit in the decimal representation of a number, then 0.d1 d2 d3 d4 . . . −

d1 d2 d3 dn 1 2 1 3 1 ∙∙∙ 1 n 1 ∙∙∙ 10 10 10 10

Therefore some infinite sums, or infinite series as they are called, have a meaning. But we must define carefully what the sum of an infinite series is. Returning to the series in Equation 3, we denote by sn the sum of the first n terms of the series. Thus s1 − 12 − 0.5 s2 − 12 1 14 − 0.75 s3 − 12 1 14 1 18 − 0.875 1 s4 − 12 1 14 1 18 1 16 − 0.9375 1 1 s5 − 12 1 14 1 18 1 16 1 32 − 0.96875 1 1 1 s6 − 12 1 14 1 18 1 16 1 32 1 64 − 0.984375 1 1 1 1 s7 − 12 1 14 1 18 1 16 1 32 1 64 1 128 − 0.9921875

f 1 s10 − 12 1 14 1 ∙ ∙ ∙ 1 1024 < 0.99902344

f s16 −

1 1 1 1 1 ∙ ∙ ∙ 1 16 < 0.99998474 2 4 2

Observe that as we add more and more terms, the partial sums become closer and closer to 1. In fact, it can be shown that by taking n large enough (that is, by adding sufficiently many terms of the series), we can make the partial sum sn as close as we please to the number 1. It therefore seems reasonable to say that the sum of the infinite series is 1 and to write 1 1 1 1 1 1 1 ∙∙∙ 1 n 1 ∙∙∙ − 1 2 4 8 2 Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

8

A PREVIEW OF CALCULUS

In other words, the reason the sum of the series is 1 is that lim sn − 1

nl`

In Chapter 11 we will discuss these ideas further. We will then use Newton’s idea of combining infinite series with differential and integral calculus.

Summary We have seen that the concept of a limit arises in trying to find the area of a region, the slope of a tangent to a curve, the velocity of a car, or the sum of an infinite series. In each case the common theme is the calculation of a quantity as the limit of other, easily calculated quantities. It is this basic idea of a limit that sets calculus apart from other areas of mathematics. In fact, we could define calculus as the part of mathematics that deals with limits. After Sir Isaac Newton invented his version of calculus, he used it to explain the motion of the planets around the sun. Today calculus is used in calculating the orbits of satellites and spacecraft, in predicting population sizes, in estimating how fast oil prices rise or fall, in forecasting weather, in measuring the cardiac output of the heart, in calculating life insurance premiums, and in a great variety of other areas. We will explore some of these uses of calculus in this book. In order to convey a sense of the power of the subject, we end this preview with a list of some of the questions that you will be able to answer using calculus: 1. How can we explain the fact, illustrated in Figure 12, that the angle of elevation from an observer up to the highest point in a rainbow is 42°? (See page 285.)

rays from sun

138°

2. How can we explain the shapes of cans on supermarket shelves? (See page 343.) 3. Where is the best place to sit in a movie theater? (See page 465.)

rays from sun

42°

4. How can we design a roller coaster for a smooth ride? (See page 182.) 5. How far away from an airport should a pilot start descent? (See page 208.)

observer

FIGURE 12

6. How can we fit curves together to design shapes to represent letters on a laser printer? (See page 657.) 7. How can we estimate the number of workers that were needed to build the Great Pyramid of Khufu in ancient Egypt? (See page 460.) 8. Where should an infielder position himself to catch a baseball thrown by an outfielder and relay it to home plate? (See page 465.) 9. Does a ball thrown upward take longer to reach its maximum height or to fall back to its original height? (See page 609.) 10. How can we explain the fact that planets and satellites move in elliptical orbits? (See page 876.) 11. How can we distribute water flow among turbines at a hydroelectric station so as to maximize the total energy production? (See page 980.) 12. If a marble, a squash ball, a steel bar, and a lead pipe roll down a slope, which of them reaches the bottom first? (See page 1052.)

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1 Often a graph is the best way to represent a function because it conveys so much information at a glance. Shown is a graph of the vertical ground acceleration created by the 2011 earthquake near Tohoku, Japan. The earthquake had a magnitude of 9.0 on the Richter scale and was so powerful that it moved northern Japan 2.4 meters closer to North America.

Functions and Models

Pictura Collectus/Alamy

(cm/s@) 2000 1000 0

time _1000 _2000 0

50

100

150

200 Seismological Society of America

The fundamenTal objecTs ThaT we deal with in calculus are functions. This chapter prepares the way for calculus by discussing the basic ideas concerning functions, their graphs, and ways of transforming and combining them. We stress that a function can be represented in different ways: by an equation, in a table, by a graph, or in words. We look at the main types of functions that occur in calculus and describe the process of using these functions as mathematical models of real-world phenomena.

9 Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

10

Chapter 1

Functions and Models

Functions arise whenever one quantity depends on another. Consider the following four situations. A. The area A of a circle depends on the radius r of the circle. The rule that connects r and A is given by the equation A − r 2. With each positive number r there is associated one value of A, and we say that A is a function of r. Year

Population (millions)

1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010

1650 1750 1860 2070 2300 2560 3040 3710 4450 5280 6080 6870

B. The human population of the world P depends on the time t. The table gives estimates of the world population Pstd at time t, for certain years. For instance, Ps1950d < 2,560,000,000 But for each value of the time t there is a corresponding value of P, and we say that P is a function of t. C. The cost C of mailing an envelope depends on its weight w. Although there is no simple formula that connects w and C, the post office has a rule for determining C when w is known. D. The vertical acceleration a of the ground as measured by a seismograph during an earthquake is a function of the elapsed time t. Figure 1 shows a graph generated by seismic activity during the Northridge earthquake that shook Los Angeles in 1994. For a given value of t, the graph provides a corresponding value of a. a

{cm/s@}

100

50

5

FIGURE 1 Vertical ground acceleration during the Northridge earthquake

10

15

20

25

30

t (seconds)

_50 Calif. Dept. of Mines and Geology

Each of these examples describes a rule whereby, given a number (r, t, w, or t), another number (A, P, C, or a) is assigned. In each case we say that the second number is a function of the first number. A function f is a rule that assigns to each element x in a set D exactly one element, called f sxd, in a set E. We usually consider functions for which the sets D and E are sets of real numbers. The set D is called the domain of the function. The number f sxd is the value of f at x and is read “ f of x.” The range of f is the set of all possible values of f sxd as x varies throughout the domain. A symbol that represents an arbitrary number in the domain of a function f is called an independent variable. A symbol that represents a number in the range of f is called a dependent variable. In Example A, for instance, r is the independent variable and A is the dependent variable.

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SeCtion 1.1

x (input)

f

ƒ (output)

FIGURE 2 Machine diagram for a function f

x

ƒ a

f(a)

f

D

11

Four Ways to Represent a Function

It’s helpful to think of a function as a machine (see Figure 2). If x is in the domain of the function f, then when x enters the machine, it’s accepted as an input and the machine produces an output f sxd according to the rule of the function. Thus we can think of the domain as the set of all possible inputs and the range as the set of all possible outputs. The preprogrammed functions in a calculator are good examples of a function as a machine. For example, the square root key on your calculator computes such a function. You press the key labeled s (or s x ) and enter the input x. If x , 0, then x is not in the domain of this function; that is, x is not an acceptable input, and the calculator will indicate an error. If x > 0, then an approximation to s x will appear in the display. Thus the s x key on your calculator is not quite the same as the exact mathematical function f defined by f sxd − s x . Another way to picture a function is by an arrow diagram as in Figure 3. Each arrow connects an element of D to an element of E. The arrow indicates that f sxd is associated with x, f sad is associated with a, and so on. The most common method for visualizing a function is its graph. If f is a function with domain D, then its graph is the set of ordered pairs hsx, f sxdd | x [ Dj

E

(Notice that these are input-output pairs.) In other words, the graph of f consists of all points sx, yd in the coordinate plane such that y − f sxd and x is in the domain of f. The graph of a function f gives us a useful picture of the behavior or “life history” of a function. Since the y-coordinate of any point sx, yd on the graph is y − f sxd, we can read the value of f sxd from the graph as being the height of the graph above the point x (see Figure 4). The graph of f also allows us to picture the domain of f on the x-axis and its range on the y-axis as in Figure 5.

FIGURE 3 Arrow diagram for f

y

y

{ x, ƒ}

range

ƒ

y  ƒ(x)

f(2) f(1) 0

1

2

x

x

x

0

domain

FIGURE 4 y

example 1 The graph of a function f is shown in Figure 6. (a) Find the values of f s1d and f s5d. (b) What are the domain and range of f ? Solution

1 0

FIGURE 5

1

x

FIGURE 6 The notation for intervals is given in Appendix A.

(a) We see from Figure 6 that the point s1, 3d lies on the graph of f, so the value of f at 1 is f s1d − 3. (In other words, the point on the graph that lies above x − 1 is 3 units above the x-axis.) When x − 5, the graph lies about 0.7 units below the x-axis, so we estimate that f s5d < 20.7. (b) We see that f sxd is defined when 0 < x < 7, so the domain of f is the closed interval f0, 7g. Notice that f takes on all values from 22 to 4, so the range of f is h y | 22 < y < 4j − f22, 4g

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12

Chapter 1

Functions and Models

y

example 2 Sketch the graph and find the domain and range of each function. (a) fsxd − 2x 2 1 (b) tsxd − x 2 Solution

y=2x-1 0 -1

x

1 2

FIGURE 7 y (2, 4)

y=≈ (_1, 1)

(a) The equation of the graph is y − 2x 2 1, and we recognize this as being the equation of a line with slope 2 and y-intercept 21. (Recall the slope-intercept form of the equation of a line: y − mx 1 b. See Appendix B.) This enables us to sketch a portion of the graph of f in Figure 7. The expression 2x 2 1 is defined for all real numbers, so the domain of f is the set of all real numbers, which we denote by R. The graph shows that the range is also R. (b) Since ts2d − 2 2 − 4 and ts21d − s21d2 − 1, we could plot the points s2, 4d and s21, 1d, together with a few other points on the graph, and join them to produce the graph (Figure 8). The equation of the graph is y − x 2, which represents a parabola (see Appendix C). The domain of t is R. The range of t consists of all values of tsxd, that is, all numbers of the form x 2. But x 2 > 0 for all numbers x and any positive number y is a square. So the range of t is hy | y > 0j − f0, `d. This can also be seen from Figure 8. ■

1 0

1

x

example 3 If f sxd − 2x 2 2 5x 1 1 and h ± 0, evaluate

f sa 1 hd 2 f sad . h

Solution We first evaluate f sa 1 hd by replacing x by a 1 h in the expression for f sxd:

FIGURE 8

f sa 1 hd − 2sa 1 hd2 2 5sa 1 hd 1 1 − 2sa 2 1 2ah 1 h 2 d 2 5sa 1 hd 1 1 − 2a 2 1 4ah 1 2h 2 2 5a 2 5h 1 1 The expression

Then we substitute into the given expression and simplify: f sa 1 hd 2 f sad s2a 2 1 4ah 1 2h 2 2 5a 2 5h 1 1d 2 s2a 2 2 5a 1 1d − h h

f sa 1 hd 2 f sad h in Example 3 is called a difference quotient and occurs frequently in calculus. As we will see in Chapter 2, it represents the average rate of change of f sxd between x − a and x − a 1 h.

2a 2 1 4ah 1 2h 2 2 5a 2 5h 1 1 2 2a 2 1 5a 2 1 h

4ah 1 2h 2 2 5h − 4a 1 2h 2 5 h

representations of Functions There are four possible ways to represent a function: ● ● ● ●

verbally numerically visually algebraically

(by a description in words) (by a table of values) (by a graph) (by an explicit formula)

If a single function can be represented in all four ways, it’s often useful to go from one representation to another to gain additional insight into the function. (In Example 2, for instance, we started with algebraic formulas and then obtained the graphs.) But certain functions are described more naturally by one method than by another. With this in mind, let’s reexamine the four situations that we considered at the beginning of this section.

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SeCtion 1.1

t (years since 1900)

Population (millions)

0 10 20 30 40 50 60 70 80 90 100 110

1650 1750 1860 2070 2300 2560 3040 3710 4450 5280 6080 6870

A. The most useful representation of the area of a circle as a function of its radius is probably the algebraic formula Asrd − r 2, though it is possible to compile a table of values or to sketch a graph (half a parabola). Because a circle has to have a positive radius, the domain is hr | r . 0j − s0, `d, and the range is also s0, `d. B. We are given a description of the function in words: Pstd is the human population of the world at time t. Let’s measure t so that t − 0 corresponds to the year 1900. The table of values of world population provides a convenient representation of this function. If we plot these values, we get the graph (called a scatter plot) in Figure 9. It too is a useful representation; the graph allows us to absorb all the data at once. What about a formula? Of course, it’s impossible to devise an explicit formula that gives the exact human population Pstd at any time t. But it is possible to find an expression for a function that approximates Pstd. In fact, using methods explained in Section 1.2, we obtain the approximation Pstd < f std − s1.43653 3 10 9 d ∙ s1.01395d t Figure 10 shows that it is a reasonably good “fit.” The function f is called a mathematical model for population growth. In other words, it is a function with an explicit formula that approximates the behavior of our given function. We will see, however, that the ideas of calculus can be applied to a table of values; an explicit formula is not necessary.

P

P

5x10'

0

13

Four Ways to Represent a Function

5x10'

20

40 60 80 Years since 1900

FIGURE 9

100

120

t

0

20

40 60 80 Years since 1900

100

120

t

FIGURE 10

A function defined by a table of values is called a tabular function.

w (grams)

Cswd (dollars)

0 , w < 20 20 , w < 30 30 , w < 40 40 , w < 50 ∙ ∙ ∙

2.90 5.50 7.00 8.50 ∙ ∙ ∙

The function P is typical of the functions that arise whenever we attempt to apply calculus to the real world. We start with a verbal description of a function. Then we may be able to construct a table of values of the function, perhaps from instrument readings in a scientific experiment. Even though we don’t have complete knowledge of the values of the function, we will see throughout the book that it is still possible to perform the operations of calculus on such a function. C. Again the function is described in words: Let Cswd be the cost of mailing a large envelope with weight w. The rule that the Hong Kong Post Office used as of 2015 for mail to Taiwan is as follows: the cost is \$2.90 for 20 g or less, \$5.50 for 30 g or less, and \$1.50 for each additional 10 g or part thereof. The table of values shown in the margin is the most convenient representation for this function, though it is possible to sketch a graph (see Example 10). D. The graph shown in Figure 1 is the most natural representation of the vertical acceleration function astd. It’s true that a table of values could be compiled, and it is even possible to devise an approximate formula. But everything a geologist needs to

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14

Chapter 1

Functions and Models

know— amplitudes and patterns — can be seen easily from the graph. (The same is true for the patterns seen in electrocardiograms of heart patients and polygraphs for lie-detection.) In the next example we sketch the graph of a function that is defined verbally.

T

example 4 When you turn on a hot-water faucet, the temperature T of the water depends on how long the water has been running. Draw a rough graph of T as a function of the time t that has elapsed since the faucet was turned on.

0

Solution The initial temperature of the running water is close to room temperature because the water has been sitting in the pipes. When the water from the hot-water tank starts flowing from the faucet, T increases quickly. In the next phase, T is constant at the temperature of the heated water in the tank. When the tank is drained, T decreases to the temperature of the water supply. This enables us to make the rough sketch of T as a function of t in Figure 11. ■

t

FIGURE 11

In the following example we start with a verbal description of a function in a physical situation and obtain an explicit algebraic formula. The ability to do this is a useful skill in solving calculus problems that ask for the maximum or minimum values of quantities.

example 5 A rectangular storage container with an open top has a volume of 10 m3. The length of its base is twice its width. Material for the base costs \$10 per square meter; material for the sides costs \$6 per square meter. Express the cost of materials as a function of the width of the base. Solution We draw a diagram as in Figure 12 and introduce notation by letting w and 2w be the width and length of the base, respectively, and h be the height. The area of the base is s2wdw − 2w 2, so the cost, in dollars, of the material for the base is 10s2w 2 d. Two of the sides have area wh and the other two have area 2wh, so the cost of the material for the sides is 6f2swhd 1 2s2whdg. The total cost is therefore

h w

C − 10s2w 2 d 1 6f2swhd 1 2s2whdg − 20 w 2 1 36 wh

2w

FIGURE 12

To express C as a function of w alone, we need to eliminate h and we do so by using the fact that the volume is 10 m3. Thus w s2wdh − 10

which gives

10 5 2 − 2w w2

h−

Substituting this into the expression for C, we have PS In setting up applied functions as in Example 5, it may be useful to review the principles of problem solving as discussed on page 71, particularly Step 1: Understand the Problem.

S D

C − 20w 2 1 36w

5

w

2

− 20w 2 1

180 w

Therefore the equation Cswd − 20w 2 1

180 w

w.0

expresses C as a function of w.

example 6 Find the domain of each function. (a) f sxd − sx 1 2

(b) tsxd −

1 x2 2 x

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SeCtion 1.1

domain convention If a function is given by a formula and the domain is not stated explicitly, the convention is that the domain is the set of all numbers for which the formula makes sense and defines a real number.

Four Ways to Represent a Function

15

Solution

(a) Because the square root of a negative number is not defined (as a real number), the domain of f consists of all values of x such that x 1 2 > 0. This is equivalent to x > 22, so the domain is the interval f22, `d. (b) Since 1 1 tsxd − 2 − x 2x xsx 2 1d and division by 0 is not allowed, we see that tsxd is not defined when x − 0 or x − 1. Thus the domain of t is hx | x ± 0, x ± 1j which could also be written in interval notation as s2`, 0d ø s0, 1d ø s1, `d

y

The graph of a function is a curve in the xy-plane. But the question arises: Which curves in the xy-plane are graphs of functions? This is answered by the following test.

x=a (a, b)

The Vertical Line Test A curve in the xy-plane is the graph of a function of x if and only if no vertical line intersects the curve more than once. 0

x

a

(a) This curve represents a function. y (a, c)

x=a

(a, b)

0

a

x

(b) This curve doesn’t represent a function.

FIGURE 13

The reason for the truth of the Vertical Line Test can be seen in Figure 13. If each vertical line x − a intersects a curve only once, at sa, bd, then exactly one function value is defined by f sad − b. But if a line x − a intersects the curve twice, at sa, bd and sa, cd, then the curve can’t represent a function because a function can’t assign two different values to a. For example, the parabola x − y 2 2 2 shown in Figure 14(a) is not the graph of a function of x because, as you can see, there are vertical lines that intersect the parabola twice. The parabola, however, does contain the graphs of two functions of x. Notice that the equation x − y 2 2 2 implies y 2 − x 1 2, so y − 6sx 1 2 . Thus the upper and lower halves of the parabola are the graphs of the functions f sxd − s x 1 2 [from Example 6(a)] and tsxd − 2s x 1 2 . [See Figures 14(b) and (c).] We observe that if we reverse the roles of x and y, then the equation x − hsyd − y 2 2 2 does define x as a function of y (with y as the independent variable and x as the dependent variable) and the parabola now appears as the graph of the function h. y

y

y

_2 (_2, 0)

FIGURE 14

0

x

_2 0

(a) x=¥-2

(b) y=œ„„„„ x+2

x

0

x

(c) y=_œ„„„„ x+2

piecewise Deﬁned Functions The functions in the following four examples are defined by different formulas in different parts of their domains. Such functions are called piecewise defined functions.

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

16

Chapter 1

Functions and Models

example 7 A function f is defined by f sxd −

H

1 2 x if x < 21 x2 if x . 21

Evaluate f s22d, f s21d, and f s0d and sketch the graph. Solution Remember that a function is a rule. For this particular function the rule is the following: First look at the value of the input x. If it happens that x < 21, then the value of f sxd is 1 2 x. On the other hand, if x . 21, then the value of f sxd is x 2.

Since 22 < 21, we have f s22d − 1 2 s22d − 3. Since 21 < 21, we have f s21d − 1 2 s21d − 2. y

Since 0 . 21, we have f s0d − 0 2 − 0.

1

_1

0

1

x

FIGURE 15

How do we draw the graph of f ? We observe that if x < 21, then f sxd − 1 2 x, so the part of the graph of f that lies to the left of the vertical line x − 21 must coincide with the line y − 1 2 x, which has slope 21 and y-intercept 1. If x . 21, then f sxd − x 2, so the part of the graph of f that lies to the right of the line x − 21 must coincide with the graph of y − x 2, which is a parabola. This enables us to sketch the graph in Figure 15. The solid dot indicates that the point s21, 2d is included on the graph; the open dot indicates that the point s21, 1d is excluded from the graph. ■ The next example of a piecewise defined function is the absolute value function. Recall that the absolute value of a number a, denoted by | a |, is the distance from a to 0 on the real number line. Distances are always positive or 0, so we have

For a more extensive review of absolute values, see Appendix A.

|a| > 0

for every number a

For example,

| 3 | − 3 | 23 | − 3 | 0 | − 0 | s2 2 1 | − s2 2 1 | 3 2  | −  2 3 In general, we have

|a| − a | a | − 2a

if a > 0 if a , 0

(Remember that if a is negative, then 2a is positive.)

example 8 Sketch the graph of the absolute value function f sxd − | x |.

y

Solution From the preceding discussion we know that

y=| x |

|x| − 0

FIGURE 16

x

H

x if x > 0 2x if x , 0

Using the same method as in Example 7, we see that the graph of f coincides with the line y − x to the right of the y-axis and coincides with the line y − 2x to the left of the y-axis (see Figure 16). ■

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SeCtion 1.1

Four Ways to Represent a Function

17

y

example 9 Find a formula for the function f graphed in Figure 17.

1

Solution The line through s0, 0d and s1, 1d has slope m − 1 and y-intercept b − 0, so its equation is y − x. Thus, for the part of the graph of f that joins s0, 0d to s1, 1d, we have

0

x

1

f sxd − x

FIGURE 17

if 0 < x < 1

The line through s1, 1d and s2, 0d has slope m − 21, so its point-slope form is

Point-slope form of the equation of a line: y 2 y1 − msx 2 x 1 d

y 2 0 − s21dsx 2 2d So we have

See Appendix B.

or

f sxd − 2 2 x

y−22x

if 1 , x < 2

We also see that the graph of f coincides with the x-axis for x . 2. Putting this information together, we have the following three-piece formula for f :

H

x if 0 < x < 1 f sxd − 2 2 x if 1 , x < 2 0 if x . 2

example 10 In Example C at the beginning of this section we considered the cost Cswd of mailing a large envelope with weight w. In effect, this is a piecewise defined function because, from the table of values on page 13, we have

C 10.00 8.00 6.00

Cswd −

4.00 2.00

0

10

20

30

40

50

FIGURE 18

w

2.90 5.50 7.00 8.50 ∙ ∙ ∙

if if if if

0 , w < 20 20 , w < 30 30 , w < 40 40 , w < 50

The graph is shown in Figure 18. You can see why functions similar to this one are called step functions—they jump from one value to the next. Such functions will be ■ studied in Chapter 2.

Symmetry If a function f satisfies f s2xd − f sxd for every number x in its domain, then f is called an even function. For instance, the function f sxd − x 2 is even because

y

f s2xd − s2xd2 − x 2 − f sxd f(_x)

ƒ _x

0

FIGURE 19 An even function

x

x

The geometric significance of an even function is that its graph is symmetric with respect to the y-axis (see Figure 19). This means that if we have plotted the graph of f for x > 0, we obtain the entire graph simply by reflecting this portion about the y-axis. If f satisfies f s2xd − 2f sxd for every number x in its domain, then f is called an odd function. For example, the function f sxd − x 3 is odd because f s2xd − s2xd3 − 2x 3 − 2f sxd

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18

Chapter 1

Functions and Models

The graph of an odd function is symmetric about the origin (see Figure 20). If we already have the graph of f for x > 0, we can obtain the entire graph by rotating this portion through 1808 about the origin.

y

_x

ƒ

0 x

x

example 11 Determine whether each of the following functions is even, odd, or neither even nor odd. (a) f sxd − x 5 1 x (b) tsxd − 1 2 x 4 (c) hsxd − 2x 2 x 2 Solution

f s2xd − s2xd5 1 s2xd − s21d5x 5 1 s2xd

(a)

FIGURE 20 An odd function

− 2x 5 2 x − 2sx 5 1 xd − 2f sxd Therefore f is an odd function. (b) So t is even.

ts2xd − 1 2 s2xd4 − 1 2 x 4 − tsxd

(c)

hs2xd − 2s2xd 2 s2xd2 − 22x 2 x 2

Since hs2xd ± hsxd and hs2xd ± 2hsxd, we conclude that h is neither even nor odd. ■ The graphs of the functions in Example 11 are shown in Figure 21. Notice that the graph of h is symmetric neither about the y-axis nor about the origin.

1

y

y

y

1

f

g

h

1 1

1

_1

x

x

1

x

_1

FIGURE 21

( b)

(a)

(c)

increasing and Decreasing Functions The graph shown in Figure 22 rises from A to B, falls from B to C, and rises again from C to D. The function f is said to be increasing on the interval fa, bg, decreasing on fb, cg, and increasing again on fc, dg. Notice that if x 1 and x 2 are any two numbers between a and b with x 1 , x 2, then f sx 1 d , f sx 2 d. We use this as the defining property of an increasing function. y

B

D

y=ƒ C f(x™) A

FIGURE 22

0 a x¡

f(x¡)

x™

b

c

d

x

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

SeCtion 1.1

19

Four Ways to Represent a Function

A function f is called increasing on an interval I if f sx 1 d , f sx 2 d

whenever x 1 , x 2 in I

y

It is called decreasing on I if

y=≈

f sx 1 d . f sx 2 d

In the definition of an increasing function it is important to realize that the inequality f sx 1 d , f sx 2 d must be satisfied for every pair of numbers x 1 and x 2 in I with x 1 , x 2. You can see from Figure 23 that the function f sxd − x 2 is decreasing on the interval s2`, 0g and increasing on the interval f0, `d.

x

0

whenever x 1 , x 2 in I

FIGURE 23

1.1 EXERCISES 1. If f sxd − x 1 s2 2 x and tsud − u 1 s2 2 u , is it true that f − t? 2. If f sxd −

x2 2 x x21

tsxd − x

and

is it true that f − t? 3. The graph of a function f is given. (a) State the value of f s1d. (b) Estimate the value of f s21d. (c) For what values of x is f sxd − 1? (d) Estimate the value of x such that f sxd − 0. (e) State the domain and range of f. (f) On what interval is f increasing? y

(c) (d) (e) (f)

Estimate the solution of the equation f sxd − 21. On what interval is f decreasing? State the domain and range of f. State the domain and range of t.

5. Figure 1 was recorded by an instrument operated by the California Department of Mines and Geology at the University Hospital of the University of Southern California in Los Angeles. Use it to estimate the range of the vertical ground acceleration function at USC during the Northridge earthquake. 6. In this section we discussed examples of ordinary, everyday functions: Population is a function of time, postage cost is a function of weight, water temperature is a function of time. Give three other examples of functions from everyday life that are described verbally. What can you say about the domain and range of each of your functions? If possible, sketch a rough graph of each function. 7–10 Determine whether the curve is the graph of a function of x. If it is, state the domain and range of the function.

1 0

x

1

7.

8.

y

1

1

4. The graphs of f and t are given.

0

y

1

0

x

1

x

1

x

y

g f 2

0

9. 2

x

10.

y 1 0

y 1

1

x

0

(a) State the values of f s24d and ts3d. (b) For what values of x is f sxd − tsxd?

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

20

Chapter 1

Functions and Models

11. Shown is a graph of the global average temperature T during the 20th century. Estimate the following. (a) The global average temperature in 1950 (b) The year when the average temperature was 14.2°C (c) The year when the temperature was smallest? Largest? (d) The range of T T (•C)

y

A

B

C

100

0

20

15. The graph shows the power consumption for a day in September in San Francisco. (P is measured in megawatts; t is measured in hours starting at midnight.) (a) What was the power consumption at 6 am? At 6 pm? (b) When was the power consumption the lowest? When was it the highest? Do these times seem reasonable?

14

13 1900

2000 t

1950

Source: Adapted from Globe and Mail [Toronto], 5 Dec. 2009. Print.

12. Trees grow faster and form wider rings in warm years and grow more slowly and form narrower rings in cooler years. The figure shows ring widths of a Siberian pine from 1500 to 2000. (a) What is the range of the ring width function? (b) What does the graph tend to say about the temperature of the earth? Does the graph reflect the volcanic eruptions of the mid-19th century?

Ring width (mm)

R

P 800 600 400 200 0

3

6

9

12

15

18

21

t

Pacific Gas & Electric

16. Sketch a rough graph of the number of hours of daylight as a function of the time of year. 17. Sketch a rough graph of the outdoor temperature as a function of time during a typical spring day.

1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

18. Sketch a rough graph of the market value of a new car as a function of time for a period of 20 years. Assume the car is well maintained. 19. Sketch the graph of the amount of a particular brand of coffee sold by a store as a function of the price of the coffee. 1500

1600

1700

1800

1900

2000 t

Year Source: Adapted from G. Jacoby et al., “Mongolian Tree Rings and 20thCentury Warming,” Science 273 (1996): 771–73.

13. You put some ice cubes in a glass, fill the glass with cold water, and then let the glass sit on a table. Describe how the temperature of the water changes as time passes. Then sketch a rough graph of the temperature of the water as a function of the elapsed time. 14. Three runners compete in a 100-meter race. The graph depicts the distance run as a function of time for each runner. Describe in words what the graph tells you about this race. Who won the race? Did each runner finish the race?

20. You place a frozen pie in an oven and bake it for an hour. Then you take it out and let it cool before eating it. Describe how the temperature of the pie changes as time passes. Then sketch a rough graph of the temperature of the pie as a function of time. 21. A homeowner mows the lawn every Wednesday afternoon. Sketch a rough graph of the height of the grass as a function of time over the course of a four-week period. 22. An airplane takes off from an airport and lands an hour later at another airport, 400 km away. If t represents the time in minutes since the plane has left the terminal building, let xstd be the horizontal distance traveled and ystd be the altitude of the plane. (a) Sketch a possible graph of xstd. (b) Sketch a possible graph of ystd.

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SeCtion 1.1

(c) Sketch a possible graph of the ground speed. (d) Sketch a possible graph of the vertical velocity. 23. Temperature readings T (in °C) were recorded every three hours from midnight to 3:00 pm in Montréal on June 13, 2004. The time t was measured in hours from midnight.

30. f sxd −

0

3

6

9

12

15

T

21.5

19.8

20.0

22.2

24.8

25.8

x14 x2 2 9

24. Researchers measured the blood alcohol concentration (BAC) of eight adult male subjects after rapid consumption of 30 mL of ethanol (corresponding to two standard alcoholic drinks). The table shows the data they obtained by averaging the BAC (in mgymL) of the eight men. (a) Use the readings to sketch the graph of the BAC as a function of t. (b) Use your graph to describe how the effect of alcohol varies with time. t (hours)

BAC

t (hours)

BAC

0 0.2 0.5 0.75 1.0 1.25 1.5

0 0.25 0.41 0.40 0.33 0.29 0.24

1.75 2.0 2.25 2.5 3.0 3.5 4.0

0.22 0.18 0.15 0.12 0.07 0.03 0.01

Source: Adapted from P. Wilkinson et al., “Pharmacokinetics of Ethanol after Oral Administration in the Fasting State,” Journal of Pharmacokinetics and Biopharmaceutics 5 (1977): 207–24.

25. If f sxd − 3x 2 2 x 1 2, find f s2d, f s22d, f sad, f s2ad, f sa 1 1d, 2 f sad, f s2ad, f sa 2 d, [ f sad] 2, and f sa 1 hd. 26. A spherical balloon with radius r centimeters has volume Vsrd − 43 r 3. Find a function that represents the amount of air required to inflate the balloon from a radius of r centimeters to a radius of r 1 1 centimeters. 27–30 Evaluate the difference quotient for the given function. Simplify your answer. 27. f sxd − 4 1 3x 2 x 2,

f s3 1 hd 2 f s3d h

28. f sxd − x 3,

f sa 1 hd 2 f sad h

1 29. f sxd − , x

f sxd 2 f sad x2a

32. f sxd −

1 sx 2 5x 4

2x 3 2 5 x 1x26 2

34. tstd − s3 2 t 2 s2 1 t

3 33. f std − s 2t 2 1

35. hsxd − (a) Use the readings to sketch a rough graph of T as a function of t. (b) Use your graph to estimate the temperature at 11:00 am.

f sxd 2 f s1d x21

x13 , x11

31–37 Find the domain of the function. 31. f sxd −

t

21

Four Ways to Represent a Function

2

36. f sud −

37. Fs pd − s2 2 s p

u11 1 11 u11

38. Find the domain and range and sketch the graph of the function hsxd − s4 2 x 2 . 39–40 Find the domain and sketch the graph of the function. 39. f sxd − 1.6x 2 2.4

40. tstd −

t2 2 1 t11

41–44 Evaluate f s23d, f s0d, and f s2d for the piecewise defined function. Then sketch the graph of the function. 41. f sxd −

42. f sxd − 43. f sxd −

44. f sxd −

H H H H

x 1 2 if x , 0 1 2 x if x > 0

3 2 21 x if x , 2 2x 2 5 if x > 2 x 1 1 if x < 21 x2 if x . 21

21 if x < 1 7 2 2x if x . 1

45–50 Sketch the graph of the function.

| |

|

45. f sxd − x 1 x

|

47. tstd − 1 2 3t 49. f sxd −

H| | x 1

46. f sxd − x 1 2

|

|

|| |

|

48. hstd − t 1 t 1 1

| | | |

if x < 1 if x . 1

50. tsxd −

|| x | 2 1|

51–56 Find an expression for the function whose graph is the given curve. 51. The line segment joining the points s1, 23d and s5, 7d 52. The line segment joining the points s25, 10d and s7, 210d 53. The bottom half of the parabola x 1 s y 2 1d2 − 0 54. The top half of the circle x 2 1 s y 2 2d 2 − 4

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22

Chapter 1

55.

Functions and Models

56.

y

1

1 0

64. A cell phone plan has a basic charge of \$35 a month. The plan includes 400 free minutes and charges 10 cents for each additional minute of usage. Write the monthly cost C as a function of the number x of minutes used and graph C as a function of x for 0 < x < 600.

y

0

x

1

1

x

57–61 Find a formula for the described function and state its domain. 57. A rectangle has perimeter 20 m. Express the area of the rectangle as a function of the length of one of its sides. 2

58. A rectangle has area 16 m . Express the perimeter of the rectangle as a function of the length of one of its sides. 59. Express the area of an equilateral triangle as a function of the length of a side. 60. A closed rectangular box with volume 6 m3 has length twice the width. Express the height of the box as a function of the width. 61. An open rectangular box with volume 2 m3 has a square base. Express the surface area of the box as a function of the length of a side of the base. 62. A Norman window has the shape of a rectangle surmounted by a semicircle. If the perimeter of the window is 10 m, express the area A of the window as a function of the width x of the window.

65. In a certain province the maximum speed permitted on freeways is 100 kmyh and the minimum speed is 50 kmyh. The fine for violating these limits is \$10 for every kilometer per hour above the maximum speed or below the minimum speed. Express the amount of the fine F as a function of the driving speed x and graph Fsxd for 0 < x < 180. 66. An electricity company charges its customers a base rate of \$10 a month, plus 6 cents per kilowatt-hour (kWh) for the first 1200 kWh and 7 cents per kWh for all usage over 1200 kWh. Express the monthly cost E as a function of the amount x of electricity used. Then graph the function E for 0 < x < 2000. 67. In a certain country, income tax is assessed as follows. There is no tax on income up to \$10,000. Any income over \$10,000 is taxed at a rate of 10%, up to an income of \$20,000. Any income over \$20,000 is taxed at 15%. (a) Sketch the graph of the tax rate R as a function of the income I. (b) How much tax is assessed on an income of \$14,000? On \$26,000? (c) Sketch the graph of the total assessed tax T as a function of the income I. 68. The functions in Example 10 and Exercise 67 are called step functions because their graphs look like stairs. Give two other examples of step functions that arise in everyday life. 69–70 Graphs of f and t are shown. Decide whether each function is even, odd, or neither. Explain your reasoning. 69.

70.

y

y

g f

f x

x

x

63. A box with an open top is to be constructed from a rectangular piece of cardboard with dimensions 12 cm by 20 cm 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. 20 x

x

x

x

x

x

12 x

x

g

71. (a) If the point s5, 3d is on the graph of an even function, what other point must also be on the graph? (b) If the point s5, 3d is on the graph of an odd function, what other point must also be on the graph? 72. A function f has domain f25, 5g and a portion of its graph is shown. (a) Complete the graph of f if it is known that f is even. (b) Complete the graph of f if it is known that f is odd.

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SeCtion 1.2 Mathematical Models: A Catalog of Essential Functions

y

75. f sxd −

x x11

23

| |

76. f sxd − x x

77. f sxd − 1 1 3x 2 2 x 4 78. f sxd − 1 1 3x 3 2 x 5 _5

0

5

x

79. If f and t are both even functions, is f 1 t even? If f and t are both odd functions, is f 1 t odd? What if f is even and t is odd? Justify your answers.

73–78 Determine whether f is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually.

80. If f and t are both even functions, is the product ft even? If f and t are both odd functions, is ft odd? What if f is even and t is odd? Justify your answers.

x2 74. f sxd − 4 x 11

x 73. f sxd − 2 x 11

A mathematical model is a mathematical description (often by means of a function or an equation) of a real-world phenomenon such as the size of a population, the demand for a product, the speed of a falling object, the concentration of a product in a chemical reaction, the life expectancy of a person at birth, or the cost of emission reductions. The purpose of the model is to understand the phenomenon and perhaps to make predictions about future behavior. Figure 1 illustrates the process of mathematical modeling. Given a real-world problem, our first task is to formulate a mathematical model by identifying and naming the independent and dependent variables and making assumptions that simplify the phenomenon enough to make it mathematically tractable. We use our knowledge of the physical situation and our mathematical skills to obtain equations that relate the variables. In situations where there is no physical law to guide us, we may need to collect data (either from a library or the Internet or by conducting our own experiments) and examine the data in the form of a table in order to discern patterns. From this numerical representation of a function we may wish to obtain a graphical representation by plotting the data. The graph might even suggest a suitable algebraic formula in some cases.

Real-world problem

Formulate

Mathematical model

Solve

Mathematical conclusions

Interpret

Real-world predictions

Test

FIGURE 1 The modeling process

The second stage is to apply the mathematics that we know (such as the calculus that will be developed throughout this book) to the mathematical model that we have formulated in order to derive mathematical conclusions. Then, in the third stage, we take those mathematical conclusions and interpret them as information about the original real-world phenomenon by way of offering explanations or making predictions. The final step is to test our predictions by checking against new real data. If the predictions don’t compare well with reality, we need to refine our model or to formulate a new model and start the cycle again. A mathematical model is never a completely accurate representation of a physical situation—it is an idealization. A good model simplifies reality enough to permit math-

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24

Chapter 1

Functions and Models

ematical calculations but is accurate enough to provide valuable conclusions. It is important to realize the limitations of the model. In the end, Mother Nature has the final say. There are many different types of functions that can be used to model relationships observed in the real world. In what follows, we discuss the behavior and graphs of these functions and give examples of situations appropriately modeled by such functions.

linear Models The coordinate geometry of lines is reviewed in Appendix B.

When we say that y is a linear function of x, we mean that the graph of the function is a line, so we can use the slope-intercept form of the equation of a line to write a formula for the function as y − f sxd − mx 1 b where m is the slope of the line and b is the y-intercept. A characteristic feature of linear functions is that they grow at a constant rate. For instance, Figure 2 shows a graph of the linear function f sxd − 3x 2 2 and a table of sample values. Notice that whenever x increases by 0.1, the value of f sxd increases by 0.3. So f sxd increases three times as fast as x. Thus the slope of the graph y − 3x 2 2, namely 3, can be interpreted as the rate of change of y with respect to x. y

y=3x-2

0

1

x

_2

x

f sxd − 3x 2 2

1.0 1.1 1.2 1.3 1.4 1.5

1.0 1.3 1.6 1.9 2.2 2.5

FIGURE 2

example 1 (a) As dry air moves upward, it expands and cools. If the ground temperature is 20°C and the temperature at a height of 1 km is 10°C, express the temperature T (in °C) as a function of the height h (in kilometers), assuming that a linear model is appropriate. (b) Draw the graph of the function in part (a). What does the slope represent? (c) What is the temperature at a height of 2.5 km? Solution

(a) Because we are assuming that T is a linear function of h, we can write T − mh 1 b We are given that T − 20 when h − 0, so 20 − m ? 0 1 b − b In other words, the y-intercept is b − 20. We are also given that T − 10 when h − 1, so 10 − m ? 1 1 20 The slope of the line is therefore m − 10 2 20 − 210 and the required linear function is T − 210h 1 20

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25

SECTION 1.2 Mathematical Models: A Catalog of Essential Functions

(b) The graph is sketched in Figure 3. The slope is m − 210°Cykm, and this represents the rate of change of temperature with respect to height. (c) At a height of h − 2.5 km, the temperature is

T 20

T=_10h+20 10

0

T − 210s2.5d 1 20 − 25°C 1

h

3

If there is no physical law or principle to help us formulate a model, we construct an empirical model, which is based entirely on collected data. We seek a curve that “fits” the data in the sense that it captures the basic trend of the data points.

FIGURE 3

ExamplE 2 Table 1 lists the average carbon dioxide level in the atmosphere, measured in parts per million at Mauna Loa Observatory from 1980 to 2012. Use the data in Table 1 to find a model for the carbon dioxide level. SOLUTION We use the data in Table 1 to make the scatter plot in Figure 4, where t represents time (in years) and C represents the CO2 level (in parts per million, ppm).

C (ppm) 400

Table 1 Year

CO 2 level (in ppm)

1980 1982 1984 1986 1988 1990 1992 1994 1996

338.7 341.2 344.4 347.2 351.5 354.2 356.3 358.6 362.4

Year

CO 2 level (in ppm)

1998 2000 2002 2004 2006 2008 2010 2012

366.5 369.4 373.2 377.5 381.9 385.6 389.9 393.8

390 380 370 360 350 340 1980

1985

1990

1995

2000

2005

2010

t

FIGURE 4 Scatter plot for the average CO2 level

Notice that the data points appear to lie close to a straight line, so it’s natural to choose a linear model in this case. But there are many possible lines that approximate these data points, so which one should we use? One possibility is the line that passes through the first and last data points. The slope of this line is 393.8 2 338.7 55.1 − − 1.721875 < 1.722 2012 2 1980 32 We write its equation as C 2 338.7 − 1.722st 2 1980d or 1

C − 1.722t 2 3070.86

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26

CHAPTER 1

Functions and Models

Equation 1 gives one possible linear model for the carbon dioxide level; it is graphed in Figure 5. C (ppm) 400 390 380 370 360 350

FIGURE 5

340

Linear model through first and last data points A computer or graphing calculator finds the regression line by the method of least squares, which is to minimize the sum of the squares of the vertical distances between the data points and the line. The details are explained in Section 14.7.

1980

1985

1990

1995

2000

2005

2010

t

Notice that our model gives values higher than most of the actual CO2 levels. A better linear model is obtained by a procedure from statistics called linear regression. If we use a graphing calculator, we enter the data from Table 1 into the data editor and choose the linear regression command. (With Maple we use the fit[leastsquare] command in the stats package; with Mathematica we use the Fit command.) The machine gives the slope and y-intercept of the regression line as m − 1.71262

b − 23054.14

So our least squares model for the CO2 level is 2

C − 1.71262t 2 3054.14

In Figure 6 we graph the regression line as well as the data points. Comparing with Figure 5, we see that it gives a better fit than our previous linear model. C (ppm) 400 390 380 370 360 350 340

FIGURE 6 The regression line

1980

1985

1990

1995

2000

2005

2010

t

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

SECTION 1.2 Mathematical Models: A Catalog of Essential Functions

27

ExamplE 3 Use the linear model given by Equation 2 to estimate the average CO2 level for 1987 and to predict the level for the year 2020. According to this model, when will the CO2 level exceed 420 parts per million? SOLUTION Using Equation 2 with t − 1987, we estimate that the average CO2 level in

1987 was Cs1987d − s1.71262ds1987d 2 3054.14 < 348.84 This is an example of interpolation because we have estimated a value between observed values. (In fact, the Mauna Loa Observatory reported that the average CO2 level in 1987 was 348.93 ppm, so our estimate is quite accurate.) With t − 2020, we get Cs2020d − s1.71262ds2020d 2 3054.14 < 405.35 So we predict that the average CO2 level in the year 2020 will be 405.4 ppm. This is an example of extrapolation because we have predicted a value outside the time frame of observations. Consequently, we are far less certain about the accuracy of our prediction. Using Equation 2, we see that the CO2 level exceeds 420 ppm when 1.71262t 2 3054.14 . 420 Solving this inequality, we get t.

3474.14 < 2028.55 1.71262

We therefore predict that the CO2 level will exceed 420 ppm by the year 2029. This prediction is risky because it involves a time quite remote from our observations. In fact, we see from Figure 6 that the trend has been for CO2 levels to increase rather more rapidly in recent years, so the level might exceed 420 ppm well before 2029. ■

y 2

Polynomials A function P is called a polynomial if 0

1

x

Psxd − a n x n 1 a n21 x n21 1 ∙ ∙ ∙ 1 a 2 x 2 1 a 1 x 1 a 0

(a) y=≈+x+1

where n is a nonnegative integer and the numbers a 0 , a 1, a 2 , . . . , a n are constants called the coefficients of the polynomial. The domain of any polynomial is R − s2`, `d. If the leading coefficient a n ± 0, then the degree of the polynomial is n. For example, the function Psxd − 2x 6 2 x 4 1 25 x 3 1 s2

y 2

1

x

(b) y=_2≈+3x+1

FIGURE 7 The graphs of quadratic functions are parabolas.

is a polynomial of degree 6. A polynomial of degree 1 is of the form Psxd − mx 1 b and so it is a linear function. A polynomial of degree 2 is of the form Psxd − ax 2 1 bx 1 c and is called a quadratic function. Its graph is always a parabola obtained by shifting the parabola y − ax 2, as we will see in the next section. The parabola opens upward if a . 0 and downward if a , 0. (See Figure 7.) A polynomial of degree 3 is of the form Psxd − ax 3 1 bx 2 1 cx 1 d

a±0

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28

CHAPTER 1

Functions and Models

and is called a cubic function. Figure 8 shows the graph of a cubic function in part (a) and graphs of polynomials of degrees 4 and 5 in parts (b) and (c). We will see later why the graphs have these shapes. y

y

1

2

20 1

0

FIGURE 8

y

x

1

x

(a) y=˛-x+1

x

1

(b) y=x\$-3≈+x

(c) y=3x%-25˛+60x

Polynomials are commonly used to model various quantities that occur in the natural and social sciences. For instance, in Section 3.7 we will explain why economists often use a polynomial Psxd to represent the cost of producing x units of a commodity. In the following example we use a quadratic function to model the fall of a ball. Table 2 Time (seconds)

Height (meters)

0 1 2 3 4 5 6 7 8 9

450 445 431 408 375 332 279 216 143 61

ExamplE 4 A ball is dropped from the upper observation deck of the CN Tower, 450 m above the ground, and its height h above the ground is recorded at 1-second intervals in Table 2. Find a model to fit the data and use the model to predict the time at which the ball hits the ground. SOLUTION We draw a scatter plot of the data in Figure 9 and observe that a linear model is inappropriate. But it looks as if the data points might lie on a parabola, so we try a quadratic model instead. Using a graphing calculator or computer algebra system (which uses the least squares method), we obtain the following quadratic model:

h − 449.36 1 0.96t 2 4.90t 2

3 h (meters)

h

400

400

200

200

0

2

4

6

8

t (seconds)

0

2

4

6

8

FIGURE 9

FIGURE 10

Scatter plot for a falling ball

Quadratic model for a falling ball

t

In Figure 10 we plot the graph of Equation 3 together with the data points and see that the quadratic model gives a very good fit. The ball hits the ground when h − 0, so we solve the quadratic equation 24.90t 2 1 0.96t 1 449.36 − 0

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29

SECTION 1.2 Mathematical Models: A Catalog of Essential Functions

20.96 6 ss0.96d2 2 4s24.90d s449.36d 2s24.90d

The positive root is t < 9.67, so we predict that the ball will hit the ground after about 9.7 seconds. ■

Power Functions A function of the form f sxd − x a, where a is a constant, is called a power function. We consider several cases. (i ) a − n, where n is a positive integer The graphs of f sxd − x n for n − 1, 2, 3, 4, and 5 are shown in Figure 11. (These are polynomials with only one term.) We already know the shape of the graphs of y − x (a line through the origin with slope 1) and y − x 2 [a parabola, see Example 1.1.2(b)]. y

y=≈

y

y=x

y

1

1

0

1

x

0

y=x#

y

x

0

1

x

0

y=x%

y

1

1

1

y=x\$

1

1

x

0

1

FIGURE 11 Graphs of f sxd − x n for n − 1, 2, 3, 4, 5

The general shape of the graph of f sxd − x n depends on whether n is even or odd. If n is even, then f sxd − x n is an even function and its graph is similar to the parabola y − x 2. If n is odd, then f sxd − x n is an odd function and its graph is similar to that of y − x 3. Notice from Figure 12, however, that as n increases, the graph of y − x n becomes flatter near 0 and steeper when | x | > 1. (If x is small, then x 2 is smaller, x 3 is even smaller, x 4 is smaller still, and so on.) y

A family of functions is a collection of functions whose equations are related. Figure 12 shows two families of power functions, one with even powers and one with odd powers.

y

y=x\$ y=x^

y=x# y=≈

(_1, 1)

(1, 1)

(1, 1) y=x%

0

x

(_1, _1)

FIGURE 12

0

x

(i i) a − 1yn, where n is a positive integer n x is a root function. For n − 2 it is the square root The function f sxd − x 1yn − s function f sxd − sx , whose domain is f0, `d and whose graph is the upper half of the

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x

30

CHAPTER 1

Functions and Models n x is parabola x − y 2. [See Figure 13(a).] For other even values of n, the graph of y − s 3 similar to that of y − sx . For n − 3 we have the cube root function f sxd − sx whose domain is R (recall that every real number has a cube root) and whose graph is shown n 3 in Figure 13(b). The graph of y − s x for n odd sn . 3d is similar to that of y − s x.

y

y (1, 1) 0

FIGURE 13 Graphs of root functions y

y=∆ 1 0

x

1

(1, 1) x

0

x (a) ƒ=œ„

x

x (b) ƒ=Œ„

(iii) a − 21 The graph of the reciprocal function f sxd − x 21 − 1yx is shown in Figure 14. Its graph has the equation y − 1yx, or xy − 1, and is a hyperbola with the coordinate axes as its asymptotes. This function arises in physics and chemistry in connection with Boyle’s Law, which says that, when the temperature is constant, the volume V of a gas is inversely proportional to the pressure P: V−

C P

where C is a constant. Thus the graph of V as a function of P (see Figure 15) has the same general shape as the right half of Figure 14. Power functions are also used to model species-area relationships (Exercises 30–31), illumination as a function of distance from a light source (Exercise 29), and the period of revolution of a planet as a function of its distance from the sun (Exercise 32).

FIGURE 14 The reciprocal function V

Rational Functions A rational function f is a ratio of two polynomials: 0

f sxd −

P

FIGURE 15 Volume as a function of pressure at constant temperature

where P and Q are polynomials. The domain consists of all values of x such that Qsxd ± 0. A simple example of a rational function is the function f sxd − 1yx, whose domain is hx | x ± 0j; this is the reciprocal function graphed in Figure 14. The function

y

f sxd −

2

FIGURE 16 ƒ=

2x 4 2 x 2 1 1 x2 2 4

is a rational function with domain hx | x ± 62j. Its graph is shown in Figure 16.

20 0

Psxd Qsxd

2x\$-≈+1 ≈-4

x

Algebraic Functions A function f is called an algebraic function if it can be constructed using algebraic operations (such as addition, subtraction, multiplication, division, and taking roots) starting with polynomials. Any rational function is automatically an algebraic function. Here are two more examples: f sxd − sx 2 1 1

tsxd −

x 4 2 16x 2 3 1 sx 2 2ds x11 x 1 sx

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31

SECTION 1.2 Mathematical Models: A Catalog of Essential Functions

When we sketch algebraic functions in Chapter 4, we will see that their graphs can assume a variety of shapes. Figure 17 illustrates some of the possibilities. y

y

y

1

1

2

1

_3

x

0

0

(a) ƒ=xœ„„„„ x+3

FIGURE 17

x

5

x

1

(c) h(x)=x@?#(x-2)@

An example of an algebraic function occurs in the theory of relativity. The mass of a particle with velocity v is m − f svd −

m0 s1 2 v 2yc 2

where m 0 is the rest mass of the particle and c − 3.0 3 10 5 kmys is the speed of light in a vacuum.

Trigonometric Functions Trigonometry and the trigonometric functions are reviewed on Reference Page 2 and also in Appendix D. In calculus the convention is that radian measure is always used (except when otherwise indicated). For example, when we use the function f sxd − sin x, it is understood that sin x means the sine of the angle whose radian measure is x. Thus the graphs of the sine and cosine functions are as shown in Figure 18.

The Reference Pages are located at the back of the book.

y _ _π

π 2

y 3π 2

1 0 _1

π 2

π

_π 2π

5π 2

_

π 2

1 π 0

x _1

(a) ƒ=sin x

π 2

3π 3π 2

5π 2

x

FIGURE 18

Notice that for both the sine and cosine functions the domain is s2`, `d and the range is the closed interval f21, 1g. Thus, for all values of x, we have 21 < sin x < 1

21 < cos x < 1

or, in terms of absolute values,

| sin x | < 1

| cos x | < 1

Also, the zeros of the sine function occur at the integer multiples of ; that is, sin x − 0

when

x − n n an integer

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32

CHAPTER 1

Functions and Models

An important property of the sine and cosine functions is that they are periodic functions and have period 2. This means that, for all values of x, sinsx 1 2d − sin x

cossx 1 2d − cos x

The periodic nature of these functions makes them suitable for modeling repetitive phenomena such as tides, vibrating springs, and sound waves. For instance, in Example 1.3.4 we will see that a reasonable model for the number of hours of daylight in Philadelphia t days after January 1 is given by the function

F

Lstd − 12 1 2.8 sin

2 st 2 80d 365

G

1 ? 1 2 2 cos x SOLUTION This function is defined for all values of x except for those that make the denominator 0. But

ExamplE 5 What is the domain of the function f sxd −

1 2 2 cos x − 0

&?

cos x −

1 2

&?

x−

5 1 2n or x − 1 2n 3 3

where n is any integer (because the cosine function has period 2). So the domain of f is the set of all real numbers except for the ones noted above. ■ y

The tangent function is related to the sine and cosine functions by the equation tan x −

1 _

3π 2

0

_π _ π 2

π 2

3π 2

π

x

sin x cos x

and its graph is shown in Figure 19. It is undefined whenever cos x − 0, that is, when x − 6y2, 63y2, . . . . Its range is s2`, `d. Notice that the tangent function has period : tansx 1 d − tan x

The remaining three trigonometric functions (cosecant, secant, and cotangent) are the reciprocals of the sine, cosine, and tangent functions. Their graphs are shown in Appendix D.

FIGURE 19 y − tan x

y

Exponential Functions

y

1 0

1

(a) y=2®

FIGURE 20

for all x

x

1 0

1

(b) y=(0.5)®

x

The exponential functions are the functions of the form f sxd − b x, where the base b is a positive constant. The graphs of y − 2 x and y − s0.5d x are shown in Figure 20. In both cases the domain is s2`, `d and the range is s0, `d. Exponential functions will be studied in detail in Section 1.4, and we will see that they are useful for modeling many natural phenomena, such as population growth (if b . 1) and radioactive decay (if b , 1d.

Logarithmic Functions The logarithmic functions f sxd − log b x, where the base b is a positive constant, are the inverse functions of the exponential functions. They will be studied in Section 1.5. Figure

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33

SECTION 1.2 Mathematical Models: A Catalog of Essential Functions

y

21 shows the graphs of four logarithmic functions with various bases. In each case the domain is s0, `d, the range is s2`, `d, and the function increases slowly when x . 1.

y=log™ x y=log£ x

1

0

x

1

y=log∞ x y=log¡¸ x

ExamplE 6 Classify the following functions as one of the types of functions that we have discussed. (a) f sxd − 5 x (b) tsxd − x 5 11x (c) hsxd − (d) ustd − 1 2 t 1 5t 4 1 2 sx SOLUTION

(a) f sxd − 5 x is an exponential function. (The x is the exponent.) (b) tsxd − x 5 is a power function. (The x is the base.) We could also consider it to be a polynomial of degree 5.

FIGURE 21

(c) hsxd −

11x is an algebraic function. 1 2 sx

(d) ustd − 1 2 t 1 5t 4 is a polynomial of degree 4.

1.2

EXERCISES

1–2 Classify each function as a power function, root function, polynomial (state its degree), rational function, algebraic function, trigonometric function, exponential function, or logarithmic function.

(c) hsxd −

F

(d) ustd − 1 2 1.1t 1 2.54t 2

g

(f ) wsd − sin  cos 2

(e) vstd − 5 t 2. (a) y −  x

f

(b) y − x

(c) y − x 2 s2 2 x 3 d

(d) y − tan t 2 cos t

x

sx 2 1 3 x 11s 3

s 11s

(f ) y −

3–4 Match each equation with its graph. Explain your choices. (Don’t use a computer or graphing calculator.) 3. (a) y − x 2

y

4

2x 3 1 2 x2

3 (d) y − s x

(c) y − x 3

(b) y − 3 x

4. (a) y − 3x

(b) tsxd − sx

1. (a) f sxd − log 2 x

(e) y −

(b) y − x 5

G

5–6 Find the domain of the function.

(c) y − x 8 5. f sxd −

y

g

cos x 1 2 sin x

6. tsxd −

1 1 2 tan x

h

0

f

x

7. (a) Find an equation for the family of linear functions with slope 2 and sketch several members of the family. (b) Find an equation for the family of linear functions such that f s2d − 1 and sketch several members of the family. (c) Which function belongs to both families? 8. What do all members of the family of linear functions f sxd − 1 1 msx 1 3d have in common? Sketch several members of the family.

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34

CHAPTER 1

Functions and Models

9. What do all members of the family of linear functions f sxd − c 2 x have in common? Sketch several members of the family. 10. Find expressions for the quadratic functions whose graphs are shown. y

y (_2, 2)

f

(0, 1) (4, 2)

0

x

g 0

3

x

(1, _2.5)

11. Find an expression for a cubic function f if f s1d − 6 and f s21d − f s0d − f s2d − 0. 12. Recent studies indicate that the average surface temperature of the earth has been rising steadily. Some scientists have modeled the temperature by the linear function T − 0.02t 1 8.50, where T is temperature in °C and t represents years since 1900. (a) What do the slope and T-intercept represent? (b) Use the equation to predict the average global surface temperature in 2100. 13. If the recommended adult dosage for a drug is D (in mg), then to determine the appropriate dosage c for a child of age a, pharmacists use the equation c − 0.0417Dsa 1 1d. Suppose the dosage for an adult is 200 mg. (a) Find the slope of the graph of c. What does it represent? (b) What is the dosage for a newborn? 14. The manager of a weekend flea market knows from past experience that if he charges x dollars for a rental space at the market, then the number y of spaces he can rent is given by the equation y − 200 2 4x. (a) Sketch a graph of this linear function. (Remember that the rental charge per space and the number of spaces rented can’t be negative quantities.) (b) What do the slope, the y-intercept, and the x-intercept of the graph represent? 15. The relationship between the Fahrenheit sFd and Celsius sCd temperature scales is given by the linear function F − 95 C 1 32. (a) Sketch a graph of this function. (b) What is the slope of the graph and what does it represent? What is the F-intercept and what does it represent? 16. Kelly leaves Winnipeg at 2:00 pm and drives at a constant speed west along the Trans-Canada highway. He passes Brandon, 210 km from Winnipeg, at 4:00 pm. (a) Express the distance traveled in terms of the time elapsed. (b) Draw the graph of the equation in part (a). (c) What is the slope of this line? What does it represent?

17. Biologists have noticed that the chirping rate of crickets of a certain species is related to temperature, and the relationship appears to be very nearly linear. A cricket produces 112 chirps per minute at 20°C and 180 chirps per minute at 29°C. (a) Find a linear equation that models the temperature T as a function of the number of chirps per minute N. (b) What is the slope of the graph? What does it represent? (c) If the crickets are chirping at 150 chirps per minute, estimate the temperature. 18. The manager of a furniture factory finds that it costs \$2200 to manufacture 100 chairs in one day and \$4800 to produce 300 chairs in one day. (a) Express the cost as a function of the number of chairs produced, assuming that it is linear. Then sketch the graph. (b) What is the slope of the graph and what does it represent? (c) What is the y-intercept of the graph and what does it represent? 19. At the surface of the ocean, the water pressure is the same as the air pressure above the water, 1.05 kgycm2. Below the surface, the water pressure increases by 0.10 kgycm2 for every meter of descent. (a) Express the water pressure as a function of the depth below the ocean surface. (b) At what depth is the pressure 7 kgycm2? 20. The monthly cost of driving a car depends on the number of kilometers driven. Lynn found that in May it cost her \$380 to drive 768 km and in June it cost her \$460 to drive 1280 km. (a) Express the monthly cost C as a function of the distance driven d, assuming that a linear relationship gives a suitable model. (b) Use part (a) to predict the cost of driving 2400 km per month. (c) Draw the graph of the linear function. What does the slope represent? (d) What does the y-intercept represent? (e) Why does a linear function give a suitable model in this situation? 21–22 For each scatter plot, decide what type of function you might choose as a model for the data. Explain your choices. 21. (a)

y

0

(b)

x

y

0

x

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35

SECTION 1.2 Mathematical Models: A Catalog of Essential Functions

22. (a) y

(b) y

0

x

0

x

; 23. The table shows (lifetime) peptic ulcer rates (per 100 population) for various family incomes as reported by the National Health Interview Survey.

Income

Ulcer rate (per 100 population)

\$4,000 \$6,000 \$8,000 \$12,000 \$16,000 \$20,000 \$30,000 \$45,000 \$60,000

14.1 13.0 13.4 12.5 12.0 12.4 10.5 9.4 8.2

; 25. Anthropologists use a linear model that relates human femur (thighbone) length to height. The model allows an anthropologist to determine the height of an individual when only a partial skeleton (including the femur) is found. Here we find the model by analyzing the data on femur length and height for the eight males given in the following table. (a) Make a scatter plot of the data. (b) Find and graph the regression line that models the data. (c) An anthropologist finds a human femur of length 53 cm. How tall was the person? Femur length (cm)

Height (cm)

Femur length (cm)

Height (cm)

50.1 48.3 45.2 44.7

178.5 173.6 164.8 163.7

44.5 42.7 39.5 38.0

168.3 165.0 155.4 155.8

; 26. When laboratory rats are exposed to asbestos fibers, some of them develop lung tumors. The table lists the results of several experiments by different scientists. (a) Find the regression line for the data. (b) Make a scatter plot and graph the regression line. Does the regression line appear to be a suitable model for the data? (c) What does the y-intercept of the regression line represent?

(a) Make a scatter plot of these data and decide whether a linear model is appropriate. (b) Find and graph a linear model using the first and last data points. (c) Find and graph the least squares regression line. (d) Use the linear model in part (c) to estimate the ulcer rate for an income of \$25,000. (e) According to the model, how likely is someone with an income of \$80,000 to suffer from peptic ulcers? (f ) Do you think it would be reasonable to apply the model to someone with an income of \$200,000? ; 24. Biologists have observed that the chirping rate of crickets of a certain species appears to be related to temperature. The table shows the chirping rates for various temperatures. Temperature (°C)

Chirping rate (chirpsymin)

Temperature (°C)

Chirping rate (chirpsymin)

20 22 24 26 28

113 128 143 158 173

30 32 34 36

188 203 218 233

(a) Make a scatter plot of the data. (b) Find and graph the regression line. (c) Use the linear model in part (b) to estimate the chirping rate at 40°C.

Asbestos Percent of mice exposure that develop (fibersymL) lung tumors 50 400 500 900 1100

2 6 5 10 26

Asbestos Percent of mice exposure that develop (fibersymL) lung tumors 1600 1800 2000 3000

42 37 38 50

; 27. The table shows world average daily oil consumption from 1985 to 2010 measured in thousands of barrels per day. (a) Make a scatter plot and decide whether a linear model is appropriate. (b) Find and graph the regression line. (c) Use the linear model to estimate the oil consumption in 2002 and 2012. Years since 1985

Thousands of barrels of oil per day

0 5 10 15 20 25

60,083 66,533 70,099 76,784 84,077 87,302

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36

CHAPTER 1

Functions and Models

; 28. The table shows the percentage of the population of Argentina that has lived in rural areas from 1995 to 2000. Find a model for the data and use it to estimate the rural percentage in 1988 and 2002.

Year

Percentage rural

Year

Percentage rural

1955 1960 1965 1970 1975

30.4 26.4 23.6 21.1 19.0

1980 1985 1990 1995 2000

17.1 15.0 13.0 11.7 10.5

29. Many physical quantities are connected by inverse square laws, that is, by power functions of the form f sxd − kx 22. In particular, the illumination of an object by a light source is inversely proportional to the square of the distance from the source. Suppose that after dark you are in a room with just one lamp and you are trying to read a book. The light is too dim and so you move halfway to the lamp. How much brighter is the light? 30. It makes sense that the larger the area of a region, the larger the number of species that inhabit the region. Many ecologists have modeled the species-area relation with a power function and, in particular, the number of species S of bats living in caves in central Mexico has been related to the surface area A of the caves by the equation S − 0.7A0.3. (a) The cave called Misión Imposible near Puebla, Mexico, has a surface area of A − 60 m2. How many species of bats would you expect to find in that cave? (b) If you discover that four species of bats live in a cave, estimate the area of the cave.

(b) The Caribbean island of Dominica has area 291 mi 2. How many species of reptiles and amphibians would you expect to find on Dominica? Island

A

N

Saba Monserrat Puerto Rico Jamaica Hispaniola Cuba

4 40 3,459 4,411 29,418 44,218

5 9 40 39 84 76

; 32. The table shows the mean (average) distances d of the planets from the sun (taking the unit of measurement to be the distance from the earth to the sun) and their periods T (time of revolution in years). (a) Fit a power model to the data. (b) Kepler’s Third Law of Planetary Motion states that “ The square of the period of revolution of a planet is proportional to the cube of its mean distance from the sun.” Does your model corroborate Kepler’s Third Law?

; 31. The table shows the number N of species of reptiles and amphibians inhabiting Caribbean islands and the area A of the island in square miles. (a) Use a power function to model N as a function of A.

Planet Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune

d

T

0.387 0.723 1.000 1.523 5.203 9.541 19.190 30.086

0.241 0.615 1.000 1.881 11.861 29.457 84.008 164.784

In this section we start with the basic functions we discussed in Section 1.2 and obtain new functions by shifting, stretching, and reflecting their graphs. We also show how to combine pairs of functions by the standard arithmetic operations and by composition.

Transformations of Functions By applying certain transformations to the graph of a given function we can obtain the graphs of related functions. This will give us the ability to sketch the graphs of many functions quickly by hand. It will also enable us to write equations for given graphs. Let’s first consider translations. If c is a positive number, then the graph of y − f sxd 1 c is just the graph of y − f sxd shifted upward a distance of c units (because each y-coordi-

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SECTION 1.3

37

New Functions from Old Functions

nate is increased by the same number c). Likewise, if tsxd − f sx 2 cd, where c . 0, then the value of t at x is the same as the value of f at x 2 c (c units to the left of x). Therefore the graph of y − f sx 2 cd is just the graph of y − f sxd shifted c units to the right (see Figure 1). Vertical and Horizontal Shifts Suppose c . 0. To obtain the graph of

y − f sxd 1 c, shift the graph of y − f sxd a distance c units upward y − f sxd 2 c, shift the graph of y − f sxd a distance c units downward y − f sx 2 cd, shift the graph of y − f sxd a distance c units to the right y − f sx 1 cd, shift the graph of y − f sxd a distance c units to the left

y

y

y=ƒ+c

y=f(x+c)

c

y =ƒ

y=cƒ (c>1)

y=f(_x)

y=f(x-c)

y=ƒ c 0

y= 1c ƒ

c x

c

x

0

y=ƒ-c y=_ƒ

FIGURE 1 Translating the graph of f

FIGURE 2 Stretching and reflecting the graph of f

Now let’s consider the stretching and reflecting transformations. If c . 1, then the graph of y − cf sxd is the graph of y − f sxd stretched by a factor of c in the vertical direction (because each y-coordinate is multiplied by the same number c). The graph of y − 2f sxd is the graph of y − f sxd reflected about the x-axis because the point sx, yd is replaced by the point sx, 2yd. (See Figure 2 and the following chart, where the results of other stretching, shrinking, and reflecting transformations are also given.) Vertical and Horizontal Stretching and Reflecting Suppose c . 1. To obtain the

graph of y − cf sxd, stretch the graph of y − f sxd vertically by a factor of c y − s1ycd f sxd, shrink the graph of y − f sxd vertically by a factor of c y − f scxd, shrink the graph of y − f sxd horizontally by a factor of c y − f sxycd, stretch the graph of y − f sxd horizontally by a factor of c y − 2f sxd, reflect the graph of y − f sxd about the x-axis y − f s2xd, reflect the graph of y − f sxd about the y-axis

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38

CHAPTER 1

Functions and Models

Figure 3 illustrates these stretching transformations when applied to the cosine function with c − 2. For instance, in order to get the graph of y − 2 cos x we multiply the y-coordinate of each point on the graph of y − cos x by 2. This means that the graph of y − cos x gets stretched vertically by a factor of 2. y

y=2 cos x

y

2

y=cos x

2

1

1 y= 2

0

y=cos 1 x

y=cos 2x

2

1

cos x x

1

0

x

y=cos x

FIGURE 3

ExamplE 1 Given the graph of y − sx , use transformations to graph y − sx 2 2, y − sx 2 2 , y − 2sx , y − 2sx , and y − s2x . SOLUTION The graph of the square root function y − sx , obtained from Figure 1.2.13(a), is shown in Figure 4(a). In the other parts of the figure we sketch y − sx 2 2 by shifting 2 units downward, y − sx 2 2 by shifting 2 units to the right, y − 2sx by reflecting about the x-axis, y − 2sx by stretching vertically by a factor of 2, and y − s2x by reflecting about the y-axis. y

y

y

y

y

y

1 0

x

1

x

0

0

x

2

x

0

x

0

x

0

_2

(a) y=œ„x

(b) y=œ„-2 x

(d) y=_œ„x

(c) y=œ„„„„ x-2

(f) y=œ„„ _x

(e) y=2œ„x

FIGURE 4

ExamplE 2 Sketch the graph of the function f sxd − x 2 1 6x 1 10. SOLUTION Completing the square, we write the equation of the graph as

y − x 2 1 6x 1 10 − sx 1 3d2 1 1 This means we obtain the desired graph by starting with the parabola y − x 2 and shifting 3 units to the left and then 1 unit upward (see Figure 5). y

y

1

(_3, 1) 0

FIGURE 5

(a) y=≈

x

_3

_1

0

(b) y=(x+3)@+1

x

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

SECTION 1.3

39

New Functions from Old Functions

ExamplE 3 Sketch the graphs of the following functions. (a) y − sin 2x (b) y − 1 2 sin x SOLUTION

(a) We obtain the graph of y − sin 2x from that of y − sin x by compressing horizontally by a factor of 2. (See Figures 6 and 7.) Thus, whereas the period of y − sin x is 2, the period of y − sin 2x is 2y2 − . y

y

y=sin x

1 0

π 2

π

y=sin 2x

1 x

0 π π 4

x

π

2

FIGURE 7

FIGURE 6

(b) To obtain the graph of y − 1 2 sin x, we again start with y − sin x. We reflect about the x-axis to get the graph of y − 2sin x and then we shift 1 unit upward to get y − 1 2 sin x. (See Figure 8.) y

y=1-sin x

2 1 0

FIGURE 8

π 2

π

3π 2

x

ExamplE 4 Figure 9 shows graphs of the number of hours of daylight as functions of the time of the year at several latitudes. Given that Ankara, Turkey, is located at approximately 408N latitude, find a function that models the length of daylight at Ankara. 20 18 16 14 12

20° N 30° N 40° N 50° N

Hours 10 8

FIGURE 9

6

Graph of the length of daylight from March 21 through December 21 at various latitudes

4

Source: Adapted from L. Harrison, Daylight, Twilight, Darkness and Time (New York: Silver, Burdett, 1935), 40.

0

60° N

2 Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec.

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40

CHAPTER 1

Functions and Models

SOLUTION Notice that each curve resembles a shifted and stretched sine function. By looking at the blue curve we see that, at the latitude of Ankara, daylight lasts about 14.8 hours on June 21 and 9.2 hours on December 21, so the amplitude of the curve (the factor by which we have to stretch the sine curve vertically) is 12 s14.8 2 9.2d − 2.8. By what factor do we need to stretch the sine curve horizontally if we measure the time t in days? Because there are about 365 days in a year, the period of our model should be 365. But the period of y − sin t is 2, so the horizontal stretching factor is 2y365. We also notice that the curve begins its cycle on March 21, the 80th day of the year, so we have to shift the curve 80 units to the right. In addition, we shift it 12 units upward. Therefore we model the length of daylight in Ankara on the tth day of the year by the function

F

Lstd − 12 1 2.8 sin

y

_1

0

1

x

(a) y=≈-1

2 st 2 80d 365

G

Another transformation of some interest is taking the absolute value of a function. If y − | f sxd|, then according to the definition of absolute value, y − f sxd when f sxd > 0 and y − 2f sxd when f sxd , 0. This tells us how to get the graph of y − | f sxd| from the graph of y − f sxd: The part of the graph that lies above the x-axis remains the same; the part that lies below the x-axis is reflected about the x-axis.

ExamplE 5 Sketch the graph of the function y − | x 2 2 1 |.

y

SOLUTION We first graph the parabola y − x 2 2 1 in Figure 10(a) by shifting the _1

0

1

(b) y=| ≈-1 |

FIGURE 10

x

parabola y − x 2 downward 1 unit. We see that the graph lies below the x-axis when 21 , x , 1, so we reflect that part of the graph about the x-axis to obtain the graph of y − | x 2 2 1| in Figure 10(b). ■

Combinations of Functions Two functions f and t can be combined to form new functions f 1 t, f 2 t, ft, and fyt in a manner similar to the way we add, subtract, multiply, and divide real numbers. The sum and difference functions are defined by s f 1 tdsxd − f sxd 1 tsxd

s f 2 tdsxd − f sxd 2 tsxd

If the domain of f is A and the domain of t is B, then the domain of f 1 t is the intersection A > B because both f sxd and tsxd have to be defined. For example, the domain of f sxd − sx is A − f0, `d and the domain of tsxd − s2 2 x is B − s2`, 2g, so the domain of s f 1 tdsxd − sx 1 s2 2 x is A > B − f0, 2g. Similarly, the product and quotient functions are defined by s ftdsxd − f sxd tsxd

SD

f f sxd sxd − t tsxd

The domain of ft is A > B, but we can’t divide by 0 and so the domain of fyt is hx [ A > B | tsxd ± 0j. For instance, if f sxd − x 2 and tsxd − x 2 1, then the domain of the rational function s fytdsxd − x 2ysx 2 1d is h x | x ± 1j, or s2`, 1d ø s1, `d. There is another way of combining two functions to obtain a new function. For example, suppose that y − f sud − su and u − tsxd − x 2 1 1. Since y is a function of u and u is, in turn, a function of x, it follows that y is ultimately a function of x.

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SECTION 1.3

New Functions from Old Functions

41

We compute this by substitution: y − f sud − f stsxdd − f sx 2 1 1d − sx 2 1 1 x (input)

g

f•g

The procedure is called composition because the new function is composed of the two given functions f and t. In general, given any two functions f and t, we start with a number x in the domain of t and calculate tsxd. If this number tsxd is in the domain of f, then we can calculate the value of f s tsxdd. Notice that the output of one function is used as the input to the next function. The result is a new function hsxd − f s tsxdd obtained by substituting t into f. It is called the composition (or composite) of f and t and is denoted by f 8 t (“ f circle t”). Definition Given two functions f and t, the composite function f 8 t (also called the composition of f and t) is defined by

f

s f 8 tdsxd − f s tsxdd

FIGURE 11

The f 8 t machine is composed of the t machine (first) and then the f machine.

The domain of f 8 t is the set of all x in the domain of t such that tsxd is in the domain of f. In other words, s f 8 tdsxd is defined whenever both tsxd and f s tsxdd are defined. Figure 11 shows how to picture f 8 t in terms of machines.

ExamplE 6 If f sxd − x 2 and tsxd − x 2 3, find the composite functions f 8 t and t 8 f. SOLUTION We have

s f 8 tdsxd − f s tsxdd − f sx 2 3d − sx 2 3d2 s t 8 f dsxd − ts f sxdd − tsx 2 d − x 2 2 3

NOTE You can see from Example 6 that, in general, f 8 t ± t 8 f. Remember, the notation f 8 t means that the function t is applied first and then f is applied second. In Example 6, f 8 t is the function that first subtracts 3 and then squares; t 8 f is the function that first squares and then subtracts 3.

ExamplE 7 If f sxd − sx and tsxd − s2 2 x , find each of the following functions and their domains. (a) f 8 t (b) t 8 f (c) f 8 f (d) t 8 t SOLUTION

(a)

4 s f 8 tdsxd − f stsxdd − f (s2 2 x ) − ss2 2 x − s 22x

The domain of f 8 t is hx | 2 2 x > 0j − hx | x < 2j − s2`, 2g. (b)

If 0 < a < b, then a 2 < b 2.

s t 8 f dsxd − ts f sxdd − t (sx ) − s2 2 sx

For sx to be defined we must have x > 0. For s2 2 sx to be defined we must have 2 2 sx > 0, that is, sx < 2, or x < 4. Thus we have 0 < x < 4, so the domain of t 8 f is the closed interval f0, 4g. (c)

4 s f 8 f dsxd − f s f sxdd − f (sx ) − ssx − s x

The domain of f 8 f is f0, `d.

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42

CHAPTER 1

Functions and Models

st 8 tdsxd − tstsxdd − t (s2 2 x ) − s2 2 s2 2 x

(d)

This expression is defined when both 2 2 x > 0 and 2 2 s2 2 x > 0. The first inequality means x < 2, and the second is equivalent to s2 2 x < 2, or 2 2 x < 4, or x > 22. Thus 22 < x < 2, so the domain of t 8 t is the closed interval f22, 2g. ■ It is possible to take the composition of three or more functions. For instance, the composite function f 8 t 8 h is found by first applying h, then t, and then f as follows: s f 8 t 8 hdsxd − f stshsxddd

ExamplE 8 Find f 8 t 8 h if f sxd − xysx 1 1d, tsxd − x 10, and hsxd − x 1 3. SOLUTION

s f 8 t 8 hdsxd − f stshsxddd − f s tsx 1 3dd − f ssx 1 3d10 d −

sx 1 3d10 sx 1 3d10 1 1

So far we have used composition to build complicated functions from simpler ones. But in calculus it is often useful to be able to decompose a complicated function into simpler ones, as in the following example.

ExamplE 9 Given Fsxd − cos2sx 1 9d, find functions f , t, and h such that F − f 8 t 8 h. SOLUTION Since Fsxd − fcossx 1 9dg 2, the formula for F says: First add 9, then take

the cosine of the result, and finally square. So we let hsxd − x 1 9 Then

tsxd − cos x

f sxd − x 2

s f 8 t 8 hdsxd − f stshsxddd − f s tsx 1 9dd − f scossx 1 9dd − fcossx 1 9dg 2 − Fsxd

1.3

EXERCISES

1. Suppose the graph of f is given. Write equations for the graphs that are obtained from the graph of f as follows. (a) Shift 3 units upward. (b) Shift 3 units downward. (c) Shift 3 units to the right. (d) Shift 3 units to the left. (e) Reflect about the x-axis. (f ) Reflect about the y-axis. (g) Stretch vertically by a factor of 3. (h) Shrink vertically by a factor of 3. 2. Explain how each graph is obtained from the graph of y − f sxd. (a) y − f sxd 1 8 (b) y − f sx 1 8d (c) y − 8 f sxd (d) y − f s8xd (e) y − 2f sxd 2 1 (f ) y − 8 f s 81 xd 3. The graph of y − f sxd is given. Match each equation with its graph and give reasons for your choices.

(a) y − f sx 2 4d (c) y − 13 f sxd (e) y − 2 f sx 1 6d

(b) y − f sxd 1 3 (d) y − 2f sx 1 4d

y

@

6

3

!

f

#

\$ _6

_3

%

0

3

6

x

_3

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SECTION 1.3

4. The graph of f is given. Draw the graphs of the following functions. (a) y − f sxd 2 3 (b) y − f sx 1 1d (c) y −

1 2

f sxd

(d) y − 2f sxd

New Functions from Old Functions

9–24 Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations. 9. y − 2x 2

y

10. y − sx 2 3d2

x

1

5. The graph of f is given. Use it to graph the following functions. (a) y − f s2xd (b) y − f s 21xd (c) y − f s2xd

(d) y − 2f s2xd

12. y − 1 2

13. y − 2 cos 3x

14. y − 2 sx 1 1

15. y − x 2 2 4x 1 5

16. y − 1 1 sin  x

17. y − 2 2 sx

18. y − 3 2 2 cos x

19. y − sin ( 21 x )

20. y −

21. y − 12 s1 2 cos xd

22. y − 1 2 2 sx 1 3

|

23. y − sx 2 1

y 1 0

x

1

6–7 The graph of y − s3x 2 x 2 is given. Use transformations to create a function whose graph is as shown. y

y=œ„„„„„„ 3x-≈

1.5 0

6.

x

3

7.

y

y

3 _4

_1 0

x _1 _2.5

0

2

5

x

8. (a) How is the graph of y − 2 sin x related to the graph of y − sin x? Use your answer and Figure 6 to sketch the graph of y − 2 sin x. (b) How is the graph of y − 1 1 sx related to the graph of y − sx ? Use your answer and Figure 4(a) to sketch the graph of y − 1 1 sx .

1 x

11. y − x 3 1 1

2

0

43

|

2 22 x

|

24. y − cos  x

|

25. The city of New Delhi, India, is located at latitude 30°N. Use Figure 9 to find a function that models the number of hours of daylight at New Delhi as a function of the time of year. To check the accuracy of your model, use the fact that on March 31 the sun rises at 6:13 am and sets at 6:39 pm in New Delhi. 26. A variable star is one whose brightness alternately increases and decreases. For the most visible variable star, Delta Cephei, the time between periods of maximum brightness is 5.4 days, the average brightness (or magnitude) of the star is 4.0, and its brightness varies by 60.35 magnitude. Find a function that models the brightness of Delta Cephei as a function of time. 27. Some of the highest tides in the world occur in the Bay of Fundy on the Atlantic Coast of Canada. At Hopewell Cape the water depth at low tide is about 2.0 m and at high tide it is about 12.0 m. The natural period of oscillation is about 12 hours and on June 30, 2009, high tide occurred at 6:45 am. Find a function involving the cosine function that models the water depth Dstd (in meters) as a function of time t (in hours after midnight) on that day. 28. In a normal respiratory cycle the volume of air that moves into and out of the lungs is about 500 mL. The reserve and residue volumes of air that remain in the lungs occupy about 2000 mL and a single respiratory cycle for an average human takes about 4 seconds. Find a model for the total volume of air Vstd in the lungs as a function of time. 29. (a) How is the graph of y − f ( x ) related to the graph of f ? (b) Sketch the graph of y − sin x . (c) Sketch the graph of y − s x .

| | | | | |

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44

CHAPTER 1

Functions and Models

30. Use the given graph of f to sketch the graph of y − 1yf sxd. Which features of f are the most important in sketching y − 1yf sxd? Explain how they are used. y

1 0

52. Use the table to evaluate each expression. (a) f s ts1dd (b) ts f s1dd (d) ts ts1dd (e) s t 8 f ds3d x

1

2

3

4

5

6

f sxd

3

1

4

2

2

5

tsxd

6

3

2

1

2

3

x

1

31–32 Find (a) f 1 t, (b) f 2 t, (c) f t, and (d) fyt and state their domains. 31. f sxd − x 3 1 2x 2,

tsxd − 3x 2 2 1

32. f sxd − s3 2 x ,

tsxd − sx 2 2 1

53. Use the given graphs of f and t to evaluate each expression, or explain why it is undefined. (a) f s ts2dd (b) ts f s0dd (c) s f 8 tds0d (d) s t 8 f ds6d (e) s t 8 tds22d (f ) s f 8 f ds4d y

g

33–38 Find the functions (a) f 8 t, (b) t 8 f , (c) f 8 f , and (d) t 8 t and their domains. 33. f sxd − 3x 1 5,

tsxd − x 2 1 x

34. f sxd − x 3 2 2,

tsxd − 1 2 4x

37. f sxd − x 1

1 , x

38. f sxd − sx ,

tsxd −

x

2

54. Use the given graphs of f and t to estimate the value of f s tsxdd for x − 25, 24, 23, . . . , 5. Use these estimates to sketch a rough graph of f 8 t.

tsxd − x 2 1 1

36. f sxd − sin x,

f

2

0

tsxd − 4x 2 3

35. f sxd − sx 1 1,

(c) f s f s1dd (f ) s f 8 tds6d

x11 x12

y

3 tsxd − s 12x

g 1

39–42 Find f 8 t 8 h. 39. f sxd − 3x 2 2, tsxd − sin x,

0

40. f sxd − x 2 4 ,

tsxd − 2 x, hsxd − sx

41. f sxd − sx 2 3 ,

tsxd − x 2,

|

42. f sxd − tan x,

|

tsxd −

x , x21

45. Fsxd −

3 x s 3 11s x

47. vstd − secst 2 d tanst 2 d

hsxd − x 3 1 2 55. A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of 60 cmys. (a) Express the radius r of this circle as a function of the time t (in seconds). (b) If A is the area of this circle as a function of the radius, find A 8 r and interpret it.

3 hsxd − s x

44. Fsxd − cos2 x 46. Gsxd − 48. ustd −

Î 3

x 11x

tan t 1 1 tan t

49–51 Express the function in the form f 8 t 8 h. 49. Rsxd − ssx 2 1 51. Sstd − sin2scos td

x

f

43–48 Express the function in the form f 8 t. 43. Fsxd − s2 x 1 x 2 d 4

1

hsxd − x 2

8 50. Hsxd − s 21 x

| |

56. A spherical balloon is being inflated and the radius of the balloon is increasing at a rate of 2 cmys. (a) Express the radius r of the balloon as a function of the time t (in seconds). (b) If V is the volume of the balloon as a function of the radius, find V 8 r and interpret it. 57. A ship is moving at a speed of 30 kmyh parallel to a straight shoreline. The ship is 6 km from shore and it passes a lighthouse at noon. (a) Express the distance s between the lighthouse and the ship

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

SECTION 1.4 Exponential Functions

as a function of d, the distance the ship has traveled since noon; that is, find f so that s − f sdd. (b) Express d as a function of t, the time elapsed since noon; that is, find t so that d − tstd. (c) Find f 8 t. What does this function represent? 58. An airplane is flying at a speed of 350 kmyh at an altitude of 1 km and passes directly over a radar station at time t − 0. (a) Express the horizontal distance d (in kilometers) that the plane has flown as a function of t. (b) Express the distance s between the plane and the radar station as a function of d. (c) Use composition to express s as a function of t. 59. The Heaviside function H is defined by Hstd −

H

0 if t , 0 1 if t > 0

It is used in the study of electric circuits to represent the sudden surge of electric current, or voltage, when a switch is instantaneously turned on. (a) Sketch the graph of the Heaviside function. (b) Sketch the graph of the voltage Vstd in a circuit if the switch is turned on at time t − 0 and 120 volts are applied instantaneously to the circuit. Write a formula for Vstd in terms of Hstd. (c) Sketch the graph of the voltage Vstd in a circuit if the switch is turned on at time t − 5 seconds and 240 volts are applied instantaneously to the circuit. Write a formula for Vstd in terms of Hstd. (Note that starting at t − 5 corresponds to a translation.)

45

represents a gradual increase in voltage or current in a circuit. (a) Sketch the graph of the ramp function y − tHstd. (b) Sketch the graph of the voltage Vstd in a circuit if the switch is turned on at time t − 0 and the voltage is gradually increased to 120 volts over a 60-second time interval. Write a formula for Vstd in terms of Hstd for t < 60. (c) Sketch the graph of the voltage Vstd in a circuit if the switch is turned on at time t − 7 seconds and the voltage is gradually increased to 100 volts over a period of 25 seconds. Write a formula for Vstd in terms of Hstd for t < 32. 61. Let f and t be linear functions with equations f sxd − m1 x 1 b1 and tsxd − m 2 x 1 b 2. Is f 8 t also a linear function? If so, what is the slope of its graph? 62. If you invest x dollars at 4% interest compounded annually, then the amount Asxd of the investment after one year is Asxd − 1.04x. Find A 8 A, A 8 A 8 A, and A 8 A 8 A 8 A. What do these compositions represent? Find a formula for the composition of n copies of A. 63. (a) If tsxd − 2x 1 1 and hsxd − 4x 2 1 4x 1 7, find a function f such that f 8 t − h. (Think about what operations you would have to perform on the formula for t to end up with the formula for h.) (b) If f sxd − 3x 1 5 and hsxd − 3x 2 1 3x 1 2, find a function t such that f 8 t − h. 64. If f sxd − x 1 4 and hsxd − 4x 2 1, find a function t such that t 8 f − h. 65. Suppose t is an even function and let h − f 8 t. Is h always an even function? 66. Suppose t is an odd function and let h − f 8 t. Is h always an odd function? What if f is odd? What if f is even?

60. The Heaviside function defined in Exercise 59 can also be used to define the ramp function y − ctHstd, which

The function f sxd − 2 x is called an exponential function because the variable, x, is the exponent. It should not be confused with the power function tsxd − x 2, in which the variable is the base. In general, an exponential function is a function of the form f sxd − b x In Appendix G we present an alternative approach to the exponential and logarithmic functions using integral calculus.

where b is a positive constant. Let’s recall what this means. If x − n, a positive integer, then bn − b ? b ? ∙ ∙ ∙ ? b n factors 0

If x − 0, then b − 1, and if x − 2n, where n is a positive integer, then b 2n −

1 bn

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

46

CHAPTER 1

Functions and Models

If x is a rational number, x − pyq, where p and q are integers and q . 0, then

y

q q b x − b pyq − s b − ss bd p

1 0

1

x

FIGURE 1 Representation of y − 2 x, x rational

p

But what is the meaning of b x if x is an irrational number? For instance, what is meant by 2 s3 or 5? To help us answer this question we first look at the graph of the function y − 2 x, where x is rational. A representation of this graph is shown in Figure 1. We want to enlarge the domain of y − 2 x to include both rational and irrational numbers. There are holes in the graph in Figure 1 corresponding to irrational values of x. We want to fill in the holes by defining f sxd − 2 x, where x [ R, so that f is an increasing function. In particular, since the irrational number s3 satisfies 1.7 , s3 , 1.8 2 1.7 , 2 s3 , 2 1.8

we must have

and we know what 21.7 and 21.8 mean because 1.7 and 1.8 are rational numbers. Similarly, if we use better approximations for s3 , we obtain better approximations for 2 s3: 1.73 , s3 , 1.74

?

2 1.73 , 2 s3 , 2 1.74

1.732 , s3 , 1.733

?

2 1.732 , 2 s3 , 2 1.733

1.7320 , s3 , 1.7321

?

2 1.7320 , 2 s3 , 2 1.7321

1.73205 , s3 , 1.73206 ? 2 1.73205 , 2 s3 , 2 1.73206 . . . . . . . . . . . . A proof of this fact is given in J. Marsden and A. Weinstein, Calculus Unlimited (Menlo Park, CA: Benjamin/Cummings, 1981).

It can be shown that there is exactly one number that is greater than all of the numbers 2 1.7,

2 1.73,

2 1.732,

2 1.7320,

2 1.73205,

...

2 1.733,

2 1.7321,

2 1.73206,

...

and less than all of the numbers 2 1.8,

2 1.74,

s3

We define 2 to be this number. Using the preceding approximation process we can compute it correct to six decimal places: 2 s3 < 3.321997 Similarly, we can define 2 x (or b x, if b . 0) where x is any irrational number. Figure 2 shows how all the holes in Figure 1 have been filled to complete the graph of the function f sxd − 2 x, x [ R. y

1

FIGURE 2 y − 2 x, x real

0

1

x

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SECTION 1.4 Exponential Functions

47

The graphs of members of the family of functions y − b x are shown in Figure 3 for various values of the base b. Notice that all of these graphs pass through the same point s0, 1d because b 0 − 1 for b ± 0. Notice also that as the base b gets larger, the exponential function grows more rapidly (for x . 0). ” 4 ’®

” 2 ’®

1

1

y

10®

If 0 , b , 1, then b x approaches 0 as x becomes large. If b . 1, then b x approaches 0 as x decreases through negative values. In both cases the x-axis is a horizontal asymptote. These matters are discussed in Section 2.6.

1.5®

0

FIGURE 3

1

x

You can see from Figure 3 that there are basically three kinds of exponential functions y − b x. If 0 , b , 1, the exponential function decreases; if b − 1, it is a constant; and if b . 1, it increases. These three cases are illustrated in Figure 4. Observe that if b ± 1, then the exponential function y − b x has domain R and range s0, `d. Notice also that, since s1ybd x − 1yb x − b 2x, the graph of y − s1ybd x is just the reflection of the graph of y − b x about the y-axis. y

y

y

1

(0, 1)

(0, 1) 0

FIGURE 4

(a) y=b®, 0 0. So the graph of f 21 is the right half of the parabola y − 2x 2 2 1 and this seems reasonable from Figure 10. ■

y=f –!(x)

FIGURE 10

Logarithmic Functions If b . 0 and b ± 1, the exponential function f sxd − b x is either increasing or decreasing and so it is one-to-one by the Horizontal Line Test. It therefore has an inverse function f 21, which is called the logarithmic function with base b and is denoted by log b. If we use the formulation of an inverse function given by (3), f 21sxd − y &?

f syd − x

then we have 6

log b x − y &? b y − x

Thus, if x . 0, then log b x is the exponent to which the base b must be raised to give x. For example, log10 0.001 − 23 because 1023 − 0.001. The cancellation equations (4), when applied to the functions f sxd − b x and 21 f sxd − log b x, become log b sb x d − x for every x [ R 7

b log x − x for every x . 0 b

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60

CHAPTER 1

y

Functions and Models

y=x

y=b®, b>1 0

x

The logarithmic function log b has domain s0, `d and range R. Its graph is the reflection of the graph of y − b x about the line y − x. Figure 11 shows the case where b . 1. (The most important logarithmic functions have base b . 1.) The fact that y − b x is a very rapidly increasing function for x . 0 is reflected in the fact that y − log b x is a very slowly increasing function for x . 1. Figure 12 shows the graphs of y − log b x with various values of the base b . 1. Since log b 1 − 0, the graphs of all logarithmic functions pass through the point s1, 0d. y

y=log b x, b>1

y=log™ x y=log£ x

FIGURE 11

1

0

1

x

y=log∞ x y=log¡¸ x

FIGURE 12

The following properties of logarithmic functions follow from the corresponding properties of exponential functions given in Section 1.4. Laws of Logarithms If x and y are positive numbers, then 1. log b sxyd − log b x 1 log b y 2. log b

SD x y

− log b x 2 log b y

3. log b sx r d − r log b x

(where r is any real number)

ExamplE 6 Use the laws of logarithms to evaluate log 2 80 2 log 2 5. SOLUTION Using Law 2, we have

log 2 80 2 log 2 5 − log 2

S D 80 5

because 2 4 − 16.

− log 2 16 − 4 ■

Natural Logarithms Notation for logarithms Most textbooks in calculus and the sciences, as well as calculators, use the notation ln x for the natural logarithm and log x for the “common logarithm,” log10 x. In the more advanced mathematical and scientific literature and in computer languages, however, the notation log x usually denotes the natural logarithm.

Of all possible bases b for logarithms, we will see in Chapter 3 that the most convenient choice of a base is the number e, which was defined in Section 1.4. The logarithm with base e is called the natural logarithm and has a special notation: log e x − ln x If we put b − e and replace log e with “ln” in (6) and (7), then the defining properties of the natural logarithm function become

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SECTION 1.5 Inverse Functions and Logarithms

8

ln x − y

&? e y − x

lnse x d − x

x[R

ln x

x.0

9

e

61

−x

In particular, if we set x − 1, we get ln e − 1

ExamplE 7 Find x if ln x − 5. SOLUTION 1 From (8) we see that

ln x − 5

means

e5 − x

Therefore x − e 5. (If you have trouble working with the “ln” notation, just replace it by log e. Then the equation becomes log e x − 5; so, by the definition of logarithm, e 5 − x.) SOLUTION 2 Start with the equation

ln x − 5 and apply the exponential function to both sides of the equation: e ln x − e 5 But the second cancellation equation in (9) says that e ln x − x. Therefore x − e 5.

ExamplE 8 Solve the equation e 523x − 10. SOLUTION We take natural logarithms of both sides of the equation and use (9):

lnse 523x d − ln 10 5 2 3x − ln 10 3x − 5 2 ln 10 x − 13 s5 2 ln 10d Since the natural logarithm is found on scientific calculators, we can approximate the solution: to four decimal places, x < 0.8991. ■

ExamplE 9 Express ln a 1 12 ln b as a single logarithm. SOLUTION Using Laws 3 and 1 of logarithms, we have

ln a 1 12 ln b − ln a 1 ln b 1y2 − ln a 1 lnsb − ln ( asb )

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62

CHAPTER 1

Functions and Models

The following formula shows that logarithms with any base can be expressed in terms of the natural logarithm. 10 Change of Base Formula For any positive number b sb ± 1d, we have log b x −

ln x ln b

PROOF Let y − log b x. Then, from (6), we have b y − x. Taking natural logarithms of

both sides of this equation, we get y ln b − ln x. Therefore y−

ln x ln b

Scientific calculators have a key for natural logarithms, so Formula 10 enables us to use a calculator to compute a logarithm with any base (as shown in the following example). Similarly, Formula 10 allows us to graph any logarithmic function on a graphing calculator or computer (see Exercises 43 and 44).

ExamplE 10 Evaluate log 8 5 correct to six decimal places. SOLUTION Formula 10 gives

log 8 5 −

ln 5 < 0.773976 ln 8

y

y=´

Graph and Growth of the Natural Logarithm

y=x

1

y=ln x

0 x

1

The graphs of the exponential function y − e x and its inverse function, the natural logarithm function, are shown in Figure 13. Because the curve y − e x crosses the y-axis with a slope of 1, it follows that the reflected curve y − ln x crosses the x-axis with a slope of 1. In common with all other logarithmic functions with base greater than 1, the natural logarithm is an increasing function defined on s0, `d and the y-axis is a vertical asymptote. (This means that the values of ln x become very large negative as x approaches 0.)

ExamplE 11 Sketch the graph of the function y − lnsx 2 2d 2 1. FIGURE 13 The graph of y − ln x is the reflection of the graph of y − e x about the line y − x. y

SOLUTION We start with the graph of y − ln x as given in Figure 13. Using the transformations of Section 1.3, we shift it 2 units to the right to get the graph of y − lnsx 2 2d and then we shift it 1 unit downward to get the graph of y − lnsx 2 2d 2 1. (See Figure 14.) y

y=ln x 0

(1, 0)

y

x=2

x=2 y=ln(x-2)-1

y=ln(x-2) x

0

2

(3, 0)

x

0

2

x (3, _1)

FIGURE 14

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63

SECTION 1.5 Inverse Functions and Logarithms

Although ln x is an increasing function, it grows very slowly when x . 1. In fact, ln x grows more slowly than any positive power of x. To illustrate this fact, we compare approximate values of the functions y − ln x and y − x 1y2 − sx in the following table and we graph them in Figures 15 and 16. You can see that initially the graphs of y − sx and y − ln x grow at comparable rates, but eventually the root function far surpasses the logarithm.

x

1

2

5

10

50

100

500

1000

10,000

100,000

ln x

0

0.69

1.61

2.30

3.91

4.6

6.2

6.9

9.2

11.5

sx

1

1.41

2.24

3.16

7.07

10.0

22.4

31.6

100

316

ln x sx

0

0.49

0.72

0.73

0.55

0.46

0.28

0.22

0.09

0.04

y

y 20

x y=œ„ 1 0

x y=œ„

y=ln x

y=ln x x

1

FIGURE 15

0

1000 x

FIGURE 16

Inverse Trigonometric Functions When we try to find the inverse trigonometric functions, we have a slight difficulty: Because the trigonometric functions are not one-to-one, they don’t have inverse functions. The difficulty is overcome by restricting the domains of these functions so that they become one-to-one. You can see from Figure 17 that the sine function y − sin x is not one-to-one (use the Horizontal Line Test). But the function f sxd − sin x, 2y2 < x < y2, is one-toone (see Figure 18). The inverse function of this restricted sine function f exists and is denoted by sin 21 or arcsin. It is called the inverse sine function or the arcsine function. y

y

y=sin x _ π2 _π

FIGURE 17

0

π 2

π

0

x

π 2

FIGURE 18

y − sin x, 22 < x < 2

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x

64

CHAPTER 1

Functions and Models

Since the definition of an inverse function says that f 21sxd − y &?

f syd − x

we have sin21x − y &? sin y − x

sin21x ±

1 sin x

and 2

1)

&? sec y − x

and y [ f0, y2d ø f, 3y2d

&? cot y − x

and

y − cot21x sx [ Rd _1

0

π

x

The choice of intervals for y in the definitions of csc21 and sec21 is not universally agreed upon. For instance, some authors use y [ f0, y2d ø sy2, g in the definition of sec21. [You can see from the graph of the secant function in Figure 26 that both this choice and the one in (11) will work.]

FIGURE 26 y − sec x

1. (a) What is a one-to-one function? (b) How can you tell from the graph of a function whether it is one-to-one? 2. (a) Suppose f is a one-to-one function with domain A and range B. How is the inverse function f 21 defined? What is the domain of f 21? What is the range of f 21? (b) If you are given a formula for f, how do you find a formula for f 21? (c) If you are given the graph of f, how do you find the graph of f 21? 3–14 A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one. 3.

4.

x

1

2

3

4

5

6

f sxd

1.5

2.0

3.6

5.3

2.8

2.0

x

1

2

3

4

5

6

f sxd

1.0

1.9

2.8

3.5

3.1

2.9

5.

y [ s0, d

6.

y

10. f sxd − x 4 2 16

9. f sxd − 2x 2 3 11. tsxd − 1 2 sin x

3 12. tsxd − s x

13. f std is the height of an arrow t seconds after it is launched upward. 14. f std is your shoe size at age t. 15. Assume that f is a one-to-one function. (a) If f s4d − 7, what is f 21s7d? (b) If f 21s8d − 2, what is f s2d? 16. If f sxd − x 5 1 x 3 1 x, find f 21s3d and f s f 21s2dd. 17. If tsxd − 3 1 x 1 e x, find t21s4d. 18. The graph of f is given. (a) Why is f one-to-one? (b) What are the domain and range of f 21? (c) What is the value of f 21s2d? (d) Estimate the value of f 21s0d. y

y 1 x 0

x

7.

8.

y

x

1

x

y

x

19. The formula C − 59 sF 2 32d, where F > 2459.67, expresses the Celsius temperature C as a function of the Fahrenheit temperature F. Find a formula for the inverse function and interpret it. What is the domain of the inverse function?

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SECTION 1.5 Inverse Functions and Logarithms

20. In the theory of relativity, the mass of a particle with speed  is m0 m − f svd − s1 2 v 2yc 2 where m 0 is the rest mass of the particle and c is the speed of light in a vacuum. Find the inverse function of f and explain its meaning. 21–26 Find a formula for the inverse of the function. 4x 2 1 21. f sxd − 1 1 s2 1 3x 22. f sxd − 2x 1 3 x e 23. f sxd − e 2x21 24. y − 1 1 2e x 26. y −

12e 1 1 e2x

28. f sxd − 1 1 e2x

y

39. ln 20 2 2 ln 2 41.

1 3

3

40. ln b 1 2 ln c 2 3 ln d 1 2

lnsx 1 2d 1 fln x 2 lnsx 2 1 3x 1 2d2 g

43. y − log 1.5 x,

29–30 Use the given graph of f to sketch the graph of f 21. 29.

39–41 Express the given quantity as a single logarithm.

; 43–44 Use Formula 10 to graph the given functions on a common screen. How are these graphs related?

21 21 ; 27–28 Find an explicit formula for f and use it to graph f , f, and the line y − x on the same screen. To check your work, see whether the graphs of f and f 21 are reflections about the line.

27. f sxd − s4 x 1 3

3

(b) e lnsln e d

38. (a) e2ln 2

42. Use Formula 10 to evaluate each logarithm correct to six decimal places. (a) log 5 10 (b) log 3 57

2x

25. y − lnsx 1 3d

67

30. y

44. y − ln x,

y − ln x,

y − log 10 x,

y − log 10 x, x

y−e,

y − log 50 x

y − 10 x

45. Suppose that the graph of y − log 2 x is drawn on a coordinate grid where the unit of measurement is a centimeter. How many kilometers to the right of the origin do we have to move before the height of the curve reaches 1 m? 0.1 ; 46. Compare the functions f sxd − x and tsxd − ln x by graphing both f and t in several viewing rectangles. When does the graph of f finally surpass the graph of t?

1 1 0

0 1

2

x

x

31. Let f sxd − s1 2 x 2 , 0 < x < 1. (a) Find f 21. How is it related to f ? (b) Identify the graph of f and explain your answer to part (a). 3 1 2 x3 . 32. Let tsxd − s (a) Find t 21. How is it related to t? (b) Graph t. How do you explain your answer to part (a)? ;

33. (a) (b) (c) (d)

How is the logarithmic function y − log b x defined? What is the domain of this function? What is the range of this function? Sketch the general shape of the graph of the function y − log b x if b . 1.

34. (a) What is the natural logarithm? (b) What is the common logarithm? (c) Sketch the graphs of the natural logarithm function and the natural exponential function with a common set of axes. 35–38 Find the exact value of each expression. 35. (a) log 2 32

(b) log 8 2

1 36. (a) log 5 125

(b) lns1ye 2 d

37. (a) log 10 40 1 log 10 2.5 (b) log 8 60 2 log 8 3 2 log 8 5

47–48 Make a rough sketch of the graph of each function. Do not use a calculator. Just use the graphs given in Figures 12 and 13 and, if necessary, the transformations of Section 1.3. 47. (a) y − log 10sx 1 5d

(b) y − 2ln x

48. (a) y − lns2xd

(b) y − ln x

| |

49–50 (a) What are the domain and range of f ? (b) What is the x-intercept of the graph of f ? (c) Sketch the graph of f. 49. f sxd − ln x 1 2

50. f sxd − lnsx 2 1d 2 1

51–54 Solve each equation for x. 51. (a) e 724x − 6 2

(b) lns3x 2 10d − 2

52. (a) lnsx 2 1d − 3

(b) e 2x 2 3e x 1 2 − 0

53. (a) 2 x25 − 3

(b) ln x 1 lnsx 2 1d − 1

54. (a) lnsln xd − 1

(b) e ax − Ce bx, where a ± b

55–56 Solve each inequality for x. 55. (a) ln x , 1

(b) e x > 3

56. (a) 1 , e 3x21 , 2

(b) 1 2 2 ln x , 3

57. (a) Find the domain of f sxd − lnse x 2 3d. (b) Find f 21 and its domain.

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68

CAS

CAS

CHAPTER 1

Functions and Models

58. (a) What are the values of e ln 300 and lnse 300 d? (b) Use your calculator to evaluate e ln 300 and lnse 300 d. What do you notice? Can you explain why the calculator has trouble?

65. (a) csc21 s2

59. Graph the function f sxd − sx 3 1 x 2 1 x 1 1 and explain why it is one-to-one. Then use a computer algebra system to find an explicit expression for f 21sxd. (Your CAS will produce three possible expressions. Explain why two of them are irrelevant in this context.)

68. (a) arcsinssins5y4dd

60. (a) If tsxd − x 6 1 x 4, x > 0, use a computer algebra system to find an expression for t 21sxd. (b) Use the expression in part (a) to graph y − tsxd, y − x, and y − t 21sxd on the same screen. 61. If a bacteria population starts with 100 bacteria and doubles every three hours, then the number of bacteria after t hours is n − f std − 100 ∙ 2 ty3. (a) Find the inverse of this function and explain its meaning. (b) When will the population reach 50,000? 62. When a camera flash goes off, the batteries immediately begin to recharge the flash’s capacitor, which stores electric charge given by Qstd − Q 0 s1 2 e 2tya d (The maximum charge capacity is Q 0 and t is measured in seconds.) (a) Find the inverse of this function and explain its meaning. (b) How long does it take to recharge the capacitor to 90% of capacity if a − 2? 63–68 Find the exact value of each expression. 63. (a) cos21 s21d

(b) sin21s0.5d

64. (a) tan21 s3

(b) arctans21d

1

(b) arcsin 1

66. (a) sin

(21ys2 ) 21 cot (2s3 )

(b) cos21 (s3 y2 )

67. (a)

(b) sec21 2

21

5 (b) cos (2 sin21 (13 ))

69. Prove that cosssin21 xd − s1 2 x 2 . 70–72 Simplify the expression. 70. tanssin21 xd

71. sinstan21 xd

72. sins2 arccos xd

; 73-74 Graph the given functions on the same screen. How are these graphs related? 73. y − sin x, 2y2 < x < y2; 74. y − tan x, 2y2 , x , y2;

y − sin21x; 21

y − tan x;

y−x y−x

75. Find the domain and range of the function tsxd − sin21s3x 1 1d 21 ; 76. (a) Graph the function f sxd − sinssin xd and explain the appearance of the graph. (b) Graph the function tsxd − sin21ssin xd. How do you explain the appearance of this graph?

77. (a) If we shift a curve to the left, what happens to its reflection about the line y − x? In view of this geometric principle, find an expression for the inverse of tsxd − f sx 1 cd, where f is a one-to-one function. (b) Find an expression for the inverse of hsxd − f scxd, where c ± 0.

REvIEw

CONCEpT CHECK 1. (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How can you tell whether a given curve is the graph of a function? 2. Discuss four ways of representing a function. Illustrate your discussion with examples. 3. (a) What is an even function? How can you tell if a function is even by looking at its graph? Give three examples of an even function. (b) What is an odd function? How can you tell if a function is odd by looking at its graph? Give three examples of an odd function.

Answers to the Concept Check can be found on the back endpapers.

4. What is an increasing function? 5. What is a mathematical model? 6. Give an example of each type of function. (a) Linear function (b) Power function (c) Exponential function (d) Quadratic function (e) Polynomial of degree 5 (f ) Rational function 7. Sketch by hand, on the same axes, the graphs of the following functions. (a) f sxd − x (b) tsxd − x 2 (c) hsxd − x 3

(d) jsxd − x 4

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CHAPTER 1

8. Draw, by hand, a rough sketch of the graph of each function. (b) y − tan x (c) y − e x (a) y − sin x (d) y − ln x (e) y − 1yx (f ) y − x (g) y − sx (h) y − tan21 x

| |

9. Suppose that f has domain A and t has domain B. (a) What is the domain of f 1 t? (b) What is the domain of f t? (c) What is the domain of fyt? 10. How is the composite function f 8 t defined? What is its domain? 11. Suppose the graph of f is given. Write an equation for each of the graphs that are obtained from the graph of f as follows. (a) Shift 2 units upward. (b) Shift 2 units downward. (c) Shift 2 units to the right. (d) Shift 2 units to the left. (e) Reflect about the x-axis.

(f ) (g) (h) (i) ( j)

Review

Reflect about the y-axis. Stretch vertically by a factor of 2. Shrink vertically by a factor of 2. Stretch horizontally by a factor of 2. Shrink horizontally by a factor of 2.

12. (a) What is a one-to-one function? How can you tell if a function is one-to-one by looking at its graph? (b) If f is a one-to-one function, how is its inverse function f 21 defined? How do you obtain the graph of f 21 from the graph of f ? 13. (a) How is the inverse sine function f sxd − sin21 x defined? What are its domain and range? (b) How is the inverse cosine function f sxd − cos21 x defined? What are its domain and range? (c) How is the inverse tangent function f sxd − tan21 x defined? What are its domain and range?

TRUE-FaLSE qUIz Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. 1. If f is a function, then f ss 1 td − f ssd 1 f std. 2. If f ssd − f std, then s − t. 3. If f is a function, then f s3xd − 3 f sxd. 4. If x 1 , x 2 and f is a decreasing function, then f sx 1 d . f sx 2 d. 5. A vertical line intersects the graph of a function at most once. 6. If f and t are functions, then f 8 t − t 8 f.

7. If f is one-to-one, then f 21sxd − 8. You can always divide by e x.

1 . f sxd

9. If 0 , a , b, then ln a , ln b. 10. If x . 0, then sln xd6 − 6 ln x. ln x x 11. If x . 0 and a . 1, then − ln . ln a a 12. tan21s21d − 3y4 sin21x cos21x 14. If x is any real number, then sx 2 − x. 13. tan21x −

EXERCISES 1. Let f be the function whose graph is given.

(f ) Is f one-to-one? Explain. (g) Is f even, odd, or neither even nor odd? Explain. 2. The graph of t is given.

y

f

y

g

1 1

x 1 0 1

(a) (b) (c) (d) (e)

Estimate the value of f s2d. Estimate the values of x such that f sxd − 3. State the domain of f. State the range of f. On what interval is f increasing?

69

(a) (b) (c) (d) (e)

x

State the value of ts2d. Why is t one-to-one? Estimate the value of t21s2d. Estimate the domain of t21. Sketch the graph of t21.

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70

CHAPTER 1

Functions and Models

21. The table gives the population of Indonesia (in millions) for the years 1950–2000. Decide what type of model is appropriate and use your model to predict the population of Indonesia in 2010.

3. If f sxd − x 2 2 2x 1 3, evaluate the difference quotient f sa 1 hd 2 f sad h 4. Sketch a rough graph of the yield of a crop as a function of the amount of fertilizer used. 5–8 Find the domain and range of the function. Write your answer in interval notation. 5. f sxd − 2ys3x 2 1d

6. tsxd − s16 2 x 4

7. hsxd − lnsx 1 6d

8. Fstd − 1 1 sin 2t

9. Suppose that the graph of f is given. Describe how the graphs of the following functions can be obtained from the graph of f. (a) y − f sxd 1 8 (b) y − f sx 1 8d (c) y − 1 1 2 f sxd (d) y − f sx 2 2d 2 2 (e) y − 2f sxd (f ) y − f 21sxd

Year

Population

Year

Population

1950 1955 1960 1965 1970 1975

80 86 96 107 120 134

1980 1985 1990 1995 2000

150 166 182 197 212

22. A small-appliance manufacturer finds that it costs \$9000 to produce 1000 toaster ovens a week and \$12,000 to produce 1500 toaster ovens a week. (a) Express the cost as a function of the number of toaster ovens produced, assuming that it is linear. Then sketch the graph. (b) What is the slope of the graph and what does it represent? (c) What is the y-intercept of the graph and what does it represent?

10. The graph of f is given. Draw the graphs of the following functions. (a) y − f sx 2 8d (b) y − 2f sxd (c) y − 2 2 f sxd (d) y − 12 f sxd 2 1 21 (e) y − f sxd (f ) y − f 21sx 1 3d

23. If f sxd − 2x 1 ln x, find f 21s2d.

y

24. Find the inverse function of f sxd −

25. Find the exact value of each expression. (a) e 2 ln 3 (b) log10 25 1 log10 4

1 0

x11 . 2x 1 1

1

(c) tansarcsin 12 d

x

26. Solve each equation for x. (a) e x − 5 x (c) ee − 2

11–16 Use transformations to sketch the graph of the function. 11. y − sx 2 2d3

12. y − 2 sx

13. y − x 2 2 2 x 1 2

14. y − lnsx 1 1d

15. f sxd − 2cos 2x

16. f sxd −

H

17. Determine whether f is even, odd, or neither even nor odd. (a) f sxd − 2x 5 2 3x 2 1 2 (b) f sxd − x 3 2 x 7 2 (c) f sxd − e2x (d) f sxd − 1 1 sin x

19. If f sxd − ln x and tsxd − x 2 2 9, find the functions (a) f 8 t, (b) t 8 f , (c) f 8 f , (d) t 8 t, and their domains. 20. Express the function Fsxd − 1ysx 1 sx as a composition of three functions.

(b) ln x − 2 (d) tan21x − 1

27. The half-life of palladium-100, 100 Pd, is four days. (So half of any given quantity of 100 Pd will disintegrate in four days.) The initial mass of a sample is one gram. (a) Find the mass that remains after 16 days. (b) Find the mass mstd that remains after t days. (c) Find the inverse of this function and explain its meaning. (d) When will the mass be reduced to 0.01g?

2x if x , 0 e x 2 1 if x > 0

18. Find an expression for the function whose graph consists of the line segment from the point s22, 2d to the point s21, 0d together with the top half of the circle with center the origin and radius 1.

(d) sinscos21s54dd

28. The population of a certain species in a limited environment with initial population 100 and carrying capacity 1000 is Pstd −

;

100,000 100 1 900e2t

where t is measured in years. (a) Graph this function and estimate how long it takes for the population to reach 900. (b) Find the inverse of this function and explain its meaning. (c) Use the inverse function to find the time required for the population to reach 900. Compare with the result of part (a).

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principles of problem Solving 1 UNDERSTAND THE PROBLEM

There are no hard and fast rules that will ensure success in solving problems. However, it is possible to outline some general steps in the problem-solving process and to give some principles that may be useful in the solution of certain problems. These steps and principles are just common sense made explicit. They have been adapted from George Polya’s book How To Solve It. The first step is to read the problem and make sure that you understand it clearly. Ask yourself the following questions: What is the unknown? What are the given quantities? What are the given conditions? For many problems it is useful to draw a diagram and identify the given and required quantities on the diagram. Usually it is necessary to introduce suitable notation In choosing symbols for the unknown quantities we often use letters such as a, b, c, m, n, x, and y, but in some cases it helps to use initials as suggestive symbols; for instance, V for volume or t for time.

2 THINK OF A PLAN

Find a connection between the given information and the unknown that will enable you to calculate the unknown. It often helps to ask yourself explicitly: “How can I relate the given to the unknown?” If you don’t see a connection immediately, the following ideas may be helpful in devising a plan. Try to Recognize Something Familiar Relate the given situation to previous knowledge. Look at the unknown and try to recall a more familiar problem that has a similar unknown. Try to Recognize Patterns Some problems are solved by recognizing that some kind of pattern is occurring. The pattern could be geometric, or numerical, or algebraic. If you can see regularity or repetition in a problem, you might be able to guess what the continuing pattern is and then prove it. Use Analogy Try to think of an analogous problem, that is, a similar problem, a related problem, but one that is easier than the original problem. If you can solve the similar, simpler problem, then it might give you the clues you need to solve the original, more difficult problem. For instance, if a problem involves very large numbers, you could first try a similar problem with smaller numbers. Or if the problem involves three-dimensional geometry, you could look for a similar problem in two-dimensional geometry. Or if the problem you start with is a general one, you could first try a special case. Introduce Something Extra It may sometimes be necessary to introduce something new, an auxiliary aid, to help make the connection between the given and the unknown. For instance, in a problem where a diagram is useful the auxiliary aid could be a new line drawn in a diagram. In a more algebraic problem it could be a new unknown that is related to the original unknown. Take Cases We may sometimes have to split a problem into several cases and give a different argument for each of the cases. For instance, we often have to use this strategy in dealing with absolute value.

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Work Backward Sometimes it is useful to imagine that your problem is solved and work backward, step by step, until you arrive at the given data. Then you may be able to reverse your steps and thereby construct a solution to the original problem. This procedure is commonly used in solving equations. For instance, in solving the equation 3x 2 5 − 7, we suppose that x is a number that satisfies 3x 2 5 − 7 and work backward. We add 5 to each side of the equation and then divide each side by 3 to get x − 4. Since each of these steps can be reversed, we have solved the problem. Establish Subgoals In a complex problem it is often useful to set subgoals (in which the desired situation is only partially fulfilled). If we can first reach these subgoals, then we may be able to build on them to reach our final goal. Indirect Reasoning Sometimes it is appropriate to attack a problem indirectly. In using proof by contradiction to prove that P implies Q, we assume that P is true and Q is false and try to see why this can’t happen. Somehow we have to use this information and arrive at a contradiction to what we absolutely know is true. Mathematical Induction In proving statements that involve a positive integer n, it is frequently helpful to use the following principle. principle of Mathematical Induction Let Sn be a statement about the positive integer n. Suppose that

1. S1 is true. 2. Sk11 is true whenever Sk is true. Then Sn is true for all positive integers n. This is reasonable because, since S1 is true, it follows from condition 2 swith k − 1d that S2 is true. Then, using condition 2 with k − 2, we see that S3 is true. Again using condition 2, this time with k − 3, we have that S4 is true. This procedure can be followed indefinitely. 3 CARRY OUT THE PLAN

In Step 2 a plan was devised. In carrying out that plan we have to check each stage of the plan and write the details that prove that each stage is correct.

4 LOOK BACK

Having completed our solution, it is wise to look back over it, partly to see if we have made errors in the solution and partly to see if we can think of an easier way to solve the problem. Another reason for looking back is that it will familiarize us with the method of solution and this may be useful for solving a future problem. Descartes said, “Every problem that I solved became a rule which served afterwards to solve other problems.” These principles of problem solving are illustrated in the following examples. Before you look at the solutions, try to solve these problems yourself, referring to these Principles of Problem Solving if you get stuck. You may find it useful to refer to this section from time to time as you solve the exercises in the remaining chapters of this book. ExamplE 1 Express the hypotenuse h of a right triangle with area 25 m2 as a function of its perimeter P.

PS Understand the problem

SOLUTION Let’s first sort out the information by identifying the unknown quantity and

the data: Unknown: hypotenuse h Given quantities: perimeter P, area 25 m 2 72 Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

PS Draw a diagram

It helps to draw a diagram and we do so in Figure 1. h b

FIGURE 1 a PS Connect the given with the unknown PS Introduce something extra

In order to connect the given quantities to the unknown, we introduce two extra variables a and b, which are the lengths of the other two sides of the triangle. This enables us to express the given condition, which is that the triangle is right-angled, by the Pythagorean Theorem: h2 − a2 1 b2 The other connections among the variables come by writing expressions for the area and perimeter: 25 − 12 ab P−a1b1h Since P is given, notice that we now have three equations in the three unknowns a, b, and h: 1

h2 − a2 1 b2

2

25 − 12 ab

3

PS Relate to the familiar

P−a1b1h

Although we have the correct number of equations, they are not easy to solve in a straightforward fashion. But if we use the problem-solving strategy of trying to recognize something familiar, then we can solve these equations by an easier method. Look at the right sides of Equations 1, 2, and 3. Do these expressions remind you of anything familiar? Notice that they contain the ingredients of a familiar formula: sa 1 bd2 − a 2 1 2ab 1 b 2 Using this idea, we express sa 1 bd2 in two ways. From Equations 1 and 2 we have sa 1 bd2 − sa 2 1 b 2 d 1 2ab − h 2 1 4s25d From Equation 3 we have sa 1 bd2 − sP 2 hd2 − P 2 2 2Ph 1 h 2

Thus

h 2 1 100 − P 2 2 2Ph 1 h 2 2Ph − P 2 2 100 h−

P 2 2 100 2P

This is the required expression for h as a function of P.

As the next example illustrates, it is often necessary to use the problem-solving principle of taking cases when dealing with absolute values. ExamplE 2 Solve the inequality | x 2 3 | 1 | x 1 2 | , 11. SOLUTION Recall the definition of absolute value:

|x| −

H

x if x > 0 2x if x , 0 73

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It follows that

|x 2 3| − −

H H

x23 2sx 2 3d

if x 2 3 > 0 if x 2 3 , 0

x23 2x 1 3

if x > 3 if x , 3

x12 2sx 1 2d

if x 1 2 > 0 if x 1 2 , 0

x12 2x 2 2

if x > 22 if x , 22

Similarly

|x 1 2| − −

PS Take cases

H H

These expressions show that we must consider three cases: x , 22

22 < x , 3

x>3

CASE I If x , 22, we have

| x 2 3 | 1 | x 1 2 | , 11 2x 1 3 2 x 2 2 , 11 22x , 10 x . 25 CASE II If 22 < x , 3, the given inequality becomes 2x 1 3 1 x 1 2 , 11 5 , 11

(always true)

CASE III If x > 3, the inequality becomes x 2 3 1 x 1 2 , 11 2x , 12 x,6 Combining cases I, II, and III, we see that the inequality is satisfied when 25 , x , 6. So the solution is the interval s25, 6d. ■ In the following example we first guess the answer by looking at special cases and recognizing a pattern. Then we prove our conjecture by mathematical induction. In using the Principle of Mathematical Induction, we follow three steps: Step 1 Prove that Sn is true when n − 1. Step 2 Assume that Sn is true when n − k and deduce that Sn is true when n − k 1 1. Step 3 Conclude that Sn is true for all n by the Principle of Mathematical Induction. 74 Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

ExamplE 3 If f0sxd − xysx 1 1d and fn11 − f0 8 fn for n − 0, 1, 2, . . . , find a formula for fnsxd. PS Analogy: Try a similar, simpler problem

SOLUTION We start by finding formulas for fnsxd for the special cases n − 1, 2, and 3.

S D

x f1sxd − s f0 8 f0dsxd − f0( f0sxd) − f0 x11

x x x11 x11 x − − − x 2x 1 1 2x 1 1 11 x11 x11

S

x f2sxd − s f0 8 f1 dsxd − f0( f1sxd) − f0 2x 1 1

D

x x 2x 1 1 2x 1 1 x − − − x 3x 1 1 3x 1 1 11 2x 1 1 2x 1 1

S

x f3sxd − s f0 8 f2 dsxd − f0( f2sxd) − f0 3x 1 1

D

x x 3x 1 1 3x 1 1 x − − − x 4x 1 1 4x 1 1 11 3x 1 1 3x 1 1

PS Look for a pattern

We notice a pattern: The coefficient of x in the denominator of fnsxd is n 1 1 in the three cases we have computed. So we make the guess that, in general, 4

fnsxd −

x sn 1 1dx 1 1

To prove this, we use the Principle of Mathematical Induction. We have already verified that (4) is true for n − 1. Assume that it is true for n − k, that is, fksxd −

Then

x sk 1 1dx 1 1

S

x fk11sxd − s f0 8 fk dsxd − f0( fksxd) − f0 sk 1 1dx 1 1

D

x x sk 1 1d x 1 1 sk 1 1dx 1 1 x − − − x sk 1 2d x 1 1 sk 1 2d x 1 1 11 sk 1 1dx 1 1 sk 1 1dx 1 1 This expression shows that (4) is true for n − k 1 1. Therefore, by mathematical induction, it is true for all positive integers n.

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1. One of the legs of a right triangle has length 4 cm. Express the length of the altitude perpendicular to the hypotenuse as a function of the length of the hypotenuse.

problems

2. The altitude perpendicular to the hypotenuse of a right triangle is 12 cm. Express the length of the hypotenuse as a function of the perimeter.

|

| | | Solve the inequality | x 2 1 | 2 | x 2 3 | > 5.

3. Solve the equation 2x 2 1 2 x 1 5 − 3. 4.

|

| | | Sketch the graph of the function tsxd − | x 2 1 | 2 | x 2 4 |. Draw the graph of the equation x 1 | x | − y 1 | y |.

5. Sketch the graph of the function f sxd − x 2 2 4 x 1 3 . 6. 7.

2

2

8. Sketch the region in the plane consisting of all points sx, yd such that

|x 2 y| 1 |x| 2 |y| < 2 9. The notation maxha, b, . . .j means the largest of the numbers a, b, . . . . Sketch the graph of each function. (a) f sxd − maxhx, 1yxj (b) f sxd − maxhsin x, cos xj (c) f sxd − maxhx 2, 2 1 x, 2 2 xj 10. Sketch the region in the plane defined by each of the following equations or inequalities. (a) maxhx, 2yj − 1 (b) 21 < maxhx, 2yj < 1 (c) maxhx, y 2 j − 1 11. Evaluate slog 2 3dslog 3 4dslog 4 5d ∙ ∙ ∙ slog 31 32d. 12. (a) Show that the function f sxd − ln ( x 1 sx 2 1 1 ) is an odd function. (b) Find the inverse function of f. 13. Solve the inequality lnsx 2 2 2x 2 2d < 0. 14. Use indirect reasoning to prove that log 2 5 is an irrational number. 15. A driver sets out on a journey. For the first half of the distance she drives at the leisurely pace of 60 kmyh; she drives the second half at 120 kmyh. What is her average speed on this trip? 16. Is it true that f 8 s t 1 hd − f 8 t 1 f 8 h? 17. Prove that if n is a positive integer, then 7 n 2 1 is divisible by 6. 18. Prove that 1 1 3 1 5 1 ∙ ∙ ∙ 1 s2n 2 1d − n 2. 19. If f0sxd − x 2 and fn11sxd − f0s fnsxdd for n − 0, 1, 2, . . . , find a formula for fnsxd. 1 and fn11 − f0 8 fn for n − 0, 1, 2, . . . , find an expression for fnsxd and 22x use mathematical induction to prove it. (b) Graph f0 , f1, f2 , f3 on the same screen and describe the effects of repeated composition.

20. (a) If f0sxd − ;

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2

Limits and Derivatives

The maximum sustainable swimming speed S of salmon depends on the water temperature T. Exercise 58 in Section 2.7 asks you to analyze how S varies as T changes by estimating the derivative of S with respect to T. © Jody Ann / Shutterstock.com

In A PREVIEW OF CALCULUS (page 1) we saw how the idea of a limit underlies the various branches of calculus. It is therefore appropriate to begin our study of calculus by investigating limits and their properties. The special type of limit that is used to find tangents and velocities gives rise to the central idea in differential calculus, the derivative.

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78

Chapter 2

Limits and Derivatives

In this section we see how limits arise when we attempt to find the tangent to a curve or the velocity of an object.

t

the tangent problem

(a) P C

t

l

The word tangent is derived from the Latin word tangens, which means “touching.” Thus a tangent to a curve is a line that touches the curve. In other words, a tangent line should have the same direction as the curve at the point of contact. How can this idea be made precise? For a circle we could simply follow Euclid and say that a tangent is a line that intersects the circle once and only once, as in Figure 1(a). For more complicated curves this definition is inadequate. Figure l(b) shows two lines l and t passing through a point P on a curve C. The line l intersects C only once, but it certainly does not look like what we think of as a tangent. The line t, on the other hand, looks like a tangent but it intersects C twice. To be specific, let’s look at the problem of trying to find a tangent line t to the parabola y − x 2 in the following example.

(b)

ExamplE 1 Find an equation of the tangent line to the parabola y − x 2 at the

FIGURE 1

point Ps1, 1d. SOLUtION We will be able to find an equation of the tangent line t as soon as we know its slope m. The difficulty is that we know only one point, P, on t, whereas we need two points to compute the slope. But observe that we can compute an approximation to m by choosing a nearby point Qsx, x 2 d on the parabola (as in Figure 2) and computing the slope mPQ of the secant line PQ. [A secant line, from the Latin word secans, meaning cutting, is a line that cuts (intersects) a curve more than once.] We choose x ± 1 so that Q ± P. Then

y

Q {x, ≈} y=≈

t

P (1, 1) x

0

mPQ − FIGURE 2

x2 2 1 x21

For instance, for the point Qs1.5, 2.25d we have x

mPQ

2 1.5 1.1 1.01 1.001

3 2.5 2.1 2.01 2.001

x

mPQ

0 0.5 0.9 0.99 0.999

1 1.5 1.9 1.99 1.999

mPQ −

2.25 2 1 1.25 − − 2.5 1.5 2 1 0.5

The tables in the margin show the values of mPQ for several values of x close to 1. The closer Q is to P, the closer x is to 1 and, it appears from the tables, the closer mPQ is to 2. This suggests that the slope of the tangent line t should be m − 2. We say that the slope of the tangent line is the limit of the slopes of the secant lines, and we express this symbolically by writing lim mPQ − m

Q lP

and

lim

xl1

x2 2 1 −2 x21

Assuming that the slope of the tangent line is indeed 2, we use the point-slope form of the equation of a line [y 2 y1 − msx 2 x 1d, see Appendix B] to write the equation of the tangent line through s1, 1d as y 2 1 − 2sx 2 1d

or

y − 2x 2 1

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79

SeCtION 2.1 The Tangent and Velocity Problems

Figure 3 illustrates the limiting process that occurs in this example. As Q approaches P along the parabola, the corresponding secant lines rotate about P and approach the tangent line t. y

y

y

Q t

t

t Q

Q P

P

0

P

0

x

0

x

x

Q approaches P from the right y

y

y

t

Q

t

P

t

P

P

Q 0

0

x

FIGURE 3 TEC In Visual 2.1 you can see how the process in Figure 3 works for additional functions.

t

Q

0.00 0.02 0.04 0.06 0.08 0.10

100.00 81.87 67.03 54.88 44.93 36.76

Q 0

x

Q approaches P from the left

Many functions that occur in science are not described by explicit equations; they are defined by experimental data. The next example shows how to estimate the slope of the tangent line to the graph of such a function.

ExamplE 2 The flash unit on a camera operates by storing charge on a capacitor and releasing it suddenly when the flash is set off. The data in the table describe the charge Q remaining on the capacitor (measured in microcoulombs) at time t (measured in seconds after the flash goes off ). Use the data to draw the graph of this function and estimate the slope of the tangent line at the point where t − 0.04. [Note: The slope of the tangent line represents the electric current flowing from the capacitor to the flash bulb (measured in microamperes).] SOLUtION In Figure 4 we plot the given data and use them to sketch a curve that approximates the graph of the function. Q (microcoulombs) 100 90 80 70 60 50

FIGURE 4

x

0

0.02

0.04

0.06

0.08

0.1

t (seconds)

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80

Chapter 2

Limits and Derivatives

Given the points Ps0.04, 67.03d and Rs0.00, 100.00d on the graph, we find that the slope of the secant line PR is mPR −

R

mPR

(0.00, 100.00) (0.02, 81.87) (0.06, 54.88) (0.08, 44.93) (0.10, 36.76)

2824.25 2742.00 2607.50 2552.50 2504.50

100.00 2 67.03 − 2824.25 0.00 2 0.04

The table at the left shows the results of similar calculations for the slopes of other secant lines. From this table we would expect the slope of the tangent line at t − 0.04 to lie somewhere between 2742 and 2607.5. In fact, the average of the slopes of the two closest secant lines is 1 2 s2742

2 607.5d − 2674.75

So, by this method, we estimate the slope of the tangent line to be about 2675. Another method is to draw an approximation to the tangent line at P and measure the sides of the triangle ABC, as in Figure 5. Q (microcoulombs) 100 90 80

A P

70 60 50

FIGURE 5

0

B

C

0.02

0.04

0.06

0.08

0.1

t (seconds)

This gives an estimate of the slope of the tangent line as The physical meaning of the answer in Example 2 is that the electric current flowing from the capacitor to the flash bulb after 0.04 seconds is about 2670 microamperes.

2

| AB | < 2 80.4 2 53.6 − 2670 0.06 2 0.02 | BC |

the Velocity problem If you watch the speedometer of a car as you travel in city traffic, you see that the speed doesn’t stay the same for very long; that is, the velocity of the car is not constant. We assume from watching the speedometer that the car has a definite velocity at each moment, but how is the “instantaneous” velocity defined? Let’s investigate the example of a falling ball.

ExamplE 3 Suppose that a ball is dropped from the upper observation deck of the CN Tower in Toronto, 450 m above the ground. Find the velocity of the ball after 5 seconds. SOLUtION Through experiments carried out four centuries ago, Galileo discovered that the distance fallen by any freely falling body is proportional to the square of the time it has been falling. (This model for free fall neglects air resistance.) If the distance fallen

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SeCtION 2.1 The Tangent and Velocity Problems

81

after t seconds is denoted by sstd and measured in meters, then Galileo’s law is expressed by the equation sstd − 4.9t 2 The difficulty in finding the velocity after 5 seconds is that we are dealing with a single instant of time st − 5d, so no time interval is involved. However, we can approximate the desired quantity by computing the average velocity over the brief time interval of a tenth of a second from t − 5 to t − 5.1: change in position time elapsed

Steve Allen / Stockbyte / Getty Images

average velocity −

ss5.1d 2 ss5d 0.1

4.9s5.1d2 2 4.9s5d2 − 49.49 mys 0.1

The following table shows the results of similar calculations of the average velocity over successively smaller time periods.

The CN Tower in Toronto was the tallest freestanding building in the world for 32 years.

Average velocity smysd

Time interval 5 1

23–28 Use a table of values to estimate the value of the limit. If you have a graphing device, use it to confirm your result graphically. ln x 2 ln 4 1 1 p9 23. lim 24. lim xl4 p l 21 1 1 p 15 x24

1 1 sin x if x , 0 12. f sxd − cos x if 0 < x <  sin x if x .

25. lim

H H

l0

sin 3 tan 2

26. lim

tl0

27. lim1 x x ; 13–14 Use the graph of the function f to state the value of each limit, if it exists. If it does not exist, explain why. (a) lim2 f sxd (b) lim1 f sxd (c) lim f sxd xl0

13. f sxd −

xl0

1 1 1 e 1yx

xl0

14. f sxd −

x2 1 x sx 3 1 x 2

15–18 Sketch the graph of an example of a function f that satisfies all of the given conditions. 15. lim2 f sxd − 2, xl1

16. lim2 f sxd − 1, xl0

lim1 f sxd − 1,

xl2

lim f sxd − 22,

x l 11

lim1 f sxd − 21,

xl0

f s2d − 1,

f s1d − 2 lim2 f sxd − 0,

xl2

f s0d is undefined

5t 2 1 t

28. lim1 x 2 ln x

x l0

x l0

2 ; 29. (a) By graphing the function f sxd − scos 2x 2 cos xdyx and zooming in toward the point where the graph crosses the y-axis, estimate the value of lim x l 0 f sxd. (b) Check your answer in part (a) by evaluating f sxd for values of x that approach 0.

; 30. (a) Estimate the value of lim

xl0

sin x sin x

by graphing the function f sxd − ssin xdyssin xd. State your answer correct to two decimal places. (b) Check your answer in part (a) by evaluating f sxd for values of x that approach 0.

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94

Chapter 2

Limits and Derivatives

(b) Evaluate f sxd for x − 0.04, 0.02, 0.01, 0.005, 0.003, and 0.001. Guess again.

31–43 Determine the infinite limit. x11 x11 31. lim1 32. lim2 x l5 x 2 5 x l5 x 2 5 33. lim

x l1

22x sx 2 1d2

34. lim2 x l3

lim

xl0

1 sec x x

38. lim2 cot x x l

39. lim 2 x csc x

40. lim2

x l2

xl2

41. lim1

x 2 2 2x 2 8 x 2 2 5x 1 6

42. lim1

S

x l2

xl0

x 2 2 2x x 2 2 4x 1 4 ;

D

1 2 ln x x

43. lim sln x 2 2 x22 d xl0

44. (a) Find the vertical asymptotes of the function y− ;

x l1

;

x2 1 1 3x 2 2x 2

1 1 and lim1 3 x l1 x 2 1 x3 2 1

(a) by evaluating f sxd − 1ysx 3 2 1d for values of x that approach 1 from the left and from the right, (b) by reasoning as in Example 9, and (c) from a graph of f.

; 46. (a) By graphing the function f sxd − stan 4xdyx and zooming in toward the point where the graph crosses the y-axis, estimate the value of lim x l 0 f sxd. (b) Check your answer in part (a) by evaluating f sxd for values of x that approach 0. 47. (a) Estimate the value of the limit lim x l 0 s1 1 xd1yx to five decimal places. Does this number look familiar? (b) Illustrate part (a) by graphing the function ; y − s1 1 xd1yx.

|

|

x ; 48. (a) Graph the function f sxd − e 1 ln x 2 4 for 0 < x < 5. Do you think the graph is an accurate representation of f ? (b) How would you get a graph that represents f better?

49. (a) Evaluate the function f sxd − x 2 2 s2 xy1000d for x − 1, 0.8, 0.6, 0.4, 0.2, 0.1, and 0.05, and guess the value of lim

xl0

S

x2 2

2x 1000

tan x 2 x . x3

(c) Evaluate hsxd for successively smaller values of x until you finally reach a value of 0 for hsxd. Are you still confident that your guess in part (b) is correct? Explain why you eventually obtained 0 values. (In Section 4.4 a method for evaluating this limit will be explained.) (d) Graph the function h in the viewing rectangle f21, 1g by f0, 1g. Then zoom in toward the point where the graph crosses the y-axis to estimate the limit of hsxd as x approaches 0. Continue to zoom in until you observe distortions in the graph of h. Compare with the results of part (c).

; 51. Graph the function f sxd − sinsyxd of Example 4 in the viewing rectangle f21, 1g by f21, 1g. Then zoom in toward the origin several times. Comment on the behavior of this function. 52. Consider the function f sxd − tan

(b) Confirm your answer to part (a) by graphing the function. 45. Determine lim2

(b) Guess the value of lim

36. lim1 lnssin xd

x l5

xlsy2d1

sx sx 2 3d 5

xl0

35. lim1 lnsx 2 2 25d 37.

50. (a) Evaluate hsxd − stan x 2 xdyx 3 for x − 1, 0.5, 0.1, 0.05, 0.01, and 0.005.

D

1 . x

(a) Show that f sxd − 0 for x −

1 1 1 , , ,...  2 3

(b) Show that f sxd − 1 for x −

4 4 4 , , ,...  5 9

(c) What can you conclude about lim1 tan xl0

1 ? x

; 53. Use a graph to estimate the equations of all the vertical asymptotes of the curve y − tans2 sin xd

2 < x <

Then find the exact equations of these asymptotes. 54. In the theory of relativity, the mass of a particle with velocity v is m0 m− s1 2 v 2yc 2 where m 0 is the mass of the particle at rest and c is the speed of light. What happens as v l c2? ; 55. (a) Use numerical and graphical evidence to guess the value of the limit lim xl1

x3 2 1 sx 2 1

(b) How close to 1 does x have to be to ensure that the function in part (a) is within a distance 0.5 of its limit?

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SeCtION 2.3 Calculating Limits Using the Limit Laws

95

In Section 2.2 we used calculators and graphs to guess the values of limits, but we saw that such methods don’t always lead to the correct answer. In this section we use the following properties of limits, called the Limit Laws, to calculate limits.

Limit Laws Suppose that c is a constant and the limits lim f sxd

xla

and

lim tsxd

xla

exist. Then 1. lim f f sxd 1 tsxdg − lim f sxd 1 lim tsxd xla

xla

xla

2. lim f f sxd 2 tsxdg − lim f sxd 2 lim tsxd xla

xla

xla

3. lim fcf sxdg − c lim f sxd xla

xla

4. lim f f sxd tsxdg − lim f sxd ? lim tsxd xla

xla

5. lim

lim f sxd f sxd xla − tsxd lim tsxd

xla

xla

if lim tsxd ± 0 xla

xla

Sum Law Difference Law Constant Multiple Law Product Law Quotient Law

These five laws can be stated verbally as follows: 1. The limit of a sum is the sum of the limits. 2. The limit of a difference is the difference of the limits. 3. The limit of a constant times a function is the constant times the limit of the function. 4. The limit of a product is the product of the limits. 5. The limit of a quotient is the quotient of the limits (provided that the limit of the denominator is not 0). It is easy to believe that these properties are true. For instance, if f sxd is close to L and tsxd is close to M, it is reasonable to conclude that f sxd 1 tsxd is close to L 1 M. This gives us an intuitive basis for believing that Law 1 is true. In Section 2.4 we give a precise definition of a limit and use it to prove this law. The proofs of the remaining laws are given in Appendix F.

y

f 1

0

g

FIGURE 1

1

x

ExamplE 1 Use the Limit Laws and the graphs of f and t in Figure 1 to evaluate the following limits, if they exist. f sxd (a) lim f f sxd 1 5tsxdg (b) lim f f sxd tsxdg (c) lim x l 22 xl1 x l 2 tsxd SOLUtION

(a) From the graphs of f and t we see that lim f sxd − 1

x l 22

and

lim tsxd − 21

x l 22

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96

Chapter 2

Limits and Derivatives

Therefore we have lim f f sxd 1 5tsxdg − lim f sxd 1 lim f5tsxdg

(by Limit Law 1)

− lim f sxd 1 5 lim tsxd

(by Limit Law 3)

x l 22

x l 22

x l 22

x l 22

x l 22

− 1 1 5s21d − 24 (b) We see that lim x l 1 f sxd − 2. But lim x l 1 tsxd does not exist because the left and right limits are different: lim tsxd − 22

lim tsxd − 21

x l 12

x l 11

So we can’t use Law 4 for the desired limit. But we can use Law 4 for the one-sided limits: lim f f sxdtsxdg − lim2 f sxd ? lim2 tsxd − 2 ? s22d − 24

x l 12

x l1

x l1

lim f f sxdtsxdg − lim1 f sxd ? lim1 tsxd − 2 ? s21d − 22

x l 11

x l1

x l1

The left and right limits aren’t equal, so lim x l 1 f f sxdtsxdg does not exist. (c) The graphs show that lim f sxd < 1.4

xl2

lim tsxd − 0

and

xl2

Because the limit of the denominator is 0, we can’t use Law 5. The given limit does not exist because the denominator approaches 0 while the numerator approaches a nonzero ■ number. If we use the Product Law repeatedly with tsxd − f sxd, we obtain the following law. Power Law

6. lim f f sxdg n − f lim f sxdg n x la

x la

where n is a positive integer

In applying these six limit laws, we need to use two special limits: 7.

lim c − c

xla

8.

lim x − a

xla

These limits are obvious from an intuitive point of view (state them in words or draw graphs of y − c and y − x), but proofs based on the precise definition are requested in the exercises for Section 2.4. If we now put f sxd − x in Law 6 and use Law 8, we get another useful special limit. 9.

lim x n − a n

xla

where n is a positive integer

A similar limit holds for roots as follows. (For square roots the proof is outlined in Exercise 2.4.37.) 10.

n n x −s a lim s

xla

where n is a positive integer

(If n is even, we assume that a . 0.)

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SeCtION 2.3 Calculating Limits Using the Limit Laws

97

More generally, we have the following law, which is proved in Section 2.5 as a consequence of Law 10. 11.

Root Law

n n lim f sxd f sxd − s lim s x la

where n is a positive integer

x la

f sxd . 0.g f If n is even, we assume that xlim la Newton and Limits Isaac Newton was born on Christmas Day in 1642, the year of Galileo’s death. When he entered Cambridge University in 1661 Newton didn’t know much mathematics, but he learned quickly by reading Euclid and Descartes and by attending the lectures of Isaac Barrow. Cambridge was closed because of the plague in 1665 and 1666, and Newton returned home to reflect on what he had learned. Those two years were amazingly productive for at that time he made four of his major discoveries: (1) his representation of functions as sums of infinite series, including the binomial theorem; (2) his work on differential and integral calculus; (3) his laws of motion and law of universal gravitation; and (4) his prism experiments on the nature of light and color. Because of a fear of controversy and criticism, he was reluctant to publish his discoveries and it wasn’t until 1687, at the urging of the astronomer Halley, that Newton published Principia Mathematica. In this work, the greatest scientific treatise ever written, Newton set forth his version of calculus and used it to investigate mechanics, fluid dynamics, and wave motion, and to explain the motion of planets and comets. The beginnings of calculus are found in the calculations of areas and volumes by ancient Greek scholars such as Eudoxus and Archimedes. Although aspects of the idea of a limit are implicit in their “method of exhaustion,” Eudoxus and Archimedes never explicitly formulated the concept of a limit. Likewise, mathematicians such as Cavalieri, Fermat, and Barrow, the immediate precursors of Newton in the development of calculus, did not actually use limits. It was Isaac Newton who was the first to talk explicitly about limits. He explained that the main idea behind limits is that quantities “approach nearer than by any given difference.” Newton stated that the limit was the basic concept in calculus, but it was left to later mathematicians like Cauchy to clarify his ideas about limits.

ExamplE 2 Evaluate the following limits and justify each step. (a) lim s2x 2 2 3x 1 4d

(b) lim

x l5

x l 22

x 3 1 2x 2 2 1 5 2 3x

SOLUtION

(a)

lim s2x 2 2 3x 1 4d − lim s2x 2 d 2 lim s3xd 1 lim 4

x l5

x l5

x l5

x l5

(by Laws 2 and 1)

− 2 lim x 2 2 3 lim x 1 lim 4

(by 3)

− 2s5 2 d 2 3s5d 1 4

(by 9, 8, and 7)

x l5

x l5

x l5

− 39 (b) We start by using Law 5, but its use is fully justified only at the final stage when we see that the limits of the numerator and denominator exist and the limit of the denominator is not 0. x 3 1 2x 2 2 1 lim − x l 22 5 2 3x

lim sx 3 1 2x 2 2 1d

x l 22

(by Law 5)

lim s5 2 3xd

x l 22

lim x 3 1 2 lim x 2 2 lim 1

x l 22

x l 22

x l 22

lim 5 2 3 lim x

x l 22

x l 22

s22d3 1 2s22d2 2 1 5 2 3s22d

−2

(by 1, 2, and 3)

(by 9, 8, and 7)

1 11

NOTE If we let f sxd − 2x 2 2 3x 1 4, then f s5d − 39. In other words, we would have gotten the correct answer in Example 2(a) by substituting 5 for x. Similarly, direct substitution provides the correct answer in part (b). The functions in Example 2 are a polynomial and a rational function, respectively, and similar use of the Limit Laws proves that direct substitution always works for such functions (see Exercises 57 and 58). We state this fact as follows. Direct Substitution Property If f is a polynomial or a rational function and a is in the domain of f , then lim f sxd − f sad x la

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98

Chapter 2

Limits and Derivatives

Functions with the Direct Substitution Property are called continuous at a and will be studied in Section 2.5. However, not all limits can be evaluated by direct substitution, as the following examples show.

ExamplE 3 Find lim xl1

x2 2 1 . x21

SOLUtION Let f sxd − sx 2 2 1dysx 2 1d. We can’t find the limit by substituting x − 1

because f s1d isn’t defined. Nor can we apply the Quotient Law, because the limit of the denominator is 0. Instead, we need to do some preliminary algebra. We factor the numerator as a difference of squares: x2 2 1 sx 2 1dsx 1 1d − x21 x21 Notice that in Example 3 we do not have an infinite limit even though the denominator approaches 0 as x l 1. When both numerator and denominator approach 0, the limit may be infinite or it may be some finite value.

The numerator and denominator have a common factor of x 2 1. When we take the limit as x approaches 1, we have x ± 1 and so x 2 1 ± 0. Therefore we can cancel the common factor and then compute the limit by direct substitution as follows: lim

xl1

x2 2 1 sx 2 1dsx 1 1d − lim xl1 x21 x21 − lim sx 1 1d xl1

−111−2 The limit in this example arose in Example 2.1.1 when we were trying to find the tangent to the parabola y − x 2 at the point s1, 1d.

NOTE In Example 3 we were able to compute the limit by replacing the given function f sxd − sx 2 2 1dysx 2 1d by a simpler function, tsxd − x 1 1, with the same limit. This is valid because f sxd − tsxd except when x − 1, and in computing a limit as x approaches 1 we don’t consider what happens when x is actually equal to 1. In general, we have the following useful fact.

y

y=ƒ

3

If f sxd − tsxd when x ± a, then lim f sxd − lim tsxd, provided the limits exist.

2

xla

x la

1 0

1

2

3

x

ExamplE 4 Find lim tsxd where x l1

y

3 2 1 0

1

2

3

x

tsxd −

H

x 1 1 if x ± 1  if x − 1

SOLUtION Here t is defined at x − 1 and ts1d − , but the value of a limit as x approaches 1 does not depend on the value of the function at 1. Since tsxd − x 1 1 for x ± 1, we have

lim tsxd − lim sx 1 1d − 2

xl1

xl1

FIGURE 2 The graphs of the functions f (from Example 3) and t (from Example 4)

Note that the values of the functions in Examples 3 and 4 are identical except when x − 1 (see Figure 2) and so they have the same limit as x approaches 1.

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SeCtION 2.3 Calculating Limits Using the Limit Laws

ExamplE 5 Evaluate lim

hl0

99

s3 1 hd2 2 9 . h

SOLUtION If we define

Fshd −

s3 1 hd2 2 9 h

then, as in Example 3, we can’t compute lim h l 0 Fshd by letting h − 0 since Fs0d is undefined. But if we simplify Fshd algebraically, we find that Fshd −

s9 1 6h 1 h 2 d 2 9 6h 1 h 2 hs6 1 hd − − −61h h h h

(Recall that we consider only h ± 0 when letting h approach 0.) Thus lim

hl0

ExamplE 6 Find lim

tl0

s3 1 hd2 2 9 − lim s6 1 hd − 6 hl0 h

st 2 1 9 2 3 . t2

SOLUtION We can’t apply the Quotient Law immediately, since the limit of the denominator is 0. Here the preliminary algebra consists of rationalizing the numerator:

lim

tl0

st 2 1 9 2 3 st 2 1 9 2 3 st 2 1 9 1 3 − lim  2 t tl0 t2 st 2 1 9 1 3 − lim

st 2 1 9d 2 9 t (st 2 1 9 1 3)

− lim

t2 t 2 (st 2 1 9 1 3)

− lim

1 st 1 9 1 3

tl0

tl0

tl0

2

2

1 slim st 1 9d 1 3 2

t l0

Here we use several properties of limits (5, 1, 10, 7, 9).

1 1 − 313 6

This calculation confirms the guess that we made in Example 2.2.2.

Some limits are best calculated by first finding the left- and right-hand limits. The following theorem is a reminder of what we discovered in Section 2.2. It says that a twosided limit exists if and only if both of the one-sided limits exist and are equal. 1 Theorem lim f sxd − L xla

if and only if

lim f sxd − L − lim1 f sxd

x l a2

x la

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100

Chapter 2

Limits and Derivatives

When computing one-sided limits, we use the fact that the Limit Laws also hold for one-sided limits.

ExamplE 7 Show that lim | x | − 0. xl0

SOLUtION Recall that The result of Example 7 looks plausible from Figure 3. y

|x| −

H

x if x > 0 2x if x , 0

Since | x | − x for x . 0, we have lim | x | − lim1 x − 0

y=|x|

x l 01

x l0

For x , 0 we have | x | − 2x and so lim | x | − lim2 s2xd − 0

x l 02

0

x

x l0

Therefore, by Theorem 1, lim | x | − 0

FIGURE 3

xl0

ExamplE 8 Prove that lim

xl0

| x | does not exist. x

| |

| |

SOLUtION Using the facts that x − x when x . 0 and x − 2x when x , 0, we

have y

lim

|x| −

lim

|x| −

x l 01

|x|

y= x

1 0

x l 02

x

x

x

lim

x − lim1 1 − 1 x l0 x

lim

2x − lim2 s21d − 21 x l0 x

x l 01

x l 02

_1

Since the right- and left-hand limits are different, it follows from Theorem 1 that lim x l 0 | x |yx does not exist. The graph of the function f sxd − | x |yx is shown in Figure 4 and supports the one-sided limits that we found. ■

FIGURE 4

ExamplE 9 If f sxd −

H

sx 2 4 8 2 2x

if x . 4 if x , 4

determine whether lim x l 4 f sxd exists. SOLUtION Since f sxd − sx 2 4 for x . 4, we have

It is shown in Example 2.4.3 that lim x l 01 sx − 0.

lim f sxd − lim1 s x 2 4 − s4 2 4 − 0

x l 41

x l4

Since f sxd − 8 2 2x for x , 4, we have

y

lim f sxd − lim2 s8 2 2xd − 8 2 2 ? 4 − 0

x l 42

x l4

The right- and left-hand limits are equal. Thus the limit exists and 0

4

x

lim f sxd − 0

xl4

FIGURE 5

The graph of f is shown in Figure 5.

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SeCtION 2.3 Calculating Limits Using the Limit Laws

Other notations for v x b are fxg and :x;. The greatest integer function is sometimes called the floor function.

ExamplE 10 The greatest integer function is defined by v x b − the largest integer that is less than or equal to x. (For instance, v4 b − 4, v 4.8b − 4, v  b − 3, vs2 b − 1,

v212 b − 21.) Show that lim x l3 v x b does not exist.

y

SOLUtION The graph of the greatest integer function is shown in Figure 6. Since

4

v x b − 3 for 3 < x , 4, we have

3

lim v x b − lim1 3 − 3

x l 31

y=[ x ]

2

x l3

Since v x b − 2 for 2 < x , 3, we have

1 0

101

1

2

3

4

5

lim v x b − lim2 2 − 2

x

x l 32

x l3

Because these one-sided limits are not equal, lim xl3 v x b does not exist by Theorem 1. ■ The next two theorems give two additional properties of limits. Their proofs can be found in Appendix F.

FIGURE 6 Greatest integer function

2 Theorem If f sxd < tsxd when x is near a (except possibly at a) and the limits of f and t both exist as x approaches a, then lim f sxd < lim tsxd

xla

xla

3 The Squeeze Theorem If f sxd < tsxd < hsxd when x is near a (except possibly at a) and lim f sxd − lim hsxd − L

xla

y

h g

xla

lim tsxd − L

then

xla

L

f 0

FIGURE 7

a

x

The Squeeze Theorem, which is sometimes called the Sandwich Theorem or the Pinching Theorem, is illustrated by Figure 7. It says that if tsxd is squeezed between f sxd and hsxd near a, and if f and h have the same limit L at a, then t is forced to have the same limit L at a. 1 − 0. xl0 x SOLUtION First note that we cannot use

ExamplE 11 Show that lim x 2 sin

lim x 2 sin

xl0

1 1 − lim x 2 ? lim sin xl0 xl0 x x

because lim x l 0 sins1yxd does not exist (see Example 2.2.4). Instead we apply the Squeeze Theorem, and so we need to find a function f smaller than tsxd − x 2 sins1yxd and a function h bigger than t such that both f sxd and hsxd approach 0. To do this we use our knowledge of the sine function. Because the sine of Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

102

Chapter 2

Limits and Derivatives

any number lies between 21 and 1, we can write. 4

21 < sin

1 0 for all x and so, multiplying each side of the inequalities in (4) by x 2, we get y

2x 2 < x 2 sin

y=≈

1 < x2 x

as illustrated by Figure 8. We know that lim x 2 − 0

x

0

and

xl0

lim s2x 2 d − 0

xl0

y − x 2 sins1yxd

Taking f sxd − 2x 2, tsxd − x 2 sins1yxd, and hsxd − x 2 in the Squeeze Theorem, we obtain 1 lim x 2 sin − 0 xl0 x

1. Given that

4. lim sx 2 1 xds3x 2 1 6d

y=_≈

FIGURE 8

lim f sxd − 4

xl2

xl 21

lim tsxd − 22

lim hsxd − 0

xl 2

xl2

find the limits that exist. If the limit does not exist, explain why. (a) lim f f sxd 1 5tsxdg (b) lim f tsxdg 3 xl2

xl2

(c) lim sf sxd

(d) lim

xl2

(e) lim

x l2

xl2

tsxd hsxd

(f ) lim

xl2

3f sxd tsxd tsxdhsxd f sxd

2. The graphs of f and t are given. Use them to evaluate each limit, if it exists. If the limit does not exist, explain why. (a) lim f f sxd 1 tsxdg (b) lim f f sxd 2 tsxdg x l2

5. lim

t l 22

t4 2 2 2t 2 3t 1 2

(d) lim

x l 21

x l3

(e) lim fx f sxdg x l2

9. lim

xl2

Î

0

2x 2 1 1 3x 2 2

lim

x l2

x2 1 x 2 6 − lim sx 1 3d x l2 x22

is correct.

x l2

1

x

x2 1 x 2 6 x22

12. lim

x l 23

2

13. lim

x l2

x 2x16 x22

14. lim

xl4

3–9 Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). x l 22

15. lim

t l 23

17. lim

hl0

t 29 2t 1 7t 1 3 2

s25 1 hd2 2 25 h

16. lim

x 2 1 3x x 2 x 2 12

x l 21

18. lim

hl0

x 2 1 3x x 2 x 2 12 2

2

2

3. lim s3x 4 1 2x 2 2 x 1 1d

2

11–32 Evaluate the limit, if it exists.

1 x

D

x2 1 x 2 6 −x13 x22

11. lim 1

t2 2 2 t 2 3t 1 5 3

10. (a) What is wrong with the following equation?

y=ƒ

0

tl2

S

(b) In view of part (a), explain why the equation

f sxd tsxd

y

1

8. lim

xl8

x l 21

y

ul 22

3 7. lim s1 1 s x ds2 2 6x 2 1 x 3 d

(f ) f s21d 1 lim tsxd

2

6. lim su 4 1 3u 1 6

2

xl0

(c) lim f f sxd tsxdg

2x 2 1 3x 1 1 x 2 2 2x 2 3

s2 1 hd3 2 8 h

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103

SeCtION 2.3 Calculating Limits Using the Limit Laws

19. lim

x l3

21. lim

hl0

x23 x 3 2 27

20. lim

s9 1 h 2 3 h

22. lim

tl1

t4 2 1 t3 2 1

ul 2

40. Prove that lim1 sx e sinsyxd − 0. x l0

41–46 Find the limit, if it exists. If the limit does not exist, explain why. 2x 1 12 41. lim s2x 1 x 2 3 d 42. lim xl3 x l 26 x 1 6

s4u 1 1 2 3 u22

|

1 1 2 x 3 23. lim x l3 x 2 3

21

24. lim

hl0

s3 1 hd 2 3 h

43. lim 2 x l 0.5

25. lim

tl0

s1 1 t 2 s1 2 t t

26. lim

t l 21

x 2 1 2x 1 1 x4 2 1

45. lim2 x l0

27. lim

x l 16

29. lim

tl0

31. lim

hl0

4 2 sx 16x 2 x 2

S

2x 2 1 3 2 x2

| 2x

S

x 2 4x 1 4 x 4 2 3x 2 2 4

30. lim

sx 1 9 2 5 x14

x l2

1 1 2 t s1 1 t t

D

xl24

1 1 2 2 sx 1 hd2 x 32. lim hl0 h

; 33. (a) Estimate the value of x l0

S

x l0

H

sgn x −

D

21 if x , 0 20 if x − 0 21 if x . 0

|

(iv) lim sgn x xl0

|

48. Let tsxd − sgnssin xd . (a) Find each of the following limits or explain why it does not exist. (i) lim1 tsxd (ii) lim2 tsxd (iii) lim tsxd x l0

x l0

(iv) lim1 tsxd

(v)

x l

x l0

lim2 tsxd

x l

(vi) lim tsxd x l

(b) For which values of a does lim x l a tsxd not exist? (c) Sketch a graph of t.

; 35. Use the Squeeze Theorem to show that lim x l 0 sx 2 cos 20xd − 0. Illustrate by graphing the functions f sxd − 2x 2, tsxd − x 2 cos 20x, and hsxd − x 2 on the same screen.

49. Let tsxd −

x2 1 x 2 6 . x22

|

|

(a) Find (i) lim1 tsxd

Illustrate by graphing the functions f, t, and h (in the notation of the Squeeze Theorem) on the same screen. 37. If 4x 2 9 < f sxd < x 2 2 4x 1 7 for x > 0, find lim f sxd. xl4

38. If 2x < tsxd < x 2 x 1 2 for all x, evaluate lim tsxd. 2

xl1

(ii) lim2 tsxd

x l2

x l2

(b) Does lim x l 2 tsxd exist? (c) Sketch the graph of t. 50. Let f sxd −

; 36. Use the Squeeze Theorem to show that  −0 lim sx 3 1 x 2 sin x l0 x

2 39. Prove that lim x cos − 0. x l0 x

| |

x l0

xl0

s3 1 x 2 s3 x

1 1 2 x x

(a) Sketch the graph of this function. (b) Find each of the following limits or explain why it does not exist. (i) lim1 sgn x (ii) lim2 sgn x (iii) lim sgn x

to estimate the value of lim x l 0 f sxd to two decimal places. (b) Use a table of values of f sxd to estimate the limit to four decimal places. (c) Use the Limit Laws to find the exact value of the limit.

4

46. lim1

47. The signum (or sign) function, denoted by sgn, is defined by

; 34. (a) Use a graph of f sxd −

| |

22 x 21x

x l 22

| |D

|

44. lim

x l0

x s1 1 3x 2 1

by graphing the function f sxd − xyss1 1 3x 2 1d. (b) Make a table of values of f sxd for x close to 0 and guess the value of the limit. (c) Use the Limit Laws to prove that your guess is correct.

4

|

2

sx 1 hd3 2 x 3 h

lim

|

1 1 2 x x

2

28. lim

|

21

H

x2 1 1 if x , 1 sx 2 2d 2 if x > 1

(a) Find lim x l12 f sxd and lim x l11 f sxd. (b) Does lim x l1 f sxd exist? (c) Sketch the graph of f. 51. Let Bstd −

H

4 2 12 t

if t , 2

st 1 c

if t > 2

Find the value of c so that lim Bstd exists. tl2

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

104

Chapter 2

Limits and Derivatives

f sxd 2 8 − 10, find lim f sxd. xl1 x21 f sxd 60. If lim 2 − 5, find the following limits. xl0 x f sxd (a) lim f sxd (b) lim xl0 xl0 x 61. If x 2 if x is rational f sxd − 0 if x is irrational 59. If lim

52. Let tsxd −

x 3 2 2 x2 x23

if if if if

xl1

x,1 x−1 1,x 21 for all x and so 2 1 cos x . 0 everywhere. Thus the ratio

ExamplE 7 Evaluate lim

x l

f sxd −

sin x 2 1 cos x

is continuous everywhere. Hence, by the definition of a continuous function, lim

x l

sin x sin  0 − lim f sxd − f sd − − −0 x l 2 1 cos x 2 1 cos  221

Another way of combining continuous functions f and t to get a new continuous function is to form the composite function f 8 t. This fact is a consequence of the following theorem. This theorem says that a limit symbol can be moved through a function symbol if the function is continuous and the limit exists. In other words, the order of these two symbols can be reversed.

8 Theorem If f is continuous at b and lim tsxd − b, then lim f s tsxdd − f sbd. x la x la In other words,

S

D

lim f stsxdd − f lim tsxd

xla

xl a

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SeCtion 2.5 Continuity

121

Intuitively, Theorem 8 is reasonable because if x is close to a, then tsxd is close to b, and since f is continuous at b, if tsxd is close to b, then fs tsxdd is close to f sbd. A proof of Theorem 8 is given in Appendix F.

ExamplE 8 Evaluate lim arcsin x l1

S

D

1 2 sx . 12x

SoLUtion Because arcsin is a continuous function, we can apply Theorem 8:

lim arcsin

x l1

S

1 2 sx 12x

D

S S S

D

− arcsin lim

1 2 sx 12x

− arcsin lim

1 2 sx (1 2 sx ) (1 1 sx )

− arcsin lim

1 1 1 sx

− arcsin

x l1

x l1

x l1

D

D

1  − 2 6

n Let’s now apply Theorem 8 in the special case where f sxd − s x , with n being a positive integer. Then n tsxd f s tsxdd − s

and

S

D

f lim tsxd − xla

t s xd s xlim la n

If we put these expressions into Theorem 8, we get n n lim tsxd tsxd − s lim s

xla

xla

and so Limit Law 11 has now been proved. (We assume that the roots exist.) 9 Theorem If t is continuous at a and f is continuous at tsad, then the composite function f 8 t given by s f 8 tds xd − f s ts xdd is continuous at a. This theorem is often expressed informally by saying “a continuous function of a continuous function is a continuous function.” proof Since t is continuous at a, we have

xla

Since f is continuous at b − tsad, we can apply Theorem 8 to obtain lim f stsxdd − f stsadd

xla

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122

Chapter 2

Limits and Derivatives

which is precisely the statement that the function hsxd − f s tsxdd is continuous at a; that is, f 8 t is continuous at a.

ExamplE 9 Where are the following functions continuous? (a) hsxd − sinsx 2 d (b) Fsxd − lns1 1 cos xd SoLUtion

(a) We have hsxd − f s tsxdd, where 2

tsxd − x 2

_10

10

_6

FIGURE 7 y − lns1 1 cos xd

and

f sxd − sin x

Now t is continuous on R since it is a polynomial, and f is also continuous everywhere. Thus h − f 8 t is continuous on R by Theorem 9. (b) We know from Theorem 7 that f sxd − ln x is continuous and tsxd − 1 1 cos x is continuous (because both y − 1 and y − cos x are continuous). Therefore, by Theorem 9, Fsxd − f s tsxdd is continuous wherever it is defined. Now ln s1 1 cos xd is defined when 1 1 cos x . 0. So it is undefined when cos x − 21, and this happens when x − 6, 63, . . . . Thus F has discontinuities when x is an odd multiple of  and is continuous on the intervals between these values (see Figure 7). ■ An important property of continuous functions is expressed by the following theorem, whose proof is found in more advanced books on calculus. 10 The Intermediate Value Theorem Suppose that f is continuous on the closed interval fa, bg and let N be any number between f sad and f sbd, where f sad ± f sbd. Then there exists a number c in sa, bd such that f scd − N. The Intermediate Value Theorem states that a continuous function takes on every intermediate value between the function values f sad and f sbd. It is illustrated by Figure 8. Note that the value N can be taken on once [as in part (a)] or more than once [as in part (b)]. y

y

f(b)

f(b)

y=ƒ

N N

y=ƒ

f(a) 0

c b

(a)

y f(a)

y=N

f(b) a

FIGURE 9

x

0

a c¡

c™

b

x

(b)

FIGURE 8

y=ƒ

N

0

a

f(a)

b

x

If we think of a continuous function as a function whose graph has no hole or break, then it is easy to believe that the Intermediate Value Theorem is true. In geometric terms it says that if any horizontal line y − N is given between y − f sad and y − f sbd as in Figure 9, then the graph of f can’t jump over the line. It must intersect y − N somewhere. It is important that the function f in Theorem 10 be continuous. The Intermediate Value Theorem is not true in general for discontinuous functions (see Exercise 50).

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

123

SeCtion 2.5 Continuity

One use of the Intermediate Value Theorem is in locating roots of equations as in the following example.

ExamplE 10 Show that there is a root of the equation 4x 3 2 6x 2 1 3x 2 2 − 0 between 1 and 2. SoLUtion Let f sxd − 4x 3 2 6x 2 1 3x 2 2. We are looking for a solution of the given

equation, that is, a number c between 1 and 2 such that f scd − 0. Therefore we take a − 1, b − 2, and N − 0 in Theorem 10. We have f s1d − 4 2 6 1 3 2 2 − 21 , 0 f s2d − 32 2 24 1 6 2 2 − 12 . 0

and

Thus f s1d , 0 , f s2d; that is, N − 0 is a number between f s1d and f s2d. Now f is continuous since it is a polynomial, so the Intermediate Value Theorem says there is a number c between 1 and 2 such that f scd − 0. In other words, the equation 4x 3 2 6x 2 1 3x 2 2 − 0 has at least one root c in the interval s1, 2d. In fact, we can locate a root more precisely by using the Intermediate Value Theorem again. Since f s1.2d − 20.128 , 0

f s1.3d − 0.548 . 0

and

a root must lie between 1.2 and 1.3. A calculator gives, by trial and error, f s1.22d − 20.007008 , 0

f s1.23d − 0.056068 . 0

and

so a root lies in the interval s1.22, 1.23d.

We can use a graphing calculator or computer to illustrate the use of the Intermediate Value Theorem in Example 10. Figure 10 shows the graph of f in the viewing rectangle f21, 3g by f23, 3g and you can see that the graph crosses the x-axis between 1 and 2. Figure 11 shows the result of zooming in to the viewing rectangle f1.2, 1.3g by f20.2, 0.2g. 3

0.2

3

_1

_3

FIGURE 10

1.2

1.3

_0.2

FIGURE 11

In fact, the Intermediate Value Theorem plays a role in the very way these graphing devices work. A computer calculates a finite number of points on the graph and turns on the pixels that contain these calculated points. It assumes that the function is continuous and takes on all the intermediate values between two consecutive points. The computer therefore “connects the dots” by turning on the intermediate pixels. Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

124

Chapter 2

Limits and Derivatives

1. Write an equation that expresses the fact that a function f is continuous at the number 4. 2. If f is continuous on s2`, `d, what can you say about its graph? 3. (a) From the graph of f , state the numbers at which f is discontinuous and explain why. (b) For each of the numbers stated in part (a), determine whether f is continuous from the right, or from the left, or neither.

10. Explain why each function is continuous or discontinuous. (a) The temperature at a specific location as a function of time (b) The temperature at a specific time as a function of the distance due west from Paris (c) The altitude above sea level as a function of the distance due west from Paris (d) The cost of a taxi ride as a function of the distance traveled (e) The current in the circuit for the lights in a room as a function of time 11–14 Use the definition of continuity and the properties of limits to show that the function is continuous at the given number a.

y

11. f sxd − x 2 1 s7 2 x , 12. tstd −

_4

_2

0

2

4

x

6

a−4

2

t 1 5t , a−2 2t 1 1

13. psvd − 2s3v 2 1 1 , a − 1 3 14. f sxd − 3x 4 2 5x 1 s x2 1 4 ,

4. From the graph of t, state the intervals on which t is continuous. y

_3

_2

0

15 –16 Use the definition of continuity and the properties of limits to show that the function is continuous on the given interval. 15. f sxd − x 1 sx 2 4 , f4, `d 16. tsxd −

1

2

3

x

5 – 8 Sketch the graph of a function f that is continuous except for the stated discontinuity. 5. Discontinuous, but continuous from the right, at 2 6. Discontinuities at 21 and 4, but continuous from the left at 21 and from the right at 4 7. Removable discontinuity at 3, jump discontinuity at 5 8. Neither left nor right continuous at 22, continuous only from the left at 2 9. The toll T charged for driving on a certain stretch of a toll road is \$5 except during rush hours (between 7 am and 10 am and between 4 pm and 7 pm) when the toll is \$7. (a) Sketch a graph of T as a function of the time t, measured in hours past midnight. (b) Discuss the discontinuities of this function and their significance to someone who uses the road.

a−2

x21 , s2`, 22d 3x 1 6

17– 22 Explain why the function is discontinuous at the given number a. Sketch the graph of the function. 1 17. f sxd − a − 22 x12 18. f sxd −

19. f sxd −

H

1 x12 1

H

x 1 3 if x < 21 2x if x . 21

H H H

x2 2 x 20. f sxd − x 2 2 1 1

if x ± 22

a − 22

if x − 22

if x ± 1

a − 21

a−1

if x − 1

cos x if x , 0 21. f sxd − 0 if x − 0 1 2 x 2 if x . 0

a−0

2x 2 2 5x 2 3 22. f sxd − x23 6

a−3

if x ± 3 if x − 3

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

SeCtion 2.5 Continuity

H

23 –24 How would you “remove the discontinuity” of f ? In other words, how would you define f s2d in order to make f continuous at 2? x2 2 x 2 2 x22

23. f sxd −

1 1 x2 if x < 0 43. f sxd − 2 2 x if 0 , x < 2 sx 2 2d2 if x . 2

x 2 2 3x 1 2 x2 1 x 2 6

24. f sxd −

44. The gravitational force exerted by the planet Earth on a unit mass at a distance r from the center of the planet is

25 – 32 Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain. 2

2x 2 x 2 1 25. Fsxd − x2 1 1 sx 2 2 x3 2 2

2

x 11 26. Gsxd − 2x 2 2 x 2 1

27. Qsxd −

29. Astd − arcsins1 1 2td 31. Msxd −

Î

11

Fsrd −

sin t

3

1 x

28. Rstd −

e 2 1 cos t

30. Bsxd −

tan x s4 2 x 2

; 33 –34 Locate the discontinuities of the function and illustrate by graphing. 34. y − lnstan xd

35 – 38 Use continuity to evaluate the limit.

x l1

5 2 x2 11x

40. f sxd −

H H

1 2 x2 ln x

H

cx 2 1 2x if x , 2 x 3 2 cx if x > 2

46. Find the values of a and b that make f continuous everywhere.

f sxd −

x2 2 4 if x , 2 x22 ax 2 2 bx 1 3 if 2 < x , 3 2x 2 a 1 b if x > 3

47. Suppose f and t are continuous functions such that ts2d − 6 and lim x l2 f3 f sxd 1 f sxd tsxdg − 36. Find f s2d.

38. lim 3 sx

48. Let f sxd − 1yx and tsxd − 1yx 2. (a) Find s f + tds xd. (b) Is f + t continuous everywhere? Explain.

2 22x24

xl4

39 – 40 Show that f is continuous on s2`, `d. 39. f sxd −

if r > R

36. lim sinsx 1 sin xd x l

S D

37. lim ln

f sxd −

2

x l2

if r , R

45. For what value of the constant c is the function f continuous on s2`, `d? 2

35. lim x s20 2 x 2

GMr R3 GM r2

where M is the mass of Earth, R is its radius, and G is the gravitational constant. Is F a continuous function of r?

32. Nsrd − tan21 s1 1 e2r d

1 33. y − 1 1 e 1yx

125

if x < 1 if x . 1

sin x if x , y4 cos x if x > y4

49. Which of the following functions f has a removable discontinuity at a? If the discontinuity is removable, find a function t that agrees with f for x ± a and is continuous at a. (a) f sxd −

x4 2 1 , a−1 x21

(b) f sxd −

x 3 2 x 2 2 2x , x22

a−2

(c) f sxd − v sin x b , a −  41– 43 Find the numbers at which f is discontinuous. At which of these numbers is f continuous from the right, from the left, or neither? Sketch the graph of f .

H H

x 2 if x , 21 if 21 < x , 1 41. f sxd − x 1yx if x > 1 2x if x < 1 42. f sxd − 3 2 x if 1 , x < 4 if x . 4 sx

50. Suppose that a function f is continuous on [0, 1] except at 0.25 and that f s0d − 1 and f s1d − 3. Let N − 2. Sketch two possible graphs of f , one showing that f might not satisfy the conclusion of the Intermediate Value Theorem and one showing that f might still satisfy the conclusion of the Intermediate Value Theorem (even though it doesn’t satisfy the hypothesis). 51. If f sxd − x 2 1 10 sin x, show that there is a number c such that f scd − 1000. 52. Suppose f is continuous on f1, 5g and the only solutions of the equation f sxd − 6 are x − 1 and x − 4. If f s2d − 8, explain why f s3d . 6.

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126

Chapter 2

Limits and Derivatives

53– 56 Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval.

65. Prove that cosine is a continuous function.

53. x 4 1 x 2 3 − 0, s1, 2d

66. (a) Prove Theorem 4, part 3. (b) Prove Theorem 4, part 5.

54. ln x − x 2 sx ,

67. For what values of x is f continuous?

s2, 3d

55. sx − 1 2 x, s0, 1d 3

f sxd −

56. sin x − x 2 2 x, s1, 2d

H

0 if x is rational 1 if x is irrational

68. For what values of x is t continuous? 57 – 58 (a) Prove that the equation has at least one real root. (b) Use your calculator to find an interval of length 0.01 that contains a root. 57. cos x − x 3

58. ln x − 3 2 2x

H

tsxd −

0 if x is rational x if x is irrational

69. Is there a number that is exactly 1 more than its cube? 70. If a and b are positive numbers, prove that the equation

; 59– 60 (a) Prove that the equation has at least one real root. (b) Use your graphing device to find the root correct to three decimal places. 59. 100e2xy100 − 0.01x 2

a b 1 3 −0 x 3 1 2x 2 2 1 x 1x22 has at least one solution in the interval s21, 1d. 71. Show that the function

60. arctan x − 1 2 x f sxd − 61– 62 Prove, without graphing, that the graph of the function has at least two x-intercepts in the specified interval.

H

x 4 sins1yxd 0

if x ± 0 if x − 0

is continuous on s2`, `d.

| |

61. y − sin x 3, s1, 2d 62. y − x 2 2 3 1 1yx, s0, 2d

72. (a) Show that the absolute value function Fsxd − x is continuous everywhere. (b) Prove that if f is a continuous function on an interval, then so is f . (c) Is the converse of the statement in part (b) also true? In other words, if f is continuous, does it follow that f is continuous? If so, prove it. If not, find a counterexample.

| |

63. Prove that f is continuous at a if and only if lim f sa 1 hd − f sad

| |

hl0

64. To prove that sine is continuous, we need to show that lim x l a sin x − sin a for every real number a. By Exercise 63 an equivalent statement is that lim sinsa 1 hd − sin a

hl0

Use (6) to show that this is true.

73. A Tibetan monk leaves the monastery at 7:00 am and takes his usual path to the top of the mountain, arriving at 7:00 pm. The following morning, he starts at 7:00 am at the top and takes the same path back, arriving at the monastery at 7:00 pm. Use the Intermediate Value Theorem to show that there is a point on the path that the monk will cross at exactly the same time of day on both days.

In Sections 2.2 and 2.4 we investigated infinite limits and vertical asymptotes. There we let x approach a number and the result was that the values of y became arbitrarily large (positive or negative). In this section we let x become arbitrarily large (positive or negative) and see what happens to y. Let’s begin by investigating the behavior of the function f defined by f sxd −

x2 2 1 x2 1 1

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

SeCtion 2.6

as x becomes large. The table at the left gives values of this function correct to six decimal places, and the graph of f has been drawn by a computer in Figure 1.

f sxd

x 0 61 62 63 64 65 610 650 6100 61000

127

Limits at Infinity; Horizontal Asymptotes

21 0 0.600000 0.800000 0.882353 0.923077 0.980198 0.999200 0.999800 0.999998

y

y=1

0

1

y=

≈-1 ≈+1

x

FIGURE 1

As x grows larger and larger you can see that the values of f sxd get closer and closer to 1. (The graph of f approaches the horizontal line y − 1 as we look to the right.) In fact, it seems that we can make the values of f sxd as close as we like to 1 by taking x sufficiently large. This situation is expressed symbolically by writing lim

xl`

x2 2 1 −1 x2 1 1

In general, we use the notation lim f sxd − L

xl`

to indicate that the values of f sxd approach L as x becomes larger and larger. y

1 Intuitive Definition of a Limit at Infinity Let f be a function defined on some interval sa, `d. Then lim f sxd − L

y=L

y=ƒ

0

xl`

means that the values of f sxd can be made arbitrarily close to L by requiring x to be sufficiently large.

x

Another notation for lim x l ` f sxd − L is

y

f sxd l L

y=ƒ

as

xl`

The symbol ` does not represent a number. Nonetheless, the expression lim f sxd − L x l` is often read as “the limit of f sxd, as x approaches infinity, is L”

y=L 0

x

y

y=L

y=ƒ 0

x

FIGURE 2 Examples illustrating lim f sxd − L xl`

or

“the limit of f sxd, as x becomes infinite, is L”

or

“the limit of f sxd, as x increases without bound, is L”

The meaning of such phrases is given by Definition 1. A more precise definition, similar to the «,  definition of Section 2.4, is given at the end of this section. Geometric illustrations of Definition 1 are shown in Figure 2. Notice that there are many ways for the graph of f to approach the line y − L (which is called a horizontal asymptote) as we look to the far right of each graph. Referring back to Figure 1, we see that for numerically large negative values of x, the values of f sxd are close to 1. By letting x decrease through negative values without bound, we can make f sxd as close to 1 as we like. This is expressed by writing lim

x l2`

x2 2 1 −1 x2 1 1

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

128

Chapter 2

Limits and Derivatives

The general definition is as follows. y

y=ƒ

2 Definition Let f be a function defined on some interval s2`, ad. Then lim f sxd − L

x l 2`

y=L 0

x y

means that the values of f sxd can be made arbitrarily close to L by requiring x to be sufficiently large negative.

Again, the symbol 2` does not represent a number, but the expression lim f sxd − L x l 2` is often read as

y=ƒ

“the limit of f sxd, as x approaches negative infinity, is L”

y=L

0

x

FIGURE 3 Examples illustrating lim f sxd − L x l 2`

Definition 2 is illustrated in Figure 3. Notice that the graph approaches the line y − L as we look to the far left of each graph. 3 Definition The line y − L is called a horizontal asymptote of the curve y − f sxd if either lim f sxd − L or lim f sxd − L x l`

x l 2`

For instance, the curve illustrated in Figure 1 has the line y − 1 as a horizontal asymptote because x2 2 1 lim 2 −1 xl` x 1 1

y π 2

An example of a curve with two horizontal asymptotes is y − tan21x. (See Figure 4.) In fact,

0 x

4

lim tan21 x − 2

x l2`

2

lim tan21 x −

xl`

2

_ π2

so both of the lines y − 2y2 and y − y2 are horizontal asymptotes. (This follows from the fact that the lines x − 6y2 are vertical asymptotes of the graph of the tangent function.)

FIGURE 4 y − tan21x

EXAMPLE 1 Find the infinite limits, limits at infinity, and asymptotes for the function

y

f whose graph is shown in Figure 5. SoLUtion We see that the values of f sxd become large as x l 21 from both sides, so 2

lim f sxd − `

x l21

0

2

x

Notice that f sxd becomes large negative as x approaches 2 from the left, but large positive as x approaches 2 from the right. So lim f sxd − 2`

x l 22

FIGURE 5

and

lim f sxd − `

x l 21

Thus both of the lines x − 21 and x − 2 are vertical asymptotes.

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

SeCtion 2.6

Limits at Infinity; Horizontal Asymptotes

129

As x becomes large, it appears that f sxd approaches 4. But as x decreases through negative values, f sxd approaches 2. So lim f sxd − 4

and

xl`

lim f sxd − 2

x l2`

This means that both y − 4 and y − 2 are horizontal asymptotes.

EXAMPLE 2 Find lim

xl`

1 1 and lim . x l2` x x

SoLUtion Observe that when x is large, 1yx is small. For instance,

1 − 0.01 100

y

y=∆

0

FIGURE 6 lim

xl`

1 1 − 0, lim −0 x l2` x x

1 − 0.0001 10,000

1 − 0.000001 1,000,000

In fact, by taking x large enough, we can make 1yx as close to 0 as we please. Therefore, according to Definition 1, we have lim

x

xl`

1 −0 x

Similar reasoning shows that when x is large negative, 1yx is small negative, so we also have 1 lim −0 x l2` x It follows that the line y − 0 (the x-axis) is a horizontal asymptote of the curve y − 1yx. (This is an equilateral hyperbola; see Figure 6.)

Most of the Limit Laws that were given in Section 2.3 also hold for limits at infinity. It can be proved that the Limit Laws listed in Section 2.3 (with the exception of Laws 9 and 10) are also valid if “x l a” is replaced by “x l `” or “x l 2`.” In particular, if we combine Laws 6 and 11 with the results of Example 2, we obtain the following important rule for calculating limits. 5 Theorem If r . 0 is a rational number, then lim

xl`

1 −0 xr

If r . 0 is a rational number such that x r is defined for all x, then lim

x l2`

1 −0 xr

EXAMPLE 3 Evaluate lim

x l`

3x 2 2 x 2 2 5x 2 1 4x 1 1

and indicate which properties of limits are used at each stage. SoLUtion As x becomes large, both numerator and denominator become large, so it isn’t obvious what happens to their ratio. We need to do some preliminary algebra. Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

130

Chapter 2

Limits and Derivatives

To evaluate the limit at infinity of any rational function, we first divide both the numerator and denominator by the highest power of x that occurs in the denominator. (We may assume that x ± 0, since we are interested only in large values of x.) In this case the highest power of x in the denominator is x 2, so we have 3x 2 2 x 2 2 1 2 32 2 2 3x 2 x 2 2 x2 x x lim − lim − lim x l ` 5x 2 1 4x 1 1 x l` x l ` 5x 2 1 4x 1 1 4 1 51 1 2 2 x x x 2

S S

D D

1 2 2 2 x x − 4 1 lim 5 1 1 2 xl` x x lim 3 2

xl`

(by Limit Law 5)

1 2 2 lim x l` x l` x x l` − 1 lim 5 1 4 lim 1 lim x l` x l` x x l` lim 3 2 lim

y

y=0.6 0

FIGURE 7 y−

3x 2 2 x 2 2 5x 2 1 4x 1 1

1

1 x2 1 x2

(by 1, 2, and 3)

x

32020 51010

3 5

(by 7 and Theorem 5)

A similar calculation shows that the limit as x l 2` is also 35. Figure 7 illustrates the results of these calculations by showing how the graph of the given rational function approaches the horizontal asymptote y − 35 − 0.6. ■

EXAMPLE 4 Find the horizontal and vertical asymptotes of the graph of the function f sxd −

s2x 2 1 1 3x 2 5

SoLUtion Dividing both numerator and denominator by x and using the properties of limits, we have

Î

s2x 2 1 1 x s2x 2 1 1 lim − lim − lim x l ` 3x 2 5 xl` xl` 3x 2 5 x lim

xl`

− lim

xl`

2x 2 1 1 x2 3x 2 5 x

Î Î S D 21

32

5 x

1 x2

lim 2 1 lim

xl`

xl`

lim 3 2 5 lim

xl`

xl`

(since sx 2 − x for x . 0)

1 x2 1 x

s2 1 0 s2 − 325?0 3

Therefore the line y − s2 y3 is a horizontal asymptote of the graph of f. Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

SeCtion 2.6

Limits at Infinity; Horizontal Asymptotes

131

In computing the limit as x l 2`, we must remember that for x , 0, we have sx 2 − | x | − 2x. So when we divide the numerator by x, for x , 0 we get

Î

s2x 2 1 1 s2x 2 1 1 − −2 x 2 sx 2

Î

2x 2 1 1 −2 x2

21

1 x2

Therefore

s2x 1 1 lim − lim x l2` x l2` 3x 2 5 2

21

1 x2

5 32 x

Î

2 −

1 s2 x2 −2 1 3 3 2 5 lim x l2` x 2 1 lim

x l2`

Thus the line y − 2s2 y3 is also a horizontal asymptote. A vertical asymptote is likely to occur when the denominator, 3x 2 5, is 0, that is, when x − 53. If x is close to 53 and x . 53, then the denominator is close to 0 and 3x 2 5 is positive. The numerator s2x 2 1 1 is always positive, so f sxd is positive. Therefore

y

œ„2

y= 3

lim

x

x l s5y3d1

œ„2

y= _ 3

x=

s2x 2 1 1 −` 3x 2 5

(Notice that the numerator does not approach 0 as x l 5y3). If x is close to 53 but x , 53, then 3x 2 5 , 0 and so f sxd is large negative. Thus

5 3

lim

FIGURE 8 y−

Î

2

x l s5y3d2

s2 x 2 1 1

s2x 2 1 1 − 2` 3x 2 5

The vertical asymptote is x − 53. All three asymptotes are shown in Figure 8.

3x 2 5

EXAMPLE 5 Compute lim (sx 2 1 1 2 x). x l`

SoLUtion Because both sx 2 1 1 and x are large when x is large, it’s difficult to see We can think of the given function as having a denominator of 1.

what happens to their difference, so we use algebra to rewrite the function. We first multiply numerator and denominator by the conjugate radical: lim (sx 2 1 1 2 x) − lim (sx 2 1 1 2 x) ?

x l`

x l`

− lim

y

x l`

y=œ„„„„„-x ≈+1

FIGURE 9

1

sx 2 1 1d 2 x 2 1 − lim 2 1 1 1 x 2 x l ` sx 1 1 1 x sx

Notice that the denominator of this last expression (sx 2 1 1 1 x) becomes large as x l ` (it’s bigger than x). So

1 0

sx 2 1 1 1 x sx 2 1 1 1 x

x

lim (sx 2 1 1 2 x) − lim

x l`

x l`

1 −0 sx 1 1 1 x 2

Figure 9 illustrates this result.

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132

Chapter 2

Limits and Derivatives

S D

1 . x22

EXAMPLE 6 Evaluate lim1 arctan x l2

SoLUtion If we let t − 1ysx 2 2d, we know that t l ` as x l 2 1. Therefore, by the

second equation in (4), we have

S D 1 x22

lim arctan

x l 21

− lim arctan t − tl`

2

The graph of the natural exponential function y − e x has the line y − 0 (the x-axis) as a horizontal asymptote. (The same is true of any exponential function with base b . 1.) In fact, from the graph in Figure 10 and the corresponding table of values, we see that

lim e x − 0

6

x l 2`

Notice that the values of e x approach 0 very rapidly. y

y=´

1 0

x

1

x

ex

0 21 22 23 25 28 210

1.00000 0.36788 0.13534 0.04979 0.00674 0.00034 0.00005

FIGURE 10

EXAMPLE 7 Evaluate lim2 e 1yx. x l0

PS The problem-solving strategy for Examples 6 and 7 is introducing something extra (see page 71). Here, the something extra, the auxiliary aid, is the new variable t.

SoLUtion If we let t − 1yx, we know that t l 2` as x l 02. Therefore, by (6),

lim e 1yx − lim e t − 0

x l 02

t l 2`

(See Exercise 81.)

EXAMPLE 8 Evaluate lim sin x. xl`

SoLUtion As x increases, the values of sin x oscillate between 1 and 21 infinitely often and so they don’t approach any definite number. Thus lim x l` sin x does not exist. ■

infinite Limits at infinity The notation lim f sxd − `

x l`

is used to indicate that the values of f sxd become large as x becomes large. Similar meanCopyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

SeCtion 2.6

Limits at Infinity; Horizontal Asymptotes

133

ings are attached to the following symbols: lim f sxd − `

lim f sxd − 2`

x l 2`

x l`

lim f sxd − 2`

x l 2`

EXAMPLE 9 Find lim x 3 and lim x 3.

y

xl`

y=˛

0

x l2`

SoLUtion When x becomes large, x 3 also becomes large. For instance,

10 3 − 1000

x

100 3 − 1,000,000

1000 3 − 1,000,000,000

In fact, we can make x 3 as big as we like by requiring x to be large enough. Therefore we can write lim x 3 − ` xl`

FIGURE 11 lim x 3 − `, lim x 3 − 2`

xl`

Similarly, when x is large negative, so is x 3. Thus

x l2`

lim x 3 − 2`

x l2`

y

y=´

These limit statements can also be seen from the graph of y − x 3 in Figure 11.

Looking at Figure 10 we see that lim e x − `

x l`

y=˛

100

0

1

but, as Figure 12 demonstrates, y − e x becomes large as x l ` at a much faster rate than y − x 3.

x

EXAMPLE 10 Find lim sx 2 2 xd. x l`

FIGURE 12 e x is much larger than x 3 when x is large.

SoLUtion It would be wrong to write

lim sx 2 2 xd − lim x 2 2 lim x − ` 2 `

x l`

x l`

x l`

The Limit Laws can’t be applied to infinite limits because ` is not a number (` 2 ` can’t be defined). However, we can write lim sx 2 2 xd − lim xsx 2 1d − `

x l`

x l`

because both x and x 2 1 become arbitrarily large and so their product does too.

EXAMPLE 11 Find lim

xl`

x2 1 x . 32x

SoLUtion As in Example 3, we divide the numerator and denominator by the highest power of x in the denominator, which is just x:

lim

x l`

x2 1 x x11 − 2` − lim x l` 3 32x 21 x

because x 1 1 l ` and 3yx 2 1 l 0 2 1 − 21 as x l `. Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

134

Chapter 2

Limits and Derivatives

The next example shows that by using infinite limits at infinity, together with intercepts, we can get a rough idea of the graph of a polynomial without having to plot a large number of points.

EXAMPLE 12 Sketch the graph of y − sx 2 2d4sx 1 1d3sx 2 1d by finding its intercepts and its limits as x l ` and as x l 2`. SoLUtion The y-intercept is f s0d − s22d4s1d3s21d − 216 and the x-intercepts are

found by setting y − 0: x − 2, 21, 1. Notice that since sx 2 2d4 is never negative, the function doesn’t change sign at 2; thus the graph doesn’t cross the x-axis at 2. The graph crosses the axis at 21 and 1. When x is large positive, all three factors are large, so

y

0

_1

1

2

x

lim sx 2 2d4sx 1 1d3sx 2 1d − `

xl`

When x is large negative, the first factor is large positive and the second and third factors are both large negative, so _16

lim sx 2 2d4sx 1 1d3sx 2 1d − `

x l2`

FIGURE 13 y − sx 2 2d4 sx 1 1d3 sx 2 1d

Combining this information, we give a rough sketch of the graph in Figure 13.

precise Definitions Definition 1 can be stated precisely as follows. 7 Precise Definition of a Limit at Infinity Let f be a function defined on some interval sa, `d. Then lim f sxd − L xl`

means that for every « . 0 there is a corresponding number N such that if

x.N

then

| f sxd 2 L | , «

In words, this says that the values of f sxd can be made arbitrarily close to L (within a distance «, where « is any positive number) by requiring x to be sufficiently large (larger than N, where N depends on «). Graphically it says that by keeping x large enough (larger than some number N) we can make the graph of f lie between the given horizontal lines y − L 2 « and y − L 1 « as in Figure 14. This must be true no matter how small we choose «. y

y=L+∑ ∑ L ∑ y=L-∑ 0

FIGURE 14 lim f sxd − L

xl`

y=ƒ ƒ is in here

x

N when x is in here

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SeCtion 2.6

135

Limits at Infinity; Horizontal Asymptotes

Figure 15 shows that if a smaller value of « is chosen, then a larger value of N may be required.

y=ƒ y=L+∑ L y=L-∑

FIGURE 15

0

N

lim f sxd − L

x

xl`

Similarly, a precise version of Definition 2 is given by Definition 8, which is illustrated in Figure 16.

8 Definition Let f be a function defined on some interval s2`, ad. Then lim f sxd − L

x l 2`

means that for every « . 0 there is a corresponding number N such that if

x,N

then

| f sxd 2 L | , «

y

y=ƒ y=L+∑ L y=L-∑ 0

N

FIGURE 16

x

lim f sxd − L

x l2`

In Example 3 we calculated that lim

xl`

3x 2 2 x 2 2 3 − 5x 2 1 4x 1 1 5

In the next example we use a graphing device to relate this statement to Definition 7 with L − 35 − 0.6 and « − 0.1. TEC In Module 2.4y2.6 you can explore the precise definition of a limit both graphically and numerically.

EXAMPLE 13 Use a graph to find a number N such that if

x.N

then

Z

3x 2 2 x 2 2 2 0.6 5x 2 1 4x 1 1

Z

, 0.1

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136

Chapter 2

Limits and Derivatives

SoLUtion We rewrite the given inequality as

3x 2 2 x 2 2 , 0.7 5x 2 1 4x 1 1

0.5 ,

We need to determine the values of x for which the given curve lies between the horizontal lines y − 0.5 and y − 0.7. So we graph the curve and these lines in Figure 17. Then we use the cursor to estimate that the curve crosses the line y − 0.5 when x < 6.7. To the right of this number it seems that the curve stays between the lines y − 0.5 and y − 0.7. Rounding up to be safe, we can say that

1 y=0.7 y=0.5 y=

3≈-x-2 5≈+4x+1

15

0

if

x.7

Z

then

3x 2 2 x 2 2 2 0.6 5x 2 1 4x 1 1

Z

, 0.1

FIGURE 17

In other words, for « − 0.1 we can choose N − 7 (or any larger number) in Definition 7.

1 − 0. x

EXAMPLE 14 Use Definition 7 to prove that lim

xl`

SoLUtion Given « . 0, we want to find N such that

if

then

x.N

Z

1 20 x

Z

In computing the limit we may assume that x . 0. Then 1yx , « &? x . 1y«. Let’s choose N − 1y«. So if

x.N−

1 «

then

Z

1 20 x

Z

1 ,« x

Therefore, by Definition 7, lim

xl`

1 −0 x

Figure 18 illustrates the proof by showing some values of « and the corresponding values of N. y

y

y

∑=1 ∑=0.2 0

FIGURE 18

N=1

x

0

∑=0.1 N=5

x

0

N=10

x

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SeCtion 2.6

137

Limits at Infinity; Horizontal Asymptotes

Finally we note that an infinite limit at infinity can be defined as follows. The geometric illustration is given in Figure 19.

y

y=M M

0

9 Definition of an Infinite Limit at Infinity Let f be a function defined on some interval sa, `d. Then lim f sxd − `

x

N

xl`

means that for every positive number M there is a corresponding positive number N such that if x . N then f sxd . M

FIGURE 19 lim f sxd − `

xl`

Similar definitions apply when the symbol ` is replaced by 2`. (See Exercise 80.)

y

1. Explain in your own words the meaning of each of the following. (a) lim f sxd − 5 (b) lim f sxd − 3 xl`

x l 2`

1

2. (a) Can the graph of y − f sxd intersect a vertical asymptote? Can it intersect a horizontal asymptote? Illustrate by sketching graphs. (b) How many horizontal asymptotes can the graph of y − f sxd have? Sketch graphs to illustrate the possibilities. 3. For the function f whose graph is given, state the following. (a) lim f sxd (b) lim f sxd x l`

x l 2`

(c) lim f sxd

(d) lim f sxd

x l1

x l3

0

5–10 Sketch the graph of an example of a function f that satisfies all of the given conditions. 5. lim f sxd − 2`,

(e) The equations of the asymptotes

lim f sxd − 5,

xl0

6. lim f sxd − `,

lim

lim f sxd − 0,

lim f sxd − `,

x l2

1

x

lim f sxd − `,

lim f sxd − 2`,

x l 222

f s0d − 0

x l`

7. lim f sxd − 2`,

x l`

f sxd − `,

lim f sxd − 0,

x l 2`

1

lim f sxd − 25

x l 2`

x l 221

xl2

y

x

2

x l`

lim f sxd − 0,

x l 2`

lim f sxd − 2`

x l 01

x l 02

8. lim f sxd − 3, lim2 f sxd − `, lim1 f sxd − 2`, f is odd x l2

xl`

9. f s0d − 3, 4. For the function t whose graph is given, state the following. (a) lim tsxd (b) lim tsxd x l`

(c) lim tsxd x l3

(e)

lim tsxd

x l 221

x l 2`

(d) lim tsxd x l0

x l2

lim f sxd − 4,

x l 02

lim f sxd − 2`,

x l 2`

lim f sxd − 2,

x l 01

lim f sxd − 2`,

x l 42

lim f sxd − `,

x l 41

lim f sxd − 3

x l`

10. lim f sxd − 2`, xl3

lim f sxd − 2,

x l`

f s0d − 0,

f is even

(f ) The equations of the asymptotes

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138

CHAPTER 2

Limits and Derivatives

; 11. Guess the value of the limit

35. lim arctanse x d xl`

2

lim

x l`

x 2x

by evaluating the function f sxd − x 2y2 x for x − 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 50, and 100. Then use a graph of f to support your guess. ; 12. (a) Use a graph of

S D

2 f sxd − 1 2 x

xl`

e 3x 2 e23x e 3x 1 e23x

38. lim

sin2 x x2 1 1

xl`

1 2 ex 1 1 2e x

37. lim

36. lim

xl`

39. lim se22x cos xd xl`

40. lim1 tan21sln xd xl0

41. lim flns1 1 x 2 d 2 lns1 1 xdg xl `

x

42. lim flns2 1 xd 2 lns1 1 xdg xl `

to estimate the value of lim x l ` f sxd correct to two decimal places. (b) Use a table of values of f sxd to estimate the limit to four decimal places. 13–14 Evaluate the limit and justify each step by indicating the appropriate properties of limits. 13. lim

xl`

2x 2 2 7 5x 2 1 x 2 3

14. lim

xl`

Î

9x 3 1 8x 2 4 3 2 5x 1 x 3

xl`

1 2x 1 3

17. lim

x l 2`

16. lim

xl`

1 2 x 2 x2 2x 2 2 7

3x 1 5 x24 2 2 3y 2 5y 2 1 4y t 2 t st 2t 3y2 1 3t 2 5

st 1 t 2 2t 2 t 2

20. lim

21. lim

s9x 6 2 x x3 1 1

22. lim

s9x 6 2 x x3 1 1

23. lim

s1 1 4x 6 2 2 x3

24. lim

s1 1 4x 6 2 2 x3

25. lim

sx 1 3x 4x 2 1

xl`

xl`

xl`

tl `

x l 2`

x l 2`

2

xl`

28. lim (s4x 2 1 3x 1 2 x ) x l2`

30. lim

x l 2`

31. lim

xl`

33. lim sx 2 1 2x 7 d x l 2`

)

(c) lim2 f sxd

(d) lim1 f sxd

xl0 x l1

(e) Use the information from parts (a)–(d) to make a rough sketch of the graph of f. ; 45. (a) Estimate the value of lim (sx 2 1 x 1 1 1 x)

x l 2`

by graphing the function f sxd − sx 2 1 x 1 1 1 x. (b) Use a table of values of f sxd to guess the value of the limit. (c) Prove that your guess is correct. ; 46. (a) Use a graph of f sxd − s3x 2 1 8x 1 6 2 s3x 2 1 3x 1 1

47–52 Find the horizontal and vertical asymptotes of each curve. If you have a graphing device, check your work by graphing the curve and estimating the asymptotes.

( x 1 sx 2 1 2x )

x 4 2 3x 2 1 x x3 2 x 1 2

(b) lim1 f sxd

to estimate the value of lim x l ` f sxd to one decimal place. (b) Use a table of values of f sxd to estimate the limit to four decimal places. (c) Find the exact value of the limit.

x l`

x l`

2 1 2 find each of the following limits. x ln x

(a) lim f sxd

x 1 3x 4x 2 1

27. lim (s9x 2 1 x 2 3x)

29. lim (sx 2 1 ax 2 sx 2 1 bx

xl`

(c) Use the information from parts (a) and (b) to make a rough sketch of the graph of f.

2

26. lim

x l1

(b) Use a table of values to estimate lim f sxd.

xl1

19. lim

tl`

xl1

xl`

18. lim

yl`

xl0

44. For f sxd −

15–42 Find the limit or show that it does not exist. 15. lim

x find each of the following limits. ln x (i) lim1 f sxd (ii) lim2 f sxd (iii) lim1 f sxd

43. (a) For f sxd −

32. lim se2x 1 2 cos 3xd

47. y −

5 1 4x x13

48. y −

2x 2 1 1 3x 1 2x 2 1

49. y −

2x 2 1 x 2 1 x2 1 x 2 2

50. y −

1 1 x4 x2 2 x4

xl`

34. lim

x l 2`

1 1 x6 x4 1 1

2

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

SECTION 2.6

x3 2 x x 2 6x 1 5

51. y −

52. y −

2

Limits at Infinity; Horizontal Asymptotes

139

It is known that f has a removable discontinuity at x − 21 and lim x l21 f sxd − 2. Evaluate (a) f s0d (b) lim f sxd

2e x e 25 x

xl`

; 53. Estimate the horizontal asymptote of the function 3

f sxd −

60–64 Find the limits as x l ` and as x l 2`. Use this information, together with intercepts, to give a rough sketch of the graph as in Example 12.

2

3x 1 500x x 3 1 500x 2 1 100x 1 2000

60. y − 2x 3 2 x 4

by graphing f for 210 < x < 10. Then calculate the equation of the asymptote by evaluating the limit. How do you explain the discrepancy?

62. y − x 3sx 1 2d 2sx 2 1d 63. y − s3 2 xds1 1 xd 2s1 2 xd 4

; 54. (a) Graph the function f sxd −

64. y − x 2sx 2 2 1d 2sx 1 2d s2x 2 1 1 3x 2 5

s2x 1 1 3x 2 5 2

lim

s2x 1 1 3x 2 5 2

and

lim

x l 2`

;

Psxd − 3x 5 2 5x 3 1 2x

Psxd Qsxd

67. Find lim x l ` f sxd if, for all x . 1, 10e x 2 21 5sx , f sxd , 2e x sx 2 1

if the degree of P is (a) less than the degree of Q and (b) greater than the degree of Q. 56. Make a rough sketch of the curve y − x n (n an integer) for the following five cases: (i) n − 0 (ii) n . 0, n odd (iii) n . 0, n even (iv) n , 0, n odd (v) n , 0, n even Then use these sketches to find the following limits. (a) lim1 x n (b) lim2 x n x l0

68. (a) A tank contains 5000 L of pure water. Brine that contains 30 g of salt per liter of water is pumped into the tank at a rate of 25 Lymin. Show that the concentration of salt after t minutes (in grams per liter) is Cstd −

(b) What happens to the concentration as t l `?

(d) lim x n

x l`

x l 2`

69. In Chapter 9 we will be able to show, under certain assumptions, that the velocity vstd of a falling raindrop at time t is vstd − v *s1 2 e 2ttyv * d

57. Find a formula for a function f that satisfies the following conditions: lim f sxd − 0,

lim f sxd − `,

x l 32

30t 200 1 t

x l0

(c) lim x n

x l 6`

Qsxd − 3x 5

by graphing both functions in the viewing rectangles f22, 2g by f22, 2g and f210, 10g by f210,000, 10,000g. (b) Two functions are said to have the same end behavior if their ratio approaches 1 as x l `. Show that P and Q have the same end behavior.

55. Let P and Q be polynomials. Find lim

xl`

; 66. By the end behavior of a function we mean the behavior of its values as x l ` and as x l 2`. (a) Describe and compare the end behavior of the functions

(b) By calculating values of f sxd, give numerical estimates of the limits in part (a). (c) Calculate the exact values of the limits in part (a). Did you get the same value or different values for these two limits? [In view of your answer to part (a), you might have to check your calculation for the second limit.]

xl`

sin x . x (b) Graph f sxd − ssin xdyx. How many times does the graph cross the asymptote?

65. (a) Use the Squeeze Theorem to evaluate lim

How many horizontal and vertical asymptotes do you observe? Use the graph to estimate the values of the limits

x l`

61. y − x 4 2 x 6

lim f sxd − 2`,

x l0

f s2d − 0,

lim f sxd − 2`

x l 31

58. Find a formula for a function that has vertical asymptotes x − 1 and x − 3 and horizontal asymptote y − 1. 59. A function f is a ratio of quadratic functions and has a vertical asymptote x − 4 and just one x-intercept, x − 1.

;

where t is the acceleration due to gravity and v * is the terminal velocity of the raindrop. (a) Find lim t l ` vstd. (b) Graph vstd if v* − 1 mys and t − 9.8 mys2. How long does it take for the velocity of the raindrop to reach 99% of its terminal velocity?

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

140

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2xy10 and y − 0.1 on a common ; 70. (a) By graphing y − e screen, discover how large you need to make x so that e 2xy10 , 0.1. (b) Can you solve part (a) without using a graphing device?

76. (a) How large do we have to take x so that 1ysx , 0.0001? (b) Taking r − 12 in Theorem 5, we have the statement lim

xl`

; 71. Use a graph to find a number N such that if

x.N

Z

then

Z

3x2 1 1 2 1.5 , 0.05 2x 2 1 x 1 1

1 sx

−0

Prove this directly using Definition 7. 77. Use Definition 8 to prove that lim

; 72. For the limit

x l2`

lim

xl`

1 2 3x sx 2 1 1

− 23

1 − 0. x

78. Prove, using Definition 9, that lim x 3 − `. xl`

illustrate Definition 7 by finding values of N that correspond to « − 0.1 and « − 0.05. ; 73. For the limit

79. Use Definition 9 to prove that lim e x − `. xl`

80. Formulate a precise definition of 1 2 3x

lim

x l2`

sx 2 1 1

−3

lim f sxd − 2`

x l2`

illustrate Definition 8 by finding values of N that correspond to « − 0.1 and « − 0.05.

Then use your definition to prove that lim s1 1 x 3 d − 2`

; 74. For the limit

x l2`

lim sx ln x − `

81. (a) Prove that

xl`

lim f sxd − lim1 f s1ytd

illustrate Definition 9 by finding a value of N that corresponds to M − 100. 75. (a) How large do we have to take x so that 1yx 2 , 0.0001? (b) Taking r − 2 in Theorem 5, we have the statement lim

xl`

1 −0 x2

tl0

xl`

and

f s1ytd lim f sxd − t lim l 02

xl2`

if these limits exist. (b) Use part (a) and Exercise 65 to find lim x sin

x l 01

Prove this directly using Definition 7.

1 x

The problem of finding the tangent line to a curve and the problem of finding the velocity of an object both involve finding the same type of limit, as we saw in Section 2.1. This special type of limit is called a derivative and we will see that it can be interpreted as a rate of change in any of the natural or social sciences or engineering.

Tangents If a curve C has equation y − f sxd and we want to find the tangent line to C at the point Psa, f sadd, then we consider a nearby point Qsx, f sxdd, where x ± a, and compute the slope of the secant line PQ: mPQ −

f sxd 2 f sad x2a

Then we let Q approach P along the curve C by letting x approach a. If mPQ approaches Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

SECTION 2.7

141

Derivatives and Rates of Change

y

Q{ x, ƒ } ƒ-f(a)

P { a, f(a)} x-a

0

a number m, then we define the tangent t to be the line through P with slope m. (This amounts to saying that the tangent line is the limiting position of the secant line PQ as Q approaches P. See Figure 1.)

1 Definition The tangent line to the curve y − f sxd at the point Psa, f sadd is the line through P with slope

a

x

x

m − lim

xla

f sxd 2 f sad x2a

provided that this limit exists. y

t Q

In our first example we confirm the guess we made in Example 2.1.1.

Q Q

P

EXAMPLE 1 Find an equation of the tangent line to the parabola y − x 2 at the point Ps1, 1d. SOLUTION Here we have a − 1 and f sxd − x 2, so the slope is x

0

m − lim

x l1

f sxd 2 f s1d x2 2 1 − lim x l1 x 2 1 x21

FIGURE 1

− lim

x l1

sx 2 1dsx 1 1d x21

− lim sx 1 1d − 1 1 1 − 2 x l1

Point-slope form for a line through the point sx1 , y1 d with slope m:

Using the point-slope form of the equation of a line, we find that an equation of the tangent line at s1, 1d is

y 2 y1 − msx 2 x 1 d

y 2 1 − 2sx 2 1d

TEC Visual 2.7 shows an animation of Figure 2.

2

y − 2x 2 1

1.5

1.1

(1, 1)

2

We sometimes refer to the slope of the tangent line to a curve at a point as the slope of the curve at the point. The idea is that if we zoom in far enough toward the point, the curve looks almost like a straight line. Figure 2 illustrates this procedure for the curve y − x 2 in Example 1. The more we zoom in, the more the parabola looks like a line. In other words, the curve becomes almost indistinguishable from its tangent line.

(1, 1)

0

or

0.5

(1, 1)

1.5

0.9

FIGURE 2 Zooming in toward the point (1, 1) on the parabola y − x 2

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1.1

142

CHAPTER 2

Limits and Derivatives

Q { a+h, f(a+h)} y

t

There is another expression for the slope of a tangent line that is sometimes easier to use. If h − x 2 a, then x − a 1 h and so the slope of the secant line PQ is mPQ −

P { a, f(a)} f(a+h)-f(a)

h 0

a

f sa 1 hd 2 f sad h

a+h

x

FIGURE 3

(See Figure 3 where the case h . 0 is illustrated and Q is to the right of P. If it happened that h , 0, however, Q would be to the left of P.) Notice that as x approaches a, h approaches 0 (because h − x 2 a) and so the expression for the slope of the tangent line in Definition 1 becomes

2

m − lim

hl0

f sa 1 hd 2 f sad h

EXAMPLE 2 Find an equation of the tangent line to the hyperbola y − 3yx at the point s3, 1d. SOLUTION Let f sxd − 3yx. Then, by Equation 2, the slope of the tangent at s3, 1d is

m − lim

hl0

f s3 1 hd 2 f s3d h

3 3 2 s3 1 hd 21 31h 31h − lim − lim hl0 h l 0 h h − lim

y

x+3y-6=0

hl0

y=

2h 1 1 − lim 2 −2 h l 0 hs3 1 hd 31h 3

3 x

Therefore an equation of the tangent at the point s3, 1d is (3, 1)

y 2 1 − 213 sx 2 3d

x

0

which simplifies to FIGURE 4

The hyperbola and its tangent are shown in Figure 4.

position at time t=a

position at time t=a+h s

0

f(a+h)-f(a)

f(a) f(a+h)

FIGURE 5

x 1 3y 2 6 − 0 ■

Velocities In Section 2.1 we investigated the motion of a ball dropped from the CN Tower and defined its velocity to be the limiting value of average velocities over shorter and shorter time periods. In general, suppose an object moves along a straight line according to an equation of motion s − f std, where s is the displacement (directed distance) of the object from the origin at time t. The function f that describes the motion is called the position function of the object. In the time interval from t − a to t − a 1 h the change in position is f sa 1 hd 2 f sad. (See Figure 5.)

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

SECTION 2.7

Derivatives and Rates of Change

143

The average velocity over this time interval is

s

Q { a+h, f(a+h)}

average velocity −

P { a, f(a)}

displacement f sa 1 hd 2 f sad − time h

h

0

a

mPQ=

a+h

t

f(a+h)-f(a) h

which is the same as the slope of the secant line PQ in Figure 6. Now suppose we compute the average velocities over shorter and shorter time intervals fa, a 1 hg. In other words, we let h approach 0. As in the example of the falling ball, we define the velocity (or instantaneous velocity) vsad at time t − a to be the limit of these average velocities:

average velocity

FIGURE 6

3

hl0

f sa 1 hd 2 f sad h

This means that the velocity at time t − a is equal to the slope of the tangent line at P (compare Equations 2 and 3). Now that we know how to compute limits, let’s reconsider the problem of the falling ball.

EXAMPLE 3 Suppose that a ball is dropped from the upper observation deck of the CN Tower, 450 m above the ground. (a) What is the velocity of the ball after 5 seconds? (b) How fast is the ball traveling when it hits the ground? Recall from Section 2.1: The distance (in meters) fallen after t seconds is 4.9t 2.

SOLUTION We will need to find the velocity both when t − 5 and when the ball hits the ground, so it’s efficient to start by finding the velocity at a general time t. Using the equation of motion s − f std − 4.9t 2, we have

f st 1 hd 2 f std 4.9st 1 hd2 2 4.9t 2 − lim hl0 h h

vstd − lim

hl0

− lim

4.9st 2 1 2th 1 h 2 2 t 2 d 4.9s2th 1 h 2 d − lim hl0 h h

− lim

4.9hs2t 1 hd − lim 4.9s2t 1 hd − 9.8t hl0 h

hl0

hl0

(a) The velocity after 5 seconds is vs5d − s9.8ds5d − 49 mys. (b) Since the observation deck is 450 m above the ground, the ball will hit the ground at the time t when sstd − 450, that is, 4.9t 2 − 450 This gives t2 −

450 4.9

and

t−

Î

450 < 9.6 s 4.9

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144

CHAPTER 2

Limits and Derivatives

The velocity of the ball as it hits the ground is therefore

v

SÎ D Î 450 4.9

− 9.8

450 < 94 mys 4.9

Derivatives We have seen that the same type of limit arises in finding the slope of a tangent line (Equation 2) or the velocity of an object (Equation 3). In fact, limits of the form lim

h l0

f sa 1 hd 2 f sad h

arise whenever we calculate a rate of change in any of the sciences or engineering, such as a rate of reaction in chemistry or a marginal cost in economics. Since this type of limit occurs so widely, it is given a special name and notation.

4 Definition The derivative of a function f at a number a, denoted by f 9sad, is f sa 1 hd 2 f sad f 9sad − lim h l0 h if this limit exists. If we write x − a 1 h, then we have h − x 2 a and h approaches 0 if and only if x approaches a. Therefore an equivalent way of stating the definition of the derivative, as we saw in finding tangent lines, is

5

xla

f sxd 2 f sad x2a

EXAMPLE 4 Find the derivative of the function f sxd − x 2 2 8x 1 9 at the number a. SOLUTION From Definition 4 we have

Definitions 4 and 5 are equivalent, so we can use either one to compute the derivative. In practice, Definition 4 often leads to simpler computations.

h l0

f sa 1 hd 2 f sad h

− lim

fsa 1 hd2 2 8sa 1 hd 1 9g 2 fa 2 2 8a 1 9g h

− lim

a 2 1 2ah 1 h 2 2 8a 2 8h 1 9 2 a 2 1 8a 2 9 h

− lim

2ah 1 h 2 2 8h − lim s2a 1 h 2 8d h l0 h

h l0

h l0

h l0

− 2a 2 8

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SECTION 2.7

Derivatives and Rates of Change

145

We defined the tangent line to the curve y − f sxd at the point Psa, f sadd to be the line that passes through P and has slope m given by Equation 1 or 2. Since, by Definition 4, this is the same as the derivative f 9sad, we can now say the following.

The tangent line to y − f sxd at sa, f sadd is the line through sa, f sadd whose slope is equal to f 9sad, the derivative of f at a.

If we use the point-slope form of the equation of a line, we can write an equation of the tangent line to the curve y − f sxd at the point sa, f sadd:

y

y=≈-8x+9

EXAMPLE 5 Find an equation of the tangent line to the parabola y − x 2 2 8x 1 9 at 0

the point s3, 26d.

x

SOLUTION From Example 4 we know that the derivative of f sxd − x 2 2 8x 1 9 at

(3, _6)

the number a is f 9sad − 2a 2 8. Therefore the slope of the tangent line at s3, 26d is f 9s3d − 2s3d 2 8 − 22. Thus an equation of the tangent line, shown in Figure 7, is

y=_2 x

y 2 s26d − s22dsx 2 3d

FIGURE 7

or

y − 22x

Rates of Change

y

Q { ¤, ‡}

Suppose y is a quantity that depends on another quantity x. Thus y is a function of x and we write y − f sxd. If x changes from x 1 to x 2, then the change in x (also called the increment of x) is Dx − x 2 2 x 1 and the corresponding change in y is

P {⁄, ﬂ}

Îy

Dy − f sx 2d 2 f sx 1d Îx 0

The difference quotient ¤

Dy f sx 2d 2 f sx 1d − Dx x2 2 x1

x

average rate of change − mPQ instantaneous rate of change − slope of tangent at P

FIGURE 8

is called the average rate of change of y with respect to x over the interval fx 1, x 2g and can be interpreted as the slope of the secant line PQ in Figure 8. By analogy with velocity, we consider the average rate of change over smaller and smaller intervals by letting x 2 approach x 1 and therefore letting Dx approach 0. The limit of these average rates of change is called the (instantaneous) rate of change of y with respect to x at x − x1, which (as in the case of velocity) is interpreted as the slope of the tangent to the curve y − f sxd at Psx 1, f sx 1dd:

6

instantaneous rate of change − lim

Dx l 0

Dy f sx2 d 2 f sx1d − lim x lx Dx x2 2 x1 2

1

We recognize this limit as being the derivative f 9sx 1d. Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

146

CHAPTER 2

Limits and Derivatives

We know that one interpretation of the derivative f 9sad is as the slope of the tangent line to the curve y − f sxd when x − a. We now have a second interpretation: y

The derivative f 9sad is the instantaneous rate of change of y − f sxd with respect to x when x − a.

Q

P

x

FIGURE 9 The y-values are changing rapidly at P and slowly at Q.

The connection with the first interpretation is that if we sketch the curve y − f sxd, then the instantaneous rate of change is the slope of the tangent to this curve at the point where x − a. This means that when the derivative is large (and therefore the curve is steep, as at the point P in Figure 9), the y-values change rapidly. When the derivative is small, the curve is relatively flat (as at point Q) and the y-values change slowly. In particular, if s − f std is the position function of a particle that moves along a straight line, then f 9sad is the rate of change of the displacement s with respect to the time t. In other words, f 9sad is the velocity of the particle at time t − a. The speed of the particle is the absolute value of the velocity, that is, | f 9sad |. In the next example we discuss the meaning of the derivative of a function that is defined verbally.

EXAMPLE 6 A manufacturer produces bolts of a fabric with a fixed width. The cost of producing x meters of this fabric is C − f sxd dollars. (a) What is the meaning of the derivative f 9sxd? What are its units? (b) In practical terms, what does it mean to say that f 9s1000d − 9? (c) Which do you think is greater, f 9s50d or f 9s500d? What about f 9s5000d? SOLUTION

(a) The derivative f 9sxd is the instantaneous rate of change of C with respect to x; that is, f 9sxd means the rate of change of the production cost with respect to the number of meters produced. (Economists call this rate of change the marginal cost. This idea is discussed in more detail in Sections 3.7 and 4.7.) Because f 9sxd − lim

Dx l 0

DC Dx

the units for f 9sxd are the same as the units for the difference quotient DCyDx. Since DC is measured in dollars and Dx in meters, it follows that the units for f 9sxd are dollars per meter. (b) The statement that f 9s1000d − 9 means that, after 1000 meters of fabric have been manufactured, the rate at which the production cost is increasing is \$9ym. (When x − 1000, C is increasing 9 times as fast as x.) Since Dx − 1 is small compared with x − 1000, we could use the approximation Here we are assuming that the cost function is well behaved; in other words, Csxd doesn’t oscillate rapidly near x − 1000.

f 9s1000d
4

165

65. Nick starts jogging and runs faster and faster for 3 mintues, then he walks for 5 minutes. He stops at an intersection for 2 minutes, runs fairly quickly for 5 minutes, then walks for 4 minutes. (a) Sketch a possible graph of the distance s Nick has covered after t minutes. (b) Sketch a graph of dsydt. 66. When you turn on a hot-water faucet, the temperature T of the water depends on how long the water has been running. (a) Sketch a possible graph of T as a function of the time t that has elapsed since the faucet was turned on. (b) Describe how the rate of change of T with respect to t varies as t increases. (c) Sketch a graph of the derivative of T. 67. Let  be the tangent line to the parabola y − x 2 at the point s1, 1d. The angle of inclination of  is the angle  that  makes with the positive direction of the x-axis. Calculate  correct to the nearest degree.

(b) Sketch the graph of f. (c) Where is f discontinuous? (d) Where is f not differentiable?

2

Review

REvIEw

CONCEPT CHECK

Answers to the Concept Check can be found on the back endpapers.

1. Explain what each of the following means and illustrate with a sketch. (a) lim f sxd − L (b) lim1 f sxd − L (c) lim2 f sxd − L x la

(d) lim f sxd − ` x la

x la

x la

(e) lim f sxd − L x l`

2. Describe several ways in which a limit can fail to exist. Illustrate with sketches. 3. State the following Limit Laws. (a) Sum Law (b) Difference Law (c) Constant Multiple Law (d) Product Law (e) Quotient Law (f ) Power Law (g) Root Law

7. (a) What does it mean for f to be continuous at a? (b) What does it mean for f to be continuous on the interval s2`, `d? What can you say about the graph of such a function? 8. (a) Give examples of functions that are continuous on f21, 1g. (b) Give an example of a function that is not continuous on f0, 1g. 9. What does the Intermediate Value Theorem say? 10. Write an expression for the slope of the tangent line to the curve y − f sxd at the point sa, f sadd.

4. What does the Squeeze Theorem say?

11. Suppose an object moves along a straight line with position f std at time t. Write an expression for the instantaneous velocity of the object at time t − a. How can you interpret this velocity in terms of the graph of f ?

5. (a) What does it mean to say that the line x − a is a vertical asymptote of the curve y − f sxd? Draw curves to illustrate the various possibilities. (b) What does it mean to say that the line y − L is a horizontal asymptote of the curve y − f sxd? Draw curves to illustrate the various possibilities.

12. If y − f sxd and x changes from x 1 to x 2, write expressions for the following. (a) The average rate of change of y with respect to x over the interval fx 1, x 2 g. (b) The instantaneous rate of change of y with respect to x at x − x 1.

6. Which of the following curves have vertical asymptotes? Which have horizontal asymptotes? (a) y − x 4 (b) y − sin x (c) y − tan x (d) y − tan21x (e) y − e x (f ) y − ln x (g) y − 1yx (h) y − sx

13. Define the derivative f 9sad. Discuss two ways of interpreting this number. 14. Define the second derivative of f. If f std is the position function of a particle, how can you interpret the second derivative?

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166

Chapter 2

Limits and Derivatives

15. (a) What does it mean for f to be differentiable at a? (b) What is the relation between the differentiability and continuity of a function? (c) Sketch the graph of a function that is continuous but not differentiable at a − 2.

16. Describe several ways in which a function can fail to be differentiable. Illustrate with sketches.

TRUE-FALSE QUIZ Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. 1. lim

x l4

S

2x 8 2 x24 x24

D

2x 8 − lim 2 lim x l4 x 2 4 x l4 x 2 4

lim sx 2 3d x23 xl1 − lim sx 2 1 2x 2 4d x 2 1 2x 2 4

17. If f is continuous at 5 and f s5d − 2 and f s4d − 3, then lim x l 2 f s4x 2 2 11d − 2.

xl1

18. If f is continuous on f21, 1g and f s21d − 4 and f s1d − 3, then there exists a number r such that r , 1 and f srd − .

x2 2 9 4. −x13 x23 5. lim

xl3

14. If f has domain f0, `d and has no horizontal asymptote, then lim x l ` f sxd − ` or lim x l ` f sxd − 2`.

16. If f s1d . 0 and f s3d , 0, then there exists a number c between 1 and 3 such that f scd − 0.

x l1

xl1

13. A function can have two different horizontal asymptotes.

15. If the line x − 1 is a vertical asymptote of y − f sxd, then f is not defined at 1.

lim sx 2 1 6x 2 7d x 2 1 6x 2 7 x l1 − 2. lim 2 x l 1 x 1 5x 2 6 lim sx 2 1 5x 2 6d 3. lim

12. If lim x l 0 f sxd − ` and lim x l 0 tsxd − `, then lim x l 0 f f sxd 2 tsxdg − 0.

| |

19. Let f be a function such that lim x l 0 f sxd − 6. Then there exists a positive number  such that if 0 , x , , then f sxd 2 6 , 1.

x2 2 9 − lim sx 1 3d xl3 x23

|

| |

|

6. If lim x l 5 f sxd − 2 and lim x l 5 tsxd − 0, then limx l 5 f f sxdytsxdg does not exist.

20. If f sxd . 1 for all x and lim x l 0 f sxd exists, then lim x l 0 f sxd . 1.

7. If lim x l5 f sxd − 0 and lim x l 5 tsxd − 0, then lim x l 5 f f sxdytsxdg does not exist.

21. If f is continuous at a, then f is differentiable at a. 22. If f 9srd exists, then lim x l r f sxd − f srd.

8. If neither lim x l a f sxd nor lim x l a tsxd exists, then lim x l a f f sxd 1 tsxdg does not exist.

23.

9. If lim x l a f sxd exists but lim x l a tsxd does not exist, then lim x l a f f sxd 1 tsxdg does not exist.

24. The equation x 10 2 10x 2 1 5 − 0 has a root in the interval s0, 2d.

d 2y − dx 2

S D dy dx

2

10. If lim x l 6 f f sxd tsxdg exists, then the limit must be f s6d ts6d.

25. If f is continuous at a, so is f .

11. If p is a polynomial, then lim x l b psxd − psbd.

26. If f is continuous at a, so is f .

| |

| |

EXERCISES 1. The graph of f is given.

(iv) lim f sxd

(v)

x l4

y

(vii) lim f sxd x l`

lim f sxd

x l0

(vi) lim2 f sxd x l2

(viii) lim f sxd x l 2`

(b) State the equations of the horizontal asymptotes. (c) State the equations of the vertical asymptotes. (d) At what numbers is f discontinuous? Explain. 1 0

x

1

2. Sketch the graph of a function f that satisfies all of the following conditions: lim f sxd − 22, lim f sxd − 0, lim f sxd − `, x l 2`

(a) Find each limit, or explain why it does not exist. (i) lim1 f sxd (ii) lim 1 f sxd (iii) lim f sxd x l2

x l 23

x l 23

lim f sxd − 2`,

x l 32

xl`

x l 23

lim f sxd − 2,

x l 31

f is continuous from the right at 3

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

Chapter 2

3–20 Find the limit. 3. lim e

x3 2 x

4.

x l1

x2 2 9 x l 23 x 1 2x 2 3 sh 2 1d3 1 1 7. lim h l0 h 5. lim

9. lim

r l9

6.

2

8.

sr sr 2 9d4

10.

|

11. lim

ul1

u 21 u 3 1 5u 2 2 6u

sx 2 2 9 13. lim x l ` 2x 2 6

31–32 Show that the function is continuous on its domain. State the domain. sx 2 2 9 31. hsxd − xe sin x 32. tsxd − 2 x 22

|

33–34 Use the Intermediate Value Theorem to show that there is a root of the equation in the given interval.

sx 1 6 2 x x 3 2 3x 2

12. lim

xl3

33. x 5 2 x 3 1 3x 2 5 − 0,

sx 2 2 9 14. lim x l 2` 2x 2 6 1 2 2x 2 2 x 4 16. lim x l 2` 5 1 x 2 3x 4

15. lim2 lnssin xd x l

17. lim (sx 1 4x 1 1 2 x) 18. lim e 2

x l`

xl`

19. lim1 tan21s1yxd

20. lim

x l0

xl1

34. cos sx − e 2 2, x

x 2x 2

S

1 1 1 2 x21 x 2 3x 1 2

D

; 21–22 Use graphs to discover the asymptotes of the curve. Then prove what you have discovered. 21. y −

cos x x2

22. y − sx 2 1 x 1 1 2 sx 2 2 x

23. If 2x 2 1 < f sxd < x 2 for 0 , x , 3, find lim x l1 f sxd. 24. Prove that lim x l 0 x 2 coss1yx 2 d − 0. 25–28 Prove the statement using the precise definition of a limit. 25. lim s14 2 5xd − 4

26. lim sx − 0

27. lim sx 2 2 3xd − 22

28. lim1

3

xl2

xl0

xl2

29. Let

xl4

2 −` sx 2 4

H

if x , 0 s2x f sxd − 3 2 x if 0 < x , 3 sx 2 3d2 if x . 3 (a) Evaluate each limit, if it exists. (i) lim1 f sxd

(ii) lim2 f sxd

(iii) lim f sxd

(iv) lim2 f sxd

(v) lim1 f sxd

(vi) lim f sxd

x l0 x l3

x l0 x l3

x l0

(b) Where is f discontinuous? (c) Sketch the graph of f.

x l3

35. (a) Find the slope of the tangent line to the curve y − 9 2 2x 2 at the point s2, 1d. (b) Find an equation of this tangent line. 36. Find equations of the tangent lines to the curve

at the points with x-coordinates 0 and 21. 37. The displacement (in meters) of an object moving in a straight line is given by s − 1 1 2t 1 14 t 2, where t is measured in seconds. (a) Find the average velocity over each time period. (i) f1, 3g (ii) f1, 2g (iii) f1, 1.5g (iv) f1, 1.1g (b) Find the instantaneous velocity when t − 1. 38. According to Boyle’s Law, if the temperature of a confined gas is held fixed, then the product of the pressure P and the volume V is a constant. Suppose that, for a certain gas, PV − 4000, where P is measured in pascals and V is measured in liters. (a) Find the average rate of change of P as V increases from 3 L to 4 L. (b) Express V as a function of P and show that the instantaneous rate of change of V with respect to P is inversely proportional to the square of P. 39. (a) Use the definition of a derivative to find f 9s2d, where f sxd − x 3 2 2x. (b) Find an equation of the tangent line to the curve y − x 3 2 2x at the point (2, 4). (c) Illustrate part (b) by graphing the curve and the tan; gent line on the same screen.

lim

tsxd −

2 1 2 3x

40. Find a function f and a number a such that

30. Let 2x 2 x 2 22x x24

s1, 2d

s0, 1d

y−

2

167

(a) For each of the numbers 2, 3, and 4, discover whether t is continuous from the left, continuous from the right, or continuous at the number. (b) Sketch the graph of t.

x2 2 9 lim 2 x l 3 x 1 2x 2 3 x2 2 9 lim1 2 x l 1 x 1 2x 2 3 t2 2 4 lim 3 t l2 t 2 8 42v lim v l 41 4 2 v

4

Review

if if if if

0 2

81. Let f sxd −

Give a formula for t9 and sketch the graphs of t and t9.

|

73. (a) For what values of x is the function f sxd − x 2 2 9 differentiable? Find a formula for f 9. (b) Sketch the graphs of f and f 9.

|

| |

80. The graph of any quadratic function f sxd − ax 2 1 bx 1 c is a parabola. Prove that the average of the slopes of the tangent lines to the parabola at the endpoints of any interval f p, qg equals the slope of the tangent line at the midpoint of the interval.

|

|

74. Where is the function hsxd − x 2 1 1 x 1 2 differentiable? Give a formula for h9 and sketch the graphs of h and h9. 75. Find the parabola with equation y − ax 2 1 bx whose tangent line at (1, 1) has equation y − 3x 2 2.

77. For what values of a and b is the line 2x 1 y − b tangent to the parabola y − ax 2 when x − 2? 78. Find the value of c such that the line y − the curve y − csx .

applied project

82. A tangent line is drawn to the hyperbola xy − c at a point P. (a) Show that the midpoint of the line segment cut from this tangent line by the coordinate axes is P. (b) Show that the triangle formed by the tangent line and the coordinate axes always has the same area, no matter where P is located on the hyperbola. 83. Evaluate lim

1 6 is tangent to

x 1000 2 1 . x21

84. Draw a diagram showing two perpendicular lines that intersect on the y-axis and are both tangent to the parabola y − x 2. Where do these lines intersect? 85. If c . 12, how many lines through the point s0, cd are normal lines to the parabola y − x 2? What if c < 12? 86. Sketch the parabolas y − x 2 and y − x 2 2 2x 1 2. Do you think there is a line that is tangent to both curves? If so, find its equation. If not, why not?

building a better roller coaster

f L¡

3 2x

x2 if x < 2 mx 1 b if x . 2

Find the values of m and b that make f differentiable everywhere.

xl1

76. Suppose the curve y − x 4 1 ax 3 1 bx 2 1 cx 1 d has a tangent line when x − 0 with equation y − 2x 1 1 and a tangent line when x − 1 with equation y − 2 2 3x. Find the values of a, b, c, and d.

H

P Q L™

Suppose you are asked to design the first ascent and drop for a new roller coaster. By studying photographs of your favorite coasters, you decide to make the slope of the ascent 0.8 and the slope of the drop 21.6. You decide to connect these two straight stretches y − L 1sxd and y − L 2 sxd with part of a parabola y − f sxd − a x 2 1 bx 1 c, where x and f sxd are measured in meters. For the track to be smooth there can’t be abrupt changes in direction, so you want the linear segments L 1 and L 2 to be tangent to the parabola at the transition points P and Q. (See the figure.) To simplify the equations, you decide to place the origin at P. 1. (a) Suppose the horizontal distance between P and Q is 30 m. Write equations in a, b, and c that will ensure that the track is smooth at the transition points. (b) Solve the equations in part (a) for a, b, and c to find a formula for f sxd. (c) Plot L 1, f , and L 2 to verify graphically that the transitions are smooth. ; (d) Find the difference in elevation between P and Q. 2. The solution in Problem 1 might look smooth, but it might not feel smooth because the piecewise defined function [consisting of L 1sxd for x , 0, f sxd for 0 < x < 30, and

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SECTION 3.2 The Product and Quotient Rules

183

L 2sxd for x . 30] doesn’t have a continuous second derivative. So you decide to improve the design by using a quadratic function qsxd − ax 2 1 bx 1 c only on the interval 3 < x < 27 and connecting it to the linear functions by means of two cubic functions: tsxd − kx 3 1 lx 2 1 mx 1 n

0 0, where s is measured y − 2x sin x at the point sy2, d. in centimeters and t in seconds. (Take the positive direction (b) Illustrate part (a) by graphing the curve and the tangent to be downward.) line on the same screen. (a) Find the velocity and acceleration at time t. (a) Find an equation of the tangent line to the curve (b) Graph the velocity and acceleration functions. y − 3x 1 6 cos x at the point sy3,  1 3d. (c) When does the mass pass through the equilibrium (b) Illustrate part (a) by graphing the curve and the tangent position for the first time? line on the same screen. (d) How far from its equilibrium position does the mass travel? (a) If f sxd − sec x 2 x, find f 9sxd. (e) When is the speed the greatest? (b) Check to see that your answer to part (a) is reasonable by graphing both f and f 9 for x , y2. 37. A ladder 6 m long rests against a vertical wall. Let  be the angle between the top of the ladder and the wall and let x be (a) If f sxd − e x cos x, find f 9sxd and f 99sxd. the distance from the bottom of the ladder to the wall. If the (b) Check to see that your answers to part (a) are reasonable bottom of the ladder slides away from the wall, how fast by graphing f , f 9, and f 99. does x change with respect to  when  − y3? If Hsd −  sin , find H9sd and H99sd. 38. An object with mass m is dragged along a horizontal plane by a force acting along a rope attached to the object. If the If f std − sec t, find f 0sy4d.

| |

28. ; 29. 30.

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SECTION 3.4 The Chain Rule

rope makes an angle  with the plane, then the magnitude of the force is mt F−  sin  1 cos  where  is a constant called the coefficient of friction. (a) Find the rate of change of F with respect to . (b) When is this rate of change equal to 0? (c) If m − 20 kg, t − 9.8 mys2, and  − 0.6, draw the graph of F as a function of  and use it to locate the value of  for which dFyd − 0. Is the value consistent with your answer to part (b)?

;

55. Differentiate each trigonometric identity to obtain a new (or familiar) identity. sin x 1 (a) tan x − (b) sec x − cos x cos x 1 1 cot x (c) sin x 1 cos x − csc x 56. A semicircle with diameter PQ sits on an isosceles triangle PQR to form a region shaped like a two-dimensional ice-cream cone, as shown in the figure. If Asd is the area of the semicircle and Bsd is the area of the triangle, find lim

39–50 Find the limit.

l 01

sin 5x 3x tan 6t 41. lim t l 0 sin 2t sin 3x 43. lim x l 0 5x 3 2 4x

sin x sin x cos  2 1 42. lim l0 sin  sin 3x sin 5x 44. lim xl0 x2

sin  45. lim  l 0  1 tan

46. lim csc x sinssin xd

39. lim

xl0

cos  2 1 2 2 1 2 tan x 49. lim x l y4 sin x 2 cos x 47. lim

l0

Asd Bsd

40. lim

xl0

A(¨) P

10 cm

10 cm ¨

xl0

2

sinsx d x sinsx 2 1d 50. lim 2 xl1 x 1 x 2 2

Q B(¨)

R

48. lim

xl0

57. The figure shows a circular arc of length s and a chord of length d, both subtended by a central angle . Find lim

l 01

51–52 Find the given derivative by finding the first few derivatives and observing the pattern that occurs. 99

51.

d ssin xd dx 99

197

d

s d s

35

52.

d sx sin xd dx 35

53. Find constants A and B such that the function y − A sin x 1 B cos x satisfies the differential equation y99 1 y9 2 2y − sin x. 1 54. (a) Evaluate lim x sin . xl` x 1 (b) Evaluate lim x sin . xl0 x (c) Illustrate parts (a) and (b) by graphing y − x sins1yxd. ;

¨

x . s1 2 cos 2x (a) Graph f . What type of discontinuity does it appear to have at 0? (b) Calculate the left and right limits of f at 0. Do these values confirm your answer to part (a)?

; 58. Let f sxd −

Suppose you are asked to differentiate the function Fsxd − sx 2 1 1 The differentiation formulas you learned in the previous sections of this chapter do not enable you to calculate F9sxd.

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

198

CHAPTER 3

Differentiation Rules

See Section 1.3 for a review of composite functions.

Observe that F is a composite function. In fact, if we let y − f sud − su and let u − tsxd − x 2 1 1, then we can write y − Fsxd − f stsxdd, that is, F − f 8 t. We know how to differentiate both f and t, so it would be useful to have a rule that tells us how to find the derivative of F − f 8 t in terms of the derivatives of f and t. It turns out that the derivative of the composite function f 8 t is the product of the derivatives of f and t. This fact is one of the most important of the differentiation rules and is called the Chain Rule. It seems plausible if we interpret derivatives as rates of change. Regard duydx as the rate of change of u with respect to x, dyydu as the rate of change of y with respect to u, and dyydx as the rate of change of y with respect to x. If u changes twice as fast as x and y changes three times as fast as u, then it seems reasonable that y changes six times as fast as x, and so we expect that dy dy du − dx du dx The Chain Rule If t is differentiable at x and f is differentiable at tsxd, then the composite function F − f 8 t defined by Fsxd − f stsxdd is differentiable at x and F9 is given by the product F9sxd − f 9s tsxdd ? t9sxd In Leibniz notation, if y − f sud and u − tsxd are both differentiable functions, then dy dy du − dx du dx

James Gregory The first person to formulate the Chain Rule was the Scottish mathematician James Gregory (1638–1675), who also designed the first practical reflecting telescope. Gregory discovered the basic ideas of calculus at about the same time as Newton. He became the first Professor of Mathematics at the University of St. Andrews and later held the same position at the University of Edinburgh. But one year after accepting that position he died at the age of 36.

COMMENTS ON THE PROOF OF THE CHAIN RULE Let Du be the change in u correspond-

ing to a change of Dx in x, that is, Du − tsx 1 Dxd 2 tsxd Then the corresponding change in y is Dy − f su 1 Dud 2 f sud It is tempting to write dy Dy − lim Dxl 0 dx Dx 1

− lim

Dy Du ? Du Dx

− lim

Dy Du ? lim Du Dx l 0 Dx

− lim

Dy Du ? lim Du Dx l 0 Dx

Dx l 0

Dx l 0

Du l 0

(Note that Du l 0 as Dx l 0 since t is continuous.)

dy du du dx

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

SECTION 3.4 The Chain Rule

199

The only flaw in this reasoning is that in (1) it might happen that Du − 0 (even when Dx ± 0) and, of course, we can’t divide by 0. Nonetheless, this reasoning does at least suggest that the Chain Rule is true. A full proof of the Chain Rule is given at the end of this section. ■ The Chain Rule can be written either in the prime notation s f 8 td9sxd − f 9stsxdd ? t9sxd

2

or, if y − f sud and u − tsxd, in Leibniz notation: dy dy du − dx du dx

3

Equation 3 is easy to remember because if dyydu and duydx were quotients, then we could cancel du. Remember, however, that du has not been defined and duydx should not be thought of as an actual quotient.

ExamplE 1 Find F9sxd if Fsxd − sx 2 1 1. SOLUTION 1 (using Equation 2): At the beginning of this section we expressed F as Fsxd − s f 8 tdsxd − f stsxdd where f sud − su and tsxd − x 2 1 1. Since

f 9sud − 12 u21y2 −

1 2 su

and

t9sxd − 2x

F9sxd − f 9stsxdd ? t9sxd

we have

1 x ? 2x − 2 1 1 2 sx 2 1 1 sx

SOLUTION 2 (using Equation 3): If we let u − x 2 1 1 and y − su , then

dy du 1 1 x − s2xd − s2xd − 2 2 du dx 2 su 2 sx 1 1 sx 1 1

F9sxd −

When using Formula 3 we should bear in mind that dyydx refers to the derivative of y when y is considered as a function of x (called the derivative of y with respect to x), whereas dyydu refers to the derivative of y when considered as a function of u (the derivative of y with respect to u). For instance, in Example 1, y can be considered as a function of x ( y − s x 2 1 1 ) and also as a function of u ( y − su ). Note that dy x − F9sxd − 2 dx sx 1 1

whereas

dy 1 − f 9sud − du 2 su

NOTE In using the Chain Rule we work from the outside to the inside. Formula 2 says that we differentiate the outer function f [at the inner function tsxd] and then we multiply by the derivative of the inner function. d dx

f

s tsxdd

outer function

evaluated at inner function

f9

stsxdd

derivative of outer function

evaluated at inner function

?

t9sxd derivative of inner function

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

200

CHAPTER 3

Differentiation Rules

ExamplE 2 Differentiate (a) y − sinsx 2 d and (b) y − sin2x. SOLUTION

(a) If y − sinsx 2 d, then the outer function is the sine function and the inner function is the squaring function, so the Chain Rule gives dy d − dx dx

sin

sx 2 d

outer function

evaluated at inner function

cos

sx 2 d

derivative of outer function

evaluated at inner function

?

2x derivative of inner function

− 2x cossx 2 d (b) Note that sin2x − ssin xd2. Here the outer function is the squaring function and the inner function is the sine function. So dy d − ssin xd2 dx dx inner function

See Reference Page 2 or Appendix D.

2

?

derivative of outer function

ssin xd

?

evaluated at inner function

cos x derivative of inner function

The answer can be left as 2 sin x cos x or written as sin 2x (by a trigonometric identity known as the double-angle formula). ■ In Example 2(a) we combined the Chain Rule with the rule for differentiating the sine function. In general, if y − sin u, where u is a differentiable function of x, then, by the Chain Rule, dy dy du du − − cos u dx du dx dx Thus

d du ssin ud − cos u dx dx

In a similar fashion, all of the formulas for differentiating trigonometric functions can be combined with the Chain Rule. Let’s make explicit the special case of the Chain Rule where the outer function f is a power function. If y − f tsxdg n, then we can write y − f sud − u n where u − tsxd. By using the Chain Rule and then the Power Rule, we get dy dy du du − − nu n21 − nftsxdg n21 t9sxd dx du dx dx 4 The Power Rule Combined with the Chain Rule If n is any real number and u − tsxd is differentiable, then d du su n d − nu n21 dx dx Alternatively,

d ftsxdg n − nf tsxdg n21 ? t9sxd dx

Notice that the derivative in Example 1 could be calculated by taking n − 12 in Rule 4.

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

SECTION 3.4 The Chain Rule

201

ExamplE 3 Differentiate y − sx 3 2 1d100. SOLUTION Taking u − tsxd − x 3 2 1 and n − 100 in (4), we have

dy d d − sx 3 2 1d100 − 100sx 3 2 1d99 sx 3 2 1d dx dx dx − 100sx 3 2 1d99 ? 3x 2 − 300x 2sx 3 2 1d99

ExamplE 4 Find f 9sxd if f sxd − SOLUTION First rewrite f :

Thus

1 . sx 1 x 1 1 3

2

f sxd − sx 2 1 x 1 1d21y3

f 9sxd − 213 sx 2 1 x 1 1d24y3

d sx 2 1 x 1 1d dx

− 213 sx 2 1 x 1 1d24y3s2x 1 1d

ExamplE 5 Find the derivative of the function tstd −

S D t22 2t 1 1

9

SOLUTION Combining the Power Rule, Chain Rule, and Quotient Rule, we get

S D S D S D

t9std − 9

t22 2t 1 1

8

d dt

t22 2t 1 1

8

−9

s2t 1 1d ? 1 2 2st 2 2d 45st 2 2d8 − 2 s2t 1 1d s2t 1 1d10

t22 2t 1 1

ExamplE 6 Differentiate y − s2x 1 1d5sx 3 2 x 1 1d4. The graphs of the functions y and y9 in Example 6 are shown in Figure 1. Notice that y9 is large when y increases rapidly and y9 − 0 when y has a horizontal tangent. So our answer appears to be reasonable. 10

dy d d − s2x 1 1d5 sx 3 2 x 1 1d4 1 sx 3 2 x 1 1d4 s2x 1 1d5 dx dx dx − s2x 1 1d5 ? 4sx 3 2 x 1 1d3

d sx 3 2 x 1 1d dx

1 sx 3 2 x 1 1d4 ? 5s2x 1 1d4

yª _2

1

y _10

FIGURE 1

SOLUTION In this example we must use the Product Rule before using the Chain Rule:

d s2x 1 1d dx

− 4s2x 1 1d5sx 3 2 x 1 1d3s3x 2 2 1d 1 5sx 3 2 x 1 1d4s2x 1 1d4 ? 2 Noticing that each term has the common factor 2s2x 1 1d4sx 3 2 x 1 1d3, we could factor it out and write the answer as dy − 2s2x 1 1d4sx 3 2 x 1 1d3s17x 3 1 6x 2 2 9x 1 3d dx

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

202

CHAPTER 3

Differentiation Rules

ExamplE 7 Differentiate y − e sin x. SOLUTION Here the inner function is tsxd − sin x and the outer function is the exponential function f sxd − e x. So, by the Chain Rule, More generally, the Chain Rule gives

dy d d − se sin x d − e sin x ssin xd − e sin x cos x dx dx dx

d u du se d − e u dx dx

We can use the Chain Rule to differentiate an exponential function with any base b . 0. Recall from Section 1.5 that b − e ln b. So b x − se ln b d x − e sln bdx and the Chain Rule gives d d d sb x d − se sln bdx d − e sln bdx sln bdx dx dx dx − e sln bdx ∙ ln b − b x ln b because ln b is a constant. So we have the formula Don’t confuse Formula 5 (where x is the exponent) with the Power Rule (where x is the base):

d sb x d − b x ln b dx

5

d sx n d − nx n21 dx

In particular, if b − 2, we get d s2 x d − 2 x ln 2 dx

6 In Section 3.1 we gave the estimate

d s2 x d < s0.69d2 x dx This is consistent with the exact formula (6) because ln 2 < 0.693147. The reason for the name “Chain Rule” becomes clear when we make a longer chain by adding another link. Suppose that y − f sud, u − tsxd, and x − hstd, where f , t, and h are differentiable functions. Then, to compute the derivative of y with respect to t, we use the Chain Rule twice: dy dy dx dy du dx − − dt dx dt du dx dt

ExamplE 8 If f sxd − sinscosstan xdd, then f 9sxd − cosscosstan xdd

d cosstan xd dx

− cosscosstan xddf2sinstan xdg

d stan xd dx

− 2cosscosstan xdd sinstan xd sec2x Notice that we used the Chain Rule twice.

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

SECTION 3.4 The Chain Rule

203

ExamplE 9 Differentiate y − e sec 3. SOLUTION The outer function is the exponential function, the middle function is the secant function, and the inner function is the tripling function. So we have

dy d − e sec 3 ssec 3d d d d s3d d

− e sec 3 sec 3 tan 3 − 3e sec 3 sec 3 tan 3

How to Prove the Chain Rule Recall that if y − f sxd and x changes from a to a 1 Dx, we define the increment of y as Dy − f sa 1 Dxd 2 f sad According to the definition of a derivative, we have lim

Dx l 0

So if we denote by « the difference between the difference quotient and the derivative, we obtain lim « − lim

Dx l 0

«−

But

Dx l 0

S

D

?

Dy − f 9sad Dx 1 « Dx

If we define « to be 0 when Dx − 0, then « becomes a continuous function of Dx. Thus, for a differentiable function f, we can write 7

Dy − f 9sad Dx 1 « Dx

where

« l 0 as Dx l 0

and « is a continuous function of Dx. This property of differentiable functions is what enables us to prove the Chain Rule. PROOF OF THE CHAIN RULE Suppose u − tsxd is differentiable at a and y − f sud is differentiable at b − tsad. If Dx is an increment in x and Du and Dy are the corresponding increments in u and y, then we can use Equation 7 to write

8

Du − t9sad Dx 1 «1 Dx − ft9sad 1 «1 g Dx

where «1 l 0 as Dx l 0. Similarly 9

Dy − f 9sbd Du 1 «2 Du − f f 9sbd 1 «2 g Du

where «2 l 0 as Du l 0. If we now substitute the expression for Du from Equation 8 into Equation 9, we get Dy − f f 9sbd 1 «2 gft9sad 1 «1 g Dx

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

204

CHAPTER 3

Differentiation Rules

Dy − f f 9sbd 1 «2 gft9sad 1 «1 g Dx

so

As Dx l 0, Equation 8 shows that Du l 0. So both «1 l 0 and «2 l 0 as Dx l 0. Therefore dy Dy − lim − lim f f 9sbd 1 «2 g ft9sad 1 «1 g Dx l 0 Dx Dx l 0 dx − f 9sbd t9sad − f 9stsadd t9sad This proves the Chain Rule.

3.4 EXERCISES 1–6 Write the composite function in the form f s tsxdd. [Identify the inner function u − tsxd and the outer function y − f sud.] Then find the derivative dyydx. 1. y − sin 4x

2. y − s4 1 3x

3. y − s1 2 x 2 d10

4. y − tanssin xd

5. y − e sx

6. y − s2 2 e x

7–46 Find the derivative of the function. 7. Fsxd − s5x 6 1 2x 3 d 4

8. Fsxd − s1 1 x 1 x 2 d 99

9. f sxd − s5x 1 1

10. f sxd −

1 3 x2 2 1 s

11. f sd − coss 2 d

12. tsd − cos2

13. y − x 2e 23x

14. f std − t sin t

15. f std − e at sin bt

16. tsxd − e x

2

19. hstd − st 1 1d2y3 s2t 2 2 1d3

21. y −

Î

22. y − x 1

23. y − e tan  25. tsud −

S

24. f std − 2 t u3 2 1 u3 1 1

D

27. rstd − 10 2st 29. Hsrd −

23

S D

x x11

sr 2 2 1d 3 s2r 1 1d 5

8

26. sstd −

33. Gsxd − 4 Cyx

34. Us yd −

35. y − sinstan 2xd

36. y − sec 2 smd

37. y − cot 2ssin d

38. y − s1 1 xe22x

39. f std − tanssecscos tdd

40. y − e sin 2x 1 sinse 2x d

41. f std − tanse t d 1 e tan t

42. y − sinssinssin xdd

43. tsxd − s2ra rx 1 nd p

44. y − 2 3

45. y − cos ssinstan xd

46. y − fx 1 sx 1 sin2 xd3 g 4

st 1 1 3

S D y4 1 1 y2 1 1

5

4x

47–50 Find y9 and y 99.

18. tsxd − sx 2 1 1d3 sx 2 1 2d6 20. Fstd − s3t 2 1d s2t 1 1d

32. Fstd −

2x

17. f sxd − s2x 2 3d4 sx 2 1 x 1 1d5

4

t2

31. Fstd − e t sin 2t

1 x

48. y −

49. y − s1 2 sec t

50. y − e e

x

5

3

Î

1 s1 1 tan xd 2

47. y − cosssin 3d

1 1 sin t 1 1 cos t

28. f szd − e zysz21d 30. Jsd − tan 2 snd

51–54 Find an equation of the tangent line to the curve at the given point. 51. y − 2 x, s0, 1d

52. y − s1 1 x 3 , s2, 3d

53. y − sinssin xd, s, 0d

54. y − xe 2x , s0, 0d

2

55. (a) Find an equation of the tangent line to the curve y − 2ys1 1 e2x d at the point s0, 1d. (b) Illustrate part (a) by graphing the curve and the tangent line ; on the same screen.

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SECTION 3.4 The Chain Rule

| |

56. (a) The curve y − x ys2 2 x 2 is called a bullet-nose curve. Find an equation of the tangent line to this curve at the point s1, 1d. (b) Illustrate part (a) by graphing the curve and the tangent ; line on the same screen.

66. If f is the function whose graph is shown, let hsxd − f s f sxdd and tsxd − f sx 2 d. Use the graph of f to estimate the value of each derivative. (a) h9s2d (b) t9s2d y

57. (a) If f sxd − x s2 2 x , find f 9sxd. (b) Check to see that your answer to part (a) is reasonable by comparing the graphs of f and f 9. 2

;

; 58. The function f sxd − sinsx 1 sin 2xd, 0 < x < , arises in applications to frequency modulation (FM) synthesis. (a) Use a graph of f produced by a calculator to make a rough sketch of the graph of f 9. (b) Calculate f 9sxd and use this expression, with a calculator, to graph f 9. Compare with your sketch in part (a).

205

y=ƒ

1 0

1

x

67. If tsxd − sf sxd , where the graph of f is shown, evaluate t9s3d. y

59. Find all points on the graph of the function f sxd − 2 sin x 1 sin2x at which the tangent line is horizontal. 60. At what point on the curve y − s1 1 2x is the tangent line perpendicular to the line 6x 1 2y − 1? 61. If Fsxd − f stsxdd, where f s22d − 8, f 9s22d − 4, f 9s5d − 3, ts5d − 22, and t9s5d − 6, find F9s5d. 62. If hsxd − s4 1 3f sxd , where f s1d − 7 and f 9s1d − 4, find h9s1d.

f

1 0

1

x

68. Suppose f is differentiable on R and  is a real number. Let Fsxd − f sx  d and Gsxd − f f sxdg . Find expressions for (a) F9sxd and (b) G9sxd. 69. Suppose f is differentiable on R. Let Fsxd − f se x d and Gsxd − e f sxd. Find expressions for (a) F9sxd and (b) G9sxd.

63. A table of values for f , t, f 9, and t9 is given. x

f sxd

tsxd

f 9sxd

t9sxd

1 2 3

3 1 7

2 8 2

4 5 7

6 7 9

70. Let tsxd − e cx 1 f sxd and hsxd − e kx f sxd, where f s0d − 3, f 9s0d − 5, and f 99s0d − 22. (a) Find t9s0d and t99s0d in terms of c. (b) In terms of k, find an equation of the tangent line to the graph of h at the point where x − 0. 71. Let rsxd − f s tshsxddd, where hs1d − 2, ts2d − 3, h9s1d − 4, t9s2d − 5, and f 9s3d − 6. Find r9s1d.

(a) If hsxd − f stsxdd, find h9s1d. (b) If Hsxd − ts f sxdd, find H9s1d. 64. Let f and t be the functions in Exercise 63. (a) If Fsxd − f s f sxdd, find F9s2d. (b) If Gsxd − tstsxdd, find G9s3d. 65. If f and t are the functions whose graphs are shown, let usxd − f s tsxdd, vsxd − ts f sxdd, and w sxd − ts tsxdd. Find each derivative, if it exists. If it does not exist, explain why. (a) u9s1d (b) v9s1d (c) w9s1d

72. If t is a twice differentiable function and f sxd − x tsx 2 d, find f 99 in terms of t, t9, and t99. 73. If Fsxd − f s3f s4 f sxddd, where f s0d − 0 and f 9s0d − 2, find F9s0d. 74. If Fsxd − f sx f sx f sxddd, where f s1d − 2, f s2d − 3, f 9s1d − 4, f 9s2d − 5, and f 9s3d − 6, find F9s1d. 75. Show that the function y − e 2x sA cos 3x 1 B sin 3xd satisfies the differential equation y99 2 4y9 1 13y − 0.

y

f

76. For what values of r does the function y − e rx satisfy the differential equation y99 2 4y9 1 y − 0? g

77. Find the 50th derivative of y − cos 2x.

1 0

1

x

78. Find the 1000th derivative of f sxd − xe2x.

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206

CHAPTER 3

Differentiation Rules

79. The displacement of a particle on a vibrating string is given ; by the equation sstd − 10 1 14 sins10 td where s is measured in centimeters and t in seconds. Find the velocity of the particle after t seconds. 85. 80. If the equation of motion of a particle is given by s − A cosst 1 d, the particle is said to undergo simple harmonic motion. (a) Find the velocity of the particle at time t. (b) When is the velocity 0? 81. A Cepheid variable star is a star whose brightness alternately increases and decreases. The most easily visible such star is Delta Cephei, for which the interval between times of maximum brightness is 5.4 days. The average brightness of this star is 4.0 and its brightness changes by 60.35. In view of these data, the brightness of Delta Cephei at time t, where t is measured in days, has been modeled by the function

S D

Bstd − 4.0 1 0.35 sin

2 t 5.4

82. In Example 1.3.4 we arrived at a model for the length of daylight (in hours) in Ankara, Turkey, on the tth day of the year:

F

2 st 2 80d 365

The average blood alcohol concentration (BAC) of eight male subjects was measured after consumption of 15 mL of ethanol (corresponding to one alcoholic drink). The resulting data were modeled by the concentration function Cstd − 0.0225te 20.0467t where t is measured in minutes after consumption and C is measured in mgymL. (a) How rapidly was the BAC increasing after 10 minutes? (b) How rapidly was it decreasing half an hour later? Source: Adapted from P. Wilkinson et al., “Pharmacokinetics of Ethanol after Oral Administration in the Fasting State,” Journal of Pharmacokinetics and Biopharmaceutics 5 (1977): 207–24.

86. In Section 1.4 we modeled the world population from 1900 to 2010 with the exponential function Pstd − s1436.53d ? s1.01395d t

(a) Find the rate of change of the brightness after t days. (b) Find, correct to two decimal places, the rate of increase after one day.

Lstd − 12 1 2.8 sin

(c) Graph p for the case a − 10, k − 0.5 with t measured in hours. Use the graph to estimate how long it will take for 80% of the population to hear the rumor.

G

Use this model to compare how the number of hours of daylight is increasing in Ankara on March 21 and May 21. ; 83. The motion of a spring that is subject to a frictional force or a damping force (such as a shock absorber in a car) is often modeled by the product of an exponential function and a sine or cosine function. Suppose the equation of motion of a point on such a spring is sstd − 2e21.5t sin 2t where s is measured in centimeters and t in seconds. Find the velocity after t seconds and graph both the position and velocity functions for 0 < t < 2. 84. Under certain circumstances a rumor spreads according to the equation 1 pstd − 1 1 ae 2k t where pstd is the proportion of the population that has heard the rumor at time t and a and k are positive constants. [In Section 9.4 we will see that this is a reasonable equation for pstd.] (a) Find lim t l ` pstd. (b) Find the rate of spread of the rumor.

where t − 0 corresponds to the year 1900 and Pstd is measured in millions. According to this model, what was the rate of increase of world population in 1920? In 1950? In 2000? 87. A particle moves along a straight line with displacement sstd, velocity vstd, and acceleration astd. Show that dv astd − vstd ds Explain the difference between the meanings of the derivatives dvydt and dvyds. 88. Air is being pumped into a spherical weather balloon. At any time t, the volume of the balloon is Vstd and its radius is rstd. (a) What do the derivatives dVydr and dVydt represent? (b) Express dVydt in terms of drydt. ; 89. The flash unit on a camera operates by storing charge on a capacitor and releasing it suddenly when the flash is set off. The following data describe the charge Q remaining on the capacitor (measured in microcoulombs, mC) at time t (measured in seconds). t

0.00

0.02

0.04

0.06

0.08

0.10

Q

100.00

81.87

67.03

54.88

44.93

36.76

(a) Use a graphing calculator or computer to find an exponential model for the charge. (b) The derivative Q9std represents the electric current (measured in microamperes, mA) flowing from the capacitor to the flash bulb. Use part (a) to estimate the current when t − 0.04 s. Compare with the result of Example 2.1.2.

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SECTION 3.4 The Chain Rule

; 90. The table gives the population of Mexico (in millions) in the 20th-century census years. Year

Population

Year

Population

1900 1910 1920 1930 1940 1950

13.6 15.2 14.3 16.6 19.7 25.8

1960 1970 1980 1990 2000

34.9 48.2 66.8 81.2 97.5

(a) Use a graphing calculator or computer to fit an exponential function to the data. Graph the data points and the exponential model. How good is the fit? (b) Estimate the rates of population growth in 1950 and 1960 by averaging slopes of secant lines. (c) Use the exponential model in part (a) to find a model for the rate of growth of the Mexican population in the 20th century. (d) Use your model in part (c) to estimate the rates of growth in 1950 and 1960. Compare with your estimates in part (b). CAS

CAS

91. Computer algebra systems have commands that differentiate functions, but the form of the answer may not be convenient and so further commands may be necessary to simplify the answer. (a) Use a CAS to find the derivative in Example 5 and compare with the answer in that example. Then use the simplify command and compare again. (b) Use a CAS to find the derivative in Example 6. What happens if you use the simplify command? What happens if you use the factor command? Which form of the answer would be best for locating horizontal tangents? 92. (a) Use a CAS to differentiate the function f sxd −

Î

x4 2 x 1 1 x4 1 x 1 1

and to simplify the result. (b) Where does the graph of f have horizontal tangents? (c) Graph f and f 9 on the same screen. Are the graphs consistent with your answer to part (b)? 93. Use the Chain Rule to prove the following. (a) The derivative of an even function is an odd function. (b) The derivative of an odd function is an even function.

207

94. Use the Chain Rule and the Product Rule to give an alternative proof of the Quotient Rule. [Hint: Write f sxdytsxd − f sxdf tsxdg 21.] 95. (a) If n is a positive integer, prove that d ssinn x cos nxd − n sinn21x cossn 1 1dx dx (b) Find a formula for the derivative of y − cosnx cos nx that is similar to the one in part (a). 96. Suppose y − f sxd is a curve that always lies above the x-axis and never has a horizontal tangent, where f is differentiable everywhere. For what value of y is the rate of change of y 5 with respect to x eighty times the rate of change of y with respect to x? 97. Use the Chain Rule to show that if  is measured in degrees, then d  ssin d − cos  d 180 (This gives one reason for the convention that radian measure is always used when dealing with trigonometric functions in calculus: the differentiation formulas would not be as simple if we used degree measure.) 98. (a) Write x − sx 2 and use the Chain Rule to show that

| |

d x − dx

| |

|

x

|x|

|

(b) If f sxd − sin x , find f 9sxd and sketch the graphs of f and f 9. Where is f not differentiable? (c) If tsxd − sin x , find t9sxd and sketch the graphs of t and t9. Where is t not differentiable?

| |

99. If y − f sud and u − tsxd, where f and t are twice differentiable functions, show that d2y d2y − dx 2 du 2

S D du dx

2

1

dy d 2u du dx 2

100. If y − f sud and u − tsxd, where f and t possess third derivatives, find a formula for d 3 yydx 3 similar to the one given in Exercise 99.

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208

CHAPTER 3

Differentiation Rules

APPLIED PROJECT

WHERE SHOULD A PILOT START DESCENT? An approach path for an aircraft landing is shown in the figure and satisfies the following conditions:

y

y=P(x)

0

(i) The cruising altitude is h when descent starts at a horizontal distance , from touchdown at the origin. (ii) The pilot must maintain a constant horizontal speed v throughout descent. (iii) The absolute value of the vertical acceleration should not exceed a constant k (which is much less than the acceleration due to gravity).

h

x

1. Find a cubic polynomial Psxd − ax 3 1 bx 2 1 cx 1 d that satisfies condition (i) by imposing suitable conditions on Psxd and P9sxd at the start of descent and at touchdown. 2. Use conditions (ii) and (iii) to show that 6h v 2 0, where t is measured in seconds and s in meters. (a) Find the velocity at time t. (b) What is the velocity after 1 second? (c) When is the particle at rest? (d) When is the particle moving in the positive direction? (e) Find the total distance traveled during the first 6 seconds. (f) Draw a diagram like Figure 2 to illustrate the motion of the particle. (g) Find the acceleration at time t and after 1 second. ; (h) Graph the position, velocity, and acceleration functions for 0 < t < 6. (i) When is the particle speeding up? When is it slowing down? 9t t2 1 9

1. f std − t 3 2 8t 2 1 24t

2. f std −

3. f std − sinsty2d

4. f std − t 2e 2t

5. Graphs of the velocity functions of two particles are shown, where t is measured in seconds. When is each particle speeding up? When is it slowing down? Explain. (a) √ (b) √

0

1

t

0

1

t

6. Graphs of the position functions of two particles are shown, where t is measured in seconds. When is each particle speeding up? When is it slowing down? Explain. (a) s (b) s

0

1

t

0

1

t

7. The height (in meters) of a projectile shot vertically upward from a point 2 m above ground level with an initial velocity of 24.5 mys is h − 2 1 24.5t 2 4.9t 2 after t seconds. (a) Find the velocity after 2 s and after 4 s. (b) When does the projectile reach its maximum height? (c) What is the maximum height? (d) When does it hit the ground? (e) With what velocity does it hit the ground? 8. If a ball is thrown vertically upward with a velocity of 24.5 mys, then its height after t seconds is s − 24.5t 2 4.9t 2. (a) What is the maximum height reached by the ball? (b) What is the velocity of the ball when it is 29.4 m above the ground on its way up? On its way down? 9. If a rock is thrown vertically upward from the surface of Mars with velocity 15 mys, its height after t seconds is h − 15t 2 1.86t 2. (a) What is the velocity of the rock after 2 s? (b) What is the velocity of the rock when its height is 25 m on its way up? On its way down? 10. A particle moves with position function s − t 4 2 4t 3 2 20t 2 1 20t

t>0

(a) At what time does the particle have a velocity of 20 mys? (b) At what time is the acceleration 0? What is the significance of this value of t? 11. (a) A company makes computer chips from square wafers of silicon. It wants to keep the side length of a wafer very close to 15 mm and it wants to know how the area Asxd of a wafer changes when the side length x changes. Find A9s15d and explain its meaning in this situation. (b) Show that the rate of change of the area of a square with respect to its side length is half its perimeter. Try to explain geometrically why this is true by drawing a square whose side length x is increased by an amount Dx. How can you approximate the resulting change in area DA if Dx is small?

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234

CHAPTER 3

Differentiation Rules

12. (a) Sodium chlorate crystals are easy to grow in the shape of cubes by allowing a solution of water and sodium chlorate to evaporate slowly. If V is the volume of such a cube with side length x, calculate dVydx when x − 3 mm and explain its meaning. (b) Show that the rate of change of the volume of a cube with respect to its edge length is equal to half the surface area of the cube. Explain geometrically why this result is true by arguing by analogy with Exercise 11(b). 13. (a) Find the average rate of change of the area of a circle with respect to its radius r as r changes from (i) 2 to 3 (ii) 2 to 2.5 (iii) 2 to 2.1 (b) Find the instantaneous rate of change when r − 2. (c) Show that the rate of change of the area of a circle with respect to its radius (at any r) is equal to the circumference of the circle. Try to explain geometrically why this is true by drawing a circle whose radius is increased by an amount Dr. How can you approximate the resulting change in area DA if Dr is small? 14. A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of 60 cmys. Find the rate at which the area within the circle is increasing after (a) 1 s, (b) 3 s, and (c) 5 s. What can you conclude? 15. A spherical balloon is being inflated. Find the rate of increase of the surface area sS − 4r 2 d with respect to the radius r when r is (a) 20 cm, (b) 40 cm, and (c) 60 cm. What conclusion can you make? 4 3 3 r ,

16. (a) The volume of a growing spherical cell is V − where the radius r is measured in micrometers (1 μm − 1026 m). Find the average rate of change of V with respect to r when r changes from (i) 5 to 8 μm (ii) 5 to 6 μm (iii) 5 to 5.1 μm (b) Find the instantaneous rate of change of V with respect to r when r − 5 μm. (c) Show that the rate of change of the volume of a sphere with respect to its radius is equal to its surface area. Explain geometrically why this result is true. Argue by analogy with Exercise 13(c). 17. The mass of the part of a metal rod that lies between its left end and a point x meters to the right is 3x 2 kg. Find the linear density (see Example 2) when x is (a) 1 m, (b) 2 m, and (c) 3 m. Where is the density the highest? The lowest? 18. If a tank holds 5000 liters of water, which drains from the bottom of the tank in 40 minutes, then Torricelli’s Law gives the volume V of water remaining in the tank after t minutes as 1 td V − 5000 s1 2 40

2

0 < t < 40

Find the rate at which water is draining from the tank after (a) 5 min, (b) 10 min, (c) 20 min, and (d) 40 min. At what time is the water flowing out the fastest? The slowest? Summarize your findings.

19. The quantity of charge Q in coulombs (C) that has passed through a point in a wire up to time t (measured in seconds) is given by Qstd − t 3 2 2t 2 1 6t 1 2. Find the current when (a) t − 0.5 s and (b) t − 1 s. [See Example 3. The unit of current is an ampere (1 A − 1 Cys).] At what time is the current lowest? 20. Newton’s Law of Gravitation says that the magnitude F of the force exerted by a body of mass m on a body of mass M is F−

GmM r2

where G is the gravitational constant and r is the distance between the bodies. (a) Find dFydr and explain its meaning. What does the minus sign indicate? (b) Suppose it is known that the earth attracts an object with a force that decreases at the rate of 2 Nykm when r − 20,000 km. How fast does this force change when r − 10,000 km? 21. The force F acting on a body with mass m and velocity v is the rate of change of momentum: F − sdydtdsmvd. If m is constant, this becomes F − ma, where a − dvydt is the acceleration. But in the theory of relativity the mass of a particle varies with v as follows: m − m 0 ys1 2 v 2yc 2 , where m 0 is the mass of the particle at rest and c is the speed of light. Show that F−

m0a s1 2 v 2yc 2 d3y2

22. Some of the highest tides in the world occur in the Bay of Fundy on the Atlantic Coast of Canada. At Hopewell Cape the water depth at low tide is about 2.0 m and at high tide it is about 12.0 m. The natural period of oscillation is a little more than 12 hours and on June 30, 2009, high tide occurred at 6:45 am. This helps explain the following model for the water depth D (in meters) as a function of the time t (in hours after midnight) on that day: Dstd − 7 1 5 cosf0.503st 2 6.75dg How fast was the tide rising (or falling) at the following times? (a) 3:00 am (b) 6:00 am (c) 9:00 am (d) Noon 23. Boyle’s Law states that when a sample of gas is compressed at a constant temperature, the product of the pressure and the volume remains constant: PV − C. (a) Find the rate of change of volume with respect to pressure. (b) A sample of gas is in a container at low pressure and is steadily compressed at constant temperature for 10 minutes. Is the volume decreasing more rapidly at the beginning or the end of the 10 minutes? Explain. (c) Prove that the isothermal compressibility (see Example 5) is given by  − 1yP.

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SECTION 3.7

24. If, in Example 4, one molecule of the product C is formed from one molecule of the reactant A and one molecule of the reactant B, and the initial concentrations of A and B have a common value fAg − fBg − a molesyL, then fCg − a 2ktysakt 1 1d where k is a constant. (a) Find the rate of reaction at time t. (b) Show that if x − fCg, then dx − ksa 2 xd2 dt

Rates of Change in the Natural and Social Sciences

235

(e) In Section 1.1 we modeled Pstd with the exponential function f std − s1.43653 3 10 9 d ? s1.01395d t Use this model to find a model for the rate of population growth. (f) Use your model in part (e) to estimate the rate of growth in 1920 and 1980. Compare with your estimates in parts (a) and (d). (g) Estimate the rate of growth in 1985. ; 28. The table shows how the average age of first marriage of Japanese women has varied since 1950.

(c) What happens to the concentration as t l `? (d) What happens to the rate of reaction as t l `? (e) What do the results of parts (c) and (d) mean in practical terms? 25. In Example 6 we considered a bacteria population that doubles every hour. Suppose that another population of bacteria triples every hour and starts with 400 bacteria. Find an expression for the number n of bacteria after t hours and use it to estimate the rate of growth of the bacteria population after 2.5 hours. 26. The number of yeast cells in a laboratory culture increases rapidly initially but levels off eventually. The population is modeled by the function a n − f std − 1 1 be20.7t where t is measured in hours. At time t − 0 the population is 20 cells and is increasing at a rate of 12 cellsyhour. Find the values of a and b. According to this model, what happens to the yeast population in the long run? ; 27. The table gives the population of the world Pstd, in millions, where t is measured in years and t − 0 corresponds to the year 1900.

t

Population (millions)

t

Population (millions)

0 10 20 30 40 50

1650 1750 1860 2070 2300 2560

60 70 80 90 100 110

3040 3710 4450 5280 6080 6870

(a) Estimate the rate of population growth in 1920 and in 1980 by averaging the slopes of two secant lines. (b) Use a graphing device to find a cubic function (a thirddegree polynomial) that models the data. (c) Use your model in part (b) to find a model for the rate of population growth. (d) Use part (c) to estimate the rates of growth in 1920 and 1980. Compare with your estimates in part (a).

t

Astd

t

Astd

1950 1955 1960 1965 1970 1975 1980

23.0 23.8 24.4 24.5 24.2 24.7 25.2

1985 1990 1995 2000 2005 2010

25.5 25.9 26.3 27.0 28.0 28.8

(a) Use a graphing calculator or computer to model these data with a fourth-degree polynomial. (b) Use part (a) to find a model for A9std. (c) Estimate the rate of change of marriage age for women in 1990. (d) Graph the data points and the models for A and A9. 29. Refer to the law of laminar flow given in Example 7. Consider a blood vessel with radius 0.01 cm, length 3 cm, pressure difference 3000 dynesycm2, and viscosity  − 0.027. (a) Find the velocity of the blood along the centerline r − 0, at radius r − 0.005 cm, and at the wall r − R − 0.01 cm. (b) Find the velocity gradient at r − 0, r − 0.005, and r − 0.01. (c) Where is the velocity the greatest? Where is the velocity changing most? 30. The frequency of vibrations of a vibrating violin string is given by 1 T f− 2L 

Î

where L is the length of the string, T is its tension, and  is its linear density. [See Chapter 11 in D. E. Hall, Musical Acoustics, 3rd ed. (Pacific Grove, CA: Brooks/Cole, 2002).] (a) Find the rate of change of the frequency with respect to (i) the length (when T and  are constant), (ii) the tension (when L and  are constant), and (iii) the linear density (when L and T are constant). (b) The pitch of a note (how high or low the note sounds) is determined by the frequency f . (The higher the frequency, the higher the pitch.) Use the signs of the

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derivatives in part (a) to determine what happens to the pitch of a note (i) when the effective length of a string is decreased by placing a finger on the string so a shorter portion of the string vibrates, (ii) when the tension is increased by turning a tuning peg, (iii) when the linear density is increased by switching to another string. 31. Suppose that the cost (in dollars) for a company to produce x pairs of a new line of jeans is Csxd − 2000 1 3x 1 0.01x 2 1 0.0002x 3 (a) Find the marginal cost function. (b) Find C9s100d and explain its meaning. What does it predict? (c) Compare C9s100d with the cost of manufacturing the 101st pair of jeans. 32. The cost function for a certain commodity is Csqd − 84 1 0.16q 2 0.0006q 2 1 0.000003q 3 (a) Find and interpret C9s100d. (b) Compare C9s100d with the cost of producing the 101st item. 33. If psxd is the total value of the production when there are x workers in a plant, then the average productivity of the workforce at the plant is psxd Asxd − x (a) Find A9sxd. Why does the company want to hire more workers if A9sxd . 0? (b) Show that A9sxd . 0 if p9sxd is greater than the average productivity. 34. If R denotes the reaction of the body to some stimulus of strength x, the sensitivity S is defined to be the rate of change of the reaction with respect to x. A particular example is that when the brightness x of a light source is increased, the eye reacts by decreasing the area R of the pupil. The experimental formula R−

;

40 1 24x 0.4 1 1 4x 0.4

has been used to model the dependence of R on x when R is measured in square millimeters and x is measured in appropriate units of brightness. (a) Find the sensitivity. (b) Illustrate part (a) by graphing both R and S as functions of x. Comment on the values of R and S at low levels of brightness. Is this what you would expect? 35. Patients undergo dialysis treatment to remove urea from their blood when their kidneys are not functioning properly. Blood is diverted from the patient through a machine that filters out urea. Under certain conditions, the duration of dialysis required, given that the initial urea concentration is c . 1, is

given by the equation

S

t − ln

3c 1 s9c 2 2 8c 2

D

Calculate the derivative of t with respect to c and interpret it. 36. Invasive species often display a wave of advance as they colonize new areas. Mathematical models based on random dispersal and reproduction have demonstrated that the speed with which such waves move is given by the function f srd − 2 sDr , where r is the reproductive rate of individuals and D is a parameter quantifying dispersal. Calculate the derivative of the wave speed with respect to the reproductive rate r and explain its meaning. 37. The gas law for an ideal gas at absolute temperature T (in kelvins), pressure P (in atmospheres), and volume V (in liters) is PV − nRT, where n is the number of moles of the gas and R − 0.0821 is the gas constant. Suppose that, at a certain instant, P − 8.0 atm and is increasing at a rate of 0.10 atmymin and V − 10 L and is decreasing at a rate of 0.15 Lymin. Find the rate of change of T with respect to time at that instant if n − 10 mol. 38. In a fish farm, a population of fish is introduced into a pond and harvested regularly. A model for the rate of change of the fish population is given by the equation

S

D

dP Pstd Pstd 2 Pstd − r0 1 2 dt Pc where r0 is the birth rate of the fish, Pc is the maximum population that the pond can sustain (called the carrying capacity), and  is the percentage of the population that is harvested. (a) What value of dPydt corresponds to a stable population? (b) If the pond can sustain 10,000 fish, the birth rate is 5%, and the harvesting rate is 4%, find the stable population level. (c) What happens if  is raised to 5%? 39. In the study of ecosystems, predator-prey models are often used to study the interaction between species. Consider populations of tundra wolves, given by Wstd, and caribou, given by Cstd, in northern Canada. The interaction has been modeled by the equations dC − aC 2 bCW dt

dW − 2cW 1 dCW dt

(a) What values of dCydt and dWydt correspond to stable populations? (b) How would the statement “The caribou go extinct” be represented mathematically? (c) Suppose that a − 0.05, b − 0.001, c − 0.05, and d − 0.0001. Find all population pairs sC, W d that lead to stable populations. According to this model, is it possible for the two species to live in balance or will one or both species become extinct?

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SECTION 3.8

Exponential Growth and Decay

237

In many natural phenomena, quantities grow or decay at a rate proportional to their size. For instance, if y − f std is the number of individuals in a population of animals or bacteria at time t, then it seems reasonable to expect that the rate of growth f 9std is proportional to the population f std; that is, f 9std − kf std for some constant k. Indeed, under ideal conditions (unlimited environment, adequate nutrition, immunity to disease) the mathematical model given by the equation f 9std − kf std predicts what actually happens fairly accurately. Another example occurs in nuclear physics where the mass of a radioactive substance decays at a rate proportional to the mass. In chemistry, the rate of a unimolecular first-order reaction is proportional to the concentration of the substance. In finance, the value of a savings account with continuously compounded interest increases at a rate proportional to that value. In general, if ystd is the value of a quantity y at time t and if the rate of change of y with respect to t is proportional to its size ystd at any time, then dy − ky dt

1

where k is a constant. Equation 1 is sometimes called the law of natural growth (if k . 0d or the law of natural decay (if k , 0). It is called a differential equation because it involves an unknown function y and its derivative dyydt. It’s not hard to think of a solution of Equation 1. This equation asks us to find a function whose derivative is a constant multiple of itself. We have met such functions in this chapter. Any exponential function of the form ystd − Ce kt, where C is a constant, satisfies y9std − Cske kt d − ksCe kt d − kystd We will see in Section 9.4 that any function that satisfies dyydt − ky must be of the form y − Ce kt. To see the significance of the constant C, we observe that ys0d − Ce k?0 − C Therefore C is the initial value of the function. 2 Theorem The only solutions of the differential equation dyydt − ky are the exponential functions ystd − ys0de kt

Population Growth What is the significance of the proportionality constant k? In the context of population growth, where Pstd is the size of a population at time t, we can write 3 The quantity

dP − kP dt

or

1 dP −k P dt

1 dP P dt

is the growth rate divided by the population size; it is called the relative growth rate.

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According to (3), instead of saying “the growth rate is proportional to population size” we could say “the relative growth rate is constant.” Then (2) says that a population with constant relative growth rate must grow exponentially. Notice that the relative growth rate k appears as the coefficient of t in the exponential function Ce kt. For instance, if dP − 0.02P dt and t is measured in years, then the relative growth rate is k − 0.02 and the population grows at a relative rate of 2% per year. If the population at time 0 is P0, then the expression for the population is Pstd − P0 e 0.02t

ExamplE 1 Use the fact that the world population was 2560 million in 1950 and 3040 million in 1960 to model the population of the world in the second half of the 20th century. (Assume that the growth rate is proportional to the population size.) What is the relative growth rate? Use the model to estimate the world population in 1993 and to predict the population in the year 2020. SOLUTION We measure the time t in years and let t − 0 in the year 1950. We measure the population Pstd in millions of people. Then Ps0d − 2560 and Ps10d − 3040. Since we are assuming that dPydt − kP, Theorem 2 gives

Pstd − Ps0de kt − 2560e kt Ps10d − 2560e 10k − 3040 k−

1 3040 ln < 0.017185 10 2560

The relative growth rate is about 1.7% per year and the model is Pstd − 2560e 0.017185t We estimate that the world population in 1993 was Ps43d − 2560e 0.017185s43d < 5360 million The model predicts that the population in 2020 will be Ps70d − 2560e 0.017185s70d < 8524 million The graph in Figure 1 shows that the model is fairly accurate to the end of the 20th century (the dots represent the actual population), so the estimate for 1993 is quite reliable. But the prediction for 2020 is riskier. P 6000

P=2560e 0.017185t

Population (in millions)

FIGURE 1 A model for world population growth in the second half of the 20th century

0

20

Years since 1950

40

t

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SECTION 3.8

Exponential Growth and Decay

239

Radioactive Decay Radioactive substances decay by spontaneously emitting radiation. If mstd is the mass remaining from an initial mass m0 of the substance after time t, then the relative decay rate 1 dm 2 m dt has been found experimentally to be constant. (Since dmydt is negative, the relative decay rate is positive.) It follows that dm − km dt where k is a negative constant. In other words, radioactive substances decay at a rate proportional to the remaining mass. This means that we can use (2) to show that the mass decays exponentially: mstd − m0 e kt Physicists express the rate of decay in terms of half-life, the time required for half of any given quantity to decay.

ExamplE 2 The half-life of radium-226 is 1590 years. (a) A sample of radium-226 has a mass of 100 mg. Find a formula for the mass of the sample that remains after t years. (b) Find the mass after 1000 years correct to the nearest milligram. (c) When will the mass be reduced to 30 mg? SOLUTION

(a) Let mstd be the mass of radium-226 (in milligrams) that remains after t years. Then dmydt − km and ms0d − 100, so (2) gives mstd − ms0de kt − 100e kt In order to determine the value of k, we use the fact that ms1590d − 12 s100d. Thus 100e 1590k − 50

so

e 1590k − 12

1590k − ln 12 − 2ln 2

and

k−2

ln 2 1590

mstd − 100e2sln 2dty1590

Therefore

We could use the fact that e ln 2 − 2 to write the expression for mstd in the alternative form mstd − 100 3 2 2ty1590 (b) The mass after 1000 years is ms1000d − 100e2sln 2d1000y1590 < 65 mg (c) We want to find the value of t such that mstd − 30, that is, 100e2sln 2dty1590 − 30

or

e2sln 2dty1590 − 0.3

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We solve this equation for t by taking the natural logarithm of both sides: 150

2

ln 2 t − ln 0.3 1590

m=100e_(ln 2)t/1590

Thus

t − 21590

m=30 4000

0

FIGURE 2

ln 0.3 < 2762 years ln 2

As a check on our work in Example 2, we use a graphing device to draw the graph of mstd in Figure 2 together with the horizontal line m − 30. These curves intersect when t < 2800, and this agrees with the answer to part (c).

Newton’s Law of Cooling Newton’s Law of Cooling states that the rate of cooling of an object is proportional to the temperature difference between the object and its surroundings, provided that this difference is not too large. (This law also applies to warming.) If we let Tstd be the temperature of the object at time t and Ts be the temperature of the surroundings, then we can formulate Newton’s Law of Cooling as a differential equation: dT − ksT 2 Ts d dt where k is a constant. This equation is not quite the same as Equation 1, so we make the change of variable ystd − Tstd 2 Ts . Because Ts is constant, we have y9std − T 9std and so the equation becomes dy − ky dt We can then use (2) to find an expression for y, from which we can find T.

ExamplE 3 A bottle of soda pop at room temperature (22°C) is placed in a refrigerator where the temperature is 7°C. After half an hour the soda pop has cooled to 16°C. (a) What is the temperature of the soda pop after another half hour? (b) How long does it take for the soda pop to cool to 10°C? SOlUTION (a) Let Tstd be the temperature of the soda after t minutes. The surrounding temperature is Ts − 78C, so Newton’s Law of Cooling states that dT − ksT 2 7d dt If we let y − T 2 7, then ys0d − Ts0d 2 7 − 22 2 7 − 15, so y is a solution of the initial-value problem, dy − ky dt

ys0d − 15

and by (2) we have ystd − ys0de kt − 15e kt We are given that Ts30d − 16, so ys30d − 16 2 7 − 9 and 15e 30k − 9

e 30k − 35

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SECTION 3.8

Exponential Growth and Decay

241

Taking logarithms, we have k−

ln ( 35 ) < 20.01703 30

Thus ystd − 15e 20.01703t Tstd − 7 1 15e 20.01703t Ts60d − 7 1 15e 20.01703s60d < 12.4 So after another half hour the pop has cooled to about 12°C. (b) We have Tstd − 10 when 7 1 15e 20.01703t − 10 3 e 20.01703t − 15 − 15

t−

T

ln ( 15 ) < 94.5 20.01703

22

The pop cools to 10°C after about 1 hour 35 minutes. 7 0

Notice that in Example 3, we have 30

60

90

t

lim Tstd − lim s7 1 15e 20.01703t d − 7 1 15  0 − 7

tl`

FIGURE 3

tl`

which is to be expected. The graph of the temperature function is shown in Figure 3.

Continuously Compounded Interest ExamplE 4 If \$1000 is invested at 6% interest, compounded annually, then after 1 year the investment is worth \$1000s1.06d − \$1060, after 2 years it’s worth \$f1000s1.06dg1.06 − \$1123.60, and after t years it’s worth \$1000s1.06dt. In general, if an amount A0 is invested at an interest rate r sr − 0.06 in this example), then after t years it’s worth A0 s1 1 rd t. Usually, however, interest is compounded more frequently, say, n times a year. Then in each compounding period the interest rate is ryn and there are nt compounding periods in t years, so the value of the investment is

S D

A0 1 1

r n

nt

For instance, after 3 years at 6% interest a \$1000 investment will be worth \$1000s1.06d3 − \$1191.02 with annual compounding \$1000s1.03d6 − \$1194.05 with semiannual compounding \$1000s1.015d12 − \$1195.62 with quarterly compounding \$1000s1.005d36 − \$1196.68 with monthly compounding

S

\$1000 1 1

0.06 365

D

365 ? 3

− \$1197.20 with daily compounding

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Differentiation Rules

You can see that the interest paid increases as the number of compounding periods snd increases. If we let n l `, then we will be compounding the interest continuously and the value of the investment will be

S FS F S F S

r n

D

− lim A0

11

r n

− A0 lim

11

r n

− A0 lim

11

1 m

Astd − lim A0 1 1 nl`

nl`

nl`

ml `

nt

DG DG DG nyr

rt

nyr

rt

m

rt

(where m − nyr)

But the limit in this expression is equal to the number e (see Equation 3.6.6). So with continuous compounding of interest at interest rate r, the amount after t years is Astd − A0 e rt If we differentiate this equation, we get dA − rA0 e rt − rAstd dt which says that, with continuous compounding of interest, the rate of increase of an investment is proportional to its size. Returning to the example of \$1000 invested for 3 years at 6% interest, we see that with continuous compounding of interest the value of the investment will be As3d − \$1000e s0.06d3 − \$1197.22 Notice how close this is to the amount we calculated for daily compounding, \$1197.20. But the amount is easier to compute if we use continuous compounding. ■

3.8 EXERCISES 1. A population of protozoa develops with a constant relative growth rate of 0.6028 per member per day. On day zero the population consists of two members. Find the population size after seven days. 2. A common inhabitant of human intestines is the bacterium Escherichia coli, named after the German pediatrician Theodor Escherich, who identified it in 1885. A cell of this bacterium in a nutrient-broth medium divides into two cells every 20 minutes. The initial population of a culture is 50 cells. (a) Find the relative growth rate. (b) Find an expression for the number of cells after t hours.

(c) Find the number of cells after 6 hours. (d) Find the rate of growth after 6 hours. (e) When will the population reach a million cells? 3. A bacteria culture initially contains 100 cells and grows at a rate proportional to its size. After an hour the population has increased to 420. (a) Find an expression for the number of bacteria after t hours. (b) Find the number of bacteria after 3 hours. (c) Find the rate of growth after 3 hours. (d) When will the population reach 10,000?

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SECTION 3.8

4. A bacteria culture grows with constant relative growth rate. The bacteria count was 400 after 2 hours and 25,600 after 6 hours. (a) What is the relative growth rate? Express your answer as a percentage. (b) What was the initial size of the culture? (c) Find an expression for the number of bacteria after t hours. (d) Find the number of cells after 4.5 hours. (e) Find the rate of growth after 4.5 hours. (f) When will the population reach 50,000? 5. The table gives estimates of the world population, in millions, from 1750 to 2000. Year

Population

Year

Population

1750 1800 1850

790 980 1260

1900 1950 2000

1650 2560 6080

(a) Use the exponential model and the population figures for 1750 and 1800 to predict the world population in 1900 and 1950. Compare with the actual figures. (b) Use the exponential model and the population figures for 1850 and 1900 to predict the world population in 1950. Compare with the actual population. (c) Use the exponential model and the population figures for 1900 and 1950 to predict the world population in 2000. Compare with the actual population and try to explain the discrepancy. 6. The table gives the population of Indonesia, in millions, for the second half of the 20th century. Year

Population

1950 1960 1970 1980 1990 2000

83 100 122 150 182 214

(a) Assuming the population grows at a rate proportional to its size, use the census figures for 1950 and 1960 to predict the population in 1980. Compare with the actual figure. (b) Use the census figures for 1960 and 1980 to predict the population in 2000. Compare with the actual population. (c) Use the census figures for 1980 and 2000 to predict the population in 2010 and compare with the actual population of 243 million. (d) Use the model in part (c) to predict the population in 2020. Do you think the prediction will be too high or too low? Why?

Exponential Growth and Decay

243

7. Experiments show that if the chemical reaction N2O5 l 2NO 2 1 12 O 2 takes place at 45 8C, the rate of reaction of dinitrogen pentoxide is proportional to its concentration as follows: 2

dfN2O5g − 0.0005fN2O5g dt

(See Example 3.7.4.) (a) Find an expression for the concentration fN2O5g after t seconds if the initial concentration is C. (b) How long will the reaction take to reduce the concentration of N2O5 to 90% of its original value? 8. Strontium-90 has a half-life of 28 days. (a) A sample has a mass of 50 mg initially. Find a formula for the mass remaining after t days. (b) Find the mass remaining after 40 days. (c) How long does it take the sample to decay to a mass of 2 mg? (d) Sketch the graph of the mass function. 9. The half-life of cesium-137 is 30 years. Suppose we have a 100-mg sample. (a) Find the mass that remains after t years. (b) How much of the sample remains after 100 years? (c) After how long will only 1 mg remain? 10. A sample of tritium-3 decayed to 94.5% of its original amount after a year. (a) What is the half-life of tritium-3? (b) How long would it take the sample to decay to 20% of its original amount? 11. Scientists can determine the age of ancient objects by the method of radiocarbon dating. The bombardment of the upper atmosphere by cosmic rays converts nitrogen to a radioactive isotope of carbon, 14C, with a half-life of about 5730 years. Vegetation absorbs carbon dioxide through the atmosphere and animal life assimilates 14C through food chains. When a plant or animal dies, it stops replacing its carbon and the amount of 14 C begins to decrease through radioactive decay. Therefore the level of radioactivity must also decay exponentially. A discovery revealed a parchment fragment that had about 74% as much 14C radioactivity as does plant material on the earth today. Estimate the age of the parchment. 12. Dinosaur fossils are too old to be reliably dated using carbon-14. (See Exercise 11.) Suppose we had a 68-millionyear-old dinosaur fossil. What fraction of the living dinosaur’s 14C would be remaining today? Suppose the minimum detectable amount is 0.1%. What is the maximum age of a fossil that we could date using 14C? 13. Dinosaur fossils are often dated by using an element other than carbon, such as potassium-40, that has a longer half-life (in this case, approximately 1.25 billion years). Suppose the minimum detectable amount is 0.1% and a dinosaur is dated

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244

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with 40 K to be 68 million years old. Is such a dating possible? In other words, what is the maximum age of a fossil that we could date using 40 K? 14. A curve passes through the point s0, 5d and has the property that the slope of the curve at every point P is twice the y-coordinate of P. What is the equation of the curve? 15. A roast turkey is taken from an oven when its temperature has reached 85°C and is placed on a table in a room where the temperature is 22°C. (a) If the temperature of the turkey is 65 8C after half an hour, what is the temperature after 45 minutes? (b) When will the turkey have cooled to 40 8C? 16. In a murder investigation, the temperature of the corpse was 32.5°C at 1:30 pm and 30.3°C an hour later. Normal body temperature is 37.0°C and the temperature of the surroundings was 20.0°C. When did the murder take place?

19. The rate of change of atmospheric pressure P with respect to altitude h is proportional to P, provided that the temperature is constant. At 15°C the pressure is 101.3 kPa at sea level and 87.14 kPa at h − 1000 m. (a) What is the pressure at an altitude of 3000 m? (b) What is the pressure at the top of Mount McKinley, at an altitude of 6187 m? 20. (a) If \$1000 is borrowed at 8% interest, find the amounts due at the end of 3 years if the interest is compounded (i) annually, (ii) quarterly, (iii) monthly, (iv) weekly, (v) daily, (vi) hourly, and (vii) continuously. (b) Suppose \$1000 is borrowed and the interest is com; pounded continuously. If Astd is the amount due after t years, where 0 < t < 3, graph Astd for each of the interest rates 6%, 8%, and 10% on a common screen.

17. When a cold drink is taken from a refrigerator, its temperature is 5°C. After 25 minutes in a 20°C room its temperature has increased to 10°C. (a) What is the temperature of the drink after 50 minutes? (b) When will its temperature be 15°C?

21. (a) If \$3000 is invested at 5% interest, find the value of the investment at the end of 5 years if the interest is compounded (i) annually, (ii) semiannually, (iii) monthly, (iv) weekly, (v) daily, and (vi) continuously. (b) If Astd is the amount of the investment at time t for the case of continuous compounding, write a differential equation and an initial condition satisfied by Astd.

18. A freshly brewed cup of coffee has temperature 958C in a 20°C room. When its temperature is 70°C, it is cooling at a rate of 1°C per minute. When does this occur?

22. (a) How long will it take an investment to double in value if the interest rate is 6% compounded continuously? (b) What is the equivalent annual interest rate?

APPLIED PROJECT

CONTROLLING RED BLOOD CELL LOSS DURING SURGERY A typical volume of blood in the human body is about 5 L. A certain percentage of that volume (called the hematocrit) consists of red blood cells (RBCs); typically the hematocrit is about 45% in males. Suppose that a surgery takes four hours and a male patient bleeds 2.5 L of blood. During surgery the patient’s blood volume is maintained at 5 L by injection of saline solution, which mixes quickly with the blood but dilutes it so that the hematocrit decreases as time passes.

1. Assuming that the rate of RBC loss is proportional to the volume of RBCs, determine the patient’s volume of RBCs by the end of the operation. 2. A procedure called acute normovolemic hemodilution (ANH) has been developed to minimize RBC loss during surgery. In this procedure blood is extracted from the patient before the operation and replaced with saline solution. This dilutes the patient’s blood, resulting in fewer RBCs being lost during the bleeding. The extracted blood is then returned to the patient after surgery. Only a certain amount of blood can be extracted, however, because the RBC concentration can never be allowed to drop below 25% during surgery. What is the maximum amount of blood that can be extracted in the ANH procedure for the surgery described in this project? 3. What is the RBC loss without the ANH procedure? What is the loss if the procedure is carried out with the volume calculated in Problem 2?

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SECTION 3.9

Related Rates

245

If we are pumping air into a balloon, both the volume and the radius of the balloon are increasing and their rates of increase are related to each other. But it is much easier to measure directly the rate of increase of the volume than the rate of increase of the radius. In a related rates problem the idea is to compute the rate of change of one quantity in terms of the rate of change of another quantity (which may be more easily measured). The procedure is to find an equation that relates the two quantities and then use the Chain Rule to differentiate both sides with respect to time.

ExamplE 1 Air is being pumped into a spherical balloon so that its volume increases at a rate of 100 cm3ys. How fast is the radius of the balloon increasing when the diameter is 50 cm? PS According to the Principles of Problem Solving discussed on page 71, the first step is to understand the problem. This includes reading the problem carefully, identifying the given and the unknown, and introducing suitable notation.

SOLUTION We start by identifying two things:

the given information: the rate of increase of the volume of air is 100 cm3ys and the unknown: the rate of increase of the radius when the diameter is 50 cm In order to express these quantities mathematically, we introduce some suggestive notation: Let V be the volume of the balloon and let r be its radius. The key thing to remember is that rates of change are derivatives. In this problem, the volume and the radius are both functions of the time t. The rate of increase of the volume with respect to time is the derivative dVydt, and the rate of increase of the radius is drydt. We can therefore restate the given and the unknown as follows:

PS The second stage of problem solving is to think of a plan for connecting the given and the unknown.

Given:

dV − 100 cm3ys dt

Unknown:

dr dt

when r − 25 cm

In order to connect dVydt and drydt, we first relate V and r by the formula for the volume of a sphere: V − 43 r 3 In order to use the given information, we differentiate each side of this equation with respect to t. To differentiate the right side, we need to use the Chain Rule: dV dV dr dr − − 4r 2 dt dr dt dt Now we solve for the unknown quantity:

Notice that, although dVydt is constant, drydt is not constant.

dr 1 dV − dt 4r 2 dt

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If we put r − 25 and dVydt − 100 in this equation, we obtain dr 1 1 100 − − dt 4s25d2 25 The radius of the balloon is increasing at the rate of 1ys25d < 0.0127 cmys.

ExamplE 2 A ladder 5 m long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 1 mys, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 3 m from the wall?

wall

5

y

x

ground

FIGURE 1

SOLUTION We first draw a diagram and label it as in Figure 1. Let x meters be the distance from the bottom of the ladder to the wall and y meters the distance from the top of the ladder to the ground. Note that x and y are both functions of t (time, measured in seconds). We are given that dxydt − 1 mys and we are asked to find dyydt when x − 3 m (see Figure 2). In this problem, the relationship between x and y is given by the Pythagorean Theorem: x 2 1 y 2 − 25

Differentiating each side with respect to t using the Chain Rule, we have dy dt

=?

2x

dx dy 1 2y −0 dt dt

y

and solving this equation for the desired rate, we obtain x dx dt

dy x dx −2 dt y dt =1

When x − 3, the Pythagorean Theorem gives y − 4 and so, substituting these values and dxydt − 1, we have

FIGURE 2

dy 3 3 − 2 s1d − 2 mys dt 4 4 The fact that dyydt is negative means that the distance from the top of the ladder to the ground is decreasing at a rate of 34 mys. In other words, the top of the ladder is sliding down the wall at a rate of 34 mys. ■

ExamplE 3 A water tank has the shape of an inverted circular cone with base radius 2 m and height 4 m. If water is being pumped into the tank at a rate of 2 m 3ymin, find the rate at which the water level is rising when the water is 3 m deep. 2

r 4 h

SOLUTION We first sketch the cone and label it as in Figure 3. Let V, r, and h be the volume of the water, the radius of the surface, and the height of the water at time t, where t is measured in minutes. We are given that dVydt − 2 m 3ymin and we are asked to find dhydt when h is 3 m. The quantities V and h are related by the equation

V − 13 r 2h FIGURE 3

but it is very useful to express V as a function of h alone. In order to eliminate r, we use

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SECTION 3.9

Related Rates

247

the similar triangles in Figure 3 to write r 2 − h 4 and the expression for V becomes V−

SD

h 1  3 2

r−

2

h−

h 2

3 h 12

Now we can differentiate each side with respect to t: dV  2 dh − h dt 4 dt so

dh 4 dV − dt h 2 dt

Substituting h − 3 m and dVydt − 2 m 3ymin, we have dh 4 8 − ?2− dt s3d2 9 The water level is rising at a rate of 8ys9d < 0.28 mymin. PS Look back: What have we learned from Examples 1–3 that will help us solve future problems?

WARNING A common error is to substitute the given numerical information (for quantities that vary with time) too early. This should be done only after the differentiation. (Step 7 follows Step 6.) For instance, in Example 3 we dealt with general values of h until we finally substituted h − 3 at the last stage. (If we had put h − 3 earlier, we would have gotten dVydt − 0, which is clearly wrong.)

Problem Solving Strategy It is useful to recall some of the problem-solving principles from page 71 and adapt them to related rates in light of our experience in Examples 1–3: 1. Read the problem carefully. 2. Draw a diagram if possible. 3. Introduce notation. Assign symbols to all quantities that are functions of time. 4. Express the given information and the required rate in terms of derivatives. 5. Write an equation that relates the various quantities of the problem. If necessary, use the geometry of the situation to eliminate one of the variables by substitution (as in Example 3). 6. Use the Chain Rule to differentiate both sides of the equation with respect to t. 7. Substitute the given information into the resulting equation and solve for the unknown rate. The following examples are further illustrations of the strategy.

C

x

y z B

FIGURE 4

A

ExamplE 4 Car A is traveling west at 90 kmyh and car B is traveling north at 100 kmyh. Both are headed for the intersection of the two roads. At what rate are the cars approaching each other when car A is 60 m and car B is 80 m from the intersection? SOLUTION We draw Figure 4, where C is the intersection of the roads. At a given time t, let x be the distance from car A to C, let y be the distance from car B to C, and let z be the distance between the cars, where x, y, and z are measured in kilometers. We are given that dxydt − 290 kmyh and dyydt − 2100 kmyh. (The derivatives are negative because x and y are decreasing.) We are asked to find dzydt. The equation

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248

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Differentiation Rules

that relates x, y, and z is given by the Pythagorean Theorem: z2 − x 2 1 y 2 Differentiating each side with respect to t, we have 2z

dz dx dy − 2x 1 2y dt dt dt dz 1 − dt z

S

x

dx dy 1y dt dt

D

When x − 0.06 km and y − 0.08 km, the Pythagorean Theorem gives z − 0.1 km, so dz 1 − f0.06s290d 1 0.08s2100dg dt 0.1 − 2134 kmyh The cars are approaching each other at a rate of 134 kmyh.

ExamplE 5 A man walks along a straight path at a speed of 1.5 mys. A searchlight is located on the ground 6 m from the path and is kept focused on the man. At what rate is the searchlight rotating when the man is 8 m from the point on the path closest to the searchlight? SOLUTION We draw Figure 5 and let x be the distance from the man to the point on the path closest to the searchlight. We let  be the angle between the beam of the searchlight and the perpendicular to the path. We are given that dxydt − 1.5 mys and are asked to find dydt when x − 8. The equation that relates x and  can be written from Figure 5: x

x − tan  6

6

x − 6 tan

¨

Differentiating each side with respect to t, we get dx d − 6 sec2 dt dt

FIGURE 5

so

d 1 dx − cos2 dt 6 dt −

1 1 cos2 s1.5d − cos2 6 4

When x − 8, the length of the beam is 10, so cos  − 35 and d 1 − dt 4

SD 3 5

2

9 − 0.09 100

The searchlight is rotating at a rate of 0.09 radys.

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SECTION 3.9

Related Rates

249

3.9 EXERCISES 1. If V is the volume of a cube with edge length x and the cube expands as time passes, find dVydt in terms of dxydt.

3 cmys. How fast is the x-coordinate of the point changing at that instant?

2. (a) If A is the area of a circle with radius r and the circle expands as time passes, find dAydt in terms of drydt. (b) Suppose oil spills from a ruptured tanker and spreads in a circular pattern. If the radius of the oil spill increases at a constant rate of 1 mys, how fast is the area of the spill increasing when the radius is 30 m?

13–16 (a) What quantities are given in the problem? (b) What is the unknown? (c) Draw a picture of the situation for any time t. (d) Write an equation that relates the quantities. (e) Finish solving the problem.

3. Each side of a square is increasing at a rate of 6 cmys. At what rate is the area of the square increasing when the area of the square is 16 cm2?

13. A plane flying horizontally at an altitude of 2 km and a speed of 800 kmyh passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is 3 km away from the station.

4. The length of a rectangle is increasing at a rate of 8 cmys and its width is increasing at a rate of 3 cmys. When the length is 20 cm and the width is 10 cm, how fast is the area of the rectangle increasing? 5. A cylindrical tank with radius 5 m is being filled with water at a rate of 3 m3ymin. How fast is the height of the water increasing? 6. The radius of a sphere is increasing at a rate of 4 mmys. How fast is the volume increasing when the diameter is 80 mm? 7. The radius of a spherical ball is increasing at a rate of 2 cmymin. At what rate is the surface area of the ball increasing when the radius is 8 cm? 8. The area of a triangle with sides of lengths a and b and contained angle  is A − 12 ab sin  (a) If a − 2 cm, b − 3 cm, and  increases at a rate of 0.2 radymin, how fast is the area increasing when  − y3? (b) If a − 2 cm, b increases at a rate of 1.5 cmymin, and  increases at a rate of 0.2 radymin, how fast is the area increasing when b − 3 cm and  − y3? (c) If a increases at a rate of 2.5 cmymin, b increases at a rate of 1.5 cmymin, and  increases at a rate of 0.2 radymin, how fast is the area increasing when a − 2 cm, b − 3 cm, and  − y3? 9. Suppose y − s2x 1 1 , where x and y are functions of t. (a) If dxydt − 3, find dyydt when x − 4. (b) If dyydt − 5, find dxydt when x − 12.

14. If a snowball melts so that its surface area decreases at a rate of 1 cm2ymin, find the rate at which the diameter decreases when the diameter is 10 cm. 15. A street light is mounted at the top of a 6-meter-tall pole. A man 2 m tall walks away from the pole with a speed of 1.5 mys along a straight path. How fast is the tip of his shadow moving when he is 10 m from the pole? 16. At noon, ship A is 150 km west of ship B. Ship A is sailing east at 35 kmyh and ship B is sailing north at 25 kmyh. How fast is the distance between the ships changing at 4:00 pm? 17. Two cars start moving from the same point. One travels south at 30 kmyh and the other travels west at 72 kmyh. At what rate is the distance between the cars increasing two hours later? 18. A spotlight on the ground shines on a wall 12 m away. If a man 2 m tall walks from the spotlight toward the building at a speed of 1.6 mys, how fast is the length of his shadow on the building decreasing when he is 4 m from the building? 19. A man starts walking north at 1.2 mys from a point P. Five minutes later a woman starts walking south at 1.6 mys from a point 200 m due east of P. At what rate are the people moving apart 15 min after the woman starts walking? 20. A baseball diamond is a square with side 90 ft. A batter hits the ball and runs toward first base with a speed of 24 ftys. (a) At what rate is his distance from second base decreasing when he is halfway to first base? (b) At what rate is his distance from third base increasing at the same moment?

10. Suppose 4x 2 1 9y 2 − 36, where x and y are functions of t. (a) If dyydt − 13, find dxydt when x − 2 and y − 23 s5 . (b) If dxydt − 3, find dy ydt when x − 22 and y − 23 s5 . 11. If x 2 1 y 2 1 z 2 − 9, dxydt − 5, and dyydt − 4, find dzydt when sx, y, zd − s2, 2, 1d. 12. A particle is moving along a hyperbola xy − 8. As it reaches the point s4, 2d, the y-coordinate is decreasing at a rate of

90 ft

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250

Chapter 3

Differentiation Rules

21. The altitude of a triangle is increasing at a rate of 1 cmymin while the area of the triangle is increasing at a rate of 2 cm 2ymin. At what rate is the base of the triangle changing when the altitude is 10 cm and the area is 100 cm2?

the shape of a cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is 3 m high?

22. A boat is pulled into a dock by a rope attached to the bow of the boat and passing through a pulley on the dock that is 1 m higher than the bow of the boat. If the rope is pulled in at a rate of 1 mys, how fast is the boat approaching the dock when it is 8 m from the dock?

23. At noon, ship A is 100 km west of ship B. Ship A is sailing south at 35 kmyh and ship B is sailing north at 25 kmyh. How fast is the distance between the ships changing at 4:00 pm? 24. A particle moves along the curve y − 2 sinsxy2d. As the particle passes through the point ( 13 , 1), its x-coordinate increases at a rate of s10 cmys. How fast is the distance from the particle to the origin changing at this instant? 25. Water is leaking out of an inverted conical tank at a rate of 10,000 cm 3ymin at the same time that water is being pumped into the tank at a constant rate. The tank has height 6 m and the diameter at the top is 4 m. If the water level is rising at a rate of 20 cmymin when the height of the water is 2 m, find the rate at which water is being pumped into the tank. 26. A trough is 6 m long and its ends have the shape of isosceles triangles that are 1 m across at the top and have a height of 50 cm. If the trough is being filled with water at a rate of 1.2 m 3ymin, how fast is the water level rising when the water is 30 cm deep? 27. A water trough is 10 m long and a cross-section has the shape of an isosceles trapezoid that is 30 cm wide at the bottom, 80 cm wide at the top, and has height 50 cm. If the trough is being filled with water at the rate of 0.2 m 3ymin, how fast is the water level rising when the water is 30 cm deep? 28. A swimming pool is 5 m wide,10 m long, 1 m deep at the shallow end, and 3 m deep at its deepest point. A crosssection is shown in the figure. If the pool is being filled at a rate of 0.1 m 3ymin, how fast is the water level rising when the depth at the deepest point is 1 m? 1 2 1.5

3

4

1.5

29. Gravel is being dumped from a conveyor belt at a rate of 3 m 3ymin, and its coarseness is such that it forms a pile in

30. A kite 50 m above the ground moves horizontally at a speed of 2 mys. At what rate is the angle between the string and the horizontal decreasing when 100 m of string has been let out? 31. The sides of an equilateral triangle are increasing at a rate of 10 cmymin. At what rate is the area of the triangle increasing when the sides are 30 cm long? 32. How fast is the angle between the ladder and the ground changing in Example 2 when the bottom of the ladder is 3 m from the wall? 33. The top of a ladder slides down a vertical wall at a rate of 0.15 mys. At the moment when the bottom of the ladder is 3 m from the wall, it slides away from the wall at a rate of 0.2 mys. How long is the ladder? 34. According to the model we used to solve Example 2, what happens as the top of the ladder approaches the ground? Is the model appropriate for small values of y? 35. If the minute hand of a clock has length r (in centimeters), find the rate at which it sweeps out area as a function of r. ; 36. A faucet is filling a hemispherical basin of diameter 60 cm with water at a rate of 2 Lymin. Find the rate at which the water is rising in the basin when it is half full. [Use the following facts: 1 L is 1000 cm3. The volume of the portion of a sphere with radius r from the bottom to a height h is V −  (rh 2 2 13 h 3), as we will show in Chapter 6.] 37. Boyle’s Law states that when a sample of gas is compressed at a constant temperature, the pressure P and volume V satisfy the equation PV − C, where C is a constant. Suppose that at a certain instant the volume is 600 cm3, the pressure is 150 kPa, and the pressure is increasing at a rate of 20 kPaymin. At what rate is the volume decreasing at this instant? 38. When air expands adiabatically (without gaining or losing heat), its pressure P and volume V are related by the equation PV 1.4 − C, where C is a constant. Suppose that at a certain instant the volume is 400 cm3 and the pressure is 80 kPa and is decreasing at a rate of 10 kPaymin. At what rate is the volume increasing at this instant?

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SeCtion 3.10 Linear Approximations and Differentials

39. If two resistors with resistances R1 and R2 are connected in parallel, as in the figure, then the total resistance R, measured in ohms (V), is given by 1 1 1 − 1 R R1 R2 If R1 and R2 are increasing at rates of 0.3 Vys and 0.2 Vys, respectively, how fast is R changing when R1 − 80 V and R2 − 100 V?

R™

40. Brain weight B as a function of body weight W in fish has been modeled by the power function B − 0.007W 2y3, where B and W are measured in grams. A model for body weight as a function of body length L (measured in centimeters) is W − 0.12L2.53. If, over 10 million years, the average length of a certain species of fish evolved from 15 cm to 20 cm at a constant rate, how fast was this species’ brain growing when the average length was 18 cm? 41. Two sides of a triangle have lengths 12 m and 15 m. The angle between them is increasing at a rate of 2 8ymin. How fast is the length of the third side increasing when the angle between the sides of fixed length is 60°? 42. Two carts, A and B, are connected by a rope 12 m long that passes over a pulley P (see the figure). The point Q is on the floor 4 m directly beneath P and between the carts. Cart A is being pulled away from Q at a speed of 0.5 mys. How fast is cart B moving toward Q at the instant when cart A is 3 m from Q?

4m B Q

43. A television camera is positioned 1200 m from the base of a rocket launching pad. The angle of elevation of the camera has to change at the correct rate in order to keep the rocket in sight. Also, the mechanism for focusing the camera has to take into account the increasing distance from the camera to the rising rocket. Let’s assume the rocket rises vertically and its speed is 200 mys when it has risen 900 m. (a) How fast is the distance from the television camera to the rocket changing at that moment? (b) If the television camera is always kept aimed at the rocket, how fast is the camera’s angle of elevation changing at that same moment? 44. A lighthouse is located on a small island 3 km away from the nearest point P on a straight shoreline and its light makes four revolutions per minute. How fast is the beam of light moving along the shoreline when it is 1 km from P? 45. A plane flies horizontally at an altitude of 5 km and passes directly over a tracking telescope on the ground. When the angle of elevation is y3, this angle is decreasing at a rate of y6 radymin. How fast is the plane traveling at that time? 46. A Ferris wheel with a radius of 10 m is rotating at a rate of one revolution every 2 minutes. How fast is a rider rising when his seat is 16 m above ground level? 47. A plane flying with a constant speed of 300 kmyh passes over a ground radar station at an altitude of 1 km and climbs at an angle of 308. At what rate is the distance from the plane to the radar station increasing a minute later? 48. Two people start from the same point. One walks east at 4 kmyh and the other walks northeast at 2 kmyh. How fast is the distance between the people changing after 15 minutes? 49. A runner sprints around a circular track of radius 100 m at a constant speed of 7 mys. The runner’s friend is standing at a distance 200 m from the center of the track. How fast is the distance between the friends changing when the distance between them is 200 m?

P

A

251

50. The minute hand on a watch is 8 mm long and the hour hand is 4 mm long. How fast is the distance between the tips of the hands changing at one o’clock?

We have seen that a curve lies very close to its tangent line near the point of tangency. In fact, by zooming in toward a point on the graph of a differentiable function, we noticed that the graph looks more and more like its tangent line. (See Figure 2.7.2.) This observation is the basis for a method of finding approximate values of functions. The idea is that it might be easy to calculate a value f sad of a function, but difficult (or even impossible) to compute nearby values of f . So we settle for the easily computed

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252

Chapter 3

Differentiation Rules

values of the linear function L whose graph is the tangent line of f at sa, f sadd. (See Figure 1.) In other words, we use the tangent line at sa, f sadd as an approximation to the curve y − f sxd when x is near a. An equation of this tangent line is

y

y=ƒ y=L(x)

{a, f(a)}

0

x

1

FIGURE 1

is called the linear approximation or tangent line approximation of f at a. The linear function whose graph is this tangent line, that is,

2

is called the linearization of f at a.

EXAMPLE 1 Find the linearization of the function f sxd − sx 1 3 at a − 1 and use it to approximate the numbers s3.98 and s4.05 . Are these approximations overestimates or underestimates? SoLUtion The derivative of f sxd − sx 1 3d1y2 is

f 9sxd − 12 sx 1 3d21y2 −

1 2sx 1 3

and so we have f s1d − 2 and f 9s1d − 14. Putting these values into Equation 2, we see that the linearization is Lsxd − f s1d 1 f 9s1dsx 2 1d − 2 1 14 sx 2 1d −

7 x 1 4 4

The corresponding linear approximation (1) is sx 1 3
0 y − tanh21x

&? tanh y − x

The remaining inverse hyperbolic functions are defined similarly (see Exercise 28).

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262

CHAPTER 3

Differentiation Rules

We can sketch the graphs of sinh21, cosh21, and tanh21 in Figures 8, 9, and 10 by using Figures 1, 2, and 3. y

y y

0 0

x

_1 0

FIGURE 8 y − sinh 21 x domain − R

range − R

1

x

x

1

FIGURE 9 y − cosh 21 x

FIGURE 10 y − tanh 21 x

domain − f1, `d

domain − s21, 1d

range − f0, `d

range − R

Since the hyperbolic functions are defined in terms of exponential functions, it’s not surprising to learn that the inverse hyperbolic functions can be expressed in terms of logarithms. In particular, we have:

Formula 3 is proved in Example 3. The proofs of Formulas 4 and 5 are requested in Exercises 26 and 27.

3

sinh21x − lns x 1 sx 2 1 1 d

x[R

4

cosh21x − lns x 1 sx 2 2 1 d

x>1

5

tanh21x − 12 ln

S D 11x 12x

21 , x , 1

ExamplE 3 Show that sinh21x − lns x 1 sx 2 1 1 d. SOLUTION Let y − sinh21x. Then

x − sinh y −

e y 2 e2y 2

e y 2 2x 2 e2y − 0

so or, multiplying by e y,

e 2y 2 2xe y 2 1 − 0 This is really a quadratic equation in e y: se y d2 2 2xse y d 2 1 − 0 Solving by the quadratic formula, we get ey −

2x 6 s4x 2 1 4 − x 6 sx 2 1 1 2

Note that e y . 0, but x 2 sx 2 1 1 , 0 (because x , sx 2 1 1). Thus the minus sign is inadmissible and we have e y − x 1 sx 2 1 1

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SECTION 3.11

Hyperbolic Functions

263

y − lnse y d − lns x 1 sx 2 1 1 d

Therefore

sinh21x − lns x 1 sx 2 1 1 d

This shows that

(See Exercise 25 for another method.)

6 Derivatives of Inverse Hyperbolic Functions

Notice that the formulas for the derivatives of tanh21x and coth21x appear to be identical. But the domains of these functions have no numbers in common: tanh21x is defined for x , 1, whereas coth21x is defined for x . 1.

| | | |

d 1 ssinh21xd − dx s1 1 x 2

d 1 scsch21xd − 2 dx x | | sx 2 1 1

d 1 scosh21xd − 2 dx sx 2 1

d 1 ssech21xd − 2 dx x s1 2 x 2

d 1 stanh21xd − dx 1 2 x2

d 1 scoth21xd − dx 1 2 x2

The inverse hyperbolic functions are all differentiable because the hyperbolic functions are differentiable. The formulas in Table 6 can be proved either by the method for inverse functions or by differentiating Formulas 3, 4, and 5.

ExamplE 4 Prove that

d 1 ssinh21xd − . dx s1 1 x 2

SOLUTION 1 Let y − sinh21x. Then sinh y − x. If we differentiate this equation implic-

itly with respect to x, we get cosh y

dy −1 dx

Since cosh2 y 2 sinh2 y − 1 and cosh y > 0, we have cosh y − s1 1 sinh2 y , so dy 1 1 1 − − − 2 dx cosh y s1 1 sinh y s1 1 x 2 SOLUTION 2 From Equation 3 (proved in Example 3), we have

d d ssinh21xd − lns x 1 sx 2 1 1 d dx dx 1

d s x 1 sx 2 1 1 d x 1 sx 2 1 1 dx

1 x 1 sx 2 1 1

S

11

x sx 1 1

sx 2 1 1 1 x ( x 1 sx 2 1 1 ) sx 2 1 1

1 sx 2 1 1

2

D

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264

CHAPTER 3

Differentiation Rules

ExamplE 5 Find

d ftanh21ssin xdg. dx

SOLUTION Using Table 6 and the Chain Rule, we have

d 1 d ftanh21ssin xdg − ssin xd 2 dx 1 2 ssin xd dx −

1 cos x cos x − − sec x 2 1 2 sin x cos2x

3.11 EXERCISES 1–6 Find the numerical value of each expression. 1. (a) sinh 0

(b) cosh 0

2. (a) tanh 0

(b) tanh 1

3. (a) coshsln 5d

(b) cosh 5

4. (a) sinh 4

(b) sinhsln 4d

5. (a) sech 0

(b) cosh21 1

6. (a) sinh 1

(b) sinh21 1

21. If cosh x − 53 and x . 0, find the values of the other hyperbolic functions at x. 22. (a) Use the graphs of sinh, cosh, and tanh in Figures 1–3 to draw the graphs of csch, sech, and coth. (b) Check the graphs that you sketched in part (a) by using a ; graphing device to produce them. 23. Use the definitions of the hyperbolic functions to find each of the following limits. (a) lim tanh x (b) lim tanh x xl`

(d) lim sinh x

(e) lim sech x

(f) lim coth x

(g) lim1 coth x

(h) lim2 coth x

(i)

( j) lim

xl`

7–19 Prove the identity. 7. sinhs2xd − 2sinh x (This shows that sinh is an odd function.) 8. coshs2xd − cosh x (This shows that cosh is an even function.) 9. cosh x 1 sinh x − e x 2x

10. cosh x 2 sinh x − e

x l2`

(c) lim sinh x xl` x l0

lim csch x

x l2`

x l2` xl` x l0

xl`

sinh x ex

24. Prove the formulas given in Table 1 for the derivatives of the functions (a) cosh, (b) tanh, (c) csch, (d) sech, and (e) coth.

12. coshsx 1 yd − cosh x cosh y 1 sinh x sinh y

25. Give an alternative solution to Example 3 by letting y − sinh21x and then using Exercise 9 and Example 1(a) with x replaced by y.

13. coth2x 2 1 − csch2x

26. Prove Equation 4.

tanh x 1 tanh y 14. tanhsx 1 yd − 1 1 tanh x tanh y

27. Prove Equation 5 using (a) the method of Example 3 and (b) Exercise 18 with x replaced by y.

11. sinhsx 1 yd − sinh x cosh y 1 cosh x sinh y

15. sinh 2x − 2 sinh x cosh x 16. cosh 2x − cosh2x 1 sinh2x 17. tanhsln xd − 18.

x2 2 1 x2 1 1

1 1 tanh x − e 2x 1 2 tanh x

19. scosh x 1 sinh xdn − cosh nx 1 sinh nx (n any real number)

28. For each of the following functions (i) give a definition like those in (2), (ii) sketch the graph, and (iii) find a formula similar to Equation 3. (a) csch 21 (b) sech21 (c) coth21 29. Prove the formulas given in Table 6 for the derivatives of the following functions. (a) cosh21 (b) tanh21 (c) csch21 21 21 (d) sech (e) coth 30–45 Find the derivative. Simplify where possible.

20. If tanh x − 12 13 , find the values of the other hyperbolic functions at x.

30. f sxd − e x cosh x 31. f sxd − tanh sx

32. tsxd − sinh 2 x

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SECTION 3.11

33. hsxd − sinhsx 2 d

Hyperbolic Functions

265

y

34. hsxd − lnscosh xd

35. tsxd − coshsln xd

¨ 5

36. y − sech x s1 1 ln sech xd 1 1 sinh t 1 2 sinh t

37. y − e cosh 3x

38. f std −

39. tstd − t coth st 2 1 1

40. y − sinhscosh xd

41. Gsxd −

1 2 cosh x 1 1 cosh x

43. y − x sinh21sxy3d 2 s9 1 x 2

d2y t − dx 2 T

44. y − sech21 se2x d 45. y − coth21 ssec xd

d dx

Î 4

1 1 tanh x − 12 e xy2. 1 2 tanh x

d 47. Show that arctanstanh xd − sech 2x. dx 48. The Gateway Arch in St. Louis was designed by Eero Saarinen and was constructed using the equation y − 211.49 2 20.96 cosh 0.03291765x

;

for the central curve of the arch, where x and y are measured in meters and x < 91.20. (a) Graph the central curve. (b) What is the height of the arch at its center? (c) At what points is the height 100 m? (d) What is the slope of the arch at the points in part (c)?

| |

49. If a water wave with length L moves with velocity v in a body of water with depth d, then v−

Î

S D

tL 2d tanh 2 L

where t is the acceleration due to gravity. (See Figure 5.) Explain why the approximation v
0, where b and c are positive constants. (a) Find the velocity and acceleration functions. (b) Show that the particle always moves in the positive direction.

f

0

; 82. (a) Graph the function f sxd − x 2 2 sin x in the viewing rectangle f0, 8g by f22, 8g. (b) On which interval is the average rate of change larger: f1, 2g or f2, 3g? (c) At which value of x is the instantaneous rate of change larger: x − 2 or x − 5? (d) Check your visual estimates in part (c) by computing f 9sxd and comparing the numerical values of f 9s2d and f 9s5d.

Î

f sxd tsxd

89. A particle moves on a vertical line so that its coordinate at time t is y − t 3 2 12t 1 3, t > 0. (a) Find the velocity and acceleration functions. (b) When is the particle moving upward and when is it moving downward? (c) Find the distance that the particle travels in the time interval 0 < t < 3. (d) Graph the position, velocity, and acceleration functions ; for 0 < t < 3. (e) When is the particle speeding up? When is it slowing down? 90. The volume of a right circular cone is V − 13 r 2h, where r is the radius of the base and h is the height. (a) Find the rate of change of the volume with respect to the height if the radius is constant.

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CHAPTER 3

(b) Find the rate of change of the volume with respect to the radius if the height is constant. 91. The mass of part of a wire is x s1 1 sx d kilograms, where x is measured in meters from one end of the wire. Find the linear density of the wire when x − 4 m.

(a) Find the marginal cost function. (b) Find C9s100d and explain its meaning. (c) Compare C9s100d with the cost of producing the 101st item. 93. A bacteria culture contains 200 cells initially and grows at a rate proportional to its size. After half an hour the population has increased to 360 cells. (a) Find the number of bacteria after t hours. (b) Find the number of bacteria after 4 hours. (c) Find the rate of growth after 4 hours. (d) When will the population reach 10,000? 94. Cobalt-60 has a half-life of 5.24 years. (a) Find the mass that remains from a 100-mg sample after 20 years. (b) How long would it take for the mass to decay to 1 mg? 95. Let Cstd be the concentration of a drug in the bloodstream. As the body eliminates the drug, Cstd decreases at a rate that is proportional to the amount of the drug that is present at the time. Thus C9std − 2kCstd, where k is a positive number called the elimination constant of the drug. (a) If C0 is the concentration at time t − 0, find the concentration at time t. (b) If the body eliminates half the drug in 30 hours, how long does it take to eliminate 90% of the drug? 96. A cup of hot chocolate has temperature 80°C in a room kept at 20°C. After half an hour the hot chocolate cools to 60°C. (a) What is the temperature of the chocolate after another half hour? (b) When will the chocolate have cooled to 40°C? 97. The volume of a cube is increasing at a rate of 10 cm3ymin. How fast is the surface area increasing when the length of an edge is 30 cm? 98. A paper cup has the shape of a cone with height 10 cm and radius 3 cm (at the top). If water is poured into the cup at a rate of 2 cm3ys, how fast is the water level rising when the water is 5 cm deep? 99. A balloon is rising at a constant speed of 2 mys. A boy is cycling along a straight road at a speed of 5 mys. When he passes under the balloon, it is 15 m above him. How fast is the distance between the boy and the balloon increasing 3 s later?

269

100. A waterskier skis over the ramp shown in the figure at a speed of 10 mys. How fast is she rising as she leaves the ramp?

1m

92. The cost, in dollars, of producing x units of a certain commodity is Csxd − 920 1 2x 2 0.02x 2 1 0.00007x 3

Review

5m

101. The angle of elevation of the sun is decreasing at a rate of 0.25 radyh. How fast is the shadow cast by a 400-ft-tall building increasing when the angle of elevation of the sun is y6? ; 102. (a) Find the linear approximation to f sxd − s25 2 x 2 near 3. (b) Illustrate part (a) by graphing f and the linear approximation. (c) For what values of x is the linear approximation accurate to within 0.1? 3 1 1 3x at a − 0. 103. (a) Find the linearization of f sxd − s State the corresponding linear approximation and use it 3 to give an approximate value for s 1.03 . (b) Determine the values of x for which the linear approxi; mation given in part (a) is accurate to within 0.1.

104. Evaluate dy if y − x 3 2 2x 2 1 1, x − 2, and dx − 0.2. 105. A window has the shape of a square surmounted by a semicircle. The base of the window is measured as having width 60 cm with a possible error in measurement of 0.1 cm. Use differentials to estimate the maximum error possible in computing the area of the window. 106–108 Express the limit as a derivative and evaluate. 106. lim

x l1

108. lim

x 17 2 1 x21

l y3

107. lim

hl0

4 16 1 h 2 2 s h

cos  2 0.5  2 y3

109. Evaluate lim

xl0

s1 1 tan x 2 s1 1 sin x . x3

110. Suppose f is a differentiable function such that f s tsxdd − x and f 9sxd − 1 1 f f sxdg 2. Show that t9sxd − 1ys1 1 x 2 d. 111. Find f 9sxd if it is known that d f f s2xdg − x 2 dx 112. Show that the length of the portion of any tangent line to the astroid x 2y3 1 y 2y3 − a 2y3 cut off by the coordinate axes is constant.

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Problems Plus

Before you look at the examples, cover up the solutions and try them yourself first. ExamPlE 1 How many lines are tangent to both of the parabolas y − 21 2 x 2 and y − 1 1 x 2 ? Find the coordinates of the points at which these tangents touch the parabolas. SOLUTION To gain insight into this problem, it is essential to draw a diagram. So we sketch the parabolas y − 1 1 x 2 (which is the standard parabola y − x 2 shifted 1 unit upward) and y − 21 2 x 2 (which is obtained by reflecting the first parabola about the x-axis). If we try to draw a line tangent to both parabolas, we soon discover that there are only two possibilities, as illustrated in Figure 1. Let P be a point at which one of these tangents touches the upper parabola and let a be its x-coordinate. (The choice of notation for the unknown is important. Of course we could have used b or c or x 0 or x1 instead of a. However, it’s not advisable to use x in place of a because that x could be confused with the variable x in the equation of the parabola.) Then, since P lies on the parabola y − 1 1 x 2, its y-coordinate must be 1 1 a 2. Because of the symmetry shown in Figure 1, the coordinates of the point Q where the tangent touches the lower parabola must be s2a, 2s1 1 a 2 dd. To use the given information that the line is a tangent, we equate the slope of the line PQ to the slope of the tangent line at P. We have

y

P 1

x _1

Q

FIGURE 1

mPQ −

1 1 a 2 2 s21 2 a 2 d 1 1 a2 − a 2 s2ad a

If f sxd − 1 1 x 2, then the slope of the tangent line at P is f 9sad − 2a. Thus the condition that we need to use is that 1 1 a2 − 2a a y

3≈ ≈ 1 ≈ 2

0.3≈ 0.1≈

x

0

y=ln x

FIGURE 2 y

y=c ≈ c=?

0

a

y=ln x

FIGURE 3

x

Solving this equation, we get 1 1 a 2 − 2a 2, so a 2 − 1 and a − 61. Therefore the points are (1, 2) and s21, 22d. By symmetry, the two remaining points are s21, 2d and s1, 22d. ■ ExamPlE 2 solution?

For what values of c does the equation ln x − cx 2 have exactly one

SOLUTION One of the most important principles of problem solving is to draw a diagram, even if the problem as stated doesn’t explicitly mention a geometric situation. Our present problem can be reformulated geometrically as follows: For what values of c does the curve y − ln x intersect the curve y − cx 2 in exactly one point? Let’s start by graphing y − ln x and y − cx 2 for various values of c. We know that, for c ± 0, y − cx 2 is a parabola that opens upward if c . 0 and downward if c , 0. Figure 2 shows the parabolas y − cx 2 for several positive values of c. Most of them don’t intersect y − ln x at all and one intersects twice. We have the feeling that there must be a value of c (somewhere between 0.1 and 0.3) for which the curves intersect exactly once, as in Figure 3. To find that particular value of c, we let a be the x-coordinate of the single point of intersection. In other words, ln a − ca 2, so a is the unique solution of the given equation. We see from Figure 3 that the curves just touch, so they have a common tangent line when x − a. That means the curves y − ln x and y − cx 2 have the same slope when x − a. Therefore 1 − 2ca a

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Solving the equations ln a − ca 2 and 1ya − 2ca, we get ln a − ca 2 − c ?

y

1 1 − 2c 2

Thus a − e 1y2 and

y=ln x 0

c−

x

ln a ln e 1y2 1 − − 2 a e 2e

For negative values of c we have the situation illustrated in Figure 4: All parabolas y − cx 2 with negative values of c intersect y − ln x exactly once. And let’s not forget about c − 0: The curve y − 0x 2 − 0 is just the x-axis, which intersects y − ln x exactly once. To summarize, the required values of c are c − 1ys2ed and c < 0. ■

FIGURE 4

1. Find points P and Q on the parabola y − 1 2 x 2 so that the triangle ABC formed by the x-axis and the tangent lines at P and Q is an equilateral triangle. (See the figure.)

Problems

y

A

P B

;

Q 0

C

x

2. Find the point where the curves y − x 3 2 3x 1 4 and y − 3sx 2 2 xd are tangent to each other, that is, have a common tangent line. Illustrate by sketching both curves and the common tangent. 3. Show that the tangent lines to the parabola y − ax 2 1 bx 1 c at any two points with x-coordinates p and q must intersect at a point whose x-coordinate is halfway between p and q. 4. Show that d dx 5. If f sxd − lim tlx

S

sin2 x cos2 x 1 1 1 cot x 1 1 tan x

D

− 2cos 2x

sec t 2 sec x , find the value of f 9sy4d. t2x

6. Find the values of the constants a and b such that lim

y

xl0

3 ax 1 b 2 2 5 s − x 12

7. Show that sin21stanh xd − tan21ssinh xd.

x

FIGURE FOR PROBLEM 8

8. A car is traveling at night along a highway shaped like a parabola with its vertex at the origin (see the figure). The car starts at a point 100 m west and 100 m north of the origin and travels in an easterly direction. There is a statue located 100 m east and 50 m north of the origin. At what point on the highway will the car’s headlights illuminate the statue? 9. Prove that

dn ssin4 x 1 cos4 xd − 4n21 coss4x 1 ny2d. dx n

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10. If f is differentiable at a, where a . 0, evaluate the following limit in terms of f 9sad: lim

xla

f sxd 2 f sad sx 2 sa

11. The figure shows a circle with radius 1 inscribed in the parabola y − x 2. Find the center of the circle. y

y=≈

1

1

0

x

12. Find all values of c such that the parabolas y − 4x 2 and x − c 1 2y 2 intersect each other at right angles. 13. How many lines are tangent to both of the circles x 2 1 y 2 − 4 and x 2 1 sy 2 3d 2 − 1? At what points do these tangent lines touch the circles? x 46 1 x 45 1 2 , calculate f s46ds3d. Express your answer using factorial notation: 11x n! − 1 ? 2 ? 3 ? ∙ ∙ ∙ ? sn 2 1d ? n.

14. If f sxd −

15. The figure shows a rotating wheel with radius 40 cm and a connecting rod AP with length 1.2 m. The pin P slides back and forth along the x-axis as the wheel rotates counterclockwise at a rate of 360 revolutions per minute. (a) Find the angular velocity of the connecting rod, dydt, in radians per second, when  − y3. (b) Express the distance x − OP in terms of . (c) Find an expression for the velocity of the pin P in terms of .

|

|

y

A å

¨

P (x, 0) x

O

16. Tangent lines T1 and T2 are drawn at two points P1 and P2 on the parabola y − x 2 and they intersect at a point P. Another tangent line T is drawn at a point between P1 and P2 ; it intersects T1 at Q1 and T2 at Q2. Show that

| PQ | 1 | PQ | − 1 | PP | | PP | 1

2

1

2

17. Show that dn se ax sin bxd − r ne ax sinsbx 1 nd dx n where a and b are positive numbers, r 2 − a 2 1 b 2, and  − tan21sbyad.

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18. Evaluate lim

xl

e sin x 2 1 . x2

19. Let T and N be the tangent and normal lines to the ellipse x 2y9 1 y 2y4 − 1 at any point P on the ellipse in the first quadrant. Let x T and yT be the x- and y-intercepts of T and x N and yN be the intercepts of N. As P moves along the ellipse in the first quadrant (but not on the axes), what values can x T , yT , x N, and yN take on? First try to guess the answers just by looking at the figure. Then use calculus to solve the problem and see how good your intuition is. y

yT

T

2

P xT

xN 0

3

20. Evaluate lim

xl0

x

N

yN

sins3 1 xd2 2 sin 9 . x

21. (a) Use the identity for tansx 2 yd (see Equation 14b in Appendix D) to show that if two lines L 1 and L 2 intersect at an angle , then tan  −

m 2 2 m1 1 1 m1 m 2

where m1 and m 2 are the slopes of L 1 and L 2, respectively. (b) The angle between the curves C1 and C2 at a point of intersection P is defined to be the angle between the tangent lines to C1 and C2 at P (if these tangent lines exist). Use part (a) to find, correct to the nearest degree, the angle between each pair of curves at each point of intersection. (i) y − x 2 and y − sx 2 2d2 (ii) x 2 2 y 2 − 3 and x 2 2 4x 1 y 2 1 3 − 0 22. Let Psx 1, y1d be a point on the parabola y 2 − 4px with focus Fs p, 0d. Let  be the angle between the parabola and the line segment FP, and let  be the angle between the horizontal line y − y1 and the parabola as in the figure. Prove that  − . (Thus, by a principle of geometrical optics, light from a source placed at F will be reflected along a line parallel to the x-axis. This explains why paraboloids, the surfaces obtained by rotating parabolas about their axes, are used as the shape of some automobile headlights and mirrors for telescopes.) y

å 0

∫ P(⁄, ›)

y=›

x

F(p, 0) ¥=4px

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23. Suppose that we replace the parabolic mirror of Problem 22 by a spherical mirror. Although the mirror has no focus, we can show the existence of an approximate focus. In the figure, C is a semicircle with center O. A ray of light coming in toward the mirror parallel to the axis along the line PQ will be reflected to the point R on the axis so that /PQO − /OQR (the angle of incidence is equal to the angle of reflection). What happens to the point R as P is taken closer and closer to the axis?

Q P

¨ ¨ A

R

O

24. If f and t are differentiable functions with f s0d − ts0d − 0 and t9s0d ± 0, show that C

lim

xl0

FIGURE FOR PROBLEM 23 25. Evaluate lim

xl0

CAS

f sxd f 9s0d − tsxd t9s0d

sinsa 1 2xd 2 2 sinsa 1 xd 1 sin a . x2

26. (a) The cubic function f sxd − xsx 2 2dsx 2 6d has three distinct zeros: 0, 2, and 6. Graph f and its tangent lines at the average of each pair of zeros. What do you notice? (b) Suppose the cubic function f sxd − sx 2 adsx 2 bdsx 2 cd has three distinct zeros: a, b, and c. Prove, with the help of a computer algebra system, that a tangent line drawn at the average of the zeros a and b intersects the graph of f at the third zero. 27. For what value of k does the equation e 2x − ksx have exactly one solution? 28. For which positive numbers a is it true that a x > 1 1 x for all x? 29. If y−

show that y9 −

x sa 2 2 1

2

2 sa 2 2 1

arctan

sin x a 1 sa 2 2 1 1 cos x

1 . a 1 cos x

30. Given an ellipse x 2ya 2 1 y 2yb 2 − 1, where a ± b, find the equation of the set of all points from which there are two tangents to the curve whose slopes are (a) reciprocals and (b) negative reciprocals. 31. Find the two points on the curve y − x 4 2 2x 2 2 x that have a common tangent line. 32. Suppose that three points on the parabola y − x 2 have the property that their normal lines intersect at a common point. Show that the sum of their x-coordinates is 0. 33. A lattice point in the plane is a point with integer coordinates. Suppose that circles with radius r are drawn using all lattice points as centers. Find the smallest value of r such that any line with slope 25 intersects some of these circles. 34. A cone of radius r centimeters and height h centimeters is lowered point first at a rate of 1 cmys into a tall cylinder of radius R centimeters that is partially filled with water. How fast is the water level rising at the instant the cone is completely submerged? 35. A container in the shape of an inverted cone has height 16 cm and radius 5 cm at the top. It is partially filled with a liquid that oozes through the sides at a rate proportional to the area of the container that is in contact with the liquid. (The surface area of a cone is rl, where r is the radius and l is the slant height.) If we pour the liquid into the container at a rate of 2 cm3ymin, then the height of the liquid decreases at a rate of 0.3 cmymin when the height is 10 cm. If our goal is to keep the liquid at a constant height of 10 cm, at what rate should we pour the liquid into the container?

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4

Applications of Differentiation

When we view the world around us, the light entering the eye near the center of the pupil is perceived brighter than light entering closer to the edges of the pupil. This phenomenon, known as the Stiles–Crawford effect, is explored as the pupil changes in radius in Exercise 80 on page 313. © Tatiana Makotra / Shutterstock.com

We have already investigated some of the applications of derivatives, but now that we know the differentiation rules we are in a better position to pursue the applications of differentiation in greater depth. Here we learn how derivatives affect the shape of a graph of a function and, in particular, how they help us locate maximum and minimum values of functions. Many practical problems require us to minimize a cost or maximize an area or somehow find the best possible outcome of a situation. In particular, we will be able to investigate the optimal shape of a can and to explain the location of rainbows in the sky.

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276

CHAPTER 4

Applications of Differentiation

Some of the most important applications of differential calculus are optimization problems, in which we are required to find the optimal (best) way of doing something. Here are examples of such problems that we will solve in this chapter:

• What is the shape of a can that minimizes manufacturing costs? • What is the maximum acceleration of a space shuttle? (This is an important •

question to the astronauts who have to withstand the effects of acceleration.) What is the radius of a contracted windpipe that expels air most rapidly during a cough?

• At what angle should blood vessels branch so as to minimize the energy expended by the heart in pumping blood? These problems can be reduced to finding the maximum or minimum values of a function. Let’s first explain exactly what we mean by maximum and minimum values. We see that the highest point on the graph of the function f shown in Figure 1 is the point s3, 5d. In other words, the largest value of f is f s3d − 5. Likewise, the smallest value is f s6d − 2. We say that f s3d − 5 is the absolute maximum of f and f s6d − 2 is the absolute minimum. In general, we use the following definition.

y 4 2

0

4

2

x

6

1 Definition Let c be a number in the domain D of a function f. Then f scd is the • absolute maximum value of f on D if f scd > f sxd for all x in D.

FigurE 1

• absolute minimum value of f on D if f scd < f sxd for all x in D.

y

f(d) f(a) a

0

b

c

d

e

x

FigurE 2 Abs min f sad, abs max f sdd, loc min f scd, f sed, loc max f sbd, f sdd

loc max

4

loc and abs min

loc min

2 0

FigurE 3

I

J

K

4

8

12

2 Definition The number f scd is a

• l ocal maximum value of f if f scd > f sxd when x is near c. • local minimum value of f if f scd < f sxd when x is near c.

y 6

An absolute maximum or minimum is sometimes called a global maximum or minimum. The maximum and minimum values of f are called extreme values of f . Figure 2 shows the graph of a function f with absolute maximum at d and absolute minimum at a. Note that sd, f sddd is the highest point on the graph and sa, f sadd is the lowest point. In Figure 2, if we consider only values of x near b [for instance, if we restrict our attention to the interval sa, cd], then f sbd is the largest of those values of f sxd and is called a local maximum value of f . Likewise, f scd is called a local minimum value of f because f scd < f sxd for x near c [in the interval sb, dd, for instance]. The function f also has a local minimum at e. In general, we have the following definition.

x

In Definition 2 (and elsewhere), if we say that something is true near c, we mean that it is true on some open interval containing c. For instance, in Figure 3 we see that f s4d − 5 is a local minimum because it’s the smallest value of f on the interval I. It’s not the absolute minimum because f sxd takes smaller values when x is near 12 (in the interval K, for instance). In fact f s12d − 3 is both a local minimum and the absolute minimum. Similarly, f s8d − 7 is a local maximum, but not the absolute maximum because f takes larger values near 1.

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SECTION 4.1 Maximum and Minimum Values

277

example 1 The function f sxd − cos x takes on its (local and absolute) maximum value of 1 infinitely many times, since cos 2n − 1 for any integer n and 21 < cos x < 1 for all x. (See Figure 4.) Likewise, coss2n 1 1d − 21 is its minimum value, where n is any integer. y Local and absolute maximum

0

π

x

FigurE 4 Local and absolute minimum

y − cosx

y

y=≈

0

example 2 If f sxd − x 2, then f sxd > f s0d because x 2 > 0 for all x. Therefore x

FigurE 5 Mimimum value 0, no maximum

f s0d − 0 is the absolute (and local) minimum value of f. This corresponds to the fact that the origin is the lowest point on the parabola y − x 2. (See Figure 5.) However, there is no highest point on the parabola and so this function has no maximum value. ■

example 3 From the graph of the function f sxd − x 3, shown in Figure 6, we see that this function has neither an absolute maximum value nor an absolute minimum value. In fact, it has no local extreme values either. ■

y

example 4 The graph of the function

y=˛

f sxd − 3x 4 2 16x 3 1 18x 2 0

21 < x < 4

x

FigurE 6 No mimimum, no maximum

is shown in Figure 7. You can see that f s1d − 5 is a local maximum, whereas the absolute maximum is f s21d − 37. (This absolute maximum is not a local maximum because it occurs at an endpoint.) Also, f s0d − 0 is a local minimum and f s3d − 227 is both a local and an absolute minimum. Note that f has neither a local nor an absolute maximum at x − 4. y (_1, 37)

y=3x\$-16˛+18≈

(1, 5) _1

1

2

3

4

5

x

(3, _27)

FigurE 7

We have seen that some functions have extreme values, whereas others do not. The following theorem gives conditions under which a function is guaranteed to possess extreme values.

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

278

CHAPTER 4

Applications of Differentiation

3 The Extreme Value Theorem If f is continuous on a closed interval fa, bg, then f attains an absolute maximum value f scd and an absolute minimum value f sdd at some numbers c and d in fa, bg. The Extreme Value Theorem is illustrated in Figure 8. Note that an extreme value can be taken on more than once. Although the Extreme Value Theorem is intuitively very plausible, it is difficult to prove and so we omit the proof. y

0

y

a

c

d b

0

x

y

a

c

d=b

x

0

a c¡

d

c™ b

x

FigurE 8 Functions continuous on a closed interval always attain extreme values.

Figures 9 and 10 show that a function need not possess extreme values if either hypothesis (continuity or closed interval) is omitted from the Extreme Value Theorem. y

y

3

1

0

y {c, f (c)}

{d, f (d )} 0

c

FigurE 11

d

x

1

2

x

0

2

x

FIGURE 9

FIGURE 10

This function has minimum value f(2)=0, but no maximum value.

This continuous function g has no maximum or minimum.

The function f whose graph is shown in Figure 9 is defined on the closed interval [0, 2] but has no maximum value. (Notice that the range of f is [0, 3). The function takes on values arbitrarily close to 3, but never actually attains the value 3.) This does not contradict the Extreme Value Theorem because f is not continuous. [Nonetheless, a discontinuous function could have maximum and minimum values. See Exercise 13(b).] The function t shown in Figure 10 is continuous on the open interval (0, 2) but has neither a maximum nor a minimum value. [The range of t is s1, `d. The function takes on arbitrarily large values.] This does not contradict the Extreme Value Theorem because the interval (0, 2) is not closed. The Extreme Value Theorem says that a continuous function on a closed interval has a maximum value and a minimum value, but it does not tell us how to find these extreme values. Notice in Figure 8 that the absolute maximum and minimum values that are between a and b occur at local maximum or minimum values, so we start by looking for local extreme values. Figure 11 shows the graph of a function f with a local maximum at c and a local minimum at d. It appears that at the maximum and minimum points the tangent lines are horizontal and therefore each has slope 0. We know that the derivative is the slope of the

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SECTION 4.1 Maximum and Minimum Values

279

tangent line, so it appears that f 9scd − 0 and f 9sdd − 0. The following theorem says that this is always true for differentiable functions. 4 Fermat’s Theorem If f has a local maximum or minimum at c, and if f 9scd exists, then f 9scd − 0.

Fermat Fermat’s Theorem is named after Pierre Fermat (1601–1665), a French lawyer who took up mathematics as a hobby. Despite his amateur status, Fermat was one of the two inventors of analytic geometry (Descartes was the other). His methods for finding tangents to curves and maximum and minimum values (before the invention of limits and derivatives) made him a forerunner of Newton in the creation of differential calculus.

PROOF Suppose, for the sake of definiteness, that f has a local maximum at c. Then, according to Definition 2, f scd > f sxd if x is sufficiently close to c. This implies that if h is sufficiently close to 0, with h being positive or negative, then

f scd > f sc 1 hd and therefore f sc 1 hd 2 f scd < 0

5

We can divide both sides of an inequality by a positive number. Thus, if h . 0 and h is sufficiently small, we have f sc 1 hd 2 f scd 0 h

h,0

So, taking the left-hand limit, we have f 9scd − lim

hl0

f sc 1 hd 2 f scd f sc 1 hd 2 f scd − lim2 >0 h l 0 h h

We have shown that f 9scd > 0 and also that f 9scd < 0. Since both of these inequalities must be true, the only possibility is that f 9scd − 0. We have proved Fermat’s Theorem for the case of a local maximum. The case of a local minimum can be proved in a similar manner, or we could use Exercise 78 to ■ deduce it from the case we have just proved (see Exercise 79). The following examples caution us against reading too much into Fermat’s Theorem: We can’t expect to locate extreme values simply by setting f 9sxd − 0 and solving for x. Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

280

CHAPTER 4

Applications of Differentiation

y

example 5 If f sxd − x 3, then f 9sxd − 3x 2, so f 9s0d − 0. But f has no maximum or minimum at 0, as you can see from its graph in Figure 12. (Or observe that x 3 . 0 for x . 0 but x 3 , 0 for x , 0.) The fact that f 9s0d − 0 simply means that the curve y − x 3 has a horizontal tangent at s0, 0d. Instead of having a maximum or minimum at s0, 0d, the curve crosses its horizontal tangent there. ■

y=˛

0

x

example 6 The function f sxd − | x | has its (local and absolute) minimum value at 0, but that value can’t be found by setting f 9sxd − 0 because, as was shown in Example 2.8.5, f 9s0d does not exist. (See Figure 13.) ■

FigurE 12 If f sxd − x 3, then f 9s0d − 0, but f has no maximum or minimum.

WARNING Examples 5 and 6 show that we must be careful when using Fermat’s Theorem. Example 5 demonstrates that even when f 9scd − 0 there need not be a maximum or minimum at c. (In other words, the converse of Fermat’s Theorem is false in general.) Furthermore, there may be an extreme value even when f 9scd does not exist (as in Example 6). Fermat’s Theorem does suggest that we should at least start looking for extreme values of f at the numbers c where f 9scd − 0 or where f 9scd does not exist. Such numbers are given a special name.

y

y=|x | 0

x

FigurE 13

6 Definition A critical number of a function f is a number c in the domain of f such that either f 9scd − 0 or f 9scd does not exist.

| |

If f sxd − x , then f s0d − 0 is a minimum value, but f 9s0d does not exist.

example 7 Find the critical numbers of f sxd − x 3y5s4 2 xd. SOLUTION The Product Rule gives Figure 14 shows a graph of the function f in Example 7. It supports our answer because there is a horizontal tangent when x − 1.5 fwhere f 9sxd − 0g and a vertical tangent when x − 0 fwhere f 9sxd is undefinedg. 3.5

_0.5

5

f 9sxd − x 3y5s21d 1 s4 2 xd(53 x22y5) − 2x 3y5 1

3s4 2 xd 5x 2 y5

25x 1 3s4 2 xd 12 2 8x − 5x 2y5 5x 2y5

[The same result could be obtained by first writing f sxd − 4x 3y5 2 x 8y5.] Therefore f 9sxd − 0 if 12 2 8x − 0, that is, x − 32, and f 9sxd does not exist when x − 0. Thus the critical numbers are 32 and 0. ■ In terms of critical numbers, Fermat’s Theorem can be rephrased as follows (compare Definition 6 with Theorem 4):

_2

FigurE 14

7 If f has a local maximum or minimum at c, then c is a critical number of f.

To find an absolute maximum or minimum of a continuous function on a closed interval, we note that either it is local [in which case it occurs at a critical number by (7)] or it occurs at an endpoint of the interval, as we see from the examples in Figure 8. Thus the following three-step procedure always works.

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SECTION 4.1 Maximum and Minimum Values

281

The Closed interval Method To find the absolute maximum and minimum values of a continuous function f on a closed interval fa, bg: 1. Find the values of f at the critical numbers of f in sa, bd. 2. Find the values of f at the endpoints of the interval. 3. The largest of the values from Steps 1 and 2 is the absolute maximum value; the smallest of these values is the absolute minimum value.

example 8 Find the absolute maximum and minimum values of the function 212 < x < 4

f sxd − x 3 2 3x 2 1 1

f

g

SOLUTION Since f is continuous on 2 21, 4 , we can use the Closed Interval Method:

f sxd − x 3 2 3x 2 1 1 f 9sxd − 3x 2 2 6x − 3xsx 2 2d y 20

y=˛-3≈+1 (4, 17)

Since f 9sxd exists for all x, the only critical numbers of f occur when f 9sxd − 0, that is, x − 0 or x − 2. Notice that each of these critical numbers lies in the interval s212 , 4d. The values of f at these critical numbers are

15

f s0d − 1

10

f s2d − 23

The values of f at the endpoints of the interval are

5 1 _1 0 _5

f s212 d − 18

2 3

4

f s4d − 17

x

Comparing these four numbers, we see that the absolute maximum value is f s4d − 17 and the absolute minimum value is f s2d − 23. Note that in this example the absolute maximum occurs at an endpoint, whereas the absolute minimum occurs at a critical number. The graph of f is sketched in Figure 15.

(2, _3)

FigurE 15

If you have a graphing calculator or a computer with graphing software, it is possible to estimate maximum and minimum values very easily. But, as the next example shows, calculus is needed to find the exact values.

example 9 (a) Use a graphing device to estimate the absolute minimum and maximum values of the function f sxd − x 2 2 sin x, 0 < x < 2. (b) Use calculus to find the exact minimum and maximum values.

8

SOLUTION

0 _1

FigurE 16

(a) Figure 16 shows a graph of f in the viewing rectangle f0, 2g by f21, 8g. By moving the cursor close to the maximum point, we see that the y-coordinates don’t change very much in the vicinity of the maximum. The absolute maximum value is about 6.97 and it occurs when x < 5.2. Similarly, by moving the cursor close to the minimum point, we see that the absolute minimum value is about 20.68 and it occurs when x < 1.0. It is possible to get more accurate estimates by zooming in toward the maximum and minimum points (or using a built-in maximum or minimum feature),

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282

CHAPTER 4

Applications of Differentiation

but instead let’s use calculus. (b) The function f sxd − x 2 2 sin x is continuous on f0, 2g. Since f 9sxd − 1 2 2 cos x, we have f 9sxd − 0 when cos x − 12 and this occurs when x − y3 or 5y3. The values of f at these critical numbers are f sy3d − and

f s5y3d −

2 2 sin − 2 s3 < 20.684853 3 3 3 5 5 5 2 2 sin − 1 s3 < 6.968039 3 3 3

The values of f at the endpoints are f s0d − 0

and

f s2d − 2 < 6.28

Comparing these four numbers and using the Closed Interval Method, we see that the absolute minimum value is f sy3d − y3 2 s3 and the absolute maximum value is f s5y3d − 5y3 1 s3 . The values from part (a) serve as a check on our work. ■

example 10 The Hubble Space Telescope was deployed on April 24, 1990, by the space shuttle Discovery. A model for the velocity of the shuttle during this mission, from liftoff at t − 0 until the solid rocket boosters were jettisoned at t − 126 s, is given by vstd − 0.0003968t 3 2 0.02752t 2 1 7.196t 2 0.9397 (in meters per second). Using this model, estimate the absolute maximum and minimum values of the acceleration of the shuttle between liftoff and the jettisoning of the boosters.

NASA

SOLUTION We are asked for the extreme values not of the given velocity function, but rather of the acceleration function. So we first need to differentiate to find the acceleration:

astd − v9std −

d s0.0003968t 3 2 0.02752t 2 1 7.196t 2 0.9397d dt

− 0.0011904t 2 2 0.05504t 1 7.196 We now apply the Closed Interval Method to the continuous function a on the interval 0 < t < 126. Its derivative is a9std − 0.0023808t 2 0.05504 The only critical number occurs when a9std − 0: t1 −

0.05504 < 23.12 0.0023808

Evaluating astd at the critical number and at the endpoints, we have as0d − 7.196

ast1 d < 6.56

as126d < 19.16

So the maximum acceleration is about 19.16 mys2 and the minimum acceleration is about 6.56 mys2.

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283

SECTION 4.1 Maximum and Minimum Values

1. Explain the difference between an absolute minimum and a local minimum.

(c) Sketch the graph of a function that has a local maximum at 2 and is not continuous at 2.

2. Suppose f is a continuous function defined on a closed interval fa, bg. (a) What theorem guarantees the existence of an absolute maximum value and an absolute minimum value for f ? (b) What steps would you take to find those maximum and minimum values?

12. (a) Sketch the graph of a function on [21, 2] that has an absolute maximum but no local maximum. (b) Sketch the graph of a function on [21, 2] that has a local maximum but no absolute maximum.

3–4 For each of the numbers a, b, c, d, r, and s, state whether the function whose graph is shown has an absolute maximum or minimum, a local maximum or minimum, or neither a maximum nor a minimum. 3.

4. y

y

13. (a) Sketch the graph of a function on [21, 2] that has an absolute maximum but no absolute minimum. (b) Sketch the graph of a function on [21, 2] that is discontinuous but has both an absolute maximum and an absolute minimum. 14. (a) Sketch the graph of a function that has two local maxima, one local minimum, and no absolute minimum. (b) Sketch the graph of a function that has three local minima, two local maxima, and seven critical numbers. 15–28 Sketch the graph of f by hand and use your sketch to find the absolute and local maximum and minimum values of f. (Use the graphs and transformations of Sections 1.2 and 1.3.)

0 a b

c d

r

0

s x

a

b

c d

r

s x

15. f sxd − 12 s3x 2 1d, 16. f sxd − 2 2

5–6 Use the graph to state the absolute and local maximum and minimum values of the function. 5.

6.

y

y=ƒ

0

1

1

x

0

17. f sxd − 1yx,

x 22

x>1

18. f sxd − 1yx, 1 , x , 3

y

1

1 3 x,

19. f sxd − sin x,

0 < x , y2

20. f sxd − sin x,

0 , x < y2

21. f sxd − sin x,

2y2 < x < y2

22. f std − cos t, 23y2 < t < 3y2 1

x

23. f sxd − ln x,

0,x 1

_2

2

for all x

(since x 2 > 0) so f 9sxd can never be 0. This gives a contradiction. Therefore the equation can’t have two real roots. ■ Our main use of Rolle’s Theorem is in proving the following important theorem, which was first stated by another French mathematician, Joseph-Louis Lagrange.

_3

FIGURE 2

The Mean Value Theorem is an example of what is called an existence theorem. Like the Intermediate Value Theorem, the Extreme Value Theorem, and Rolle’s Theorem, it guarantees that there exists a number with a certain property, but it doesn’t tell us how to find the number.

The Mean Value Theorem Let f be a function that satisfies the following hypotheses: 1. f is continuous on the closed interval fa, bg. 2. f is differentiable on the open interval sa, bd. Then there is a number c in sa, bd such that 1

f 9scd −

or, equivalently,

f sbd 2 f sad b2a

2

Before proving this theorem, we can see that it is reasonable by interpreting it geometrically. Figures 3 and 4 show the points Asa, f sadd and Bsb, f sbdd on the graphs of two differentiable functions. The slope of the secant line AB is 3

mAB −

f sbd 2 f sad b2a

which is the same expression as on the right side of Equation 1. Since f 9scd is the slope of the tangent line at the point sc, f scdd, the Mean Value Theorem, in the form given by Equation 1, says that there is at least one point Psc, f scdd on the graph where the slope of the tangent line is the same as the slope of the secant line AB. In other words, there is a point P where the tangent line is parallel to the secant line AB. (Imagine a line far away that stays parallel to AB while moving toward AB until it touches the graph for the first time.) y

y

P { c, f(c)}

B

P™

A

A{ a, f(a)} B { b, f(b)} 0

FIGURE 3

a

c

b

x

0

a

c™

b

x

FIGURE 4

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

SECTION 4.2 The Mean Value Theorem

y

y=ƒ h(x)

A ƒ

B 0

a

x f(a)+

b

x

f(b)-f(a) (x-a) b-a

FIGURE 5

Lagrange and the Mean Value Theorem The Mean Value Theorem was first formulated by Joseph-Louis Lagrange (1736–1813), born in Italy of a French father and an Italian mother. He was a child prodigy and became a professor in Turin at the tender age of 19. Lagrange made great contributions to number theory, theory of functions, theory of equations, and analytical and celestial mechanics. In particular, he applied calculus to the analysis of the stability of the solar system. At the invitation of Frederick the Great, he succeeded Euler at the Berlin Academy and, when Frederick died, Lagrange accepted King Louis XVI’s invitation to Paris, where he was given apartments in the Louvre and became a professor at the Ecole Polytechnique. Despite all the trappings of luxury and fame, he was a kind and quiet man, living only for science.

289

PROOF We apply Rolle’s Theorem to a new function h defined as the difference between f and the function whose graph is the secant line AB. Using Equation 3 and the point-slope equation of a line, we see that the equation of the line AB can be written as f sbd 2 f sad y 2 f sad − sx 2 ad b2a

or as

So, as shown in Figure 5, hsxd − f sxd 2 f sad 2

4

First we must verify that h satisfies the three hypotheses of Rolle’s Theorem. 1. The function h is continuous on fa, bg because it is the sum of f and a first-degree polynomial, both of which are continuous. 2. The function h is differentiable on sa, bd because both f and the first-degree polynomial are differentiable. In fact, we can compute h9 directly from Equation 4: h9sxd − f 9sxd 2

f sbd 2 f sad b2a

hsbd − f sbd 2 f sad 2

− f sbd 2 f sad 2 f f sbd 2 f sadg − 0 Therefore hsad − hsbd. Since h satisfies the hypotheses of Rolle’s Theorem, that theorem says there is a number c in sa, bd such that h9scd − 0. Therefore 0 − h9scd − f 9scd 2 and so

f 9scd −

f sbd 2 f sad b2a

f sbd 2 f sad b2a

ExamplE 3 To illustrate the Mean Value Theorem with a specific function, let’s consider f sxd − x 3 2 x, a − 0, b − 2. Since f is a polynomial, it is continuous and differentiable for all x, so it is certainly continuous on f0, 2g and differentiable on s0, 2d. Therefore, by the Mean Value Theorem, there is a number c in s0, 2d such that f s2d 2 f s0d − f 9scds2 2 0d Now f s2d − 6, f s0d − 0, and f 9sxd − 3x 2 2 1, so this equation becomes 6 − s3c 2 2 1d2 − 6c 2 2 2 which gives c 2 − 43, that is, c − 62ys3 . But c must lie in s0, 2d, so c − 2ys3 . Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

290

CHAPTER 4

y

Applications of Differentiation

Figure 6 illustrates this calculation: The tangent line at this value of c is parallel to the secant line OB. ■

y=˛- x B

ExamplE 4 If an object moves in a straight line with position function s − f std, then the average velocity between t − a and t − b is O c

FIGURE 6

2

f sbd 2 f sad b2a

x

and the velocity at t − c is f 9scd. Thus the Mean Value Theorem (in the form of Equation 1) tells us that at some time t − c between a and b the instantaneous velocity f 9scd is equal to that average velocity. For instance, if a car traveled 180 km in 2 hours, then the speedometer must have read 90 kmyh at least once. In general, the Mean Value Theorem can be interpreted as saying that there is a number at which the instantaneous rate of change is equal to the average rate of change over an interval. ■ The main significance of the Mean Value Theorem is that it enables us to obtain information about a function from information about its derivative. The next example provides an instance of this principle.

ExamplE 5 Suppose that f s0d − 23 and f 9sxd < 5 for all values of x. How large can f s2d possibly be? SOLUTION We are given that f is differentiable (and therefore continuous) everywhere. In particular, we can apply the Mean Value Theorem on the interval f0, 2g. There exists a number c such that

f s2d 2 f s0d − f 9scds2 2 0d f s2d − f s0d 1 2f 9scd − 23 1 2f 9scd

so

We are given that f 9sxd < 5 for all x, so in particular we know that f 9scd < 5. Multiplying both sides of this inequality by 2, we have 2f 9scd < 10, so f s2d − 23 1 2f 9scd < 23 1 10 − 7 The largest possible value for f s2d is 7.

The Mean Value Theorem can be used to establish some of the basic facts of differential calculus. One of these basic facts is the following theorem. Others will be found in the following sections. 5 Theorem If f 9sxd − 0 for all x in an interval sa, bd, then f is constant on sa, bd. PROOF Let x 1 and x 2 be any two numbers in sa, bd with x 1 , x 2. Since f is differentiable on sa, bd, it must be differentiable on sx 1, x 2 d and continuous on fx 1, x 2 g. By applying the Mean Value Theorem to f on the interval fx 1, x 2 g, we get a number c such that x 1 , c , x 2 and

6

f sx 2 d 2 f sx 1d − f 9scdsx 2 2 x 1d

Since f 9sxd − 0 for all x, we have f 9scd − 0, and so Equation 6 becomes f sx 2 d 2 f sx 1 d − 0

or

f sx 2 d − f sx 1 d

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SECTION 4.2 The Mean Value Theorem

291

Therefore f has the same value at any two numbers x 1 and x 2 in sa, bd. This means that f is constant on sa, bd. ■ Corollary 7 says that if two functions have the same derivatives on an interval, then their graphs must be vertical translations of each other there. In other words, the graphs have the same shape, but could be shifted up or down.

7 Corollary If f 9sxd − t9sxd for all x in an interval sa, bd, then f 2 t is constant on sa, bd; that is, f sxd − tsxd 1 c where c is a constant. PROOF Let Fsxd − f sxd 2 tsxd. Then

F9sxd − f 9sxd 2 t9sxd − 0 for all x in sa, bd. Thus, by Theorem 5, F is constant; that is, f 2 t is constant.

NOTE Care must be taken in applying Theorem 5. Let f sxd −

H

1 if x . 0 x − | x | 21 if x , 0

The domain of f is D − hx | x ± 0j and f 9sxd − 0 for all x in D. But f is obviously not a constant function. This does not contradict Theorem 5 because D is not an interval. Notice that f is constant on the interval s0, `d and also on the interval s2`, 0d.

ExamplE 6 Prove the identity tan21 x 1 cot21 x − y2. SOLUTION Although calculus isn’t needed to prove this identity, the proof using calculus is quite simple. If f sxd − tan21 x 1 cot21 x, then

f 9sxd −

1 1 2 −0 1 1 x2 1 1 x2

for all values of x. Therefore f sxd − C, a constant. To determine the value of C, we put x − 1 [because we can evaluate f s1d exactly]. Then    C − f s1d − tan21 1 1 cot21 1 − 1 − 4 4 2 21 21 Thus tan x 1 cot x − y2. ■

1. The graph of a function f is shown. Verify that f satisfies the hypotheses of Rolle’s Theorem on the interval f0, 8g. Then estimate the value(s) of c that satisfy the conclusion of Rolle’s Theorem on that interval. y

3. The graph of a function t is shown. y y=©

y=ƒ 1 0

1

x

1 0

1

x

2. Draw the graph of a function defined on f0, 8g such that f s0d − f s8d − 3 and the function does not satisfy the conclusion of Rolle’s Theorem on f0, 8g.

(a) Verify that t satisfies the hypotheses of the Mean Value Theorem on the interval f0, 8g. (b) Estimate the value(s) of c that satisfy the conclusion of the Mean Value Theorem on the interval f0, 8g. (c) Estimate the value(s) of c that satisfy the conclusion of the Mean Value Theorem on the interval f2, 6g.

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292

CHAPTER 4

Applications of Differentiation

4. Draw the graph of a function that is continuous on f0, 8g where f s0d − 1 and f s8d − 4 and that does not satisfy the conclusion of the Mean Value Theorem on f0, 8g. 5–8 Verify that the function satisfies the three hypotheses of Rolle’s Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Rolle’s Theorem. 5. f sxd − 2 x 2 2 4 x 1 5, f21, 3g

7. f sxd − sins xy2d, fy2, 3y2g

26. Suppose that 3 < f 9sxd < 5 for all values of x. Show that 18 < f s8d 2 f s2d < 30.

f 12 , 2g

27. Does there exist a function f such that f s0d − 21, f s2d − 4, and f 9sxd < 2 for all x?

9. Let f sxd − 1 2 x 2y3. Show that f s21d − f s1d but there is no number c in s21, 1d such that f 9scd − 0. Why does this not contradict Rolle’s Theorem? 10. Let f sxd − tan x. Show that f s0d − f sd but there is no number c in s0, d such that f 9scd − 0. Why does this not contradict Rolle’s Theorem? 11–14 Verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval. Then find all numbers c that satisfy the conclusion of the Mean Value Theorem.

13. f sxd − e 22x,

14. f sxd −

x , x12

f1, 4g

; 15–16 Find the number c that satisfies the conclusion of the Mean Value Theorem on the given interval. Graph the function, the secant line through the endpoints, and the tangent line at sc, f scdd. Are the secant line and the tangent line parallel? 15. f sxd − sx , f0, 4g

|

19–20 Show that the equation has exactly one real root. 20. x 3 1 e x − 0

21. Show that the equation x 3 2 15x 1 c − 0 has at most one root in the interval f22, 2g. 22. Show that the equation x 4 1 4x 1 c − 0 has at most two real roots. 23. (a) Show that a polynomial of degree 3 has at most three real roots.

for all a and b

32. If f 9sxd − c (c a constant) for all x, use Corollary 7 to show that f sxd − cx 1 d for some constant d. 33. Let f sxd − 1yx and

tsxd −

1 x 11

16. f sxd − e , f0, 2g

18. Let f sxd − 2 2 2 x 2 1 . Show that there is no value of c such that f s3d 2 f s0d − f 9scds3 2 0d. Why does this not contradict the Mean Value Theorem?

19. 2 x 1 cos x − 0

30. Suppose f is an odd function and is differentiable everywhere. Prove that for every positive number b, there exists a number c in s2b, bd such that f 9scd − f sbdyb.

2x

17. Let f sxd − s x 2 3d22. Show that there is no value of c in s1, 4d such that f s4d 2 f s1d − f 9scds4 2 1d. Why does this not contradict the Mean Value Theorem?

|

29. Show that sin x , x if 0 , x , 2.

| sin a 2 sin b | < | a 2 b |

f22, 2g

f0, 3g

28. Suppose that f and t are continuous on fa, bg and differentiable on sa, bd. Suppose also that f sad − tsad and f 9sxd , t9sxd for a , x , b. Prove that f sbd , tsbd. [Hint: Apply the Mean Value Theorem to the function h − f 2 t.]

31. Use the Mean Value Theorem to prove the inequality

11. f sxd − 2x 2 2 3x 1 1, f0, 2g 12. f sxd − x 3 2 3x 1 2,

24. (a) Suppose that f is differentiable on R and has two roots. Show that f 9 has at least one root. (b) Suppose f is twice differentiable on R and has three roots. Show that f 0 has at least one real root. (c) Can you generalize parts (a) and (b)? 25. If f s1d − 10 and f 9sxd > 2 for 1 < x < 4, how small can f s4d possibly be?

6. f sxd − x 3 2 2x 2 2 4x 1 2, f22, 2g

8. f sxd − x 1 1y x,

(b) Show that a polynomial of degree n has at most n real roots.

if x . 0 1 x

if x , 0

Show that f 9sxd − t9sxd for all x in their domains. Can we conclude from Corollary 7 that f 2 t is constant? 34. Use the method of Example 6 to prove the identity 2 sin21x − cos21s1 2 2x 2 d

x>0

x21  − 2 arctan sx 2 . x11 2 36. At 2:00 pm a car’s speedometer reads 50 kmyh. At 2:10 pm it reads 65 kmyh. Show that at some time between 2:00 and 2:10 the acceleration is exactly 90 kmyh2.

35. Prove the identity arcsin

37. Two runners start a race at the same time and finish in a tie. Prove that at some time during the race they have the same speed. [Hint: Consider f std − tstd 2 hstd, where t and h are the position functions of the two runners.] 38. A number a is called a fixed point of a function f if f sad − a. Prove that if f 9sxd ± 1 for all real numbers x, then f has at most one fixed point.

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SECTION 4.3 How Derivatives Affect the Shape of a Graph

y

293

Many of the applications of calculus depend on our ability to deduce facts about a function f from information concerning its derivatives. Because f 9sxd represents the slope of the curve y − f sxd at the point sx, f sxdd, it tells us the direction in which the curve proceeds at each point. So it is reasonable to expect that information about f 9sxd will provide us with information about f sxd.

D B

What Does f 9 Say About f ? C

A

x

0

FIGURE 1

To see how the derivative of f can tell us where a function is increasing or decreasing, look at Figure 1. (Increasing functions and decreasing functions were defined in Section 1.1.) Between A and B and between C and D, the tangent lines have positive slope and so f 9sxd . 0. Between B and C, the tangent lines have negative slope and so f 9sxd , 0. Thus it appears that f increases when f 9sxd is positive and decreases when f 9sxd is negative. To prove that this is always the case, we use the Mean Value Theorem. Increasing/Decreasing Test

Let’s abbreviate the name of this test to the I/D Test.

(a) If f 9sxd . 0 on an interval, then f is increasing on that interval. (b) If f 9sxd , 0 on an interval, then f is decreasing on that interval. PROOF

(a) Let x 1 and x 2 be any two numbers in the interval with x1 , x2. According to the definition of an increasing function (page 19), we have to show that f sx1 d , f sx2 d. Because we are given that f 9sxd . 0, we know that f is differentiable on fx1, x2 g. So, by the Mean Value Theorem, there is a number c between x1 and x2 such that f sx 2 d 2 f sx 1 d − f 9scdsx 2 2 x 1 d

1

Now f 9scd . 0 by assumption and x 2 2 x 1 . 0 because x 1 , x 2. Thus the right side of Equation 1 is positive, and so f sx 2 d 2 f sx 1 d . 0

or

f sx 1 d , f sx 2 d

This shows that f is increasing. Part (b) is proved similarly.

ExamplE 1 Find where the function f sxd − 3x 4 2 4x 3 2 12x 2 1 5 is increasing and where it is decreasing. SOLUTION We start by differentiating f :

f 9sxd − 12x 3 2 12x 2 2 24x − 12xsx 2 2dsx 1 1d

_1

0

2

x

To use the IyD Test we have to know where f 9sxd . 0 and where f 9sxd , 0. To solve these inequalities we first find where f 9sxd − 0, namely at x − 0, 2, and 21. These are the critical numbers of f , and they divide the domain into four intervals (see the number line at the left). Within each interval, f 9sxd must be always positive or always negative. (See Examples 3 and 4 in Appendix A.) We can determine which is the case for each interval from the signs of the three factors of f 9sxd, namely, 12x, x 2 2, and x 1 1, as shown in the following chart. A plus sign indicates that the given expression is positive, and a minus sign indicates that it is negative. The last col-

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294

CHAPTER 4

Applications of Differentiation

umn of the chart gives the conclusion based on the IyD Test. For instance, f 9sxd , 0 for 0 , x , 2, so f is decreasing on (0, 2). (It would also be true to say that f is decreasing on the closed interval f0, 2g.)

20

Interval _2

3

x , 21

x22

x11

f 9sxd

f

2

2

2

2

decreasing on (2`, 21)

21 , x , 0

2

2

1

1

increasing on (21, 0)

0,x,2 x.2

1 1

2 1

1 1

2 1

decreasing on (0, 2) increasing on (2, `)

_30

FIGURE 2

12x

The graph of f shown in Figure 2 confirms the information in the chart.

Local Extreme Values Recall from Section 4.1 that if f has a local maximum or minimum at c, then c must be a critical number of f (by Fermat’s Theorem), but not every critical number gives rise to a maximum or a minimum. We therefore need a test that will tell us whether or not f has a local maximum or minimum at a critical number. You can see from Figure 2 that f s0d − 5 is a local maximum value of f because f increases on s21, 0d and decreases on s0, 2d. Or, in terms of derivatives, f 9sxd . 0 for 21 , x , 0 and f 9sxd , 0 for 0 , x , 2. In other words, the sign of f 9sxd changes from positive to negative at 0. This observation is the basis of the following test. The First Derivative Test Suppose that c is a critical number of a continuous function f. (a) If f 9 changes from positive to negative at c, then f has a local maximum at c. (b) If f 9 changes from negative to positive at c, then f has a local minimum at c. (c) If f 9 is positive to the left and right of c, or negative to the left and right of c, then f has no local maximum or minimum at c. The First Derivative Test is a consequence of the IyD Test. In part (a), for instance, since the sign of f 9sxd changes from positive to negative at c, f is increasing to the left of c and decreasing to the right of c. It follows that f has a local maximum at c. It is easy to remember the First Derivative Test by visualizing diagrams such as those in Figure 3. y

y

y

y

fª(x)0

fª(x)0 fª(x)0 x

0

c

x

(c) No maximum or minimum

0

c

x

(d) No maximum or minimum

FIGURE 3

ExamplE 2 Find the local minimum and maximum values of the function f in Example 1.

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295

SECTION 4.3 How Derivatives Affect the Shape of a Graph

SOLUTION From the chart in the solution to Example 1 we see that f 9sxd changes from negative to positive at 21, so f s21d − 0 is a local minimum value by the First Derivative Test. Similarly, f 9 changes from negative to positive at 2, so f s2d − 227 is also a local minimum value. As noted previously, f s0d − 5 is a local maximum value because f 9sxd changes from positive to negative at 0. ■

ExamplE 3 Find the local maximum and minimum values of the function tsxd − x 1 2 sin x

0 < x < 2

SOLUTION As in Example 1, we start by finding the critical numbers. The derivative is:

t9sxd − 1 1 2 cos x so t9sxd − 0 when cos x − The solutions of this equation are 2y3 and 4y3. Because t is differentiable everywhere, the only critical numbers are 2y3 and 4y3. We split the domain into intervals according to the critical numbers. Within each interval, t9sxd is either always positive or always negative and so we analyze t in the following chart. 212.

The 1 signs in the chart come from the fact that t9sxd . 0 when cos x . 2 12. From the graph of y − cos x, this is true in the indicated intervals.

t9sxd − 1 1 2 cos x

Interval

t

0 , x , 2y3

1

increasing on s0, 2y3d

2y3 , x , 4y3 4y3 , x , 2

2 1

decreasing on s2y3, 4y3d increasing on s4y3, 2d

Because t9sxd changes from positive to negative at 2y3, the First Derivative Test tells us that there is a local maximum at 2y3 and the local maximum value is

6

ts2y3d −

S D

2 2 2 s3 1 2 sin − 12 3 3 3 2

2 1 s3 < 3.83 3

Likewise, t9sxd changes from negative to positive at 4y3 and so 0

2π 3

4π 3

ts4y3d −

FIGURE 4

S D

4 4 4 s3 1 2 sin − 12 2 3 3 3 2

4 2 s3 < 2.46 3

is a local minimum value. The graph of t in Figure 4 supports our conclusion.

tsxd − x 1 2 sin x

What Does f 99 Say About f ? Figure 5 shows the graphs of two increasing functions on sa, bd. Both graphs join point A to point B but they look different because they bend in different directions. How can we distinguish between these two types of behavior? B

y

B

y

g

f A

A 0

FIGURE 5

a

b

(a)

x

0

b

a

(b)

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x

296

CHAPTER 4

Applications of Differentiation

In Figure 6 tangents to these curves have been drawn at several points. In (a) the curve lies above the tangents and f is called concave upward on sa, bd. In (b) the curve lies below the tangents and t is called concave downward on sa, bd.

B

y

f

Definition If the graph of f lies above all of its tangents on an interval I, then it is called concave upward on I. If the graph of f lies below all of its tangents on I, it is called concave downward on I.

A 0

x

Figure 7 shows the graph of a function that is concave upward (abbreviated CU) on the intervals sb, cd, sd, ed, and se, pd and concave downward (CD) on the intervals sa, bd, sc, dd, and sp, qd.

(a) Concave upward

B

y

y

D g

B

P

C

A 0

x

0 a

c

CD

(b) Concave downward

FIGURE 6

b

d

CU

e

CD

CU

p

CU

q

x

CD

FIGURE 7

Let’s see how the second derivative helps determine the intervals of concavity. Looking at Figure 6(a), you can see that, going from left to right, the slope of the tangent increases. This means that the derivative f 9 is an increasing function and therefore its derivative f 0 is positive. Likewise, in Figure 6(b) the slope of the tangent decreases from left to right, so f 9 decreases and therefore f 0 is negative. This reasoning can be reversed and suggests that the following theorem is true. A proof is given in Appendix F with the help of the Mean Value Theorem. Concavity Test (a) If f 0sxd . 0 for all x in I, then the graph of f is concave upward on I. (b) If f 0sxd , 0 for all x in I, then the graph of f is concave downward on I.

ExamplE 4 Figure 8 shows a population graph for Cyprian honeybees raised in an apiary. How does the rate of population increase change over time? When is this rate highest? Over what intervals is P concave upward or concave downward? P 80

Number of bees (in thousands)

60 40 20

0

FIGURE 8

3

6

9

12

15

18

t

Time (in weeks)

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SECTION 4.3 How Derivatives Affect the Shape of a Graph

297

SOLUTION By looking at the slope of the curve as t increases, we see that the rate of increase of the population is initially very small, then gets larger until it reaches a maximum at about t − 12 weeks, and decreases as the population begins to level off. As the population approaches its maximum value of about 75,000 (called the carrying capacity), the rate of increase, P9std, approaches 0. The curve appears to be concave upward on (0, 12) and concave downward on (12, 18). ■

In Example 4, the population curve changed from concave upward to concave downward at approximately the point (12, 38,000). This point is called an inflection point of the curve. The significance of this point is that the rate of population increase has its maximum value there. In general, an inflection point is a point where a curve changes its direction of concavity. Definition A point P on a curve y − f sxd is called an inflection point if f is continuous there and the curve changes from concave upward to concave downward or from concave downward to concave upward at P. For instance, in Figure 7, B, C, D, and P are the points of inflection. Notice that if a curve has a tangent at a point of inflection, then the curve crosses its tangent there. In view of the Concavity Test, there is a point of inflection at any point where the second derivative changes sign.

ExamplE 5 Sketch a possible graph of a function f that satisfies the following conditions: sid f 9sxd . 0 on s2`, 1d, f 9sxd , 0 on s1, `d siid f 0sxd . 0 on s2`, 22d and s2, `d, f 0sxd , 0 on s22, 2d siiid lim f sxd − 22, lim f sxd − 0

y

-2

0

x l 2`

SOLUTION Condition (i) tells us that f is increasing on s2`, 1d and decreasing on s1, `d. Condition (ii) says that f is concave upward on s2`, 22d and s2, `d, and conx cave downward on s22, 2d. From condition (iii) we know that the graph of f has two horizontal asymptotes: y − 22 (to the left) and y − 0 (to the right). We first draw the horizontal asymptote y − 22 as a dashed line (see Figure 9). We then draw the graph of f approaching this asymptote at the far left, increasing to its maximum point at x − 1, and decreasing toward the x-axis as at the far right. We also make sure that the graph has inflection points when x − 22 and 2. Notice that we made the curve bend upward for x , 22 and x . 2, and bend downward when x is between 22 and 2. ■

2

1

y=_2

FIGURE 9

y

f

0

ƒ

f(c) c

Another application of the second derivative is the following test for identifying local maximum and minimum values. It is a consequence of the Concavity Test, and serves as an alternative to the First Derivative Test. The Second Derivative Test Suppose f 0 is continuous near c. (a) If f 9scd − 0 and f 0scd . 0, then f has a local minimum at c.

P f ª(c)=0

x l`

(b) If f 9scd − 0 and f 0scd , 0, then f has a local maximum at c. x

FIGURE 10 f 99scd . 0, f is concave upward

x

For instance, part (a) is true because f 0sxd . 0 near c and so f is concave upward near c. This means that the graph of f lies above its horizontal tangent at c and so f has a local minimum at c. (See Figure 10.)

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298

CHAPTER 4

Applications of Differentiation

ExamplE 6 Discuss the curve y − x 4 2 4x 3 with respect to concavity, points of inflection, and local maxima and minima. Use this information to sketch the curve. SOLUTION If f sxd − x 4 2 4x 3, then

f 9sxd − 4x 3 2 12x 2 − 4x 2sx 2 3d f 0sxd − 12x 2 2 24x − 12xsx 2 2d To find the critical numbers we set f 9sxd − 0 and obtain x − 0 and x − 3. (Note that f 9 is a polynomial and hence defined everywhere.) To use the Second Derivative Test we evaluate f 0 at these critical numbers: f 0s0d − 0

Since f 9s3d − 0 and f 0s3d . 0, f s3d − 227 is a local minimum. [In fact, the expression for f 9sxd shows that f decreases to the left of 3 and increases to the right of 3.] Since f 0s0d − 0, the Second Derivative Test gives no information about the critical number 0. But since f 9sxd , 0 for x , 0 and also for 0 , x , 3, the First Derivative Test tells us that f does not have a local maximum or minimum at 0. Since f 0sxd − 0 when x − 0 or 2, we divide the real line into intervals with these numbers as endpoints and complete the following chart.

y

y=x\$-4˛ (0, 0) 2

inflection points

3

f 0s3d − 36 . 0

x

Interval

f 99sxd − 12 x sx 2 2d

s2`, 0d

1

upward

s0, 2d s2, `d

2 1

downward upward

Concavity

(2, _16)

(3, _27)

FIGURE 11

The point s0, 0d is an inflection point since the curve changes from concave upward to concave downward there. Also s2, 216d is an inflection point since the curve changes from concave downward to concave upward there. Using the local minimum, the intervals of concavity, and the inflection points, we sketch the curve in Figure 11. ■ NOTE The Second Derivative Test is inconclusive when f 0scd − 0. In other words, at such a point there might be a maximum, there might be a minimum, or there might be neither (as in Example 6). This test also fails when f 0scd does not exist. In such cases the First Derivative Test must be used. In fact, even when both tests apply, the First Derivative Test is often the easier one to use.

ExamplE 7 Sketch the graph of the function f sxd − x 2y3s6 2 xd1y3. SOLUTION Calculation of the first two derivatives gives Use the differentiation rules to check these calculations.

f 9sxd −

42x x 1y3s6 2 xd2y3

f 0sxd −

28 x 4y3s6 2 xd5y3

Since f 9sxd − 0 when x − 4 and f 9sxd does not exist when x − 0 or x − 6, the critical numbers are 0, 4, and 6. Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

SECTION 4.3 How Derivatives Affect the Shape of a Graph

Try reproducing the graph in Figure 12 with a graphing calculator or computer. Some machines produce the complete graph, some produce only the portion to the right of the y-axis, and some produce only the portion between x − 0 and x − 6. For an explanation and cure, see Example 7 in “Graphing Calculators and Computers” at www.stewartcalculus.com. An equivalent expression that gives the correct graph is y − sx 2 d1y3 ?

|

62x 62x

||

62x

|

1y3

y 4

(4, 2%?#)

3 2

Interval

42x

x 1y3

s6 2 xd2y3

f 9sxd

f

299

x,0

1

2

1

2

decreasing on s2`, 0d

0,x,4

1

1

1

1

increasing on s0, 4d

4,x,6 x.6

2 2

1 1

1 1

2 2

decreasing on s4, 6d decreasing on s6, `d

To find the local extreme values we use the First Derivative Test. Since f 9 changes from negative to positive at 0, f s0d − 0 is a local minimum. Since f 9 changes from positive to negative at 4, f s4d − 2 5y3 is a local maximum. The sign of f 9 does not change at 6, so there is no minimum or maximum there. (The Second Derivative Test could be used at 4 but not at 0 or 6 since f 0 does not exist at either of these numbers.) Looking at the expression for f 0sxd and noting that x 4y3 > 0 for all x, we have f 0sxd , 0 for x , 0 and for 0 , x , 6 and f 0sxd . 0 for x . 6. So f is concave downward on s2`, 0d and s0, 6d and concave upward on s6, `d, and the only inflection point is s6, 0d. The graph is sketched in Figure 12. Note that the curve has vertical tangents at s0, 0d and s6, 0d because | f 9sxd | l ` as x l 0 and as x l 6. ■

ExamplE 8 Use the first and second derivatives of f sxd − e 1yx, together with asymp0

1

2

3

4

5

7 x

y=x@ ?#(6-x)!?#

totes, to sketch its graph. SOLUTION Notice that the domain of f is hx | x ± 0j, so we check for vertical asymptotes by computing the left and right limits as x l 0. As x l 01, we know that t − 1yx l `, so lim1 e 1yx − lim e t − ` tl `

xl 0

FIGURE 12

and this shows that x − 0 is a vertical asymptote. As x l 02, we have t − 1yx l 2`, so lim2 e 1yx − lim e t − 0 xl 0

TEC In Module 4.3 you can practice using information about f 9, f 0, and asymptotes to determine the shape of the graph of f.

tl2`

As x l 6`, we have 1yx l 0 and so lim e 1yx − e 0 − 1

xl 6`

This shows that y − 1 is a horizontal asymptote (both to the left and right). Now let’s compute the derivative. The Chain Rule gives f 9sxd − 2

e 1yx x2

Since e 1yx . 0 and x 2 . 0 for all x ± 0, we have f 9sxd , 0 for all x ± 0. Thus f is decreasing on s2`, 0d and on s0, `d. There is no critical number, so the function has no local maximum or minimum. The second derivative is f 0sxd − 2

x 2e 1yxs21yx 2 d 2 e 1yxs2xd e 1yxs2x 1 1d − 4 x x4

Since e 1yx . 0 and x 4 . 0, we have f 0sxd . 0 when x . 212 sx ± 0d and f 0sxd , 0 when x , 212. So the curve is concave downward on s2`, 212 d and concave upward on

s212 , 0d and on s0, `d. The inflection point is s212 , e22d. To sketch the graph of f we first draw the horizontal asymptote y − 1 (as a dashed line), together with the parts of the curve near the asymptotes in a preliminary sketch Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

300

CHAPTER 4

Applications of Differentiation

[Figure 13(a)]. These parts reflect the information concerning limits and the fact that f  is decreasing on both s2`, 0d and s0, `d. Notice that we have indicated that f sxd l 0 as x l 02 even though f s0d does not exist. In Figure 13(b) we finish the sketch by incorporating the information concerning concavity and the inflection point. In Figure 13(c) we check our work with a graphing device. y

y

y=‰ 4

inflection point y=1

y=1 _3

0

0

x

(a) Preliminary sketch

3

x

_1

(b) Finished sketch

(c) Computer confirmation

FIGURE 13

(b) At what values of x does f have a local maximum or minimum? 5. y

1–2 Use the given graph of f to find the following. (a) The open intervals on which f is increasing. (b) The open intervals on which f is decreasing. (c) The open intervals on which f is concave upward. (d) The open intervals on which f is concave downward. (e) The coordinates of the points of inflection. 1.

2.

y

y=fª(x) 0

2

4

6

x

y

6.

y y=fª(x)

1

1

0 0

1

x

0

1

3. Suppose you are given a formula for a function f. (a) How do you determine where f is increasing or decreasing? (b) How do you determine where the graph of f is concave upward or concave downward? (c) How do you locate inflection points? 4. (a) State the First Derivative Test. (b) State the Second Derivative Test. Under what circumstances is it inconclusive? What do you do if it fails?

x

2

4

6

8

x

7. In each part state the x-coordinates of the inflection points of f. Give reasons for your answers. (a) The curve is the graph of f. (b) The curve is the graph of f 9. (c) The curve is the graph of f 0. y

0

2

4

6

8

x

5–6 The graph of the derivative f 9 of a function f is shown. (a) On what intervals is f increasing or decreasing?

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SECTION 4.3 How Derivatives Affect the Shape of a Graph

8. The graph of the first derivative f 9 of a function f is shown. (a) On what intervals is f increasing? Explain. (b) At what values of x does f have a local maximum or minimum? Explain. (c) On what intervals is f concave upward or concave downward? Explain. (d) What are the x-coordinates of the inflection points of f ? Why? y y=fª(x)

27. f 9s0d − f 9s2d − f 9s4d − 0, f 9sxd . 0 if x , 0 or 2 , x , 4, f 9sxd , 0 if 0 , x , 2 or x . 4, f 0sxd . 0 if 1 , x , 3, f 0sxd , 0 if x , 1 or x . 3 28. f 9sxd . 0 for all x ± 1, vertical asymptote x − 1, f 99sxd . 0 if x , 1 or x . 3, f 99sxd , 0 if 1 , x , 3 29. f 9s5d − 0, f 9sxd , 0 when x , 5, f 9sxd . 0 when x . 5, f 99s2d − 0, f 99s8d − 0, f 99sxd , 0 when x , 2 or x . 8, f 99sxd . 0 for 2 , x , 8, lim f sxd − 3, lim f sxd − 3 xl 2`

xl`

0

2

4

6

8

x

9–18 (a) Find the intervals on which f is increasing or decreasing. (b) Find the local maximum and minimum values of f. (c) Find the intervals of concavity and the inflection points.

30. f 9s0d − f 9s4d − 0, f 9sxd − 1 if x , 21, f 9sxd . 0 if 0 , x , 2, f 9sxd , 0 if 21 , x , 0 or 2 , x , 4 or x . 4, lim2 f 9sxd − `, lim1 f 9sxd − 2`, x l2

x l2

f 99sxd . 0 if 21 , x , 2 or 2 , x , 4, f 0sxd , 0 if x . 4 31. f 9sxd . 0 if x ± 2, f 99sxd . 0 if x , 2, f 99sxd , 0 if x . 2, f has inflection point s2, 5d, lim f sxd − 8, lim f sxd − 0

9. f sxd − x 3 2 3x 2 2 9x 1 4 10. f sxd − 2x 3 2 9x 2 1 12x 2 3 11. f sxd − x 4 2 2x 2 1 3

12. f sxd −

13. f sxd − sin x 1 cos x, 0 < x < 2

x2 2 x 13

14. f sxd − cos2 x 2 2 sin x, 0 < x < 2 15. f sxd − e 2x 1 e2x

16. f sxd − x 2 ln x

17. f sxd − x 2 2 x 2 ln x

18. f sxd − sx e2x

19–21 Find the local maximum and minimum values of f using both the First and Second Derivative Tests. Which method do you prefer? x2 19. f sxd − x 5 2 5x 1 3 20. f sxd − x21 4 21. f sxd − sx 2 s x 22. (a) Find the critical numbers of f sxd − x 4sx 2 1d3. (b) What does the Second Derivative Test tell you about the behavior of f at these critical numbers? (c) What does the First Derivative Test tell you? 23. Suppose f 0 is continuous on s2`, `d. (a) If f 9s2d − 0 and f 0s2d − 25, what can you say about f ? (b) If f 9s6d − 0 and f 0s6d − 0, what can you say about f ? 24–31 Sketch the graph of a function that satisfies all of the given conditions. 24. (a) f 9sxd , 0 and f 0sxd , 0 for all x (b) f 9sxd . 0 and f 0sxd . 0 for all x

xl 2`

xl`

32. Suppose f s3d − 2, f 9s3d − 12, and f 9sxd . 0 and f 0sxd , 0 for all x. (a) Sketch a possible graph for f. (b) How many solutions does the equation f sxd − 0 have? Why? (c) Is it possible that f 9s2d − 13? Why? 33. Suppose f is a continuous function where f sxd . 0 for all x, f s0d − 4, f 9s xd . 0 if x , 0 or x . 2, f 9s xd , 0 if 0 , x , 2, f 99s21d − f 99s1d − 0, f 99s xd . 0 if x , 21 or x . 1, f 99s xd , 0 if 21 , x , 1. (a) Can f have an absolute maximum? If so, sketch a possible graph of f. If not, explain why. (b) Can f have an absolute minimum? If so, sketch a possible graph of f. If not, explain why. (c) Sketch a possible graph for f that does not achieve an absolute minimum. 34. The graph of a function y − f sxd is shown. At which point(s) are the following true? dy d 2y and are both positive. (a) dx dx 2 dy d 2y (b) and are both negative. dx dx 2 dy d 2y (c) is negative but is positive. dx dx 2 y C

25. (a) f 9sxd . 0 and f 0sxd , 0 for all x (b) f 9sxd , 0 and f 0sxd . 0 for all x

| |

A

26. f 9s1d − f 9s21d − 0, f 9sxd , 0 if x , 1, f 9sxd . 0 if 1 , x . 2, f 9sxd − 21 if x . 2, f 0sxd , 0 if 22 , x , 0, inflection point s0, 1d

| |

301

| |

0

D E

B

x

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302

CHAPTER 4

Applications of Differentiation

(e) Use the information from parts (a)–(d) to sketch the graph of f. 1 1 x2 2 4 50. f sxd − 2 49. f sxd − 1 1 2 2 x x x 14 ex 51. f sxd − sx 2 1 1 2 x 52. f sxd − 1 2 ex

35–36 The graph of the derivative f 9 of a continuous function f is shown. (a) On what intervals is f increasing? Decreasing? (b) At what values of x does f have a local maximum? Local minimum? (c) On what intervals is f concave upward? Concave downward? (d) State the x-coordinate(s) of the point(s) of inflection. (e) Assuming that f s0d − 0, sketch a graph of f.

53. f sxd − e 2x

54. f sxd − x 2 16 x 2 2 23 ln x

2

56. f sxd − e arctan x

55. f sxd − lns1 2 ln xd

35. y

y=fª(x)

57. Suppose the derivative of a function f is f 9sxd − sx 1 1d2 sx 2 3d5 sx 2 6d 4. On what interval is f increasing?

2

0

2

4

6

58. Use the methods of this section to sketch the curve y − x 3 2 3a 2x 1 2a 3, where a is a positive constant. What do the members of this family of curves have in common? How do they differ from each other?

8 x

_2

36. y

y=fª(x) 2

0

2

4

6

8 x

; 59–60 (a) Use a graph of f to estimate the maximum and minimum values. Then find the exact values. (b) Estimate the value of x at which f increases most rapidly. Then find the exact value. x11 59. f sxd − 60. f s xd − x 2 e2x sx 2 1 1 ; 61–62 (a) Use a graph of f to give a rough estimate of the intervals of concavity and the coordinates of the points of inflection. (b) Use a graph of f 0 to give better estimates.

_2

37–48 (a) Find the intervals of increase or decrease. (b) Find the local maximum and minimum values. (c) Find the intervals of concavity and the inflection points. (d) Use the information from parts (a)–(c) to sketch the graph. Check your work with a graphing device if you have one. 37. f sxd − 2x 3 2 3x 2 2 12x

38. f sxd − 2 1 3x 2 x 3

39. f sxd − 12 x 4 2 4x 2 1 3

40. tsxd − 200 1 8x 3 1 x 4

41. f sxd − 2 1 2x 2 2 x 4

42. hsxd − 5x 3 2 3x 5

43. Fsxd − x s6 2 x

44. Gsxd − 5x 2y3 2 2x 5y3

45. Csxd − x 1y3sx 1 4d

46. f sxd − lnsx 2 1 9d

47. f sd − 2 cos  1 cos 2 , 0 <  < 2 48. Ssxd − x 2 sin x, 0 < x < 4

61. f sxd − sin 2x 1 sin 4x, 2

62. f sxd − sx 2 1d sx 1 1d CAS

0 0. (b) Deduce that e x > 1 1 x 1 12 x 2 for x > 0. (c) Use mathematical induction to prove that for x > 0 and any positive integer n, ex > 1 1 x 1

xn x2 1 ∙∙∙ 1 2! n!

85. Show that a cubic function (a third-degree polynomial) always has exactly one point of inflection. If its graph has three x-intercepts x 1, x 2, and x 3, show that the x-coordinate of the inflection point is sx 1 1 x 2 1 x 3 dy3. ; 86. For what values of c does the polynomial Psxd − x 4 1 cx 3 1 x 2 have two inflection points? One inflection point? None? Illustrate by graphing P for several values of c. How does the graph change as c decreases? 87. Prove that if sc, f scdd is a point of inflection of the graph of f and f 0 exists in an open interval that contains c, then f 0scd − 0. [Hint: Apply the First Derivative Test and Fermat’s Theorem to the function t − f 9.] 4

88. Show that if f sxd − x , then f 0s0d − 0, but s0, 0d is not an inflection point of the graph of f . 89. Show that the function tsxd − x x has an inflection point at s0, 0d but t0s0d does not exist.

| |

90. Suppose that f 09 is continuous and f 9scd − f 0scd − 0, but f -scd . 0. Does f have a local maximum or minimum at c? Does f have a point of inflection at c? 91. Suppose f is differentiable on an interval I and f 9sxd . 0 for all numbers x in I except for a single number c. Prove that f is increasing on the entire interval I. 92. For what values of c is the function 1 f sxd − cx 1 2 x 13 increasing on s2`, `d? 93. The three cases in the First Derivative Test cover the situations one commonly encounters but do not exhaust all possibilities. Consider the functions f, t, and h whose values at 0 are all 0 and, for x ± 0, f sxd − x 4 sin

1 x

S D

tsxd − x 4 2 1 sin

S

hsxd − x 4 22 1 sin

1 x

D

1 x

(a) Show that 0 is a critical number of all three functions but their derivatives change sign infinitely often on both sides of 0. (b) Show that f has neither a local maximum nor a local minimum at 0, t has a local minimum, and h has a local maximum.

Suppose we are trying to analyze the behavior of the function ln x Fsxd − x21 Although F is not defined when x − 1, we need to know how F behaves near 1. In particular, we would like to know the value of the limit ln x lim 1 x l1 x 2 1 In computing this limit we can’t apply Law 5 of limits (the limit of a quotient is the quotient of the limits, see Section 2.3) because the limit of the denominator is 0. In fact, although the limit in (1) exists, its value is not obvious because both numerator and denominator approach 0 and 00 is not defined. In general, if we have a limit of the form lim

xla

f sxd tsxd

where both f sxd l 0 and tsxd l 0 as x l a, then this limit may or may not exist and is called an indeterminate form of type 00. We met some limits of this type in Chapter 2. For rational functions, we can cancel common factors: lim

x l1

x2 2 x xsx 2 1d x 1 − lim − lim − 2 x l1 x 1 1 x l1 sx 1 1dsx 2 1d x 21 2

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Section 4.4

Indeterminate Forms and l’Hospital’s Rule

305

We used a geometric argument to show that lim

xl0

sin x −1 x

But these methods do not work for limits such as (1), so in this section we introduce a systematic method, known as l’Hospital’s Rule, for the evaluation of indeterminate forms. Another situation in which a limit is not obvious occurs when we look for a horizontal asymptote of F and need to evaluate the limit ln x x21

lim

2

xl`

It isn’t obvious how to evaluate this limit because both numerator and denominator become large as x l `. There is a struggle between numerator and denominator. If the numerator wins, the limit will be ` (the numerator was increasing significantly faster than the denominator); if the denominator wins, the answer will be 0. Or there may be some compromise, in which case the answer will be some finite positive number. In general, if we have a limit of the form lim

y

xla

f g

0

a

y

x

where both f sxd l ` (or 2`) and tsxd l ` (or 2`), then the limit may or may not exist and is called an indeterminate form of type `y`. We saw in Section 2.6 that this type of limit can be evaluated for certain functions, including rational functions, by dividing numerator and denominator by the highest power of x that occurs in the denominator. For instance, 1 x2 x 21 120 1 lim − lim − − 2 x l ` 2x 1 1 xl` 1 210 2 21 2 x

y=m¡(x-a)

12

2

y=m™(x-a) 0

a

f sxd tsxd

x

This method does not work for limits such as (2), but l’Hospital’s Rule also applies to this type of indeterminate form.

FIGURE 1 Figure 1 suggests visually why l’Hospital’s Rule might be true. The first graph shows two differentiable functions f and t, each of which approaches 0 as x l a. If we were to zoom in toward the point sa, 0d, the graphs would start to look almost linear. But if the functions actually were linear, as in the second graph, then their ratio would be m1sx 2 ad m1 − m2sx 2 ad m2 which is the ratio of their derivatives. This suggests that lim

xla

f sxd f 9sxd − lim x l a t9sxd tsxd

L’Hospital’s Rule Suppose f and t are differentiable and t9sxd ± 0 on an open interval I that contains a (except possibly at a). Suppose that lim f sxd − 0

and

lim f sxd − 6`

and

xla

or that

xla

lim tsxd − 0

xla

lim tsxd − 6`

xla

(In other words, we have an indeterminate form of type 00 or `y`.) Then lim

xla

f sxd f 9sxd − lim x l a t9sxd tsxd

if the limit on the right side exists (or is ` or 2`). NOTE 1 L’Hospital’s Rule says that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives, provided that the given conditions are satisfied.

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306

chapter 4

Applications of Differentiation

It is especially important to verify the conditions regarding the limits of f and t before using l’Hospital’s Rule. L’Hospital L’Hospital’s Rule is named after a French nobleman, the Marquis de l’Hospital (1661–1704), but was discovered by a Swiss mathematician, John Bernoulli (1667–1748). You might sometimes see l’Hospital spelled as l’Hôpital, but he spelled his own name l’Hospital, as was common in the 17th century. See Exercise 83 for the example that the Marquis used to illustrate his rule. See the project on page 314 for further historical details.

NOTE 2 L’Hospital’s Rule is also valid for one-sided limits and for limits at infinity or negative infinity; that is, “x l a” can be replaced by any of the symbols x l a1, x l a2, x l `, or x l 2`. NOTE 3 For the special case in which f sad − tsad − 0, f 9 and t9 are continuous, and t9sad ± 0, it is easy to see why l’Hospital’s Rule is true. In fact, using the alternative form of the definition of a derivative, we have

f 9sxd f 9sad lim − − x l a t9sxd t9sad

f sxd 2 f sad f sxd 2 f sad x2a x2a − lim x l a tsxd 2 tsad tsxd 2 tsad lim xla x2a x2a

lim

xla

f sxd 2 f sad f sxd − lim x l a tsxd 2 tsad tsxd

− lim

xla

It is more difficult to prove the general version of l’Hospital’s Rule. See Appendix F.

ExamplE 1 Find lim

xl1

ln x . x21

SoLUtion Since

lim ln x − ln 1 − 0

and

x l1

lim sx 2 1d − 0

x l1

the limit is an indeterminate form of type 00 , so we can apply l’Hospital’s Rule: Notice that when using l’Hospital’s Rule we differentiate the numerator and denominator separately. We do not use the Quotient Rule.

d sln xd dx ln x 1yx − lim lim − lim xl1 1 xl1 x 2 1 xl1 d sx 2 1d dx − lim

xl1

The graph of the function of Example 2 is shown in Figure 2. We have noticed previously that exponential functions grow far more rapidly than power functions, so the result of Example 2 is not unexpected. See also Exercise 73. 20

ExamplE 2 Calculate lim

xl`

ex . x2

SoLUtion We have lim x l ` e x − ` and lim x l ` x 2 − `, so the limit is an indetermi-

nate form of type `y`, and l’Hospital’s Rule gives d se x d ex dx ex lim 2 − lim − lim xl` x xl` d x l ` 2x sx 2 d dx

10

FIGURE 2

Since e x l ` and 2x l ` as x l `, the limit on the right side is also indeterminate, but a second application of l’Hospital’s Rule gives

y= ´ ≈ 0

1 −1 x

lim

xl`

ex ex ex − lim − lim −` x l ` 2x xl` 2 x2

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Section 4.4

The graph of the function of Example 3 is shown in Figure 3. We have discussed previously the slow growth of logarithms, so it isn’t surprising that this ratio approaches 0 as x l `. See also Exercise 74.

ExamplE 3 Calculate lim

xl`

Indeterminate Forms and l’Hospital’s Rule

307

ln x . sx

SoLUtion Since ln x l ` and sx l ` as x l `, l’Hospital’s Rule applies:

lim

xl`

2

1yx 1yx ln x − lim 1 21y2 − lim x l ` xl` x x 1y (2sx ) s 2

Notice that the limit on the right side is now indeterminate of type 00 . But instead of applying l’Hospital’s Rule a second time as we did in Example 2, we simplify the expression and see that a second application is unnecessary:

y= ln x œ„ x 0

10,000

lim

_1

xl`

1yx ln x 2 − lim − lim −0 x l ` x 1y 2 x x l ` (s ) s sx

FIGURE 3

In both Examples 2 and 3 we evaluated limits of type `y`, but we got two different results. In Example 2, the infinite limit tells us that the numerator e x increases significantly faster than the denominator x 2, resulting in larger and larger ratios. In fact, y − e x grows more quickly than all the power functions y − x n (see Exercise 73). In Example 3 we have the opposite situation; the limit of 0 means that the denominator outpaces the numerator, and the ratio eventually approaches 0.

ExamplE 4 Find lim

xl0

tan x 2 x . (See Exercise 2.2.50.) x3

SoLUtion Noting that both tan x 2 x l 0 and x 3 l 0 as x l 0, we use l’Hospital’s

Rule: lim

xl0

tan x 2 x sec2x 2 1 − lim 3 xl0 x 3x 2

Since the limit on the right side is still indeterminate of type 00 , we apply l’Hospital’s Rule again: lim

xl0

The graph in Figure 4 gives visual confirmation of the result of Example 4. If we were to zoom in too far, however, we would get an inaccurate graph because tan x is close to x when x is small. See Exercise 2.2.50(d). 1

0

Because lim x l 0 sec2 x − 1, we simplify the calculation by writing lim

xl0

2 sec2x tan x 1 tan x 1 tan x − lim sec2 x ? lim − lim xl0 6x 3 xl0 x 3 xl 0 x

We can evaluate this last limit either by using l’Hospital’s Rule a third time or by writing tan x as ssin xdyscos xd and making use of our knowledge of trigonometric limits. Putting together all the steps, we get y=

_1

sec2x 2 1 2 sec2x tan x − lim xl0 3x 2 6x

tan x- x ˛

lim

1

xl0

tan x 2 x sec 2 x 2 1 2 sec 2 x tan x − lim − lim xl0 xl0 x3 3x 2 6x −

FIGURE 4

1 tan x 1 sec 2 x 1 lim − lim − 3 xl0 x 3 xl0 1 3

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308

chapter 4

Applications of Differentiation

ExamplE 5 Find lim2 xl

sin x . 1 2 cos x

SoLUtion If we blindly attempted to use l’Hospital’s Rule, we would get

lim

x l 2

sin x cos x − lim − 2` 1 2 cos x x l 2 sin x

This is wrong! Although the numerator sin x l 0 as x l  2, notice that the denominator s1 2 cos xd does not approach 0, so l’Hospital’s Rule can’t be applied here. The required limit is, in fact, easy to find because the function is continuous at  and the denominator is nonzero there: lim2

xl

sin x sin  0 − − −0 1 2 cos x 1 2 cos  1 2 s21d

Example 5 shows what can go wrong if you use l’Hospital’s Rule without thinking. Other limits can be found using l’Hospital’s Rule but are more easily found by other methods. (See Examples 2.3.3, 2.3.5, and 2.6.3, and the discussion at the beginning of this section.) So when evaluating any limit, you should consider other methods before using l’Hospital’s Rule.

indeterminate products If lim x l a f sxd − 0 and lim x l a tsxd − ` (or 2`), then it isn’t clear what the value of lim x l a f f sxd tsxdg, if any, will be. There is a struggle between f and t. If f wins, the answer will be 0; if t wins, the answer will be ` (or 2`). Or there may be a compromise where the answer is a finite nonzero number. This kind of limit is called an indeterminate form of type 0 ? `. We can deal with it by writing the product ft as a quotient: ft −

f 1yt

or

ft −

t 1yf

This converts the given limit into an indeterminate form of type 00 or `y` so that we can use l’Hospital’s Rule. Figure 5 shows the graph of the function in Example 6. Notice that the function is undefined at x − 0; the graph approaches the origin but never quite reaches it.

ExamplE 6 Evaluate lim1 x ln x. x l0

SoLUtion The given limit is indeterminate because, as x l 0 1, the first factor sxd

approaches 0 while the second factor sln xd approaches 2`. Writing x − 1ys1yxd, we have 1yx l ` as x l 0 1, so l’Hospital’s Rule gives

y

lim x ln x − lim1

x l 01

y=x ln x

xl0

ln x 1yx − lim1 − lim1 s2xd − 0 xl0 x l 0 21yx 2 1yx

NOTE In solving Example 6 another possible option would have been to write lim x ln x − lim1

x l 01

0

1

FIGURE 5

xl0

x 1yln x

x

This gives an indeterminate form of the type 00, but if we apply l’Hospital’s Rule we get a more complicated expression than the one we started with. In general, when we rewrite an indeterminate product, we try to choose the option that leads to the simpler limit.

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Section 4.4

Indeterminate Forms and l’Hospital’s Rule

309

indeterminate Differences If lim x l a f sxd − ` and lim x l a tsxd − `, then the limit lim f f sxd 2 tsxdg

xla

is called an indeterminate form of type ` 2 `. Again there is a contest between f and t. Will the answer be ` ( f wins) or will it be 2` (t wins) or will they compromise on a finite number? To find out, we try to convert the difference into a quotient (for instance, by using a common denominator, or rationalization, or factoring out a common factor) so that we have an indeterminate form of type 00 or `y`.

ExamplE 7 Compute lim1 x l1

S

D

1 1 2 . ln x x21

SoLUtion First notice that 1ysln xd l ` and 1ysx 2 1d l ` as x l 11, so the limit

is indeterminate of type ` 2 `. Here we can start with a common denominator: lim

x l11

S

1 1 2 ln x x21

D

− lim1 x l1

x 2 1 2 ln x sx 2 1d ln x

Both numerator and denominator have a limit of 0, so l’Hospital’s Rule applies, giving x 2 1 2 ln x lim − lim1 x l11 sx 2 1d ln x x l1

1 x x21 − lim1 x l1 1 x 2 1 1 x ln x sx 2 1d ? 1 ln x x 12

Again we have an indeterminate limit of type 00 , so we apply l’Hospital’s Rule a second time: x21 1 lim − lim1 x l11 x 2 1 1 x ln x x l1 1 1 1 x ? 1 ln x x 1 1 − lim1 − ■ x l1 2 1 ln x 2

ExamplE 8 Calculate lim se x 2 xd. xl`

SoLUtion This is an indeterminate difference because both e x and x approach infinity.

We would expect the limit to be infinity because e x l ` much faster than x. But we can verify this by factoring out x : ex 2 x − x

S D ex 21 x

The term e xyx l ` as x l ` by l’Hospital’s Rule and so we now have a product in which both factors grow large:

F S DG

lim se x 2 xd − lim x

xl`

xl`

ex 21 x

−`

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310

chapter 4

Applications of Differentiation

indeterminate powers Several indeterminate forms arise from the limit lim − f f sxdg tsxd

x la

1. lim f sxd − 0

and

2. lim f sxd − `

and

3. lim f sxd − 1

and

xla xla xla

lim tsxd − 0

type 0 0

lim tsxd − 0

type ` 0

xla xla

lim tsxd − 6` type 1`

xla

Each of these three cases can be treated either by taking the natural logarithm:

Although forms of the type 0 0, `0, and 1` are indeterminate, the form 0 ` is not indeterminate. (See Exercise 86.)

y − f f sxdg tsxd,

let

then ln y − tsxd ln f sxd

or by writing the function as an exponential: f f sxdg tsxd − e tsxd ln f sxd (Recall that both of these methods were used in differentiating such functions.) In either method we are led to the indeterminate product tsxd ln f sxd, which is of type 0 ? `.

ExamplE 9 Calculate lim1 s1 1 sin 4xdcot x. xl0

SoLUtion First notice that as x l 0 1, we have 1 1 sin 4x l 1 and cot x l `, so the

given limit is indeterminate (type 1`). Let y − s1 1 sin 4xdcot x Then

ln y − lnfs1 1 sin 4xdcot x g − cot x lns1 1 sin 4xd −

lns1 1 sin 4xd tan x

so l’Hospital’s Rule gives 4 cos 4x lns1 1 sin 4xd 1 1 sin 4x lim ln y − lim1 − lim1 −4 x l 01 xl 0 xl0 tan x sec2x So far we have computed the limit of ln y, but what we want is the limit of y. To find this we use the fact that y − e ln y: The graph of the function y − x x, x . 0, is shown in Figure 6. Notice that although 0 0 is not defined, the values of the function approach 1 as x l 01. This confirms the result of Example 10.

lim s1 1 sin 4xdcot x − lim1 y − lim1 e ln y − e 4

x l 01

xl0

xl0

ExamplE 10 Find lim1 x x. xl0

SoLUtion Notice that this limit is indeterminate since 0 x − 0 for any x . 0 but

2

x 0 − 1 for any x ± 0. (Recall that 0 0 is undefined.) We could proceed as in Example 9 or by writing the function as an exponential: x x − se ln x d x − e x ln x In Example 6 we used l’Hospital’s Rule to show that _1

2 0

FIGURE 6

lim x ln x − 0

x l 01

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Section 4.4

Indeterminate Forms and l’Hospital’s Rule

311

Therefore lim x x − lim1 e x ln x − e 0 − 1

x l 01

1–4 Given that lim tsxd − 0

lim f sxd − 0

xla

lim psxd − `

xla

7. The graph of a function f and its tangent line at 0 are shown. f sxd What is the value of lim x ? xl0 e 2 1

lim hsxd − 1

xla

xla

y

lim qs xd − `

y=x

xla

which of the following limits are indeterminate forms? For those that are not an indeterminate form, evaluate the limit where possible. 1. (a) lim

f sxd tsxd

(b) lim

f sxd psxd

(c) lim

hsxd psxd

(d) lim

psxd f sxd

(e) lim

psxd qsxd

xla

xla

xla

xla

xla

2. (a) lim f f sxdpsxdg

y=ƒ 0

x

8–68 Find the limit. Use l’Hospital’s Rule where appropriate. If there is a more elementary method, consider using it. If l’Hospital’s Rule doesn’t apply, explain why. 8. lim

x23 x2 2 9

9. lim

x 2 2 2x 2 8 x24

10. lim

x3 1 8 x12

x 3 2 2x 2 1 1 x l1 x3 2 1

12. lim

6x 2 1 5x 2 4 4x 2 1 16x 2 9

cos x 1 2 sin x

14. lim

xl3

(b) lim fhsxdpsxdg

xla

xl0

xla

(c) lim f psxdqsxdg xla

x l4

3. (a) lim f f sxd 2 psxdg

(b) lim f psxd 2 qsxdg

xla

11. lim

xla

(c) lim f psxd 1 qsxdg

x l 22

x l 1y2

xla

13.

4. (a) lim f f sxdg tsxd

(b) lim f f sxdg psxd

xl a

(c) lim fhsxdg

lim

x l sy2d1

xla

(d) lim f psxdg f sxd

psxd

xla

15. lim

xla

tl0

qsxd (f) lim spsxd

(e) lim f psxdg qsxd xla

e 2t 2 1 sin t

xla

5.

y

6.

y=1.8(x-2)

f

ln x sin x

ln x x

22. lim

ln ln x x

23. lim

t8 2 1 t5 2 1

24. lim

8t 2 5t t

f

g

25. lim

s1 1 2x 2 s1 2 4x x

26. lim

e uy10 u3

0

27. lim

ex 2 1 2 x x2

28. lim

sinh x 2 x x3

tl1

2

2 4

x

y=5 (x-2)

sx

xl`

xl0

y=1.5(x-2)

x

0

g

1 2 sin  csc

20. lim

ln x

19. lim

21. lim1

y

l y2

1 1 cos  1 2 cos

lim

l y2

5–6 Use the graphs of f and t and their tangent lines at s2, 0d to f sxd . find lim x l 2 tsxd

16. lim

18. lim

17.

xl0

1 2 sin  1 1 cos 2

tan 3x sin 2x

xl0

l

xl1

xl`

tl0

ul `

y=2-x xl0

xl0

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312

chapter 4

Applications of Differentiation

tanh x tan x

29. lim

xl0

30. lim

xl0

x 2 sin x x 2 tan x

21

67. lim1 s1 1 sin 3xd 1yx

sin x x

32. lim

sln xd x

33. lim

x3x 3 21

34. lim

cos mx 2 cos nx x2

xl`

x

xl0

xl0

x sinsx 2 1d 36. lim x l 1 2x 2 2 x 2 1

arctans2 xd 37. lim1 xl0 ln x

x 38. lim x l 0 tan21s4xd

xa 2 1 39. lim b , b±0 xl1 x 2 1

e x 2 e2x 2 2x 40. lim xl0 x 2 sin x

x a 2 ax 1 a 2 1 sx 2 1d2

xl1

x la

69. lim

S D 11

2 x

x

70. lim

xl0

71. f sxd − e x 2 1,

43. lim cot 2x sin 6x

44. lim1 sin x ln x

72. f sxd − 2x sin x,

45. lim sin 5x csc 3x

46. lim x ln 1 2

S D

x l 2`

tsxd − sec x 2 1

73. Prove that lim

1 x

47. lim x e

48. lim x

xl`

xl`

49. lim1 ln x tans xy2d x l1

51. lim

xl1

S S

53. lim1 x l0

3y2

D D

50.

lim cos x sec 5x

1 1 2 x x e 21

54. lim1

xl0

xl0

S

1 1 2 x tan21 x

xl`

56. lim1 flnsx 7 2 1d 2 lnsx 5 2 1dg x l1

57. lim1 x sx

58. lim1 stan 2 xd x

59. lim s1 2 2xd1yx

60. lim

61. lim1 x 1ys12xd

62. lim x sln 2dys1 1 ln xd

63. lim x 1yx

64. lim x e

65. lim1 s4x 1 1d cot x

66. lim s2 2 xdtansxy2d

x l1

xl`

x l0

D

75–76 What happens if you try to use l’Hospital’s Rule to find the limit? Evaluate the limit using another method. x sec x 75. lim 76. lim 2 xl ` sx 2 1 1 x l sy2d tan x x ; 77. Investigate the family of curves f sxd − e 2 cx. In particular, find the limits as x l 6` and determine the values of c for which f has an absolute minimum. What happens to the minimum points as c increases?

55. lim sx 2 ln xd

xl0

ln x −0 xp

for any number p . 0. This shows that the logarithmic function approaches infinity more slowly than any power of x.

x l sy2d2

52. lim scsc x 2 cot xd

x l0

xl`

sins1yxd

x 1 2 x21 ln x

ex −` xn

74. Prove that lim

3 2x 2

5x 2 4x 3x 2 2x

for any positive integer n. This shows that the exponential function approaches infinity faster than any power of x.

x l0

xl0

2x11

tsxd − x 3 1 4x

xl` xl0

D

; 71–72 Illustrate l’Hospital’s Rule by graphing both f sxdytsxd and f 9sxdyt9sxd near x − 0 to see that these ratios have the same limit as x l 0. Also, calculate the exact value of the limit.

cos x lnsx 2 ad lnse x 2 e a d

42. lim1

2x 2 3 2x 1 5

; 69–70 Use a graph to estimate the value of the limit. Then use l’Hospital’s Rule to find the exact value.

xl`

x 1 sin x 35. lim x l 0 x 1 cos x

41. lim

xl`

xl0

S

2

31. lim

xl0

68. lim

xl0

xl`

S D 11

a x

bx

xl`

2x

xl`

xl1

78. If an object with mass m is dropped from rest, one model for its speed v after t seconds, taking air resistance into account, is mt v− s1 2 e 2ctym d c where t is the acceleration due to gravity and c is a positive constant. (In Chapter 9 we will be able to deduce this equation from the assumption that the air resistance is proportional to the speed of the object; c is the proportionality constant.) (a) Calculate lim t l ` v. What is the meaning of this limit? (b) For fixed t, use l’Hospital’s Rule to calculate lim cl 01 v. What can you conclude about the velocity of a falling object in a vacuum?

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Section 4.4

79. If an initial amount A0 of money is invested at an interest rate r compounded n times a year, the value of the investment after t years is

S D

r A − A0 1 1 n

Indeterminate Forms and l’Hospital’s Rule

81. Some populations initally grow exponentially but eventually level off. Equations of the form Pstd −

nt

If we let n l `, we refer to the continuous compounding of interest. Use l’Hospital’s Rule to show that if interest is compounded continuously, then the amount after t years is A − A0 e rt 80. Light enters the eye through the pupil and strikes the retina, where photoreceptor cells sense light and color. W. Stanley Stiles and B. H. Crawford studied the phenomenon in which measured brightness decreases as light enters farther from the center of the pupil. (See the figure.)

M 1 1 Ae 2kt

where M, A, and k are positive constants, are called logistic equations and are often used to model such populations. (We will investigate these in detail in Chapter 9.) Here M is called the carrying capacity and represents the maximum population M 2 P0 , where P0 is the size that can be supported, and A − P0 initial population. (a) Compute lim tl ` Pstd. Explain why your answer is to be expected. (b) Compute lim Ml ` Pstd. (Note that A is defined in terms of M.) What kind of function is your result? 82. A metal cable has radius r and is covered by insulation so that the distance from the center of the cable to the exterior of the insulation is R. The velocity v of an electrical impulse in the cable is

SD SD

v − 2c

B A

2

r R

ln

r R

where c is a positive constant. Find the following limits and interpret your answers. (a) lim1 v (b) lim1 v R lr

A light beam A that enters through the center of the pupil measures brighter than a beam B entering near the edge of the pupil.

They detailed their findings of this phenomenon, known as the Stiles–Crawford effect of the first kind, in an important paper published in 1933. In particular, they observed that the amount of luminance sensed was not proportional to the area of the pupil as they expected. The percentage P of the total luminance entering a pupil of radius r mm that is sensed at the retina can be described by P−

1 2 102r r 2 ln 10

2

r l0

83. The first appearance in print of l’Hospital’s Rule was in the book Analyse des Infiniment Petits published by the Marquis de l’Hospital in 1696. This was the first calculus textbook ever published and the example that the Marquis used in that book to illustrate his rule was to find the limit of the function y−

3 aax s2a 3x 2 x 4 2 a s 4 3 a2s ax

as x approaches a, where a . 0. (At that time it was common to write aa instead of a 2.) Solve this problem. 84. The figure shows a sector of a circle with central angle . Let Asd be the area of the segment between the chord PR and the arc PR. Let Bsd be the area of the triangle PQR. Find lim  l 01 sdysd.

where  is an experimentally determined constant, typically about 0.05. (a) What is the percentage of luminance sensed by a pupil of radius 3 mm? Use  − 0.05. (b) Compute the percentage of luminance sensed by a pupil of radius 2 mm. Does it make sense that it is larger than the answer to part (a)? (c) Compute lim1 P. Is the result what you would expect? rl0 Is this result physically possible? Source: Adapted from W. Stiles and B. Crawford, “The Luminous Efficiency of Rays Entering the Eye Pupil at Different Points.” Proceedings of the Royal Society of London, Series B: Biological Sciences 112 (1933): 428–50.

313

P A(¨) B(¨ ) ¨ O

R

Q

85. Evaluate

F

S DG

lim x 2 x 2 ln

xl`

11x x

.

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314

chapter 4

Applications of Differentiation

86. Suppose f is a positive function. If lim xl a f sxd − 0 and lim xl a tsxd − `, show that

90. If f 0 is continuous, show that lim

hl0

lim f f sxdg tsxd − 0

f sx 1 hd 2 2 f sxd 1 f sx 2 hd − f 0sxd h2

xla

This shows that 0 ` is not an indeterminate form.

91. Let f sxd −

87. If f 9 is continuous, f s2d − 0, and f 9s2d − 7, evaluate f s2 1 3xd 1 f s2 1 5xd lim xl0 x 88. For what values of a and b is the following equation true? lim

xl0

S

b sin 2x 1a1 2 x3 x

D

−0

lim

f sx 1 hd 2 f sx 2 hd − f 9sxd 2h

Explain the meaning of this equation with the aid of a diagram.

WRITING PROJECT

2

if x ± 0 if x − 0

(a) Use the definition of derivative to compute f 9s0d. (b) Show that f has derivatives of all orders that are defined on R. [Hint: First show by induction that there is a polynomial pnsxd and a nonnegative integer k n such that f sndsxd − pnsxdf sxdyx kn for x ± 0.] ; 92. Let

89. If f 9 is continuous, use l’Hospital’s Rule to show that

hl0

H

e21yx 0

f sxd −

H| | x 1

x

if x ± 0 if x − 0

(a) Show that f is continuous at 0. (b) Investigate graphically whether f is differentiable at 0 by zooming in several times toward the point s0, 1d on the graph of  f . (c) Show that f is not differentiable at 0. How can you reconcile this fact with the appearance of the graphs in part (b)?

THE ORIGINS OF L’HOSPITAL’S RULE

Thomas Fisher Rare Book Library

L’Hospital’s Rule was first published in 1696 in the Marquis de l’Hospital’s calculus textbook Analyse des Infiniment Petits, but the rule was discovered in 1694 by the Swiss mathematician John (Johann) Bernoulli. The explanation is that these two mathematicians had entered into a curious business arrangement whereby the Marquis de l’Hospital bought the rights to Bernoulli’s mathematical discoveries. The details, including a translation of l’Hospital’s letter to Bernoulli proposing the arrangement, can be found in the book by Eves [1]. Write a report on the historical and mathematical origins of l’Hospital’s Rule. Start by providing brief biographical details of both men (the dictionary edited by Gillispie [2] is a good source) and outline the business deal between them. Then give l’Hospital’s statement of his rule, which is found in Struik’s sourcebook [4] and more briefly in the book of Katz [3]. Notice that l’Hospital and Bernoulli formulated the rule geometrically and gave the answer in terms of differentials. Compare their statement with the version of l’Hospital’s Rule given in Section 4.4 and show that the two statements are essentially the same. 1. Howard Eves, In Mathematical Circles (Volume 2: Quadrants III and IV) (Boston: Prindle, Weber and Schmidt, 1969), pp. 20–22. 2. C. C. Gillispie, ed., Dictionary of Scientific Biography (New York: Scribner’s, 1974). See the article on Johann Bernoulli by E. A. Fellmann and J. O. Fleckenstein in Volume II and the article on the Marquis de l’Hospital by Abraham Robinson in Volume VIII.

www.stewartcalculus.com The Internet is another source of information for this project. Click on History of Mathematics for a list of reliable websites.

3. Victor Katz, A History of Mathematics: An Introduction (New York: HarperCollins, 1993), p. 484. 4. D. J. Struik, ed., A Sourcebook in Mathematics, 1200 –1800 (Princeton, NJ: Princeton University Press, 1969), pp. 315–16.

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

SECTION 4.5 Summary of Curve Sketching

30

y=8˛-21≈+18x+2

_2

4 _10

FIGURE 1 8

y=8˛-21≈+18x+2 0

2

6

FIGURE 2

315

So far we have been concerned with some particular aspects of curve sketching: domain, range, and symmetry in Chapter 1; limits, continuity, and asymptotes in Chapter 2; derivatives and tangents in Chapters 2 and 3; and extreme values, intervals of increase and decrease, concavity, points of inflection, and l’Hospital’s Rule in this chapter. It is now time to put all of this information together to sketch graphs that reveal the important features of functions. You might ask: Why don’t we just use a graphing calculator or computer to graph a curve? Why do we need to use calculus? It’s true that modern technology is capable of producing very accurate graphs. But even the best graphing devices have to be used intelligently. It is easy to arrive at a misleading graph, or to miss important details of a curve, when relying solely on technology. (See “Graphing Calculators and Computers” at www.stewartcalculus.com, especially Examples 1, 3, 4, and 5. See also Section 4.6.) The use of calculus enables us to discover the most interesting aspects of graphs and in many cases to calculate maximum and minimum points and inflection points exactly instead of approximately. For instance, Figure 1 shows the graph of f sxd − 8x 3 2 21x 2 1 18x 1 2. At first glance it seems reasonable: It has the same shape as cubic curves like y − x 3, and it appears to have no maximum or minimum point. But if you compute the derivative, you will see that there is a maximum when x − 0.75 and a minimum when x − 1. Indeed, if we zoom in to this portion of the graph, we see that behavior exhibited in Figure 2. Without calculus, we could easily have overlooked it. In the next section we will graph functions by using the interaction between calculus and graphing devices. In this section we draw graphs by first considering the following information. We don’t assume that you have a graphing device, but if you do have one you should use it as a check on your work.

Guidelines for Sketching a Curve

y

0

x

(a) Even function: reﬂectional symmetry y

0

(b) Odd function: rotational symmetry

FIGURE 3

x

The following checklist is intended as a guide to sketching a curve y − f sxd by hand. Not every item is relevant to every function. (For instance, a given curve might not have an asymptote or possess symmetry.) But the guidelines provide all the information you need to make a sketch that displays the most important aspects of the function. A. Domain It’s often useful to start by determining the domain D of f , that is, the set of values of x for which f sxd is defined. B. Intercepts The y-intercept is f s0d and this tells us where the curve intersects the y-axis. To find the x-intercepts, we set y − 0 and solve for x. (You can omit this step if the equation is difficult to solve.) C. Symmetry (i ) If f s2xd − f sxd for all x in D, that is, the equation of the curve is unchanged when x is replaced by 2x, then f is an even function and the curve is symmetric about the y-axis. This means that our work is cut in half. If we know what the curve looks like for x > 0, then we need only reflect about the y-axis to obtain the complete curve [see Figure 3(a)]. Here are some examples: y − x 2, y − x 4, y − | x |, and y − cos x. (ii) If f s2xd − 2f sxd for all x in D, then f is an odd function and the curve is symmetric about the origin. Again we can obtain the complete curve if we know what it looks like for x > 0. [Rotate 180° about the origin; see Figure 3(b).] Some simple examples of odd functions are y − x, y − x 3, y − x 5, and y − sin x.

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CHAPTER 4

Applications of Differentiation

(iii) If f sx 1 pd − f sxd for all x in D, where p is a positive constant, then f is called a periodic function and the smallest such number p is called the period. For instance, y − sin x has period 2 and y − tan x has period . If we know what the graph looks like in an interval of length p, then we can use translation to sketch the entire graph (see Figure 4). y

period p

FIGURE 4 Periodic function: translational symmetry

a-p

0

a

a+p

a+2p

x

D. Asymptotes (i) Horizontal Asymptotes. Recall from Section 2.6 that if either lim x l ` f sxd − L or lim x l2 ` f sxd − L, then the line y − L is a horizontal asymptote of the curve y − f sxd. If it turns out that lim x l ` f sxd − ` (or 2`), then we do not have an asymptote to the right, but this fact is still useful information for sketching the curve. (ii) Vertical Asymptotes. Recall from Section 2.2 that the line x − a is a vertical asymptote if at least one of the following statements is true: 1

lim f sxd − `

x l a1

lim f sxd − 2`

x l a1

lim f sxd − `

x l a2

lim f sxd − 2`

x l a2

(For rational functions you can locate the vertical asymptotes by equating the denominator to 0 after canceling any common factors. But for other functions this method does not apply.) Furthermore, in sketching the curve it is very useful to know exactly which of the statements in (1) is true. If f sad is not defined but a is an endpoint of the domain of f, then you should compute lim xl a f sxd or lim xl a f sxd, whether or not this limit is infinite. (iii) Slant Asymptotes. These are discussed at the end of this section. Intervals of Increase or Decrease Use the I/D Test. Compute f 9sxd and find the intervals on which f 9sxd is positive ( f is increasing) and the intervals on which f 9sxd is negative ( f is decreasing). Local Maximum and Minimum Values Find the critical numbers of f [the numbers c where f 9scd − 0 or f 9scd does not exist]. Then use the First Derivative Test. If f 9 changes from positive to negative at a critical number c, then f scd is a local maximum. If f 9 changes from negative to positive at c, then f scd is a local minimum. Although it is usually preferable to use the First Derivative Test, you can use the Second Derivative Test if f 9scd − 0 and f 0scd ± 0. Then f 0scd . 0 implies that f scd is a local minimum, whereas f 0scd , 0 implies that f scd is a local maximum. Concavity and Points of Inflection Compute f 0sxd and use the Concavity Test. The curve is concave upward where f 0sxd . 0 and concave downward where f 0sxd , 0. Inflection points occur where the direction of concavity changes. Sketch the Curve Using the information in items A– G, draw the graph. Sketch the asymptotes as dashed lines. Plot the intercepts, maximum and minimum points, and inflection points. Then make the curve pass through these points, rising and falling according to E, with concavity according to G, and approaching the asymptotes. 2

E.

F.

G.

H.

1

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SECTION 4.5 Summary of Curve Sketching

317

If additional accuracy is desired near any point, you can compute the value of the derivative there. The tangent indicates the direction in which the curve proceeds.

ExamplE 1 Use the guidelines to sketch the curve y − A. The domain is hx

|x

2

2 1 ± 0j − hx

2x 2 . x2 2 1

| x ± 61j − s2`, 21d ø s21, 1d ø s1, `d

B. The x- and y-intercepts are both 0. C. Since f s2xd − f sxd, the function f is even. The curve is symmetric about the y-axis. D.

lim

x l6`

2x 2 2 − lim −2 x l6` 1 2 1yx 2 x 21 2

Therefore the line y − 2 is a horizontal asymptote. Since the denominator is 0 when x − 61, we compute the following limits: y

lim1

x l1

y=2

lim 1

x l 21

0

x=_1

x

2x 2 −` x2 2 1 2x 2 − 2` x 21

f 9sxd −

E.

2x 2 −` x 21

x l 21

2

sx 2 2 1ds4xd 2 2x 2 ? 2x 24x − 2 2 2 sx 2 1d sx 2 1d2

Since f 9sxd . 0 when x , 0 sx ± 21d and f 9sxd , 0 when x . 0 sx ± 1d, f is increasing on s2`, 21d and s21, 0d and decreasing on s0, 1d and s1, `d. F. The only critical number is x − 0. Since f 9 changes from positive to negative at 0, f s0d − 0 is a local maximum by the First Derivative Test.

y

G. y=2

lim 2

x l1

2

FIGURE 5 We have shown the curve approaching its horizontal asymptote from above in Figure 5. This is confirmed by the intervals of increase and decrease.

2x 2 − 2` x 21 2

Therefore the lines x − 1 and x − 21 are vertical asymptotes. This information about limits and asymptotes enables us to draw the preliminary sketch in Figure 5, showing the parts of the curve near the asymptotes.

x=1

Preliminary sketch

lim2

f 0sxd −

sx 2 2 1d2 s24d 1 4x ? 2sx 2 2 1d2x 12x 2 1 4 − 2 2 4 sx 2 1d sx 2 1d3

Since 12x 2 1 4 . 0 for all x, we have 0 x

x=_1

x=1

FIGURE 6 Finished sketch of y −

2x 2 x2 2 1

f 0sxd . 0 &?

x 2 2 1 . 0 &?

|x| . 1

and f 0sxd , 0 &? | x | , 1. Thus the curve is concave upward on the intervals s2`, 21d and s1, `d and concave downward on s21, 1d. It has no point of inflection since 1 and 21 are not in the domain of f. H. Using the information in E – G, we finish the sketch in Figure 6. ■

ExamplE 2 Sketch the graph of f sxd − A. Domain − hx

x2 . sx 1 1

| x 1 1 . 0j − hx | x . 21j − s21, `d

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318

CHAPTER 4

Applications of Differentiation

B. The x- and y-intercepts are both 0. C. Symmetry: None D. Since lim

xl`

x2 −` sx 1 1

there is no horizontal asymptote. Since sx 1 1 l 0 as x l 211 and f sxd is always positive, we have x2 lim 1 −` x l 21 sx 1 1 and so the line x − 21 is a vertical asymptote. E.

G. y= 0

FIGURE 7

3x 2 1 4x xs3x 1 4d sx 1 1 s2xd 2 x 2 ? 1y( 2sx 1 1 ) − 3y2 − x11 2sx 1 1d 2sx 1 1d3y2

We see that f 9sxd − 0 when x − 0 (notice that 243 is not in the domain of f ), so the only critical number is 0. Since f 9sxd , 0 when 21 , x , 0 and f 9sxd . 0 when x . 0, f is decreasing on s21, 0d and increasing on s0, `d. F. Since f 9s0d − 0 and f 9 changes from negative to positive at 0, f s0d − 0 is a local (and absolute) minimum by the First Derivative Test.

y

x=_1

f 9sxd −

≈ x+1 œ„„„„ x

f 0sxd −

2sx 1 1d3y2s6x 1 4d 2 s3x 2 1 4xd3sx 1 1d1y2 3x 2 1 8x 1 8 − 4sx 1 1d3 4sx 1 1d5y2

Note that the denominator is always positive. The numerator is the quadratic 3x 2 1 8x 1 8, which is always positive because its discriminant is b 2 2 4ac − 232, which is negative, and the coefficient of x 2 is positive. Thus f 0sxd . 0 for all x in the domain of f, which means that f is concave upward on s21, `d and there is no point of inflection. H. The curve is sketched in Figure 7. ■

ExamplE 3 Sketch the graph of f sxd − xe x. A. B. C. D.

The domain is R. The x- and y-intercepts are both 0. Symmetry: None Because both x and e x become large as x l `, we have lim x l ` xe x − `. As x l 2`, however, e x l 0 and so we have an indeterminate product that requires the use of l’Hospital’s Rule: lim xe x − lim

x l2`

x l2`

x 1 − lim s2e x d − 0 2x − x lim l2` x l2` e 2e2x

Thus the x-axis is a horizontal asymptote. E.

f 9sxd − xe x 1 e x − sx 1 1de x

Since e x is always positive, we see that f 9sxd . 0 when x 1 1 . 0, and f 9sxd , 0 when x 1 1 , 0. So f is increasing on s21, `d and decreasing on s2`, 21d. F. Because f 9s21d − 0 and f 9 changes from negative to positive at x − 21, f s21d − 2e21 < 20.37 is a local (and absolute) minimum. G.

f 0sxd − sx 1 1de x 1 e x − sx 1 2de x Since f 0sxd . 0 if x . 22 and f 0sxd , 0 if x , 22, f is concave upward on

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319

SECTION 4.5 Summary of Curve Sketching y

y=x´

s22, `d and concave downward on s2`, 22d. The inflection point is s22, 22e22 d < s22, 20.27d. H. We use this information to sketch the curve in Figure 8.

1 _2

ExamplE 4 Sketch the graph of f sxd −

_1 x (_1, _1/e)

FIGURE 8

cos x . 2 1 sin x

A. The domain is R. B. The y-intercept is f s0d − 12. The x-intercepts occur when cos x − 0, that is, x − sy2d 1 n, where n is an integer. C. f is neither even nor odd, but f sx 1 2d − f sxd for all x and so f is periodic and has period 2. Thus, in what follows, we need to consider only 0 < x < 2 and then extend the curve by translation in part H. D. Asymptotes: None E.

f 9sxd −

s2 1 sin xds2sin xd 2 cos x scos xd 2 sin x 1 1 −2 s2 1 sin xd 2 s2 1 sin xd 2

The denominator is always positive, so f 9sxd . 0 when 2 sin x 1 1 , 0 &? sin x , 221 &? 7y6 , x , 11y6. So f is increasing on s7y6, 11y6d and decreasing on s0, 7y6d and s11y6, 2d. F. From part E and the First Derivative Test, we see that the local minimum value is f s7y6d − 21ys3 and the local maximum value is f s11y6d − 1ys3 . G. If we use the Quotient Rule again and simplify, we get f 0sxd − 2

2 cos x s1 2 sin xd s2 1 sin xd 3

Because s2 1 sin xd 3 . 0 and 1 2 sin x > 0 for all x, we know that f 0sxd . 0 when cos x , 0, that is, y2 , x , 3y2. So f is concave upward on sy2, 3y2d and concave downward on s0, y2d and s3y2, 2d. The inflection points are sy2, 0d and s3y2, 0d. H. The graph of the function restricted to 0 < x < 2 is shown in Figure 9. Then we extend it, using periodicity, to the complete graph in Figure 10. y

1 2

π 2

π

11π 1 6 , œ„3

y

3π 2

1 2

2π x

π

x

1 - ’ ” 7π 6 , œ„3

FIGURE 9

FIGURE 10

ExamplE 5 Sketch the graph of y − lns4 2 x 2 d. A. The domain is hx

| 42x

2

. 0j − hx

|x

2

, 4j − hx

| | x | , 2j − s22, 2d

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320

CHAPTER 4

Applications of Differentiation

B. The y-intercept is f s0d − ln 4. To find the x-intercept we set y − lns4 2 x 2 d − 0 We know that ln 1 − 0, so we have 4 2 x 2 − 1 ? x 2 − 3 and therefore the x-intercepts are 6s3 . C . Since f s2xd − f sxd, f is even and the curve is symmetric about the y-axis. D. We look for vertical asymptotes at the endpoints of the domain. Since 4 2 x 2 l 0 1 as x l 2 2 and also as x l 221, we have lim lns4 2 x 2 d − 2`

lim lns4 2 x 2 d − 2`

x l 22

x l 221

Thus the lines x − 2 and x − 22 are vertical asymptotes. f 9sxd −

E. y (0, ln 4)

x=_2

x=2

{_ œ„3, 0}

0

Since f 9sxd . 0 when 22 , x , 0 and f 9sxd , 0 when 0 , x , 2, f is increasing on s22, 0d and decreasing on s0, 2d. F . The only critical number is x − 0. Since f 9 changes from positive to negative at 0, f s0d − ln 4 is a local maximum by the First Derivative Test.

x

{œ„ 3, 0}

22x 4 2 x2

G.

f 0sxd −

s4 2 x 2 ds22d 1 2xs22xd 28 2 2x 2 − s4 2 x 2 d2 s4 2 x 2 d2

Since f 0sxd , 0 for all x, the curve is concave downward on s22, 2d and has no inflection point. H. Using this information, we sketch the curve in Figure 11. ■

FIGURE 11 y − lns4 2 x 2 d

Slant Asymptotes Some curves have asymptotes that are oblique, that is, neither horizontal nor vertical. If

y

y=ƒ

lim f f sxd 2 smx 1 bdg − 0

xl`

ƒ-(mx+b) y=mx+b

0

x

where m ± 0, then the line y − mx 1 b is called a slant asymptote because the vertical distance between the curve y − f sxd and the line y − mx 1 b approaches 0, as in  Figure  12. (A similar situation exists if we let x l 2`.) For rational functions, slant  asymptotes occur when the degree of the numerator is one more than the degree of the denominator. In such a case the equation of the slant asymptote can be found by long division as in the following example.

FIGURE 12

ExamplE 6 Sketch the graph of f sxd − A. B. C. D.

x3 . x 11 2

The domain is R − s2`, `d. The x- and y-intercepts are both 0. Since f s2xd − 2f sxd, f is odd and its graph is symmetric about the origin. Since x 2 1 1 is never 0, there is no vertical asymptote. Since f sxd l ` as x l ` and f sxd l 2` as x l 2`, there is no horizontal asymptote. But long division

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321

SECTION 4.5 Summary of Curve Sketching

gives f sxd −

x x3 −x2 2 x 11 x 11 2

This equation suggests that y − x is a candidate for a slant asymptote. In fact, x f sxd 2 x − 2 2 −2 x 11

1 x 1 11 2 x

l0

as

x l 6`

So the line y − x is a slant asymptote. f 9sxd −

E.

sx 2 1 1ds3x 2 d 2 x 3 ? 2x x 2sx 2 1 3d − sx 2 1 1d2 sx 2 1 1d2

Since f 9sxd . 0 for all x (except 0), f is increasing on s2`, `d. F. Although f 9s0d − 0, f 9 does not change sign at 0, so there is no local maximum or minimum. f 0sxd −

G.

y=

y

˛ ≈+1

sx 2 1 1d2 s4x 3 1 6xd 2 sx 4 1 3x 2 d ? 2sx 2 1 1d2x 2xs3 2 x 2 d − 2 4 sx 1 1d sx 2 1 1d3

Since f 0sxd − 0 when x − 0 or x − 6s3 , we set up the following chart: x

3 2 x2

sx 2 1 1d3

f 99sxd

f

2

2

1

1

CU on (2`, 2s3 )

2

1

1

2

CD on (2s3 , 0)

0 , x , s3

1

1

1

1

CU on ( 0, s3 )

x . s3

1

2

1

2

CD on (s3 , `)

Interval ”œ„3,

3 œ„ 3 ’ 4

x , 2s3

0

”_œ„3,

2s3 , x , 0 x

3 3œ„ _ 4 ’

inflection points y=x

The points of inflection are (2s3 , 243 s3 ), s0, 0d, and (s3 , 34 s3 ). H. The graph of f is sketched in Figure 13.

FIGURE 13

1–54 Use the guidelines of this section to sketch the curve. 3

1. y − x 1 3x

2

3

3. y − 2 2 15x 1 9x 2 x

3

4. y − 8x 2 2 x 4 6. y − x 5 2 5x

7. y − 15 x 5 2 83 x 3 1 16x

8. y − s4 2 x 2 d 5

x2 2 4 x 2 2 2x

x 2 x2 2 2 3x 1 x 2

12. y − 1 1

13. y −

x x2 2 4

14. y −

1 x2 2 4

15. y −

x2 x 19

16. y −

sx 2 1d2 x2 1 1

17. y −

x21 x2

18. y − 1 1

2

5. y − xsx 2 4d3

9. y −

11. y −

2. y − x 1 6x 1 9x 2

10. y −

x 2 1 5x 25 2 x 2

2

1 1 1 2 x x

1 1 1 2 x x

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322

CHAPTER 4

19. y −

Applications of Differentiation

x3 x 11

20. y −

3

length, and h is Planck’s constant. Sketch the graph of E as a function of . What does the graph say about the energy?

x3 x22

21. y − 2sx 2 x

3 22. y − sx 2 4ds x

23. y − sx 2 1 x 2 2

24. y − sx 2 1 x 2 x

25. y −

x sx 1 1

26. y − x s2 2 x 2

27. y −

s1 2 x 2 x

28. y −

2

x sx 2 2 1

29. y − x 2 3x 1y3

30. y − x 5y3 2 5x 2y3

3 31. y − s x2 2 1

3 32. y − s x3 1 1

3

34. y − x 1 cos x

33. y − sin x 35. y − x tan x,

2y2 , x , y2

36. y − 2x 2 tan x,

2y2 , x , y2

37. y − sin x 1 s3 cos x, 22 < x < 2

sin x 1 1 cos x

40. y −

41. y − arctanse x d

sin x 2 1 cos x

42. y − s1 2 xde x

43. y − 1ys1 1 e 2x d 44. y − e2x sin x, 0 < x < 2 45. y −

1 1 ln x x

46. y − x 2 ln x

x 22

x

1 1 1 ae2kt

where pstd is the proportion of the population that knows the rumor at time t and a and k are positive constants. (a) When will half the population have heard the rumor? (b) When is the rate of spread of the rumor greatest? (c) Sketch the graph of p. 58. A model for the concentration at time t of a drug injected into the bloodstream is Cstd − Kse2at 2 e2bt d where a, b, and K are positive constants and b . a. Sketch the graph of the concentration function. What does the graph tell us about how the concentration varies as time passes?

48. y − e yx

49. y − lnssin xd

50. y − lns1 1 x 3d

51. y − xe21yx

52. y −

53. y − e arctan x

54. y − tan21

ln x x2

y−2

W 4 WL 3 WL 2 2 x 1 x 2 x 24EI 12EI 24EI

where E and I are positive constants. (E is Young’s modulus of elasticity and I is the moment of inertia of a crosssection of the beam.) Sketch the graph of the deflection curve. y

W

2

47. y − s1 1 e d

0 L

S D x21 x11

55. In the theory of relativity, the mass of a particle is m−

pstd −

59. The figure shows a beam of length L embedded in concrete walls. If a constant load W is distributed evenly along its length, the beam takes the shape of the deflection curve

38. y − csc x 2 2sin x, 0 , x ,  39. y −

57. A model for the spread of a rumor is given by the equation

m0 s1 2 v 2yc 2

where m 0 is the rest mass of the particle, m is the mass when the particle moves with speed v relative to the observer, and c is the speed of light. Sketch the graph of m as a function of v. 56. In the theory of relativity, the energy of a particle is E − sm 02 c 4 1 h 2 c 2y2 where m 0 is the rest mass of the particle,  is its wave

60. Coulomb’s Law states that the force of attraction between two charged particles is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. The figure shows particles with charge 1 located at positions 0 and 2 on a coordinate line and a particle with charge 21 at a position x between them. It follows from Coulomb’s Law that the net force acting on the middle particle is Fsxd − 2

k k 1 x2 sx 2 2d2

0,x,2

where k is a positive constant. Sketch the graph of the net force function. What does the graph say about the force? +1

_1

+1

0

x

2

x

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SECTION 4.6

61–64 Find an equation of the slant asymptote. Do not sketch the curve. x2 1 1 4x 3 2 10x 2 2 11x 1 1 61. y − 62. y − x11 x 2 2 3x 2x 3 2 5x 2 1 3x 63. y − x2 2 x 2 2

72. Show that the curve y − sx 2 1 4x has two slant asymptotes: y − x 1 2 and y − 2x 2 2. Use this fact to help sketch the curve. 73. Show that the lines y − sbyadx and y − 2sbyadx are slant asymptotes of the hyperbola sx 2ya 2 d 2 s y 2yb 2 d − 1.

26x 4 1 2x 3 1 3 64. y − 2x 3 2 x

74. Let f sxd − sx 3 1 1dyx. Show that lim f f sxd 2 x 2 g − 0

65–70 Use the guidelines of this section to sketch the curve. In guideline D find an equation of the slant asymptote. 65. y −

x2 x21

66. y −

1 1 5x 2 2x 2 x22

67. y −

x3 1 4 x2

68. y −

x3 sx 1 1d2

69. y − 1 1 12 x 1 e2x

x l 6`

This shows that the graph of f approaches the graph of y − x 2, and we say that the curve y − f sxd is asymptotic to the parabola y − x 2. Use this fact to help sketch the graph of f. 75. Discuss the asymptotic behavior of f sxd − sx 4 1 1dyx in the same manner as in Exercise 74. Then use your results to help sketch the graph of f.

70. y − 1 2 x 1 e 11xy3

76. Use the asymptotic behavior of f sxd − sin x 1 e2x to sketch its graph without going through the curvesketching procedure of this section.

71. Show that the curve y − x 2 tan21x has two slant asymptotes: y − x 1 y2 and y − x 2 y2. Use this fact to help sketch the curve.

You may want to read “Graphing Calculators and Computers” at www.stewartcalculus.com if you haven’t already. In particular, it explains how to avoid some of the pitfalls of graphing devices by choosing appropriate viewing rectangles.

323

Graphing with Calculus and Calculators

The method we used to sketch curves in the preceding section was a culmination of much of our study of differential calculus. The graph was the final object that we produced. In this section our point of view is completely different. Here we start with a graph produced by a graphing calculator or computer and then we refine it. We use calculus to make sure that we reveal all the important aspects of the curve. And with the use of graphing devices we can tackle curves that would be far too complicated to consider without technology. The theme is the interaction between calculus and calculators.

ExamplE 1 Graph the polynomial f sxd − 2x 6 1 3x 5 1 3x 3 2 2x 2. Use the graphs of f 9 and f 0 to estimate all maximum and minimum points and intervals of concavity. SOLUTION If we specify a domain but not a range, many graphing devices will deduce a suitable range from the values computed. Figure 1 shows the plot from one such device if we specify that 25 < x < 5. Although this viewing rectangle is useful for showing that the asymptotic behavior (or end behavior) is the same as for y − 2x 6, it is obviously hiding some finer detail. So we change to the viewing rectangle f23, 2g by f250, 100g shown in Figure 2. 100

41,000 y=ƒ y=ƒ _3 _5

5 _50

_1000

FIGURE 1

2

FIGURE 2

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324

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From this graph it appears that there is an absolute minimum value of about 215.33 when x < 21.62 (by using the cursor) and f is decreasing on s2`, 21.62d and increasing on s21.62, `d. Also there appears to be a horizontal tangent at the origin and inflection points when x − 0 and when x is somewhere between 22 and 21. Now let’s try to confirm these impressions using calculus. We differentiate and get

20 y=fª(x)

_3

2

f 9sxd − 12x 5 1 15x 4 1 9x 2 2 4x

_5

f 0sxd − 60x 4 1 60x 3 1 18x 2 4

FIGURE 3 1 y=ƒ _1

1

_1

FIGURE 4 10 _3

2 y=f ·(x)

When we graph f 9 in Figure 3 we see that f 9sxd changes from negative to positive when x < 21.62; this confirms (by the First Derivative Test) the minimum value that we found earlier. But, perhaps to our surprise, we also notice that f 9sxd changes from positive to negative when x − 0 and from negative to positive when x < 0.35. This means that f has a local maximum at 0 and a local minimum when x < 0.35, but these were hidden in Figure 2. Indeed, if we now zoom in toward the origin in Figure 4, we see what we missed before: a local maximum value of 0 when x − 0 and a local minimum value of about 20.1 when x < 0.35. What about concavity and inflection points? From Figures 2 and 4 there appear to be inflection points when x is a little to the left of 21 and when x is a little to the right of 0. But it’s difficult to determine inflection points from the graph of f , so we graph the second derivative f 0 in Figure 5. We see that f 0 changes from positive to negative when x < 21.23 and from negative to positive when x < 0.19. So, correct to two decimal places, f is concave upward on s2`, 21.23d and s0.19, `d and concave downward on s21.23, 0.19d. The inflection points are s21.23, 210.18d and s0.19, 20.05d. We have discovered that no single graph reveals all the important features of this polynomial. But Figures 2 and 4, when taken together, do provide an accurate picture. ■

_30

ExamplE 2 Draw the graph of the function

FIGURE 5

f sxd −

x 2 1 7x 1 3 x2

in a viewing rectangle that contains all the important features of the function. Estimate the maximum and minimum values and the intervals of concavity. Then use calculus to find these quantities exactly. SOLUTION Figure 6, produced by a computer with automatic scaling, is a disaster. Some graphing calculators use f210, 10g by f210, 10g as the default viewing rectangle, so let’s try it. We get the graph shown in Figure 7; it’s a major improvement. 3  10!*

10 y=ƒ _10

y=ƒ

_5

10

5 _10

FIGURE 6

FIGURE 7

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SECTION 4.6

Graphing with Calculus and Calculators

325

The y-axis appears to be a vertical asymptote and indeed it is because 10

x 2 1 7x 1 3 −` x2

lim

xl0

y=ƒ y=1 _20

20

_5

FIGURE 8

Figure 7 also allows us to estimate the x-intercepts: about 20.5 and 26.5. The exact values are obtained by using the quadratic formula to solve the equation x 2 1 7x 1 3 − 0; we get x − (27 6 s37 )y2. To get a better look at horizontal asymptotes, we change to the viewing rectangle f220, 20g by f25, 10g in Figure 8. It appears that y − 1 is the horizontal asymptote and this is easily confirmed: lim

x l 6`

2

_3

0

S

7 x 2 1 7x 1 3 3 − lim 1 1 1 2 2 x l 6` x x x

D

−1

To estimate the minimum value we zoom in to the viewing rectangle f23, 0g by f24, 2g in Figure 9. The cursor indicates that the absolute minimum value is about 23.1 when x < 20.9, and we see that the function decreases on s2`, 20.9d and s0, `d and increases on s20.9, 0d. The exact values are obtained by differentiating:

y=ƒ

f 9sxd − 2

7 6 7x 1 6 2 2 3 − 2 x x x3

_4

This shows that f 9sxd . 0 when 267 , x , 0 and f 9sxd , 0 when x , 267 and when 37 x . 0. The exact minimum value is f (2 67 ) − 2 12 < 23.08. Figure 9 also shows that an inflection point occurs somewhere between x − 21 and x − 22. We could estimate it much more accurately using the graph of the second derivative, but in this case it’s just as easy to find exact values. Since

FIGURE 9

f 0sxd −

14 18 2s7x 1 9d 3 1 4 − x x x4

we see that f 0sxd . 0 when x . 297 sx ± 0d. So f is concave upward on s297 , 0d and s0, `d and concave downward on s2`, 297 d. The inflection point is s297 , 271 27 d. The analysis using the first two derivatives shows that Figure 8 displays all the major aspects of the curve. ■

ExamplE 3 Graph the function f sxd − 10

y=ƒ _10

10

x 2sx 1 1d3 . sx 2 2d2sx 2 4d4

SOLUTION Drawing on our experience with a rational function in Example 2, let’s start by graphing f in the viewing rectangle f210, 10g by f210, 10g. From Figure 10 we have the feeling that we are going to have to zoom in to see some finer detail and also zoom out to see the larger picture. But, as a guide to intelligent zooming, let’s first take a close look at the expression for f sxd. Because of the factors sx 2 2d2 and sx 2 4d4 in the denominator, we expect x − 2 and x − 4 to be the vertical asymptotes. Indeed

_10

FIGURE 10

lim

x l2

x 2sx 1 1d3 −` sx 2 2d2sx 2 4d4

and

lim

xl4

x 2sx 1 1d3 −` sx 2 2d2sx 2 4d4

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326

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Applications of Differentiation

To find the horizontal asymptotes, we divide numerator and denominator by x 6: x 2 sx 1 1d3 ? x 2sx 1 1d3 x3 x3 − 2 4 − 2 sx 2 2d sx 2 4d sx 2 2d sx 2 4d4 ? x2 x4

S D S DS D 3

1 1 11 x x

12

2 x

2

12

4 x

4

y

_1

1

2

3

4

This shows that f sxd l 0 as x l 6`, so the x-axis is a horizontal asymptote. It is also very useful to consider the behavior of the graph near the x-intercepts using an analysis like that in Example 2.6.12. Since x 2 is positive, f sxd does not change sign at 0 and so its graph doesn’t cross the x-axis at 0. But, because of the factor sx 1 1d3, the graph does cross the x-axis at 21 and has a horizontal tangent there. Putting all this information together, but without using derivatives, we see that the curve has to look something like the one in Figure 11. Now that we know what to look for, we zoom in (several times) to produce the graphs in Figures 12 and 13 and zoom out (several times) to get Figure 14.

x

FIGURE 11

0.05

0.0001

500 y=ƒ

y=ƒ _100

1

_1.5

0.5

y=ƒ _1 _0.05

_0.0001

FIGURE 13

FIGURE 12

10 _10

FIGURE 14

We can read from these graphs that the absolute minimum is about 20.02 and occurs when x < 220. There is also a local maximum 0 and x < 600 (otherwise A , 0) So the function that we wish to maximize is Asxd − 1200x 2 2x 2

0 < x < 600

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332

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Applications of Differentiation

The derivative is A9sxd − 1200 2 4x, so to find the critical numbers we solve the equation 1200 2 4x − 0 which gives x − 300. The maximum value of A must occur either at this critical number or at an endpoint of the interval. Since As0d − 0, As300d − 180,000, and As600d − 0, the Closed Interval Method gives the maximum value as As300d − 180,000. [Alternatively, we could have observed that A0sxd − 24 , 0 for all x, so A is always concave downward and the local maximum at x − 300 must be an absolute maximum.] Thus the rectangular field should be 300 m deep and 600 m wide. ■

ExamplE 2 A cylindrical can is to be made to hold 1 L of oil. Find the dimensions that will minimize the cost of the metal to manufacture the can. SOLUTION Draw the diagram as in Figure 3, where r is the radius and h the height (both in centimeters). In order to minimize the cost of the metal, we minimize the total surface area of the cylinder (top, bottom, and sides). From Figure 4 we see that the sides are made from a rectangular sheet with dimensions 2r and h. So the surface area is

h

r

A − 2r 2 1 2rh

FIGURE 3

We would like to express A in terms of one variable, r. To eliminate h we use the fact that the volume is given as 1 L, which is equivalent to 1000 cm3. Thus

2πr r

r 2h − 1000 which gives h − 1000ysr 2 d. Substitution of this into the expression for A gives h

A − 2r 2 1 2r

Area 2{πr@}

Area (2πr)h

S D 1000 r 2

− 2r 2 1

2000 r

We know r must be positive, and there are no limitations on how large r can be. Therefore the function that we want to minimize is

FIGURE 4

Asrd − 2r 2 1

2000 r

r.0

To find the critical numbers, we differentiate: y

A9srd − 4r 2 y=A(r)

1000

0

10

FIGURE 5

r

2000 4sr 3 2 500d − r2 r2

3 Then A9srd − 0 when r 3 − 500, so the only critical number is r − s 500y . Since the domain of A is s0, `d, we can’t use the argument of Example 1 concern3 ing endpoints. But we can observe that A9srd , 0 for r , s 500y and A9srd . 0 for 3 r . s500y , so A is decreasing for all r to the left of the critical number and increas3 500y must give rise to an absolute minimum. ing for all r to the right. Thus r − s [Alternatively, we could argue that Asrd l ` as r l 0 1 and Asrd l ` as r l `, so there must be a minimum value of Asrd, which must occur at the critical number. See Figure 5.]

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SECTION 4.7

In the Applied Project on page 343 we investigate the most economical shape for a can by taking into account other manufacturing costs.

Optimization Problems

333

3 500y is The value of h corresponding to r − s

h−

Î

1000 1000 −2 2 − r s500yd2y3

3

500 − 2r

3 500y cm and the height Thus, to minimize the cost of the can, the radius should be s should be equal to twice the radius, namely, the diameter. ■

NOTE 1 The argument used in Example 2 to justify the absolute minimum is a variant of the First Derivative Test (which applies only to local maximum or minimum values) and is stated here for future reference.

First Derivative Test for Absolute Extreme Values Suppose that c is a critical number of a continuous function f defined on an interval. (a) If f 9sxd . 0 for all x , c and f 9sxd , 0 for all x . c, then f scd is the absolute maximum value of f. (b) If f 9sxd , 0 for all x , c and f 9sxd . 0 for all x . c, then f scd is the absolute minimum value of f.

TEC Module 4.7 takes you through six additional optimization problems, including animations of the physical situations.

NOTE 2 An alternative method for solving optimization problems is to use implicit differentiation. Let’s look at Example 2 again to illustrate the method. We work with the same equations A − 2r 2 1 2rh r 2h − 1000

but instead of eliminating h, we differentiate both equations implicitly with respect to r: A9 − 4r 1 2rh9 1 2h

r 2h9 1 2rh − 0

The minimum occurs at a critical number, so we set A9 − 0, simplify, and arrive at the equations 2r 1 rh9 1 h − 0 rh9 1 2h − 0 and subtraction gives 2r 2 h − 0, or h − 2r. y

ExamplE 3 Find the point on the parabola y 2 − 2x that is closest to the point s1, 4d. (1, 4)

SOLUTION The distance between the point s1, 4d and the point sx, yd is

d − ssx 2 1d2 1 sy 2 4d2

(x, y)

1 0

¥=2x

1 2 3 4

x

(See Figure 6.) But if sx, yd lies on the parabola, then x − 12 y 2, so the expression for d becomes d − s(12 y2 2 1)2 1 sy 2 4d2

FIGURE 6

(Alternatively, we could have substituted y − s2x to get d in terms of x alone.) Instead of minimizing d, we minimize its square: d 2 − f syd − s 21 y 2 2 1d 2 1 sy 2 4d2

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CHAPTER 4

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(You should convince yourself that the minimum of d occurs at the same point as the minimum of d 2, but d 2 is easier to work with.) Note that there are no restrictions on y, so the domain is all real numbers. Differentiating, we obtain f 9syd − 2s 21 y 2 2 1d y 1 2sy 2 4d − y 3 2 8 so f 9s yd − 0 when y − 2. Observe that f 9syd , 0 when y , 2 and f 9syd . 0 when y . 2, so by the First Derivative Test for Absolute Extreme Values, the absolute minimum occurs when y − 2. (Or we could simply say that because of the geometric nature of the problem, it’s obvious that there is a closest point but not a farthest point.) The corresponding value of x is x − 12 y 2 − 2. Thus the point on y 2 − 2x closest to s1, 4d is s2, 2d. [The distance between the points is d − sf s2d − s5 .] ■

ExamplE 4 A man launches his boat from point A on a bank of a straight river, 3 km wide, and wants to reach point B, 8 km downstream on the opposite bank, as quickly as possible (see Figure 7). He could row his boat directly across the river to point C and then run to B, or he could row directly to B, or he could row to some point D between C and B and then run to B. If he can row 6 kmyh and run 8 kmyh, where should he land to reach B as soon as possible? (We assume that the speed of the water is negligible compared with the speed at which the man rows.)

3 km C

A

x D

SOLUTION If we let x be the distance from C to D, then the running distance is 8 km

| DB | − 8 2 x and the Pythagorean Theorem gives the rowing distance as | AD | − sx 1 9 . We use the equation 2

time − B

distance rate

Then the rowing time is sx 2 1 9 y6 and the running time is s8 2 xdy8, so the total time T as a function of x is

FIGURE 7

Tsxd −

82x sx 2 1 9 1 6 8

The domain of this function T is f0, 8g. Notice that if x − 0, he rows to C and if x − 8, he rows directly to B. The derivative of T is T9sxd −

x 2

6sx 1 9

2

1 8

Thus, using the fact that x > 0, we have T9sxd − 0

&?

x 1 − 8 6sx 2 1 9

&?

4x − 3sx 2 1 9

&?

16x 2 − 9sx 2 1 9d

&?

7x 2 − 81

&?

x−

9 s7

The only critical number is x − 9ys7 . To see whether the minimum occurs at this critical number or at an endpoint of the domain f0, 8g, we follow the Closed Interval

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SECTION 4.7 T

335

Method by evaluating T at all three points: y=T(x)

Ts0d − 1.5 1

0

Optimization Problems

2

4

x

6

FIGURE 8

T

S D 9 s7

−11

s7 < 1.33 8

Ts8d −

s73 < 1.42 6

Since the smallest of these values of T occurs when x − 9ys7 , the absolute minimum value of T must occur there. Figure 8 illustrates this calculation by showing the graph of T. Thus the man should land the boat at a point 9ys7 km ( 0). This value of x gives a maximum value of A since As0d − 0 and Asrd − 0. Therefore the area of the largest inscribed rectangle is

S D

A

r s2

−2

r s2

Î

r2 2

r2 − r2 2

SOLUTION 2 A simpler solution is possible if we think of using an angle as a variable. Let  be the angle shown in Figure 10. Then the area of the rectangle is r ¨ r cos ¨

FIGURE 10

r sin ¨

Asd − s2r cos dsr sin d − r 2s2 sin  cos d − r 2 sin 2 We know that sin 2 has a maximum value of 1 and it occurs when 2 − y2. So Asd has a maximum value of r 2 and it occurs when  − y4. Notice that this trigonometric solution doesn’t involve differentiation. In fact, we didn’t need to use calculus at all. ■

Applications to Business and Economics In Section 3.7 we introduced the idea of marginal cost. Recall that if Csxd, the cost function, is the cost of producing x units of a certain product, then the marginal cost is the rate of change of C with respect to x. In other words, the marginal cost function is the derivative, C9sxd, of the cost function.

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336

CHAPTER 4

Applications of Differentiation

Now let’s consider marketing. Let psxd be the price per unit that the company can charge if it sells x units. Then p is called the demand function (or price function) and we would expect it to be a decreasing function of x. (More units sold corresponds to a lower price.) If x units are sold and the price per unit is psxd, then the total revenue is Rsxd − quantity 3 price − xpsxd and R is called the revenue function. The derivative R9 of the revenue function is called the marginal revenue function and is the rate of change of revenue with respect to the number of units sold. If x units are sold, then the total profit is Psxd − Rsxd 2 Csxd and P is called the profit function. The marginal profit function is P9, the derivative of the profit function. In Exercises 59 – 63 you are asked to use the marginal cost, revenue, and profit functions to minimize costs and maximize revenues and profits.

ExamplE 6 A store has been selling 200 flat-screen TVs a week at \$350 each. A market survey indicates that for each \$10 rebate offered to buyers, the number of TVs sold will increase by 20 a week. Find the demand function and the revenue function. How large a rebate should the store offer to maximize its revenue? SOLUTION If x is the number of TVs sold per week, then the weekly increase in sales is

x 2 200. For each increase of 20 units sold, the price is decreased by \$10. So for each 1 additional unit sold, the decrease in price will be 20 3 10 and the demand function is 1 psxd − 350 2 10 20 sx 2 200d − 450 2 2 x

The revenue function is Rsxd − xpsxd − 450x 2 12 x 2 Since R9sxd − 450 2 x, we see that R9sxd − 0 when x − 450. This value of x gives an absolute maximum by the First Derivative Test (or simply by observing that the graph of R is a parabola that opens downward). The corresponding price is ps450d − 450 2 12 s450d − 225 and the rebate is 350 2 225 − 125. Therefore, to maximize revenue, the store should offer a rebate of \$125. ■

1. Consider the following problem: Find two numbers whose sum is 23 and whose product is a maximum. (a) Make a table of values, like the one at the right, so that the sum of the numbers in the first two columns is always 23. On the basis of the evidence in your table, estimate the answer to the problem. (b) Use calculus to solve the problem and compare with your answer to part (a).

First number

Second number

Product

1 2 3 . . .

22 21 20 . . .

22 42 60 . . .

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SECTION 4.7

2. Find two numbers whose difference is 100 and whose product is a minimum. 3. Find two positive numbers whose product is 100 and whose sum is a minimum. 4. The sum of two positive numbers is 16. What is the smallest possible value of the sum of their squares? 5. What is the maximum vertical distance between the line y − x 1 2 and the parabola y − x 2 for 21 < x < 2? 6. What is the minimum vertical distance between the parabolas y − x 2 1 1 and y − x 2 x 2 ? 7. Find the dimensions of a rectangle with perimeter 100 m whose area is as large as possible. 8. Find the dimensions of a rectangle with area 1000 m2 whose perimeter is as small as possible. 9. A model used for the yield Y of an agricultural crop as a function of the nitrogen level N in the soil (measured in appropriate units) is Y−

kN 1 1 N2

where k is a positive constant. What nitrogen level gives the best yield? 10. The rate sin mg carbonym 3yhd at which photosynthesis takes place for a species of phytoplankton is modeled by the function P−

100 I I2 1 I 1 4

where I is the light intensity (measured in thousands of footcandles). For what light intensity is P a maximum? 11. Consider the following problem: A farmer with 300 m of fencing wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle. What is the largest possible total area of the four pens? (a) Draw several diagrams illustrating the situation, some with shallow, wide pens and some with deep, narrow pens. Find the total areas of these configurations. Does it appear that there is a maximum area? If so, estimate it. (b) Draw a diagram illustrating the general situation. Introduce notation and label the diagram with your symbols. (c) Write an expression for the total area. (d) Use the given information to write an equation that relates the variables. (e) Use part (d) to write the total area as a function of one variable. (f ) Finish solving the problem and compare the answer with your estimate in part (a). 12. Consider the following problem: A box with an open top is to be constructed from a square piece of cardboard, 3 m wide, by cutting out a square from each of the four corners and

Optimization Problems

337

bending up the sides. Find the largest volume that such a box can have. (a) Draw several diagrams to illustrate the situation, some short boxes with large bases and some tall boxes with small bases. Find the volumes of several such boxes. Does it appear that there is a maximum volume? If so, estimate it. (b) Draw a diagram illustrating the general situation. Introduce notation and label the diagram with your symbols. (c) Write an expression for the volume. (d) Use the given information to write an equation that relates the variables. (e) Use part (d) to write the volume as a function of one variable. (f ) Finish solving the problem and compare the answer with your estimate in part (a). 13. A farmer wants to fence in an area of 15,000 m2 in a rectangular field and then divide it in half with a fence parallel to one of the sides of the rectangle. How can he do this so as to minimize the cost of the fence? 14. A box with a square base and open top must have a volume of 32,000 cm3. Find the dimensions of the box that minimize the amount of material used. 15. If 1200 cm2 of material is available to make a box with a square base and an open top, find the largest possible volume of the box. 16. A rectangular storage container with an open top is to have a volume of 10 m3. The length of its base is twice the width. Material for the base costs \$10 per square meter. Material for the sides costs \$6 per square meter. Find the cost of materials for the cheapest such container. 17. Do Exercise 16 assuming the container has a lid that is made from the same material as the sides. 18. A farmer wants to fence in a rectangular plot of land adjacent to the north wall of his barn. No fencing is needed along the barn, and the fencing along the west side of the plot is shared with a neighbor who will split the cost of that portion of the fence. If the fencing costs \$30 per linear meter to install and the farmer is not willing to spend more than \$1800, find the dimensions for the plot that would enclose the most area. 19. If the farmer in Exercise 18 wants to enclose 150 square meters of land, what dimensions will minimize the cost of the fence? 20. (a) Show that of all the rectangles with a given area, the one with smallest perimeter is a square. (b) Show that of all the rectangles with a given perimeter, the one with greatest area is a square. 21. Find the point on the line y − 2x 1 3 that is closest to the origin. 22. Find the point on the curve y − sx that is closest to the point s3, 0d.

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338

CHAPTER 4

Applications of Differentiation

23. Find the points on the ellipse 4x 2 1 y 2 − 4 that are farthest away from the point s1, 0d. ; 24. Find, correct to two decimal places, the coordinates of the point on the curve y − sin x that is closest to the point s4, 2d. 25. Find the dimensions of the rectangle of largest area that can be inscribed in a circle of radius r. 26. Find the area of the largest rectangle that can be inscribed in the ellipse x 2ya 2 1 y 2yb 2 − 1. 27. Find the dimensions of the rectangle of largest area that can be inscribed in an equilateral triangle of side L if one side of the rectangle lies on the base of the triangle.

39. If you are offered one slice from a round pizza (in other words, a sector of a circle) and the slice must have a perimeter of 60 cm, what diameter pizza will reward you with the largest slice? 40. A fence 2 m tall runs parallel to a tall building at a distance of 1 m from the building. What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building? 41. A cone-shaped drinking cup is made from a circular piece of paper of radius R by cutting out a sector and joining the edges CA and CB. Find the maximum capacity of such a cup. A

B

28. Find the area of the largest trapezoid that can be inscribed in a circle of radius 1 and whose base is a diameter of the circle.

R

C

29. Find the dimensions of the isosceles triangle of largest area that can be inscribed in a circle of radius r. 30. If the two equal sides of an isosceles triangle have length a, find the length of the third side that maximizes the area of the triangle. 31. A right circular cylinder is inscribed in a sphere of radius r. Find the largest possible volume of such a cylinder. 32. A right circular cylinder is inscribed in a cone with height h and base radius r. Find the largest possible volume of such a cylinder. 33. A right circular cylinder is inscribed in a sphere of radius r. Find the largest possible surface area of such a cylinder. 34. A Norman window has the shape of a rectangle surmounted by a semicircle. (Thus the diameter of the semicircle is equal to the width of the rectangle. See Exercise 1.1.62.) If the perimeter of the window is 10 m, find the dimensions of the window so that the greatest possible amount of light is admitted. 35. The top and bottom margins of a poster are each 6 cm and the side margins are each 4 cm. If the area of printed material on the poster is fixed at 384 cm2, find the dimensions of the poster with the smallest area. 36. A poster is to have an area of 900 cm2 with 3-cm margins at the bottom and sides and a 5-cm margin at the top. What dimensions will give the largest printed area? 37. A piece of wire 10 m long is cut into two pieces. One piece is bent into a square and the other is bent into an equilateral triangle. How should the wire be cut so that the total area enclosed is (a) a maximum? (b) A minimum? 38. Answer Exercise 37 if one piece is bent into a square and the other into a circle.

42. A cone-shaped paper drinking cup is to be made to hold 27 cm3 of water. Find the height and radius of the cup that will use the smallest amount of paper. 43. A cone with height h is inscribed in a larger cone with height H so that its vertex is at the center of the base of the larger cone. Show that the inner cone has maximum volume when h − 13 H. 44. An object with mass m is dragged along a horizontal plane by a force acting along a rope attached to the object. If the rope makes an angle  with a plane, then the magnitude of the force is F−

mt  sin  1 cos

where  is a constant called the coefficient of friction. For what value of  is F smallest? 45. If a resistor of R ohms is connected across a battery of E volts with internal resistance r ohms, then the power (in watts) in the external resistor is P−

E 2R sR 1 rd 2

If E and r are fixed but R varies, what is the maximum value of the power? 46. For a fish swimming at a speed v relative to the water, the energy expenditure per unit time is proportional to v 3. It is believed that migrating fish try to minimize the total energy required to swim a fixed distance. If the fish are swimming against a current u su , vd, then the time

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SECTION 4.7

required to swim a distance L is Lysv 2 ud and the total energy E required to swim the distance is given by Esvd − av 3 ?

L

47. In a beehive, each cell is a regular hexagonal prism, open at one end with a trihedral angle at the other end as in the figure. It is believed that bees form their cells in such a way as to minimize the surface area for a given side length and height, thus using the least amount of wax in cell construction. Examination of these cells has shown that the measure of the apex angle  is amazingly consistent. Based on the geometry of the cell, it can be shown that the surface area S is given by S − 6sh 2 32 s 2 cot  1 (3s 2s3 y2) csc  where s, the length of the sides of the hexagon, and h, the height, are constants. (a) Calculate dSyd. (b) What angle should the bees prefer? (c) Determine the minimum surface area of the cell (in terms of s and h). Note: Actual measurements of the angle  in beehives have been made, and the measures of these angles seldom differ from the calculated value by more than 28. trihedral angle ¨

339

opposite A on the other side of the lake in the shortest possible time (see the figure). She can walk at the rate of 6 kmyh and row a boat at 3 kmyh. How should she proceed?

v2u

where a is the proportionality constant. (a) Determine the value of v that minimizes E. (b) Sketch the graph of E. Note: This result has been verified experimentally; migrating fish swim against a current at a speed 50% greater than the current speed.

rear of cell

Optimization Problems

B

A

¨ 3

C 3

51. An oil refinery is located on the north bank of a straight river that is 2 km wide. A pipeline is to be constructed from the refinery to storage tanks located on the south bank of the river 6 km east of the refinery. The cost of laying pipe is \$400,000ykm over land to a point P on the north bank and \$800,000ykm under the river to the tanks. To minimize the cost of the pipeline, where should P be located? ; 52. Suppose the refinery in Exercise 51 is located 1 km north of the river. Where should P be located? 53. The illumination of an object by a light source is directly proportional to the strength of the source and inversely proportional to the square of the distance from the source. If two light sources, one three times as strong as the other, are placed 4 m apart, where should an object be placed on the line between the sources so as to receive the least illumination? 54. Find an equation of the line through the point s3, 5d that cuts off the least area from the first quadrant. 55. Let a and b be positive numbers. Find the length of the shortest line segment that is cut off by the first quadrant and passes through the point sa, bd. 56. At which points on the curve y − 1 1 40x 3 2 3x 5 does the tangent line have the largest slope?

h

57. What is the shortest possible length of the line segment that is cut off by the first quadrant and is tangent to the curve y − 3yx at some point?

b

s

front of cell

48. A boat leaves a dock at 2:00 pm and travels due south at a speed of 20 kmyh. Another boat has been heading due east at 15 kmyh and reaches the same dock at 3:00 pm. At what time were the two boats closest together? 49. Solve the problem in Example 4 if the river is 5 km wide and point B is only 5 km downstream from A. 50. A woman at a point A on the shore of a circular lake with radius 3 km wants to arrive at the point C diametrically

58. What is the smallest possible area of the triangle that is cut off by the first quadrant and whose hypotenuse is tangent to the parabola y − 4 2 x 2 at some point? 59. (a) If Csxd is the cost of producing x units of a commodity, then the average cost per unit is csxd − Csxdyx. Show that if the average cost is a minimum, then the marginal cost equals the average cost. (b) If Csxd − 16,000 1 200x 1 4x 3y2, in dollars, find (i) the cost, average cost, and marginal cost at a production level of 1000 units; (ii) the production level that will minimize the average cost; and (iii) the minimum average cost.

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

340

CHAPTER 4

Applications of Differentiation

60. (a) Show that if the profit Psxd is a maximum, then the marginal revenue equals the marginal cost. (b) If Csxd − 16,000 1 500x 2 1.6x 2 1 0.004x 3 is the cost function and psxd − 1700 2 7x is the demand function, find the production level that will maximize profit. 61. A baseball team plays in a stadium that holds 55,000 spectators. With ticket prices at \$10, the average attendance had been 27,000. When ticket prices were lowered to \$8, the average attendance rose to 33,000. (a) Find the demand function, assuming that it is linear. (b) How should ticket prices be set to maximize revenue? 62. During the summer months Terry makes and sells necklaces on the beach. Last summer he sold the necklaces for \$10 each and his sales averaged 20 per day. When he increased the price by \$1, he found that the average decreased by two sales per day. (a) Find the demand function, assuming that it is linear. (b) If the material for each necklace costs Terry \$6, what should the selling price be to maximize his profit? 63. A retailer has been selling 1200 tablet computers a week at \$350 each. The marketing department estimates that an additional 80 tablets will sell each week for every \$10 that the price is lowered. (a) Find the demand function. (b) What should the price be set at in order to maximize revenue? (c) If the retailer’s weekly cost function is Csxd − 35,000 1 120x what price should it choose in order to maximize its profit? 64. A company operates 16 oil wells in a designated area. Each pump, on average, extracts 240 barrels of oil daily. The company can add more wells but every added well reduces the average daily ouput of each of the wells by 8 barrels. How many wells should the company add in order to maximize daily production? 65. Show that of all the isosceles triangles with a given perimeter, the one with the greatest area is equilateral. 66. Consider the situation in Exercise 51 if the cost of laying pipe under the river is considerably higher than the cost of laying pipe over land (\$400,000ykm). You may suspect that in some instances, the minimum distance possible under the river should be used, and P should be located 6 km from the refinery, directly across from the storage tanks. Show that this is never the case, no matter what the “under river” cost is. x2 y2 67. Consider the tangent line to the ellipse 2 1 2 − 1 a b at a point s p, qd in the first quadrant. (a) Show that the tangent line has x-intercept a 2yp and y-intercept b 2yq. (b) Show that the portion of the tangent line cut off by the coordinate axes has minimum length a 1 b.

(c) Show that the triangle formed by the tangent line and the coordinate axes has minimum area ab. CAS

68. The frame for a kite is to be made from six pieces of wood. The four exterior pieces have been cut with the lengths indicated in the figure. To maximize the area of the kite, how long should the diagonal pieces be? a

b

a

b

; 69. A point P needs to be located somewhere on the line AD so that the total length L of cables linking P to the points A, B. and C is minimized (see the figure). Express L as a function of x − AP and use the graphs of L and dLydx to estimate the minimum value of L.

| |

A

P 5m

2m B

3m D

C

70. The graph shows the fuel consumption c of a car (measured in liters per hour) as a function of the speed v of the car. At very low speeds the engine runs inefficiently, so initially c decreases as the speed increases. But at high speeds the fuel consumption increases. You can see that csvd is minimized for this car when v < 48 kmyh. However, for fuel efficiency, what must be minimized is not the consumption in gallons per hour but rather the fuel consumption in liters per kilometer. Let’s call this consumption G. Using the graph, estimate the speed at which G has its minimum value. c

0

40

80

71. Let v1 be the velocity of light in air and v2 the velocity of light in water. According to Fermat’s Principle, a ray of light will travel from a point A in the air to a point B in the water by a path ACB that minimizes the time taken. Show that sin  1 v1 − sin  2 v2

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

SECTION 4.7

where  1 (the angle of incidence) and  2 (the angle of refraction) are as shown. This equation is known as Snell’s Law. A ¨¡

341

Optimization Problems

75. An observer stands at a point P, one unit away from a track. Two runners start at the point S in the figure and run along the track. One runner runs three times as fast as the other. Find the maximum value of the observer’s angle of sight  between the runners.

C

P ¨

¨™ 1

B

72. Two vertical poles PQ and ST are secured by a rope PRS going from the top of the first pole to a point R on the ground between the poles and then to the top of the second pole as in the figure. Show that the shortest length of such a rope occurs when 1 −  2 . P S

S

76. A rain gutter is to be constructed from a metal sheet of width 30 cm by bending up one-third of the sheet on each side through an angle . How should  be chosen so that the gutter will carry the maximum amount of water?

¨ ¨¡ Q

¨™ R

¨

10 cm

10 cm

10 cm

T

73. The upper right-hand corner of a piece of paper, 30 cm by 20 cm, as in the figure, is folded over to the bottom edge. How would you fold it so as to minimize the length of the fold? In other words, how would you choose x to minimize y?

77. Where should the point P be chosen on the line segment AB so as to maximize the angle ? B

2

¨

P 3

30

y

x

5

A

20

74. A steel pipe is being carried down a hallway 3 m wide. At the end of the hall there is a right-angled turn into a narrower hallway 2 m wide. What is the length of the longest pipe that can be carried horizontally around the corner?

78. A painting in an art gallery has height h and is hung so that its lower edge is a distance d above the eye of an observer (as in the figure). How far from the wall should the observer stand to get the best view? (In other words, where should the observer stand so as to maximize the angle  subtended at his eye by the painting?) h

2

¨

d

¨

3

79. Find the maximum area of a rectangle that can be circumscribed about a given rectangle with length L and width W. [Hint: Express the area as a function of an angle .]

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

342

CHAPTER 4

Applications of Differentiation

80. The blood vascular system consists of blood vessels (arteries, arterioles, capillaries, and veins) that convey blood from the heart to the organs and back to the heart. This system should work so as to minimize the energy expended by the heart in pumping the blood. In particular, this energy is reduced when the resistance of the blood is lowered. One of Poiseuille’s Laws gives the resistance R of the blood as R−C

L r4

where L is the length of the blood vessel, r is the radius, and C is a positive constant determined by the viscosity of the blood. (Poiseuille established this law experimentally, but it also follows from Equation 8.4.2.) The figure shows a main blood vessel with radius r1 branching at an angle  into a smaller vessel with radius r 2.

fly in order to minimize the total energy expended in returning to its nesting area? (b) Let W and L denote the energy (in joules) per kilometer flown over water and land, respectively. What would a large value of the ratio WyL mean in terms of the bird’s flight? What would a small value mean? Determine the ratio WyL corresponding to the minimum expenditure of energy. (c) What should the value of WyL be in order for the bird to fly directly to its nesting area D? What should the value of WyL be for the bird to fly to B and then along the shore to D? (d) If the ornithologists observe that birds of a certain species reach the shore at a point 4 km from B, how many times more energy does it take a bird to fly over water than over land?

C

island r™ b

vascular branching A

5 km

C

¨

D

B

B

nest

13 km a

(a) Use Poiseuille’s Law to show that the total resistance of the blood along the path ABC is R−C

S

D

b csc  a 2 b cot  1 r14 r24

where a and b are the distances shown in the figure. (b) Prove that this resistance is minimized when cos  −

r24 r14

(c) Find the optimal branching angle (correct to the nearest degree) when the radius of the smaller blood vessel is two-thirds the radius of the larger vessel. 81. Ornithologists have determined that some species of birds tend to avoid flights over large bodies of water during daylight hours. It is believed that more energy is required to fly over water than over land because air generally rises over land and falls over water during the day. A bird with these tendencies is released from an island that is 5 km from the nearest point B on a straight shoreline, flies to a point C on the shoreline, and then flies along the shoreline to its nesting area D. Assume that the bird instinctively chooses a path that will minimize its energy expenditure. Points B and D are 13 km apart. (a) In general, if it takes 1.4 times as much energy to fly over water as it does over land, to what point C should the bird

; 82. Two light sources of identical strength are placed 10 m apart. An object is to be placed at a point P on a line ,, parallel to the line joining the light sources and at a distance d meters from it (see the figure). We want to locate P on , so that the intensity of illumination is minimized. We need to use the fact that the intensity of illumination for a single source is directly proportional to the strength of the source and inversely proportional to the square of the distance from the source. (a) Find an expression for the intensity Isxd at the point P. (b) If d − 5 m, use graphs of Isxd and I9sxd to show that the intensity is minimized when x − 5 m, that is, when P is at the midpoint of ,. (c) If d − 10 m, show that the intensity (perhaps surprisingly) is not minimized at the midpoint. (d) Somewhere between d − 5 m and d − 10 m there is a transitional value of d at which the point of minimal illumination abruptly changes. Estimate this value of d by graphical methods. Then find the exact value of d. P  x d

10 m

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APPLIED PROJECT The Shape of a Can

applied project

h

r

343

the shape of a can In this project we investigate the most economical shape for a can. We first interpret this to mean that the volume V of a cylindrical can is given and we need to find the height h and radius r that minimize the cost of the metal to make the can (see the figure). If we disregard any waste metal in the manufacturing process, then the problem is to minimize the surface area of the cylinder. We solved this problem in Example 4.7.2 and we found that h − 2r; that is, the height should be the same as the diameter. But if you go to your cupboard or your supermarket with a ruler, you will discover that the height is usually greater than the diameter and the ratio hyr varies from 2 up to about 3.8. Let’s see if we can explain this phenomenon. 1. The material for the cans is cut from sheets of metal. The cylindrical sides are formed by bending rectangles; these rectangles are cut from the sheet with little or no waste. But if the top and bottom discs are cut from squares of side 2r (as in the figure), this leaves considerable waste metal, which may be recycled but has little or no value to the can makers. If this is the case, show that the amount of metal used is minimized when h 8 − < 2.55 r

Discs cut from squares

2. A more efficient packing of the discs is obtained by dividing the metal sheet into hexagons and cutting the circular lids and bases from the hexagons (see the figure). Show that if this strategy is adopted, then h 4 s3 − < 2.21 r

Discs cut from hexagons

3. The values of hyr that we found in Problems 1 and 2 are a little closer to the ones that actually occur on supermarket shelves, but they still don’t account for everything. If we look more closely at some real cans, we see that the lid and the base are formed from discs with radius larger than r that are bent over the ends of the can. If we allow for this we would increase hyr. More significantly, in addition to the cost of the metal we need to incorporate the manufacturing of the can into the cost. Let’s assume that most of the expense is incurred in joining the sides to the rims of the cans. If we cut the discs from hexagons as in Problem 2, then the total cost is proportional to 4 s3 r 2 1 2rh 1 ks4r 1 hd where k is the reciprocal of the length that can be joined for the cost of one unit area of metal. Show that this expression is minimized when 3 V s − k

Î 3

2 2 hyr h ? r hyr 2 4 s3

3 ; 4. Plot sV yk as a function of x − hyr and use your graph to argue that when a can is large or joining is cheap, we should make hyr approximately 2.21 (as in Problem 2). But when the can is small or joining is costly, hyr should be substantially larger.

5. Our analysis shows that large cans should be almost square but small cans should be tall and thin. Take a look at the relative shapes of the cans in a supermarket. Is our conclusion usually true in practice? Are there exceptions? Can you suggest reasons why small cans are not always tall and thin?

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

344

CHAPTER 4

Applications of Differentiation

applied project

planes and Birds: MiniMizing energy Small birds like finches alternate between flapping their wings and keeping them folded while gliding (see Figure 1). In this project we analyze this phenomenon and try to determine how frequently a bird should flap its wings. Some of the principles are the same as for fixed-wing aircraft and so we begin by considering how required power and energy depend on the speed of airplanes.1

figUre 1 1. The power needed to propel an airplane forward at velocity v is P − Av 3 1

BL 2 v

where A and B are positive constants specific to the particular aircraft and L is the lift, the upward force supporting the weight of the plane. Find the speed that minimizes the required power. 2. The speed found in Problem 1 minimizes power but a faster speed might use less fuel. The energy needed to propel the airplane a unit distance is E − Pyv. At what speed is energy minimized? 3. Hows much faster is the speed for minimum energy than the speed for minimum power? 4. In applying the equation of Problem 1 to bird flight we split the term Av 3 into two parts: Ab v 3 for the bird’s body and Aw v 3 for its wings. Let x be the fraction of flying time spent in flapping mode. If m is the bird’s mass and all the lift occurs during flapping, then the lift is mtyx and so the power needed during flapping is

Pflap − sA b 1 Awdv 3 1

Bsmtyxd2 v

The power while wings are folded is Pfold − Abv 3. Show that the average power over an entire flight cycle is P − x Pflap 1 s1 2 xdPfold − Abv 3 1 x Awv 3 1

Bm2t 2 xv

5. For what value of x is the average power a minimum? What can you conclude if the bird flies slowly? What can you conclude if the bird flies faster and faster? 6. The average energy over a cycle is E − Pyv. What value of x minimizes E ?

1. Adapted from R. McNeill Alexander, Optima for Animals (Princeton, NJ: Princeton University Press, 1996.)

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SECTION 4.8

Newton’s Method

345

Suppose that a car dealer offers to sell you a car for \$18,000 or for payments of \$375 per month for five years. You would like to know what monthly interest rate the dealer is, in effect, charging you. To find the answer, you have to solve the equation 1 0.15

0

0.012

_0.05

figUre 1 Try to solve Equation 1 numerically using your calculator or computer. Some machines are not able to solve it. Others are successful but require you to specify a starting point for the search.

y {x ¡, f(x¡)}

y=ƒ L 0

figUre 2

r

x™ x¡

x

48xs1 1 xd60 2 s1 1 xd60 1 1 − 0

(The details are explained in Exercise 41.) How would you solve such an equation? For a quadratic equation ax 2 1 bx 1 c − 0 there is a well-known formula for the solutions. For third- and fourth-degree equations there are also formulas for the solutions, but they are extremely complicated. If f is a polynomial of degree 5 or higher, there is no such formula (see the note on page 211). Likewise, there is no formula that will enable us to find the exact roots of a transcendental equation such as cos x − x. We can find an approximate solution to Equation 1 by plotting the left side of the equation. Using a graphing device, and after experimenting with viewing rectangles, we produce the graph in Figure 1. We see that in addition to the solution x − 0, which doesn’t interest us, there is a solution between 0.007 and 0.008. Zooming in shows that the root is approximately 0.0076. If we need more accuracy we could zoom in repeatedly, but that becomes tiresome. A faster alternative is to use a calculator or computer algebra system to solve the equation numerically. If we do so, we find that the root, correct to nine decimal places, is 0.007628603. How do these devices solve equations? They use a variety of methods, but most of them make some use of Newton’s method, also called the Newton-Raphson method. We will explain how this method works, partly to show what happens inside a calculator or computer, and partly as an application of the idea of linear approximation. The geometry behind Newton’s method is shown in Figure 2. We wish to solve an equation of the form f sxd − 0, so the roots of the equation correspond to the x-intercepts of the graph of f. The root that we are trying to find is labeled r in the figure. We start with a first approximation x 1, which is obtained by guessing, or from a rough sketch of the graph of f , or from a computer-generated graph of f. Consider the tangent line L to the curve y − f sxd at the point sx 1, f sx 1dd and look at the x-intercept of L, labeled x 2. The idea behind Newton’s method is that the tangent line is close to the curve and so its x-intercept, x2, is close to the x-intercept of the curve (namely, the root r that we are seeking). Because the tangent is a line, we can easily find its x-intercept. To find a formula for x2 in terms of x1 we use the fact that the slope of L is f 9sx1 d, so its equation is y 2 f sx 1 d − f 9sx 1 dsx 2 x 1 d Since the x-intercept of L is x 2 , we know that the point sx 2 , 0d is on the line, and so 0 2 f sx 1 d − f 9sx 1 dsx 2 2 x 1 d If f 9sx 1d ± 0, we can solve this equation for x 2 : x2 − x1 2

f sx 1 d f 9sx 1 d

We use x 2 as a second approximation to r. Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

346

CHAPTER 4

Applications of Differentiation

y

Next we repeat this procedure with x 1 replaced by the second approximation x 2 , using the tangent line at sx 2 , f sx 2 dd. This gives a third approximation:

{x¡, f(x¡)}

x3 − x2 2 {x™, f(x™)}

r 0

x™ x ¡

x

If we keep repeating this process, we obtain a sequence of approximations x 1, x 2, x 3, x 4, . . . as shown in Figure 3. In general, if the nth approximation is x n and f 9sx n d ± 0, then the next approximation is given by

figUre 3

x n11 − x n 2

2

lim x n − r

nl`

y

x™

0

r

f sx n d f 9sx n d

If the numbers x n become closer and closer to r as n becomes large, then we say that the sequence converges to r and we write

Sequences were briefly introduced in A Preview of Calculus on page 5. A more thorough discussion starts in Section 11.1.

f sx 2 d f 9sx 2 d

x

Although the sequence of successive approximations converges to the desired root for functions of the type illustrated in Figure 3, in certain circumstances the sequence may not converge. For example, consider the situation shown in Figure 4. You can see that x 2 is a worse approximation than x 1. This is likely to be the case when f 9sx 1d is close to 0. It might even happen that an approximation (such as x 3 in Figure 4) falls outside the domain of f. Then Newton’s method fails and a better initial approximation x 1 should be chosen. See Exercises 31–34 for specific examples in which Newton’s method works very slowly or does not work at all.

figUre 4

EXAMPLE 1 Starting with x 1 − 2, find the third approximation x 3 to the root of the equation x 3 2 2x 2 5 − 0.

tec In Module 4.8 you can investigate how Newton’s method works for several functions and what happens when you change x 1.

SOLUTION We apply Newton’s method with

Figure 5 shows the geometry behind the first step in Newton’s method in Example 1. Since f 9s2d − 10, the tangent line to y − x 3 2 2x 2 5 at s2, 21d has equation y − 10x 2 21 so its x-intercept is x 2 − 2.1.

f sxd − x 3 2 2x 2 5

f 9sxd − 3x 2 2 2

Newton himself used this equation to illustrate his method and he chose x 1 − 2 after some experimentation because f s1d − 26, f s2d − 21, and f s3d − 16. Equation 2 becomes f sx nd x n3 2 2x n 2 5 x n11 − x n 2 − xn 2 f 9sx nd 3x n2 2 2 With n − 1 we have

1

x2 − x1 2

y=˛-2x-5 1.8

and

2.2 x™

−22

f sx1 d x13 2 2x 1 2 5 − x1 2 f 9sx1d 3x12 2 2 2 3 2 2s2d 2 5 − 2.1 3s2d2 2 2

y=10x-21

Then with n − 2 we obtain

_2

figUre 5

x3 − x2 2

s2.1d3 2 2s2.1d 2 5 x 23 2 2x 2 2 5 − 2.1 2 < 2.0946 2 3x 2 2 2 3s2.1d2 2 2

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SECTION 4.8

Newton’s Method

347

It turns out that this third approximation x 3 < 2.0946 is accurate to four decimal places.

Suppose that we want to achieve a given accuracy, say to eight decimal places, using Newton’s method. How do we know when to stop? The rule of thumb that is generally used is that we can stop when successive approximations x n and x n11 agree to eight decimal places. (A precise statement concerning accuracy in Newton’s method will be given in Exercise 11.11.39.) Notice that the procedure in going from n to n 1 1 is the same for all values of n. (It is called an iterative process.) This means that Newton’s method is particularly convenient for use with a programmable calculator or a computer. 6 EXAMPLE 2 Use Newton’s method to find s 2 correct to eight decimal places. 6 2 SOLUTION First we observe that finding s is equivalent to finding the positive root of

the equation x6 2 2 − 0 so we take f sxd − x 6 2 2. Then f 9sxd − 6x 5 and Formula 2 (Newton’s method) becomes fsx nd x n6 2 2 x n11 − x n 2 − xn 2 f 9sx nd 6x n5 If we choose x 1 − 1 as the initial approximation, then we obtain x 2 < 1.16666667 x 3 < 1.12644368 x 4 < 1.12249707 x 5 < 1.12246205 x 6 < 1.12246205 Since x 5 and x 6 agree to eight decimal places, we conclude that 6 2 < 1.12246205 s

to eight decimal places.

EXAMPLE 3 Find, correct to six decimal places, the root of the equation cos x − x. SOLUTION We first rewrite the equation in standard form:

cos x 2 x − 0 y

Therefore we let f sxd − cos x 2 x. Then f 9sxd − 2sin x 2 1, so Formula 2 becomes

y=x

y=cos x 1

figUre 6

π 2

π

x

x n11 − x n 2

cos x n 2 x n cos x n 2 x n − xn 1 2sin x n 2 1 sin x n 1 1

In order to guess a suitable value for x 1 we sketch the graphs of y − cos x and y − x in Figure 6. It appears that they intersect at a point whose x-coordinate is somewhat less

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348

CHAPTER 4

Applications of Differentiation

than 1, so let’s take x 1 − 1 as a convenient first approximation. Then, remembering to put our calculator in radian mode, we get x 2 < 0.75036387 x 3 < 0.73911289 x 4 < 0.73908513 x 5 < 0.73908513 Since x 4 and x 5 agree to six decimal places (eight, in fact), we conclude that the root of the equation, correct to six decimal places, is 0.739085. ■

1

Instead of using the rough sketch in Figure 6 to get a starting approximation for Newton’s method in Example 3, we could have used the more accurate graph that a calculator or computer provides. Figure 7 suggests that we use x1 − 0.75 as the initial approximation. Then Newton’s method gives

y=cos x

y=x

x 2 < 0.73911114

1

0

x 3 < 0.73908513

figUre 7

x 4 < 0.73908513 and so we obtain the same answer as before, but with one fewer step.

1. The figure shows the graph of a function f. Suppose that Newton’s method is used to approximate the root s of the equation f sxd − 0 with initial approximation x 1 − 6. (a) Draw the tangent lines that are used to find x 2 and x 3, and estimate the numerical values of x 2 and x 3. (b) Would x 1 − 8 be a better first approximation? Explain.

(a) x1 − 0 (d) x1 − 4

(b) x1 − 1 (e) x1 − 5

(c) x1 − 3

y

0

3

1

x

5

5. For which of the initial approximations x1 − a, b, c, and d do you think Newton’s method will work and lead to the root of the equation f sxd − 0? y

2. Follow the instructions for Exercise 1(a) but use x 1 − 1 as the starting approximation for finding the root r. 3. Suppose the tangent line to the curve y − f sxd at the point s2, 5d has the equation y − 9 2 2x. If Newton’s method is used to locate a root of the equation f sxd − 0 and the initial approximation is x1 − 2, find the second approximation x 2. 4. For each initial approximation, determine graphically what happens if Newton’s method is used for the function whose graph is shown.

a

0

b

c

d

x

6–8 Use Newton’s method with the specified initial approximation x 1 to find x 3, the third approximation to the root of the given equation. (Give your answer to four decimal places.) 6. 2 x 3 2 3x 2 1 2 − 0 ,

x 1 − 21

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SECTION 4.8

7.

2 2 x 2 1 1 − 0, x

x1 − 2

8. x 7 1 4 − 0,

x1 − 21

x n11 −

8 12. s 500

13–14 (a) Explain how we know that the given equation must have a root in the given interval. (b) Use Newton’s method to approximate the root correct to six decimal places.

4

x n11 − 2x n 2 ax n2 (This algorithm enables a computer to find reciprocals without actually dividing.) (b) Use part (a) to compute 1y1.6984 correct to six decimal places. 31. Explain why Newton’s method doesn’t work for finding the root of the equation x 3 2 3x 1 6 − 0

3

14. 22 x 1 9x 2 7x 2 11x − 0,

f3, 4g

if the initial approximation is chosen to be x 1 − 1.

15–16 Use Newton’s method to approximate the indicated root of the equation correct to six decimal places. 15. The positive root of sin x − x 2 16. The positive root of 2 cos x − x 4 17–22 Use Newton’s method to find all solutions of the equation correct to six decimal places. 17. 3 cos x − x 1 1

18. sx 1 1 − x 2 2 x

19. 2 x − 2 2 x 2

20. ln x −

21. cos x − sx

22. tan x − s1 2 x 2

1 x23

32. (a) Use Newton’s method with x 1 − 1 to find the root of the equation x 3 2 x − 1 correct to six decimal places. (b) Solve the equation in part (a) using x 1 − 0.6 as the initial approximation. (c) Solve the equation in part (a) using x 1 − 0.57. (You definitely need a programmable calculator for this part.) (d) Graph f sxd − x 3 2 x 2 1 and its tangent lines at ; x1 − 1, 0.6, and 0.57 to explain why Newton’s method is so sensitive to the value of the initial approximation. 33. Explain why Newton’s method fails when applied to the 3 equation s x − 0 with any initial approximation x 1 ± 0. Illustrate your explanation with a sketch. 34. If

; 23–28 Use Newton’s method to find all the solutions of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations. 23. 22x7 2 5x 4 1 9x 3 1 5 − 0 24. x 5 2 3x 4 1 x 3 2 x 2 2 x 1 6 − 0 x 25. 2 − s1 2 x x 11 26. cossx 2 2 xd − x 4 2

27. 4e 2x sin x − x 2 2 x 1 1 28. lnsx 2 1 2d −

3x sx 2 1 1

D

30. (a) Apply Newton’s method to the equation 1yx 2 a − 0 to derive the following reciprocal algorithm:

13. 3x 4 2 8x 3 1 2 − 0, f2, 3g 5

S

a 1 xn 1 2 xn

(b) Use part (a) to compute s1000 correct to six decimal places.

11–12 Use Newton’s method to approximate the given number correct to eight decimal places. 4 11. s 75

349

29. (a) Apply Newton’s method to the equation x 2 2 a − 0 to derive the following square-root algorithm (used by the ancient Babylonians to compute sa ):

; 9. Use Newton’s method with initial approximation x1 − 21 to find x 2, the second approximation to the root of the equation x 3 1 x 1 3 − 0. Explain how the method works by first graphing the function and its tangent line at s21, 1d. ; 10. Use Newton’s method with initial approximation x1 − 1 to find x 2, the second approximation to the root of the equation x 4 2 x 2 1 − 0. Explain how the method works by first graphing the function and its tangent line at s1, 21d.

Newton’s Method

f sxd −

H

sx 2s2x

if x > 0 if x , 0

then the root of the equation f sxd − 0 is x − 0. Explain why Newton’s method fails to find the root no matter which initial approximation x 1 ± 0 is used. Illustrate your explanation with a sketch. 35. (a) Use Newton’s method to find the critical numbers of the function f sxd − x 6 2 x 4 1 3x 3 2 2x correct to six decimal places. (b) Find the absolute minimum value of f correct to four decimal places.

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

350

CHAPTER 4

Applications of Differentiation

36. Use Newton’s method to find the absolute maximum value of the function f sxd − x cos x, 0 < x < , correct to six decimal places. 37. Use Newton’s method to find the coordinates of the inflection point of the curve y − x 2 sin x, 0 < x < , correct to six decimal places. 38. Of the infinitely many lines that are tangent to the curve y − 2sin x and pass through the origin, there is one that has the largest slope. Use Newton’s method to find the slope of that line correct to six decimal places. 39. Use Newton’s method to find the coordinates, correct to six decimal places, of the point on the parabola y − sx 2 1d 2 that is closest to the origin. 40. In the figure, the length of the chord AB is 4 cm and the length of the arc AB is 5 cm. Find the central angle , in radians, correct to four decimal places. Then give the answer to the nearest degree.

4 cm

48xs1 1 xd60 2 s1 1 xd60 1 1 − 0 Use Newton’s method to solve this equation. 42. The figure shows the sun located at the origin and the earth at the point s1, 0d. (The unit here is the distance between the centers of the earth and the sun, called an astronomical unit: 1 AU < 1.496 3 10 8 km.) There are five locations L 1, L 2, L 3, L 4, and L 5 in this plane of rotation of the earth about the sun where a satellite remains motionless with respect to the earth because the forces acting on the satellite (including the gravitational attractions of the earth and the sun) balance each other. These locations are called libration points. (A solar research satellite has been placed at one of these libration points.) If m1 is the mass of the sun, m 2 is the mass of the earth, and r − m 2ysm1 1 m 2 d, it turns out that the x-coordinate of L 1 is the unique root of the fifth-degree equation psxd − x 5 2 s2 1 rdx 4 1 s1 1 2rdx 3 2 s1 2 rdx 2

5 cm A

Replacing i by x, show that

− 1 2s1 2 rdx 1 r 2 1 − 0

B

and the x-coordinate of L 2 is the root of the equation

¨

psxd 2 2rx 2 − 0 Using the value r < 3.04042 3 10 26, find the locations of the libration points (a) L 1 and (b) L 2. 41. A car dealer sells a new car for \$18,000. He also offers to sell the same car for payments of \$375 per month for five years. What monthly interest rate is this dealer charging? To solve this problem you will need to use the formula for the present value A of an annuity consisting of n equal payments of size R with interest rate i per time period: A−

R f1 2 s1 1 i d2n g i

y L¢ sun

earth

L∞

L™

x

A physicist who knows the velocity of a particle might wish to know its position at a given time. An engineer who can measure the variable rate at which water is leaking from a tank wants to know the amount leaked over a certain time period. A biologist who knows the rate at which a bacteria population is increasing might want to deduce what the size of the population will be at some future time. In each case, the problem is to find a function F whose derivative is a known function f. If such a function F exists, it is called an antiderivative of f. definition A function F is called an antiderivative of f on an interval I if F9sxd − f sxd for all x in I.

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

SECTION 4.9

Antiderivatives

351

For instance, let f sxd − x 2. It isn’t difficult to discover an antiderivative of f if we keep the Power Rule in mind. In fact, if Fsxd − 13 x 3, then F9sxd − x 2 − f sxd. But the function Gsxd − 13 x 3 1 100 also satisfies G9sxd − x 2. Therefore both F and G are antiderivatives of f. Indeed, any function of the form Hsxd − 13 x 3 1 C, where C is a constant, is an antiderivative of f. The question arises: Are there any others? To answer this question, recall that in Section 4.2 we used the Mean Value Theorem to prove that if two functions have identical derivatives on an interval, then they must differ by a constant (Corollary 4.2.7). Thus if F and G are any two antiderivatives of f , then y

F9sxd − f sxd − G9sxd

˛

y= 3 +3 ˛

y= 3 +2

so Gsxd 2 Fsxd − C, where C is a constant. We can write this as Gsxd − Fsxd 1 C, so we have the following result.

˛

y= 3 +1 y= ˛ 0

x

3

˛

y= 3 -1 ˛

y= 3 -2

figUre 1 Members of the family of antiderivatives of f sxd − x 2

1 theorem If F is an antiderivative of f on an interval I, then the most general antiderivative of f on I is Fsxd 1 C where C is an arbitrary constant. Going back to the function f sxd − x 2, we see that the general antiderivative of f is 1 C. By assigning specific values to the constant C, we obtain a family of functions whose graphs are vertical translates of one another (see Figure 1). This makes sense because each curve must have the same slope at any given value of x. 1 3 3x

EXAMPLE 1 Find the most general antiderivative of each of the following functions. (a) f sxd − sin x (b) f sxd − 1yx (c) f sxd − x n, n ± 21 SOLUTION

(a) If Fsxd − 2cos x , then F9sxd − sin x, so an antiderivative of sin x is 2cos x. By Theorem 1, the most general antiderivative is Gsxd − 2cos x 1 C. (b) Recall from Section 3.6 that d 1 sln xd − dx x So on the interval s0, `d the general antiderivative of 1yx is ln x 1 C. We also learned that d (ln | x |) − 1x dx for all x ± 0. Theorem 1 then tells us that the general antiderivative of f sxd − 1yx is ln | x | 1 C on any interval that doesn’t contain 0. In particular, this is true on each of the intervals s2`, 0d and s0, `d. So the general antiderivative of f is Fsxd −

H

ln x 1 C1 lns2xd 1 C2

if x . 0 if x , 0

(c) We use the Power Rule to discover an antiderivative of x n. In fact, if n ± 21, then d dx

S D x n11 n11

sn 1 1dx n − xn n11

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352

CHAPTER 4

Applications of Differentiation

Therefore the general antiderivative of f sxd − x n is Fsxd −

x n11 1C n11

This is valid for n > 0 since then f sxd − x n is defined on an interval. If n is negative (but n ± 21), it is valid on any interval that doesn’t contain 0. ■ As in Example 1, every differentiation formula, when read from right to left, gives rise to an antidifferentiation formula. In Table 2 we list some particular antiderivatives. Each formula in the table is true because the derivative of the function in the right column appears in the left column. In particular, the first formula says that the antiderivative of a constant times a function is the constant times the antiderivative of the function. The second formula says that the antiderivative of a sum is the sum of the antiderivatives. (We use the notation F9− f , G9 − t.) 2 table of antidifferentiation formulas

Function

Particular antiderivative

Function

Particular antiderivative

c f sxd

cFsxd

sin x

2cos x

f sxd 1 tsxd

Fsxd 1 Gsxd

2

sec x

tan x

sec x tan x

sec x

n11

To obtain the most general antiderivative from the particular ones in Table 2, we have to add a constant (or constants), as in Example 1.

x n sn ± 21d

x n11

1 x

ln x

1 s1 2 x 2

sin21x

ex

ex

1 1 1 x2

tan21x

bx

bx ln b

cosh x

sinh x

cos x

sin x

sinh x

cosh x

| |

EXAMPLE 2 Find all functions t such that t9sxd − 4 sin x 1

2x 5 2 sx x

SOLUTION We first rewrite the given function as follows:

t9sxd − 4 sin x 1

2x 5 1 sx 2 − 4 sin x 1 2x 4 2 x x sx

Thus we want to find an antiderivative of t9sxd − 4 sin x 1 2x 4 2 x21y2 Using the formulas in Table 2 together with Theorem 1, we obtain We often use a capital letter F to represent an antiderivative of a function f. If we begin with derivative notation, f 9, an antiderivative is f, of course.

tsxd − 4s2cos xd 1 2

x5 x1y2 2 1 1C 5 2

− 24 cos x 1 25 x 5 2 2sx 1 C

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SECTION 4.9

Antiderivatives

353

In applications of calculus it is very common to have a situation as in Example 2, where it is required to find a function, given knowledge about its derivatives. An equation that involves the derivatives of a function is called a differential equation. These will be studied in some detail in Chapter 9, but for the present we can solve some elementary differential equations. The general solution of a differential equation involves an arbitrary constant (or constants) as in Example 2. However, there may be some extra conditions given that will determine the constants and therefore uniquely specify the solution.

EXAMPLE 3 Find f if f 9sxd − e x 1 20s1 1 x 2 d21 and f s0d − 22. Figure 2 shows the graphs of the function f 9 in Example 3 and its antiderivative f. Notice that f 9sxd . 0, so f is always increasing. Also notice that when f 9 has a maximum or minimum, f appears to have an inflection point. So the graph serves as a check on our calculation. 40

SOLUTION The general antiderivative of

f 9sxd − e x 1

20 1 1 x2

f sxd − e x 1 20 tan21 x 1 C

is

To determine C we use the fact that f s0d − 22: f s0d − e 0 1 20 tan21 0 1 C − 22

fª _2

3

Thus we have C − 22 2 1 − 23, so the particular solution is

f

f sxd − e x 1 20 tan21 x 2 3

_25

figUre 2

EXAMPLE 4 Find f if f 0sxd − 12x 2 1 6x 2 4, f s0d − 4, and f s1d − 1. SOLUTION The general antiderivative of f 0sxd − 12x 2 1 6x 2 4 is

f 9sxd − 12

x2 x3 16 2 4x 1 C − 4x 3 1 3x 2 2 4x 1 C 3 2

Using the antidifferentiation rules once more, we find that f sxd − 4

x4 x3 x2 13 24 1 Cx 1 D − x 4 1 x 3 2 2x 2 1 Cx 1 D 4 3 2

To determine C and D we use the given conditions that f s0d − 4 and f s1d − 1. Since f s0d − 0 1 D − 4, we have D − 4. Since f s1d − 1 1 1 2 2 1 C 1 4 − 1 we have C − 23. Therefore the required function is f sxd − x 4 1 x 3 2 2x 2 2 3x 1 4

If we are given the graph of a function f, it seems reasonable that we should be able to sketch the graph of an antiderivative F. Suppose, for instance, that we are given that Fs0d − 1. Then we have a place to start, the point s0, 1d, and the direction in which we move our pencil is given at each stage by the derivative F9sxd − f sxd. In the next example we use the principles of this chapter to show how to graph F even when we don’t have a formula for f. This would be the case, for instance, when f sxd is determined by experimental data. Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

354

CHAPTER 4

Applications of Differentiation

EXAMPLE 5 The graph of a function f is given in Figure 3. Make a rough sketch of an antiderivative F, given that Fs0d − 2.

y

y=ƒ 0

1

2

3

4

x

figUre 3 y

y=F(x)

SOLUTION We are guided by the fact that the slope of y − Fsxd is f sxd. We start at the point s0, 2d and draw F as an initially decreasing function since f sxd is negative when 0 , x , 1. Notice that f s1d − f s3d − 0, so F has horizontal tangents when x − 1 and x − 3. For 1 , x , 3, f sxd is positive and so F is increasing. We see that F has a local minimum when x − 1 and a local maximum when x − 3. For x . 3, f sxd is negative and so F is decreasing on s3, `d. Since f sxd l 0 as x l `, the graph of F becomes flatter as x l `. Also notice that F0sxd − f 9sxd changes from positive to negative at x − 2 and from negative to positive at x − 4, so F has inflection points when x − 2 and x − 4. We use this information to sketch the graph of the antiderivative in Figure 4. ■

2

Rectilinear Motion

1 0

1

figUre 4

x

Antidifferentiation is particularly useful in analyzing the motion of an object moving in a straight line. Recall that if the object has position function s − f std, then the velocity function is vstd − s9std. This means that the position function is an antiderivative of the velocity function. Likewise, the acceleration function is astd − v9std, so the velocity function is an antiderivative of the acceleration. If the acceleration and the initial values ss0d and vs0d are known, then the position function can be found by antidifferentiating twice.

EXAMPLE 6 A particle moves in a straight line and has acceleration given by astd − 6t 1 4. Its initial velocity is vs0d − 26 cmys and its initial displacement is ss0d − 9 cm. Find its position function sstd. SOLUTION Since v9std − astd − 6t 1 4, antidifferentiation gives vstd − 6

t2 1 4t 1 C − 3t 2 1 4t 1 C 2

Note that vs0d − C. But we are given that vs0d − 26, so C − 26 and vstd − 3t 2 1 4t 2 6

Since vstd − s9std, s is the antiderivative of v: sstd − 3

t3 t2 14 2 6t 1 D − t 3 1 2t 2 2 6t 1 D 3 2

This gives ss0d − D. We are given that ss0d − 9, so D − 9 and the required position function is sstd − t 3 1 2t 2 2 6t 1 9

An object near the surface of the earth is subject to a gravitational force that produces a downward acceleration denoted by t. For motion close to the ground we may assume that t is constant, its value being about 9.8 mys2 (or 32 ftys2).

EXAMPLE 7 A ball is thrown upward with a speed of 15 mys from the edge of a cliff 140 m above the ground. Find its height above the ground t seconds later. When does it reach its maximum height? When does it hit the ground?

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SECTION 4.9

355

Antiderivatives

SOLUTION The motion is vertical and we choose the positive direction to be upward. At time t the distance above the ground is sstd and the velocity vstd is decreasing. Therefore the acceleration must be negative and we have

astd −

dv − 29.8 dt

Taking antiderivatives, we have vstd − 29.8t 1 C

To determine C we use the given information that vs0d − 15. This gives 15 − 0 1 C, so vstd − 29.8t 1 15

The maximum height is reached when vstd − 0, that is, after 15y9.8 < 1.53 s. Since s9std − vstd, we antidifferentiate again and obtain sstd − 24.9t 2 1 15t 1 D Using the fact that ss0d − 140, we have 140 − 0 1 D and so Figure 5 shows the position function of the ball in Example 7. The graph corroborates the conclusions we reached: The ball reaches its maximum height after 1.5 seconds and hits the ground after about 7.1 seconds.

sstd − 24.9t 2 1 15t 1 140 The expression for sstd is valid until the ball hits the ground. This happens when sstd − 0, and therefore 2sstd − 0; that is, when 4.9t 2 2 15t 2 140 − 0 Using the quadratic formula to solve this equation, we get

200

t−

15 6 s2969 9.8

We reject the solution with the minus sign since it gives a negative value for t. Therefore the ball hits the ground after 8

0

15 1 s2969 < 7.1 s 9.8

figUre 5

1–22 Find the most general antiderivative of the function. (Check your answer by differentiation.) 1. f sxd − 4x 1 7 3. f sxd − 2x 3 2

2.

2 2 3x

f sxd − x 2 2 3x 1 2

4.

f sxd − 6x 5 2 8x 4 2 9x 2

5. f sxd − xs12 x 1 8d

6.

f sxd − sx 2 5d 2

7. f sxd − 7x 2y5 1 8x 24y5

8.

f sxd − xs2 2 xd2

1 5x

9. f sxd − s2 11. f sxd − 3sx 2 2 sx 3

10.

f sxd − e 2

12.

f sxd − sx 2 1 x sx 3

13. f sxd − 15. tstd −

1 2 2 5 x 1 1 t 1 t2 st

14. f std −

3t 4 2 t 3 1 6t 2 t4

16. rsd − sec  tan  2 2e  3

17. hsd − 2 sin  2 sec 2

18. tsv d − 2 cos v 2

19. f sxd − 2 x 1 4 sinh x

20. f sxd − 1 1 2 sin x 1 3ysx

21. f sxd −

2x 4 1 4x 3 2 x , x3

s1 2 v 2

x.0

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

356

CHAPTER 4

22. f sxd −

Applications of Differentiation

2x 2 1 5 x2 1 1

51–52 The graph of a function f is shown. Which graph is an antiderivative of f and why? 52. y

51. y ; 23–24 Find the antiderivative F of f that satisfies the given condition. Check your answer by comparing the graphs of f and F.

f

f

b

a

a x

x

b

23. f sxd − 5x 4 2 2x 5, Fs0d − 4

c c

24. f sxd − 4 2 3s1 1 x 2 d21, Fs1d − 0

53. The graph of a function is shown in the figure. Make a rough sketch of an antiderivative F, given that Fs0d − 1.

25–48 Find f. 25. f 0sxd − 20x 3 2 12x 2 1 6x 6

y

y=ƒ

4

26. f 0sxd − x 2 4x 1 x 1 1 27. f 0sxd − 2x 1 3e x

28. f 0sxd − 1yx 2

29. f -std − 12 1 sin t

30. f -std − st 2 2 cos t f s4d − 25

31. f 9sxd − 1 1 3sx , 4

2

33. f 9std − 4ys1 1 t 2 d,

35. f 9sxd − 5x

2y3

,

f s1d − 0

34. f 9std − t 1 1yt 3, t . 0,

f s1d − 6 0

f s8d − 21

36. f 9sxd − sx 1 1dysx ,

2y2 , t , y2,

f s1d − 2,

f s21d − 1

2

39. f 0sxd − 22 1 12x 2 12x , f s0d − 4, 40. f 0sxd − 8x 3 1 5,

f s1d − 0,

2

42. f 0std − t 1 1yt ,

t . 0,

43. f 0sxd − 4 1 6x 1 24x 2, 44. f 0sxd − x 3 1 sinh x, x

45. f 0sxd − e 2 2 sin x, 3 46. f 0std − s t 2 cos t,

47. f 0sxd − x 22, 48. f -sxd − cos x,

x . 0,

f s0d − 3,

f s0d − 3, f s0d − 2, f s1d − 0,

55. The graph of f 9 is shown in the figure. Sketch the graph of f if f is continuous on f0, 3g and f s0d − 21. y 2

y=fª(x)

1

f 9s0d − 4

f s2d − 3,

f s0d − 1,

f s0d − 1,

f 9s0d − 12

f 9s1d − 8

f s0d − 3,

41. f 0sd − sin  1 cos , 2

t

f s1d − 5

37. f 9std − sec t ssec t 1 tan td, f sy4d − 21 38. f 9std − 3 t 2 3yt ,

x

1

54. The graph of the velocity function of a particle is shown in the figure. Sketch the graph of a position function.

f s21d − 2

32. f 9sxd − 5x 2 3x 1 4,

0

f 9s1d − 2

0 _1

1

2

x

f s1d − 10

f s2d − 2.6 f sy2d − 0 f s1d − 2 f s2d − 0

f 9s0d − 2,

f 0s0d − 3

49. Given that the graph of f passes through the point (2, 5) and that the slope of its tangent line at sx, f sxdd is 3 2 4x, find f s1d. 50. Find a function f such that f 9sxd − x 3 and the line x 1 y − 0 is tangent to the graph of f .

; 56. (a) Use a graphing device to graph f sxd − 2x 2 3 sx . (b) Starting with the graph in part (a), sketch a rough graph of the antiderivative F that satisfies Fs0d − 1. (c) Use the rules of this section to find an expression for Fsxd. (d) Graph F using the expression in part (c). Compare with your sketch in part (b). ; 57–58 Draw a graph of f and use it to make a rough sketch of the antiderivative that passes through the origin. sin x 57. f sxd − , 22 < x < 2 1 1 x2 58. f sxd − sx 4 2 2 x 2 1 2 2 2, 23 < x < 3

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

SECTION 4.9

59–64 A particle is moving with the given data. Find the position of the particle. 59. vstd − sin t 2 cos t, ss0d − 0 60. vstd − t 2 3 st , ss4d − 8 61. astd − 2t 1 1, ss0d − 3, v s0d − 22 63. astd − 10 sin t 1 3 cos t, 2

64. astd − t 2 4t 1 6,

ss0d − 0,

v s0d − 4

ss0d − 0, ss2d − 12

ss0d − 0,

ss1d − 20

producing one item is \$562, find the cost of producing 100 items.

73. Since raindrops grow as they fall, their surface area increases and therefore the resistance to their falling increases. A raindrop has an initial downward velocity of 10 mys and its downward acceleration is a−

65. A stone is dropped from the upper observation deck (the Space Deck) of the CN Tower, 450 m above the ground. (a) Find the distance of the stone above ground level at time t. (b) How long does it take the stone to reach the ground? (c) With what velocity does it strike the ground? (d) If the stone is thrown downward with a speed of 5 mys, how long does it take to reach the ground? 66. Show that for motion in a straight line with constant acceleration a, initial velocity v 0, and initial displacement s 0, the displacement after time t is s − 12 at 2 1 v 0 t 1 s 0 67. An object is projected upward with initial velocity v 0 meters per second from a point s0 meters above the ground. Show that 2

fvstdg −

v02

2 19.6fsstd 2 s0 g

68. Two balls are thrown upward from the edge of the cliff in Example 7. The first is thrown with a speed of 15 mys and the other is thrown a second later with a speed of 8 mys. Do the balls ever pass each other? 69. A stone was dropped off a cliff and hit the ground with a speed of 40 mys. What is the height of the cliff? 70. If a diver of mass m stands at the end of a diving board with length L and linear density , then the board takes on the shape of a curve y − f sxd, where EI y 0 − mtsL 2 xd 1 12 tsL 2 xd2 E and I are positive constants that depend on the material of the board and t s, 0d is the acceleration due to gravity. (a) Find an expression for the shape of the curve. (b) Use f sLd to estimate the distance below the horizontal at the end of the board. y

0

357

72. The linear density of a rod of length 1 m is given by sxd − 1ysx , in grams per centimeter, where x is measured in centimeters from one end of the rod. Find the mass of the rod.

2

62. astd − 3 cos t 2 2 sin t,

Antiderivatives

x

71. A company estimates that the marginal cost (in dollars per item) of producing x items is 1.92 2 0.002x. If the cost of

H

9 2 0.9t 0

if 0 < t < 10 if t . 10

If the raindrop is initially 500 m above the ground, how long does it take to fall? 74. A car is traveling at 80 kmyh when the brakes are fully applied, producing a constant deceleration of 7 mys2. What is the distance traveled before the car comes to a stop? 75. What constant acceleration is required to increase the speed of a car from 50 kmyh to 80 kmyh in 5 seconds? 76. A car braked with a constant deceleration of 5 mys2, producing skid marks measuring 60 m before coming to a stop. How fast was the car traveling when the brakes were first applied? 77. A car is traveling at 100 kmyh when the driver sees an accident 80 m ahead and slams on the brakes. What constant deceleration is required to stop the car in time to avoid a pileup? 78. A model rocket is fired vertically upward from rest. Its acceleration for the first three seconds is astd − 18t, at which time the fuel is exhausted and it becomes a freely “falling” body. Fourteen seconds later, the rocket’s parachute opens, and the (downward) velocity slows linearly to 25.5 mys in 5 seconds. The rocket then “floats” to the ground at that rate. (a) Determine the position function s and the velocity function v (for all times t). Sketch the graphs of s and v. (b) At what time does the rocket reach its maximum height, and what is that height? (c) At what time does the rocket land? 79. A high-speed bullet train accelerates and decelerates at the rate of 1.2 mys2. Its maximum cruising speed is 145 kmyh. (a) What is the maximum distance the train can travel if it accelerates from rest until it reaches its cruising speed and then runs at that speed for 15 minutes? (b) Suppose that the train starts from rest and must come to a complete stop in 15 minutes. What is the maximum distance it can travel under these conditions? (c) Find the minimum time that the train takes to travel between two consecutive stations that are 72 km apart. (d) The trip from one station to the next takes 37.5 minutes. How far apart are the stations?

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

358

4

CHAPTER 4

Applications of Differentiation

review

concept checK 1. Explain the difference between an absolute maximum and a local maximum. Illustrate with a sketch. 2. (a) What does the Extreme Value Theorem say? (b) Explain how the Closed Interval Method works. 3. (a) State Fermat’s Theorem. (b) Define a critical number of f. 4. (a) State Rolle’s Theorem. (b) State the Mean Value Theorem and give a geometric interpretation. 5. (a) State the Increasing/Decreasing Test. (b) What does it mean to say that f is concave upward on an interval I? (c) State the Concavity Test. (d) What are inflection points? How do you find them? 6. (a) State the First Derivative Test. (b) State the Second Derivative Test. (c) What are the relative advantages and disadvantages of these tests? 7. (a) What does l’Hospital’s Rule say? (b) How can you use l’Hospital’s Rule if you have a product f sxd tsxd where f sxd l 0 and tsxd l ` as x l a? (c) How can you use l’Hospital’s Rule if you have a difference f sxd 2 tsxd where f sxd l ` and tsxd l ` as x l a?

Answers to the Concept Check can be found on the back endpapers.

(d) How can you use l’Hospital’s Rule if you have a power f f sxdg tsxd where f sxd l 0 and tsxd l 0 as x l a? 8. State whether each of the following limit forms is indeterminate. Where possible, state the limit. 0 ` 0 ` (a) (b) (c) (d) 0 ` ` 0 (e) ` 1 `

(f ) ` 2 `

(g) ` ? `

(h) ` ? 0

(i ) 0 0

(j ) 0 `

(k) ` 0

(l ) 1`

9. If you have a graphing calculator or computer, why do you need calculus to graph a function? 10. (a) Given an initial approximation x1 to a root of the equation f sxd − 0, explain geometrically, with a diagram, how the second approximation x 2 in Newton’s method is obtained. (b) Write an expression for x 2 in terms of x1, f sx 1 d, and f 9sx 1d. (c) Write an expression for x n11 in terms of x n , f sx n d, and f 9sx n d. (d) Under what circumstances is Newton’s method likely to fail or to work very slowly? 11. (a) What is an antiderivative of a function f ? (b) Suppose F1 and F2 are both antiderivatives of f on an interval I. How are F1 and F2 related?

trUe –false qUiz Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. 1. If f 9scd − 0, then f has a local maximum or minimum at c. 2. If f has an absolute minimum value at c, then f 9scd − 0. 3. If f is continuous on sa, bd, then f attains an absolute maximum value f scd and an absolute minimum value f sd d at some numbers c and d in sa, bd.

9. There exists a function f such that f sxd . 0, f 9sxd , 0, and f 0 sxd . 0 for all x. 10. There exists a function f such that f sxd , 0, f 9sxd , 0, and f 0 sxd . 0 for all x. 11. If f and t are increasing on an interval I, then f 1 t is increasing on I. 12. If f and t are increasing on an interval I, then f 2 t is increasing on I.

4. If f is differentiable and f s21d − f s1d, then there is a number c such that c , 1 and f 9scd − 0.

13. If f and t are increasing on an interval I, then f t is increasing on I.

5. If f 9sxd , 0 for 1 , x , 6, then f is decreasing on (1, 6).

14. If f and t are positive increasing functions on an interval I, then f t is increasing on I.

| |

6. If f 0s2d − 0, then s2, f s2dd is an inflection point of the curve y − f sxd. 7. If f 9sxd − t9sxd for 0 , x , 1, then f sxd − tsxd for 0 , x , 1. 8. There exists a function f such that f s1d − 22, f s3d − 0, and f 9sxd . 1 for all x.

15. If f is increasing and f sxd . 0 on I, then tsxd − 1yf sxd is decreasing on I. 16. If f is even, then f 9 is even. 17. If f is periodic, then f 9 is periodic.

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

CHAPTER 4

359

Review

20. If lim f sxd − 1 and lim tsxd − `, then

18. The most general antiderivative of f sxd − x 22 is

xl`

xl`

1 Fsxd − 2 1 C x

lim f f sxdg tsxd − 1

xl`

19. If f 9sxd exists and is nonzero for all x, then f s1d ± f s0d.

21. lim

xl0

x −1 ex

eXercises 1–6 Find the local and absolute extreme values of the function on the given interval. 1. f sxd − x 3 2 9x 2 1 24 x 2 2,

f0, 5g

f 9sxd − 2x if 0 , x , 1, f 9sxd − 21 if 1 , x , 3,

17. f is odd,

f 9sxd , 0 for 0 , x , 2,

f 9sxd . 0 for x . 2,

3x 2 4 , x2 1 1

f22, 2g

4. f sxd − sx 2 1 x 1 1 , 5. f sxd − x 1 2 cos x, 6. f sxd − x 2e 2x,

f is continuous and even,

f 9sxd − 1 if x . 3

2. f sxd − xs1 2 x , f21, 1g 3. f sxd −

16. f s0d − 0,

f 0sxd . 0 for 0 , x , 3,

f 0sxd , 0 for x . 3,

lim f sxd − 22

xl`

f22, 1g 18. The figure shows the graph of the derivative f 9of a function f. (a) On what intervals is f increasing or decreasing? (b) For what values of x does f have a local maximum or minimum? (c) Sketch the graph of f 0. (d) Sketch a possible graph of f.

f2, g

f21, 3g

7–14 Evaluate the limit. ex 2 1 x l 0 tan x

7. lim

2x

xl 0

e

4x

xl0

13. lim1 xl1

xl 0

S

tan 4x x 1 sin 2x 2x

22x

e 2e lnsx 1 1d

9. lim

11. lim

8. lim

10. lim

xl `

2 1 2 4x x2

x 1 2 x21 ln x

12. lim

14.

y=f ª(x)

22x

e 2e lnsx 1 1d e

4x

xl`

D

y

_2 _1

0

1

2

3

4

5

6

7

x

2 1 2 4x x2

lim stan xdcos x

x l sy2d 2

19–34 Use the guidelines of Section 4.5 to sketch the curve. 19. y − 2 2 2x 2 x 3

15–17 Sketch the graph of a function that satisfies the given conditions. 15. f s0d − 0, f 9s22d − f 9s1d − f 9s9d − 0, lim f sxd − 0,

xl`

lim f sxd − 2`,

x l6

20. y − 22 x 3 2 3x 2 1 12 x 1 5 21. y − 3x 4 2 4x 3 1 2 23. y −

1 xsx 2 3d2

24. y −

25. y −

sx 2 1d 3 x2

26. y − s1 2 x 1 s1 1 x

f 9sxd , 0 on s2`, 22d, s1, 6d, and s9, `d, f 9sxd . 0 on s22, 1d and s6, 9d, f 0sxd . 0 on s2`, 0d and s12, `d, f 0sxd , 0 on s0, 6d and s6, 12d

22. y − x 2ysx 1 8d

3 27. y − s x2 1 1

1 1 2 x2 sx 2 2d 2

28. y − x 2y3sx 2 3d 2

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

360

CHAPTER 4

29. y − e x sin x,

Applications of Differentiation 2

2cx ; 44. Investigate the family of functions f sxd − cxe . What happens to the maximum and minimum points and the inflection points as c changes? Illustrate your conclusions by graphing several members of the family.

2 < x <

30. y − 4x 2 tan x, 2y2 , x , y2 2

31. y − sin21s1yxd

32. y − e 2x2x

33. y − sx 2 2de 2x

34. y − x 1 lnsx 2 1 1d

; 35–38 Produce graphs of f that reveal all the important aspects of the curve. Use graphs of f 9 and f 0 to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points. In Exercise 35 use calculus to find these quantities exactly. 2

35. f sxd −

x 21 x3 3

36. f sxd −

x 11 x6 1 1

37. f sxd − 3x 6 2 5x 5 1 x 4 2 5x 3 2 2x 2 1 2

2

21yx in a viewing rectangle that shows all ; 39. Graph f sxd − e the main aspects of this function. Estimate the inflection points. Then use calculus to find them exactly.

CAS

46. Suppose that f is continuous on f0, 4g, f s0d − 1, and 2 < f 9sxd < 5 for all x in s0, 4d. Show that 9 < f s4d < 21. 47. By applying the Mean Value Theorem to the function f sxd − x 1y5 on the interval f32, 33g, show that 5 2,s 33 , 2.0125

48. For what values of the constants a and b is s1, 3d a point of inflection of the curve y − ax 3 1 bx 2 ? 49. Let tsxd − f sx 2 d, where f is twice differentiable for all x, f 9sxd . 0 for all x ± 0, and f is concave downward on s2`, 0d and concave upward on s0, `d. (a) At what numbers does t have an extreme value? (b) Discuss the concavity of t. 50. Find two positive integers such that the sum of the first number and four times the second number is 1000 and the product of the numbers is as large as possible.

38. f sxd − x 2 1 6.5 sin x, 25 < x < 5

CAS

45. Show that the equation 3x 1 2 cos x 1 5 − 0 has exactly one real root.

51. Show that the shortest distance from the point sx 1, y1 d to the straight line Ax 1 By 1 C − 0 is

| Ax

40. (a) Graph the function f sxd − 1ys1 1 e d . (b) Explain the shape of the graph by computing the limits of f sxd as x approaches `, 2`, 01, and 02. (c) Use the graph of f to estimate the coordinates of the inflection points. (d) Use your CAS to compute and graph f 0. (e) Use the graph in part (d) to estimate the inflection points more accurately. 41–42 Use the graphs of f, f 9, and f 0 to estimate the x-coordinates of the maximum and minimum points and inflection points of f. 41. f sxd −

cos 2 x sx 2 1 x 1 1

,

2 < x <

42. f sxd − e20.1x lnsx 2 2 1d

1

1 By1 1 C

sA2 1 B 2

1yx

52. Find the point on the hyperbola x y − 8 that is closest to the point s3, 0d. 53. Find the smallest possible area of an isosceles triangle that is circumscribed about a circle of radius r. 54. Find the volume of the largest circular cone that can be inscribed in a sphere of radius r.

|

| |

|

55. In D ABC, D lies on AB, CD  AB, AD − BD − 4 cm, and CD − 5 cm. Where should a point P be chosen on CD so that the sum PA 1 PB 1 PC is a minimum?

|

56.

|

| | | | | | Solve Exercise 55 when | CD | − 2 cm.

57. The velocity of a wave of length L in deep water is v−K

; 43. Investigate the family of functions f sxd − lnssin x 1 C d. What features do the members of this family have in common? How do they differ? For which values of C is f continuous on s2`, `d? For which values of C does f have no graph at all? What happens as C l `?

|

Î

L C 1 C L

where K and C are known positive constants. What is the length of the wave that gives the minimum velocity? 58. A metal storage tank with volume V is to be constructed in the shape of a right circular cylinder surmounted by a

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

CHAPTER 4

hemisphere. What dimensions will require the least amount of metal? 59. A hockey team plays in an arena with a seating capacity of 15,000 spectators. With the ticket price set at \$12, average attendance at a game has been 11,000. A market survey indicates that for each dollar the ticket price is lowered, average attendance will increase by 1000. How should the owners of the team set the ticket price to maximize their revenue from ticket sales? ; 60. A manufacturer determines that the cost of making x units of a commodity is

Review

71. f 0sxd − 1 2 6x 1 48x 2,

f s0d − 1,

f 9s0d − 2

72. f 0sxd − 5x 3 1 6x 2 1 2,

f s0d − 3,

f s1d − 22

361

73–74 A particle is moving with the given data. Find the position of the particle. 73. vstd − 2t 2 1ys1 1 t 2 d, ss0d − 1 74. astd − sin t 1 3 cos t, ss0d − 0, v s0d − 2

Csxd − 1800 1 25x 2 0.2x 2 1 0.001x 3 and the demand function is psxd − 48.2 2 0.03x. (a) Graph the cost and revenue functions and use the graphs to estimate the production level for maximum profit. (b) Use calculus to find the production level for maximum profit. (c) Estimate the production level that minimizes the average cost.

x ; 75. (a) If f sxd − 0.1e 1 sin x, 24 < x < 4, use a graph of f to sketch a rough graph of the antiderivative F of f that satisfies Fs0d − 0. (b) Find an expression for Fsxd. (c) Graph F using the expression in part (b). Compare with your sketch in part (a).

; 76. Investigate the family of curves given by f sxd − x 4 1 x 3 1 cx 2

61. Use Newton’s method to find the root of the equation x 5 2 x 4 1 3x 2 2 3x 2 2 − 0 in the interval f1, 2g correct to six decimal places. 62. Use Newton’s method to find all solutions of the equation sin x − x 2 2 3x 1 1 correct to six decimal places. 63. Use Newton’s method to find the absolute maximum value of the function f std − cos t 1 t 2 t 2 correct to eight decimal places. 64. Use the guidelines in Section 4.5 to sketch the curve y − x sin x, 0 < x < 2. Use Newton’s method when necessary. 65–68 Find the most general antiderivative of the function. 65. f sxd − 4 sx 2 6x 2 1 3 66. tsxd −

1 1 1 2 x x 11

67. f std − 2 sin t 2 3e t

In particular you should determine the transitional value of c at which the number of critical numbers changes and the transitional value at which the number of inflection points changes. Illustrate the various possible shapes with graphs. 77. A canister is dropped from a helicopter 500 m above the ground. Its parachute does not open, but the canister has been designed to withstand an impact velocity of 100 mys. Will it burst? 78. In an automobile race along a straight road, car A passed car B twice. Prove that at some time during the race their accelerations were equal. State the assumptions that you make. 79. A rectangular beam will be cut from a cylindrical log of radius 30 cm. (a) Show that the beam of maximal cross-sectional area is a square. (b) Four rectangular planks will be cut from the four sections of the log that remain after cutting the square

68. f sxd − x23 1 cosh x

depth

69–72 Find f. 69. f 9std − 2t 2 3 sin t, u 1 su , u

30

f s0d − 5

2

70. f 9sud −

f s1d − 3

width

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

362

CHAPTER 4

Applications of Differentiation

beam. Determine the dimensions of the planks that will have maximal cross-sectional area. (c) Suppose that the strength of a rectangular beam is proportional to the product of its width and the square of its depth. Find the dimensions of the strongest beam that can be cut from the cylindrical log. 80. If a projectile is fired with an initial velocity v at an angle of inclination  from the horizontal, then its trajectory, neglecting air resistance, is the parabola y − stan dx 2

t x2 2v 2 cos 2

0,,

2

(a) Suppose the projectile is fired from the base of a plane that is inclined at an angle ,  . 0, from the horizontal, as shown in the figure. Show that the range of the projectile, measured up the slope, is given by

Rsd −

2v 2 cos  sins 2 d t cos2

(b) Determine  so that R is a maximum. (c) Suppose the plane is at an angle  below the horizontal. Determine the range R in this case, and determine the angle at which the projectile should be fired to maximize R.

82. If a metal ball with mass m is projected in water and the force of resistance is proportional to the square of the velocity, then the distance the ball travels in time t is sstd −

Î

m ln cosh c

tc mt

where c is a positive constant. Find lim c l 01 sstd. 83. Show that, for x . 0, x , tan21x , x 1 1 x2 84. Sketch the graph of a function f such that f 9sxd , 0 for all x, f 0sxd . 0 for x . 1, f 0sxd , 0 for x , 1, and lim x l6` f f sxd 1 xg − 0.

| |

| |

85. A light is to be placed atop a pole of height h meters to illuminate a busy traffic circle, which has a radius of 20 m. The intensity of illumination I at any point P on the circle is directly proportional to the cosine of the angle  (see the figure) and inversely proportional to the square of the distance d from the source. (a) How tall should the light pole be to maximize I? (b) Suppose that the light pole is h meters tall and that a woman is walking away from the base of the pole at the rate of 1 mys. At what rate is the intensity of the light at the point on her back 1 m above the ground decreasing when she reaches the outer edge of the traffic circle? ¨

y

h

d

20

P

¨ å

R x

0

81. If an electrostatic field E acts on a liquid or a gaseous polar dielectric, the net dipole moment P per unit volume is PsEd −

1 e E 1 e 2E 2 e E 2 e 2E E

Show that lim E l 01 PsEd − 0.

86. Water is flowing at a constant rate into a spherical tank. Let Vstd be the volume of water in the tank and Hstd be the height of the water in the tank at time t. (a) What are the meanings of V9std and H9std? Are these derivatives positive, negative, or zero? (b) Is V 0std positive, negative, or zero? Explain. (c) Let t1, t 2, and t 3 be the times when the tank is one-quarter full, half full, and three-quarters full, respectively. Are the values H 0st1d, H 0st 2 d, and H 0st 3 d positive, negative, or zero? Why?

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

Problems Plus

2

1. If a rectangle has its base on the x-axis and two vertices on the curve y − e 2x , show that the rectangle has the largest possible area when the two vertices are at the points of inflection of the curve. 2. Show that sin x 2 cos x < s2 for all x.

|

|

3. Does the function f sxd − e 10 | x22 |2x have an absolute maximum? If so, find it. What about an absolute minimum? 2

| |

4. Show that x 2 y 2 s4 2 x 2 ds4 2 y 2 d < 16 for all numbers x and y such that x < 2 and y < 2.

| |

5. Show that the inflection points of the curve y − ssin xdyx lie on the curve y 2 sx 4 1 4d − 4. 6. Find the point on the parabola y − 1 2 x 2 at which the tangent line cuts from the first quadrant the triangle with the smallest area. 7. If a, b, c, and d are constants such that lim

xl0

ax 2 1 sin bx 1 sin cx 1 sin dx −8 3x 2 1 5x 4 1 7x 6

find the value of the sum a 1 b 1 c 1 d.

y

8. Evaluate lim

Q

xl`

sx 1 2d1yx 2 x 1yx sx 1 3d1yx 2 x 1yx

9. Find the highest and lowest points on the curve x 2 1 x y 1 y 2 − 12. P

|

|

10. Sketch the set of all points sx, yd such that x 1 y < e x. 11. If Psa, a 2 d is any point on the parabola y − x 2, except for the origin, let Q be the point where the normal line at P intersects the parabola again (see the figure). (a) Show that the y-coordinate of Q is smallest when a − 1ys2 . (b) Show that the line segment PQ has the shortest possible length when a − 1ys2 .

x

0

figUre for proBleM 11

12. For what values of c does the curve y − cx 3 1 e x have inflection points? 13. An isosceles triangle is circumscribed about the unit circle so that the equal sides meet at the point s0, ad on the y-axis (see the figure). Find the value of a that minimizes the lengths of the equal sides. (You may be surprised that the result does not give an equilateral triangle.). 14. Sketch the region in the plane consisting of all points sx, yd such that

|

|

2xy < x 2 y < x 2 1 y 2 15. The line y − mx 1 b intersects the parabola y − x 2 in points A and B. (See the figure.) Find the point P on the arc AOB of the parabola that maximizes the area of the triangle PAB.

figUre for proBleM 13 y

y=≈

16. ABCD is a square piece of paper with sides of length 1 m. A quarter-circle is drawn from B to D with center A. The piece of paper is folded along EF, with E on AB and F on AD, so that A falls on the quarter-circle. Determine the maximum and minimum areas that the triangle AEF can have.

B A

17. For which positive numbers a does the curve y − a x intersect the line y − x? y=mx+b

18. For what value of a is the following equation true? O

P

figUre for proBleM 15

x

lim

xl`

S D x1a x2a

x

−e

363 Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

P

19. Let f sxd − a 1 sin x 1 a 2 sin 2x 1 ∙ ∙ ∙ 1 a n sin nx, where a 1, a 2, . . . , a n are real numbers and n is a positive integer. If it is given that f sxd < sin x for all x, show that

|

¨

B(¨)

A(¨)

figUre for proBleM 20

D speed of sound=c¡

h

¨ R

S

O speed of sound=c™

figUre for proBleM 21

|

20. An arc PQ of a circle subtends a central angle  as in the figure. Let Asd be the area between the chord PQ and the arc PQ. Let Bsd be the area between the tangent lines PR, QR, and the arc. Find Asd lim  l 01 Bsd

Q

P

|

R

| | |

a 1 1 2a 2 1 ∙ ∙ ∙ 1 na n < 1

21. The speeds of sound c1 in an upper layer and c2 in a lower layer of rock and the thickness  h of the upper layer can be determined by seismic exploration if the speed of sound in the lower layer is greater than the speed in the upper layer. A dynamite charge is detonated at Q a point P and the transmitted signals are recorded at a point Q, which is a distance D from P. The first signal to arrive at Q travels along the surface and takes T1 seconds. The next signal travels from P to a point R, from R to S in the lower layer, and then to Q, taking T2 ¨ seconds. The third signal is reflected off the lower layer at the midpoint O of RS and takes T3 seconds to reach Q. (See the figure.) (a) Express T1, T2, and T3 in terms of D, h, c1, c2, and . (b) Show that T2 is a minimum when sin  − c1yc2. (c) Suppose that D − 1 km, T1 − 0.26 s, T2 − 0.32 s, and T3 − 0.34 s. Find c1, c2, and h. Note: Geophysicists use this technique when studying the structure of the earth’s crust, whether searching for oil or examining fault lines. 22. For what values of c is there a straight line that intersects the curve y − x 4 1 cx 3 1 12x 2 2 5x 1 2 in four distinct points?

d B

E

x

C r

F

23. One of the problems posed by the Marquis de l’Hospital in his calculus textbook Analyse des Infiniment Petits concerns a pulley that is attached to the ceiling of a room at a point C by a rope of length r. At another point B on the ceiling, at a distance d from C (where d . r), a rope of length , is attached and passed through the pulley at F and connected to a weight W. The weight is released and comes to rest at its equilibrium position D. (See the figure.) As l’Hospital argued, this happens when the distance ED is maximized. Show that when the system reaches equilibrium, the value of x is r (r 1 sr 2 1 8d 2 ) 4d Notice that this expression is independent of both W and ,.

|

D

figUre for proBleM 23

|

24. Given a sphere with radius r, find the height of a pyramid of minimum volume whose base is a square and whose base and triangular faces are all tangent to the sphere. What if the base of the pyramid is a regular n-gon? (A regular n-gon is a polygon with n equal sides and angles.) (Use the fact that the volume of a pyramid is 13 Ah, where A is the area of the base.) 25. Assume that a snowball melts so that its volume decreases at a rate proportional to its surface area. If it takes three hours for the snowball to decrease to half its original volume, how much longer will it take for the snowball to melt completely?

figUre for proBleM 26

26. A hemispherical bubble is placed on a spherical bubble of radius 1. A smaller hemispherical bubble is then placed on the first one. This process is continued until n chambers, including the sphere, are formed. (The figure shows the case n − 4.) Use mathematical induction to prove that the maximum height of any bubble tower with n chambers is 1 1 sn .

364 Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

5

Integrals

The photo shows Lake Lanier, which is a reservoir in Georgia, USA. In Exercise 70 in Section 5.4 you will estimate the amount of water that flowed into Lake Lanier during a certain time period. JRC, Inc. / Alamy

In chapter 2 we used the tangent and velocity problems to introduce the derivative, which is the central idea in differential calculus. In much the same way, this chapter starts with the area and distance problems and uses them to formulate the idea of a definite integral, which is the basic concept of integral calculus. We will see in Chapters 6 and 8 how to use the integral to solve problems concerning volumes, lengths of curves, population predictions, cardiac output, forces on a dam, work, consumer surplus, and baseball, among many others. There is a connection between integral calculus and differential calculus. The Fundamental Theorem of Calculus relates the integral to the derivative, and we will see in this chapter that it greatly simplifies the solution of many problems.

365 Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

366

Chapter 5

Integrals

Now is a good time to read (or reread) A Preview of Calculus (see page 1). It discusses the unifying ideas of calculus and helps put in perspective where we have been and where we are going. y

y=ƒ x=a S

x=b

a

0

x

b

In this section we discover that in trying to find the area under a curve or the distance traveled by a car, we end up with the same special type of limit.

the area problem We begin by attempting to solve the area problem: Find the area of the region S that lies under the curve y − f sxd from a to b. This means that S, illustrated in Figure 1, is bounded by the graph of a continuous function f [where f sxd > 0], the vertical lines x − a and x − b, and the x-axis. In trying to solve the area problem we have to ask ourselves: What is the meaning of the word area? This question is easy to answer for regions with straight sides. For a rectangle, the area is defined as the product of the length and the width. The area of a triangle is half the base times the height. The area of a polygon is found by dividing it into triangles (as in Figure 2) and adding the areas of the triangles.

FIGURE 1

|

S − hsx, yd a < x < b, 0 < y < f sxdj

A™ w

h

b

l

FIGURE 2

A= 21 bh

A=lw

A=A¡+A™+A£+A¢

However, it isn’t so easy to find the area of a region with curved sides. We all have an intuitive idea of what the area of a region is. But part of the area problem is to make this intuitive idea precise by giving an exact definition of area. Recall that in defining a tangent we first approximated the slope of the tangent line by slopes of secant lines and then we took the limit of these approximations. We pursue a similar idea for areas. We first approximate the region S by rectangles and then we take the limit of the areas of these rectangles as we increase the number of rectangles. The following example illustrates the procedure. y

example 1 Use rectangles to estimate the area under the parabola y − x 2 from 0 to 1

(1, 1)

(the parabolic region S illustrated in Figure 3). SOLUtION We first notice that the area of S must be somewhere between 0 and 1 because S is contained in a square with side length 1, but we can certainly do better than that. Suppose we divide S into four strips S1, S2, S3, and S4 by drawing the vertical lines x − 14, x − 12, and x − 34 as in Figure 4(a).

y=≈ S

y 0

1

y

x

(1, 1)

(1, 1)

y=≈

FIGURE 3 S¢ S™

S¡ 0

FIGURE 4

1 4

1 2

(a)

3 4

1

x

0

1 4

1 2

3 4

1

x

(b)

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

367

SeCtION 5.1 Areas and Distances

We can approximate each strip by a rectangle that has the same base as the strip and whose height is the same as the right edge of the strip [see Figure 4(b)]. In other words, the heights of these rectangles are the values of the function f sxd − x 2 at the right endpoints of the subintervals f0, 14 g, f 41 , 12 g, f 12 , 34 g, and f 34 , 1g. 2

2

2

Each rectangle has width 14 and the heights are ( 14 ) , ( 12 ) , ( 34 ) , and 12. If we let R 4 be the sum of the areas of these approximating rectangles, we get 2

2

2

R4 − 14 ? ( 14 ) 1 14 ? ( 12 ) 1 14 ? ( 34 ) 1 14 ? 12 − 15 32 − 0.46875 From Figure 4(b) we see that the area A of S is less than R 4, so A , 0.46875 Instead of using the rectangles in Figure 4(b) we could use the smaller rectangles in Figure 5 whose heights are the values of f at the left endpoints of the subintervals. (The leftmost rectangle has collapsed because its height is 0.) The sum of the areas of these approximating rectangles is

y (1, 1)

y=≈

2

2

7 L 4 − 14 ? 0 2 1 14 ? ( 14 ) 1 14 ? ( 12 ) 1 14 ? ( 34 ) − 32 − 0.21875 2

We see that the area of S is larger than L 4, so we have lower and upper estimates for A: 0

1 4

1 2

3 4

1

x

0.21875 , A , 0.46875 We can repeat this procedure with a larger number of strips. Figure 6 shows what happens when we divide the region S into eight strips of equal width.

FIGURE 5

y

y (1, 1)

(1, 1)

y=≈

0

FIGURE 6 Approximating S with eight rectangles

1 8

1

(a) Using left endpoints

x

0

1 8

1

x

(b) Using right endpoints

By computing the sum of the areas of the smaller rectangles sL 8 d and the sum of the areas of the larger rectangles sR 8 d, we obtain better lower and upper estimates for A: 0.2734375 , A , 0.3984375 n

Ln

Rn

10 20 30 50 100 1000

0.2850000 0.3087500 0.3168519 0.3234000 0.3283500 0.3328335

0.3850000 0.3587500 0.3501852 0.3434000 0.3383500 0.3338335

So one possible answer to the question is to say that the true area of S lies somewhere between 0.2734375 and 0.3984375. We could obtain better estimates by increasing the number of strips. The table at the left shows the results of similar calculations (with a computer) using n rectangles whose heights are found with left endpoints sL n d or right endpoints sR n d. In particular, we see by using 50 strips that the area lies between 0.3234 and 0.3434. With 1000 strips we narrow it down even more: A lies between 0.3328335 and 0.3338335. A good estimate is obtained by averaging these numbers: A < 0.3333335. ■

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

368

Chapter 5

Integrals

From the values in the table in Example 1, it looks as if R n is approaching increases. We confirm this in the next example.

1 3

as n

example 2 For the region S in Example 1, show that the sum of the areas of the upper approximating rectangles approaches 13, that is, lim R n − 13

nl`

SOLUtION R n is the sum of the areas of the n rectangles in Figure 7. Each rectangle has width 1yn and the heights are the values of the function f sxd − x 2 at the points 1yn, 2yn, 3yn, . . . , nyn; that is, the heights are s1ynd2, s2ynd2, s3ynd2, . . . , snynd2. Thus

y (1, 1)

y =≈

Rn −

0

1

x

SD SD SD 1 n

2

1

1 n

2

2 n

1

3 n

1 n

2

1 ∙∙∙ 1

1 1 2 ∙ s1 1 2 2 1 3 2 1 ∙ ∙ ∙ 1 n 2 d n n2

1 2 s1 1 2 2 1 3 2 1 ∙ ∙ ∙ 1 n 2 d n3

1 n

FIGURE 7

1 n

1 n

SD n n

2

Here we need the formula for the sum of the squares of the first n positive integers:

12 1 2 2 1 3 2 1 ∙ ∙ ∙ 1 n 2 −

1

nsn 1 1ds2n 1 1d 6

Perhaps you have seen this formula before. It is proved in Example 5 in Appendix E. Putting Formula 1 into our expression for R n, we get

Rn − Here we are computing the limit of the sequence hR n j. Sequences and their limits were discussed in A Preview of Calculus and will be studied in detail in Section 11.1. The idea is very similar to a limit at infinity (Section 2.6) except that in writing lim n l ` we restrict n to be a positive integer. In particular, we know that 1 lim − 0 nl ` n When we write lim n l ` Rn − 13 we mean that we can make Rn as close to 13 as we like by taking n sufficiently large.

1 nsn 1 1ds2n 1 1d sn 1 1ds2n 1 1d − 3 ∙ n 6 6n 2

Thus we have lim R n − lim

nl `

nl `

sn 1 1ds2n 1 1d 6n 2

− lim

1 6

− lim

1 6

nl`

nl`

S DS D S DS D n11 n

11

1 n

2n 1 1 n

21

1 n

1 1 ?1?2− 6 3

It can be shown that the lower approximating sums also approach 13, that is, lim L n − 13

nl`

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

369

SeCtION 5.1 Areas and Distances

From Figures 8 and 9 it appears that, as n increases, both L n and R n become better and better approximations to the area of S. Therefore we define the area A to be the limit of the sums of the areas of the approximating rectangles, that is, TEC In Visual 5.1 you can create pictures like those in Figures 8 and 9 for other values of n.

A − lim R n − lim L n − 13 nl`

nl`

y

y

n=10 R¡¸=0.385

0

y

n=50 R∞¸=0.3434

n=30 R£¸Å0.3502

1

x

0

1

x

0

1

x

1

x

FIGURE 8 Right endpoints produce upper sums because f sxd − x 2 is increasing. y

y

n=10 L¡¸=0.285

0

y

n=50 L∞¸=0.3234

n=30 L£¸Å0.3169

1

x

0

1

x

0

FIGURE 9 Left endpoints produce lower sums because f sxd − x 2 is increasing.

Let’s apply the idea of Examples 1 and 2 to the more general region S of Figure 1. We start by subdividing S into n strips S1, S2 , . . . , Sn of equal width as in Figure 10. y

y=ƒ

FIGURE 10

0

a

S™

x2

Si

. . . xi-1

Sn

xi

. . . xn-1

b

x

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370

Chapter 5

Integrals

The width of the interval fa, bg is b 2 a, so the width of each of the n strips is Dx −

b2a n

These strips divide the interval fa, bg into n subintervals fx 0 , x 1 g, fx 1, x 2 g, fx 2 , x 3 g, . . . , fx n21, x n g where x 0 − a and x n − b. The right endpoints of the subintervals are x 1 − a 1 Dx, x 2 − a 1 2 Dx, x 3 − a 1 3 Dx, ∙ ∙ ∙ Let’s approximate the ith strip Si by a rectangle with width Dx and height f sx i d, which is the value of f at the right endpoint (see Figure 11). Then the area of the ith rectangle is f sx i d Dx. What we think of intuitively as the area of S is approximated by the sum of the areas of these rectangles, which is R n − f sx 1 d Dx 1 f sx 2 d Dx 1 ∙ ∙ ∙ 1 f sx n d Dx y

Îx

f(xi)

0

a

x2

b

xi

xi-1

x

FIGURE 11

Figure 12 shows this approximation for n − 2, 4, 8, and 12. Notice that this approximation appears to become better and better as the number of strips increases, that is, as n l `. Therefore we define the area A of the region S in the following way. y

y

0

a

(a) n=2

b x

0

y

a

x2

(b) n=4

b

x

0

y

b

a

(c) n=8

x

0

b

a

x

(d) n=12

FIGURE 12

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

371

SeCtION 5.1 Areas and Distances

2 Definition The area A of the region S that lies under the graph of the continuous function f is the limit of the sum of the areas of approximating rectangles: A − lim R n − lim f f sx 1 d Dx 1 f sx 2 d Dx 1 ∙ ∙ ∙ 1 f sx n d Dxg nl`

nl`

It can be proved that the limit in Definition 2 always exists, since we are assuming that f is continuous. It can also be shown that we get the same value if we use left endpoints: A − lim L n − lim f f sx 0 d Dx 1 f sx 1 d Dx 1 ∙ ∙ ∙ 1 f sx n21 d Dxg

3

nl`

nl`

In fact, instead of using left endpoints or right endpoints, we could take the height of the ith rectangle to be the value of f at any number x*i in the ith subinterval fx i21, x i g. We call the numbers x1*, x2*, . . . , x n* the sample points. Figure 13 shows approximating rectangles when the sample points are not chosen to be endpoints. So a more general expression for the area of S is A − lim f f sx1* d Dx 1 f sx2* d Dx 1 ∙ ∙ ∙ 1 f sx*n d Dxg

4

nl`

y

Îx

f(x *) i

0

FIGURE 13

a

x¡*

x2 x™*

‹ x£*

xi-1

xi

b

xn-1

x *i

x

x n*

NOTE It can be shown that an equivalent definition of area is the following: A is the unique number that is smaller than all the upper sums and bigger than all the lower sums. We saw in Examples 1 and 2, for instance, that the area s A − 13 d is trapped between all the left approximating sums L n and all the right approximating sums Rn. The function in those examples, f sxd − x 2, happens to be increasing on f0, 1g and so the lower sums arise from left endpoints and the upper sums from right endpoints. (See Figures 8 and 9.) In general, we form lower (and upper) sums by choosing the sample points x*i so that f sx*i d is the minimum (and maximum) value of f on the ith subinterval. (See Figure 14 and Exercises 7–8.) y

FIGURE 14 Lower sums (short rectangles) and upper sums (tall rectangles)

0

a

b

x

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

372

Chapter 5

Integrals

We often use sigma notation to write sums with many terms more compactly. For instance,

This tells us to end with i=n. This tells us to add.

n

n

µ

o f sx i d Dx − f sx 1 d Dx 1 f sx 2 d Dx 1 ∙ ∙ ∙ 1 f sx n d Dx i−1

f(xi) Îx

i=m

So the expressions for area in Equations 2, 3, and 4 can be written as follows: n

If you need practice with sigma notation, look at the examples and try some of the exercises in Appendix E.

o f sx i d Dx n l ` i−1

A − lim

n

o f sx i21 d Dx n l ` i−1

A − lim

n

A − lim

o f sxi*d Dx

n l ` i−1

We can also rewrite Formula 1 in the following way: n

o i2 − i−1

nsn 1 1ds2n 1 1d 6

example 3 Let A be the area of the region that lies under the graph of f sxd − e2x between x − 0 and x − 2. (a) Using right endpoints, find an expression for A as a limit. Do not evaluate the limit. (b) Estimate the area by taking the sample points to be midpoints and using four subintervals and then ten subintervals. SOLUtION

(a) Since a − 0 and b − 2, the width of a subinterval is Dx −

220 2 − n n

So x 1 − 2yn, x 2 − 4yn, x 3 − 6yn, x i − 2iyn, and x n − 2nyn. The sum of the areas of the approximating rectangles is Rn − f sx 1 d Dx 1 f sx 2 d Dx 1 ∙ ∙ ∙ 1 f sx n d Dx − e2x 1 Dx 1 e2x 2 Dx 1 ∙ ∙ ∙ 1 e2x n Dx − e22yn

SD 2 n

1 e24yn

SD 2 n

1 ∙ ∙ ∙ 1 e22nyn

SD 2 n

According to Definition 2, the area is A − lim Rn − lim nl`

nl`

2 22yn se 1 e24yn 1 e26yn 1 ∙ ∙ ∙ 1 e22nyn d n

Using sigma notation we could write A − lim

nl`

2 n

n

o e22iyn

i−1

It is difficult to evaluate this limit directly by hand, but with the aid of a computer algebra system it isn’t hard (see Exercise 30). In Section 5.3 we will be able to find A more easily using a different method. Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

SeCtION 5.1 Areas and Distances

(b) With n − 4 the subintervals of equal width Dx − 0.5 are f0, 0.5g , f0.5, 1g , f1, 1.5g, and f1.5, 2g . The midpoints of these subintervals are x1* − 0.25, x2* − 0.75, x3* − 1.25, and x4* − 1.75, and the sum of the areas of the four approximating rectangles (see Figure 15) is

y 1

373

y=e–®

4

M4 − 0

1

2

o f sx*i d Dx i−1

x

− f s0.25d Dx 1 f s0.75d Dx 1 f s1.25d Dx 1 f s1.75d Dx FIGURE 15

− e20.25s0.5d 1 e20.75s0.5d 1 e21.25s0.5d 1 e21.75s0.5d − 12 se20.25 1 e20.75 1 e21.25 1 e21.75 d < 0.8557 So an estimate for the area is A < 0.8557

y 1

With n − 10 the subintervals are f0, 0.2g, f0.2, 0.4g, . . . , f1.8, 2g and the midpoints are * − 1.9. Thus x1* − 0.1, x2* − 0.3, x3* − 0.5, . . . , x10

y=e–®

A < M10 − f s0.1d Dx 1 f s0.3d Dx 1 f s0.5d Dx 1 ∙ ∙ ∙ 1 f s1.9d Dx 0

FIGURE 16

1

2

x

− 0.2se20.1 1 e20.3 1 e20.5 1 ∙ ∙ ∙ 1 e21.9 d < 0.8632 From Figure 16 it appears that this estimate is better than the estimate with n − 4.

the Distance problem Now let’s consider the distance problem: Find the distance traveled by an object during a certain time period if the velocity of the object is known at all times. (In a sense this is the inverse problem of the velocity problem that we discussed in Section 2.1.) If the velocity remains constant, then the distance problem is easy to solve by means of the formula distance − velocity 3 time But if the velocity varies, it’s not so easy to find the distance traveled. We investigate the problem in the following example.

example 4 Suppose the odometer on our car is broken and we want to estimate the distance driven over a 30-second time interval. We take speedometer readings every five seconds and record them in the following table: Time (s)

0

5

10

15

20

25

30

Velocity (kmyh)

27

34

38

46

51

50

45

In order to have the time and the velocity in consistent units, let’s convert the velocity readings to meters per second (1 kmyh − 1000y3600 mys): Time (s) Velocity (mys)

0

5

10

15

20

25

30

7.5

9.4

10.6

12.8

14.2

13.9

12.5

During the first five seconds the velocity doesn’t change very much, so we can estimate the distance traveled during that time by assuming that the velocity is constant. If we Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

374

Chapter 5

Integrals

take the velocity during that time interval to be the initial velocity (7.5 mys), then we obtain the approximate distance traveled during the first five seconds: 7.5 mys 3 5 s − 37.5 m Similarly, during the second time interval the velocity is approximately constant and we take it to be the velocity when t − 5 s. So our estimate for the distance traveled from t − 5 s to t − 10 s is 9.4 mys 3 5 s − 47 m If we add similar estimates for the other time intervals, we obtain an estimate for the total distance traveled: s7.5 3 5d 1 s9.4 3 5d 1 s10.6 3 5d 1 s12.8 3 5d 1 s14.2 3 5d 1 s13.9 3 5d − 342 m We could just as well have used the velocity at the end of each time period instead of the velocity at the beginning as our assumed constant velocity. Then our estimate becomes s9.4 3 5d 1 s10.6 3 5d 1 s12.8 3 5d 1 s14.2 3 5d 1 s13.9 3 5d 1 s12.5 3 5d − 367 m If we had wanted a more accurate estimate, we could have taken velocity readings every two seconds, or even every second. ■ √ 15

10

5

0

10

FIGURE 17

20

30

t

Perhaps the calculations in Example 4 remind you of the sums we used earlier to estimate areas. The similarity is explained when we sketch a graph of the velocity function of the car in Figure 17 and draw rectangles whose heights are the initial velocities for each time interval. The area of the first rectangle is 7.5 3 5 − 37.5, which is also our estimate for the distance traveled in the first five seconds. In fact, the area of each rectangle can be interpreted as a distance because the height represents velocity and the width represents time. The sum of the areas of the rectangles in Figure 17 is L 6 − 342, which is our initial estimate for the total distance traveled. In general, suppose an object moves with velocity v − f std, where a < t < b and f std > 0 (so the object always moves in the positive direction). We take velocity readings at times t0 s− ad, t1, t2 , . . . , tn s− bd so that the velocity is approximately constant on each subinterval. If these times are equally spaced, then the time between consecutive readings is Dt − sb 2 adyn. During the first time interval the velocity is approximately f st0 d and so the distance traveled is approximately f st0 d Dt. Similarly, the distance traveled during the second time interval is about f st1 d Dt and the total distance traveled during the time interval fa, bg is approximately n

f st0 d Dt 1 f st1 d Dt 1 ∙ ∙ ∙ 1 f stn21 d Dt −

o f sti21 d Dt

i−1

If we use the velocity at right endpoints instead of left endpoints, our estimate for the total distance becomes n

f st1 d Dt 1 f st2 d Dt 1 ∙ ∙ ∙ 1 f stn d Dt −

o f sti d Dt i−1

The more frequently we measure the velocity, the more accurate our estimates become,

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SeCtION 5.1 Areas and Distances

375

so it seems plausible that the exact distance d traveled is the limit of such expressions: n

n

lim o f sti d Dt o f sti21 d Dt − nl` nl` i−1 i−1

d − lim

5

We will see in Section 5.4 that this is indeed true. Because Equation 5 has the same form as our expressions for area in Equations 2 and 3, it follows that the distance traveled is equal to the area under the graph of the velocity function. In Chapter 6 we will see that other quantities of interest in the natural and social sciences—such as the work done by a variable force or the cardiac output of the heart—can also be interpreted as the area under a curve. So when we compute areas in this chapter, bear in mind that they can be interpreted in a variety of practical ways.

1. (a) By reading values from the given graph of f, use five rectangles to find a lower estimate and an upper estimate for the area under the given graph of f from x − 0 to x − 10. In each case sketch the rectangles that you use. (b) Find new estimates using ten rectangles in each case.

4. (a) Estimate the area under the graph of f sxd − sin x from x − 0 to x − y2 using four approximating rectangles and right endpoints. Sketch the graph and the rectangles. Is your estimate an underestimate or an overestimate? (b) Repeat part (a) using left endpoints.

y 4

y=ƒ

2 0

8

4

x

2. (a) Use six rectangles to find estimates of each type for the area under the given graph of f from x − 0 to x − 12. (i) L 6 (sample points are left endpoints) (ii) R 6 (sample points are right endpoints) (iii) M6 (sample points are midpoints) (b) Is L 6 an underestimate or overestimate of the true area? (c) Is R 6 an underestimate or overestimate of the true area? (d) Which of the numbers L 6, R 6, or M6 gives the best estimate? Explain.

5. (a) Estimate the area under the graph of f sxd − 1 1 x 2 from x − 21 to x − 2 using three rectangles and right endpoints. Then improve your estimate by using six rectangles. Sketch the curve and the approximating rectangles. (b) Repeat part (a) using left endpoints. (c) Repeat part (a) using midpoints. (d) From your sketches in parts (a)–(c), which appears to be the best estimate? ; 6. (a) Graph the function f sxd − x 2 2 ln x

1 y tsxd dx. b

b

a

a

8. If m < f sxd < M for a < x < b, then msb 2 ad < y f sxd dx < Msb 2 ad b

a

y

b

M

y=ƒ m 0

If f sxd > 0, then ya f sxd dx represents the area under the graph of f, so the geometric interpretation of Property 6 is simply that areas are positive. (It also follows directly from the definition because all the quantities involved are positive.) Property 7 says that a bigger function has a bigger integral. It follows from Properties 6 and 4 because f 2 t > 0. Property 8 is illustrated by Figure 16 for the case where f sxd > 0. If f is continuous, we could take m and M to be the absolute minimum and maximum values of f on the interval fa, bg. In this case Property 8 says that the area under the graph of f is greater than the area of the rectangle with height m and less than the area of the rectangle with height M.

a

FIGURE 16

b

x

prOOf Of prOperty 8 Since m < f sxd < M, Property 7 gives

y

b

a

m dx < y f sxd dx < y M dx b

a

b

a

Using Property 1 to evaluate the integrals on the left and right sides, we obtain msb 2 ad < y f sxd dx < Msb 2 ad b

a

Property 8 is useful when all we want is a rough estimate of the size of an integral without going to the bother of using the Midpoint Rule. Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

388

Chapter 5

Integrals

ExamplE 8 Use Property 8 to estimate y e2x dx. 1

2

0

2

SOLUtION Because f sxd − e2x is a decreasing function on f0, 1g, its absolute maxi-

mum value is M − f s0d − 1 and its absolute minimum value is m − f s1d − e21. Thus, by Property 8,

y

y=1

1

e21s1 2 0d < y e2x dx < 1s1 2 0d 1

y=e–x

2

2

0

e21 < y e2x dx < 1 1

or

2

0

y=1/e

Since e21 < 0.3679, we can write 0.367 < y e2x dx < 1 1

0

1

FIGURE 17

x

2

0

The result of Example 8 is illustrated in Figure 17. The integral is greater than the area of the lower rectangle and less than the area of the square.

1. Evaluate the Riemann sum for f sxd − x 2 1, 26 < x < 4, with five subintervals, taking the sample points to be right endpoints. Explain, with the aid of a diagram, what the Riemann sum represents.

4 6. The graph of t is shown. Estimate y22 tsxd dx with six subintervals using (a) right endpoints, (b) left endpoints, and (c) midpoints.

y

2. If f sxd − cos x

0 < x < 3y4

1

evaluate the Riemann sum with n − 6, taking the sample points to be left endpoints. (Give your answer correct to six decimal places.) What does the Riemann sum represent? Illustrate with a diagram.

x

1

3. If f sxd − x 2 2 4, 0 < x < 3, find the Riemann sum with n − 6, taking the sample points to be midpoints. What does the Riemann sum represent? Illustrate with a diagram. 4. (a) Find the Riemann sum for f sxd − 1yx, 1 < x < 2, with four terms, taking the sample points to be right endpoints. (Give your answer correct to six decimal places.) Explain what the Riemann sum represents with the aid of a sketch. (b) Repeat part (a) with midpoints as the sample points. 5. The graph of a function f is given. Estimate y010 f sxd dx using five subintervals with (a) right endpoints, (b) left endpoints, and (c) midpoints. y

1 0

1

x

7. A table of values of an increasing function f is shown. Use the table to find lower and upper estimates for y025 f sxd dx. x

0

5

10

15

20

25

f sxd

242

237

225

26

15

36

8. The table gives the values of a function obtained from an 9 experiment. Use them to estimate y3 f sxd dx using three equal subintervals with (a) right endpoints, (b) left endpoints, and (c) midpoints. If the function is known to be an increasing function, can you say whether your estimates are less than or greater than the exact value of the integral? x f sxd

3

4

5

23.4 22.1 20.6

6

7

8

9

0.3

0.9

1.4

1.8

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389

SeCtION 5.2 The Definite Integral

26. (a) Find an approximation to the integral y0 sx 2 2 3xd dx using a Riemann sum with right endpoints and n − 8. (b) Draw a diagram like Figure 3 to illustrate the approximation in part (a). (c) Use Theorem 4 to evaluate y04 sx 2 2 3xd dx. (d) Interpret the integral in part (c) as a difference of areas and illustrate with a diagram like Figure 4. 4

9–12 Use the Midpoint Rule with the given value of n to approximate the integral. Round the answer to four decimal places. 9.

y

y2

y

5

0

11.

1

cos4x dx, n − 4

10.

y

1

y

0

x 2e2x dx, n − 4

12.

0

sx 3 1 1 dx, n − 5 x sin2 x dx, n − 4

b2 2 a2 . 2

27. Prove that y x dx − b

CAS

13. If you have a CAS that evaluates midpoint approximations and graphs the corresponding rectangles (use RiemannSum or middlesum and middlebox commands in Maple), check the answer to Exercise 11 and illustrate with a graph. Then repeat with n − 10 and n − 20. 14. With a programmable calculator or computer (see the instructions for Exercise 5.1.9), compute the left and right Riemann sums for the function f sxd − xysx 1 1d on the interval f0, 2g with n − 100. Explain why these estimates show that 0.8946 , y

2

0

x dx , 0.9081 x11

15. Use a calculator or computer to make a table of values of right Riemann sums R n for the integral y0 sin x dx with n − 5, 10, 50, and 100. What value do these numbers appear to be approaching? 16. Use a calculator or computer to make a table of values of left and right Riemann sums L n and R n for the integral 2 y02 e2x dx with n − 5, 10, 50, and 100. Between what two numbers must the value of the integral lie? Can you make a 2 2 similar statement for the integral y21 e2x dx ? Explain.

a

28. Prove that y x 2 dx − b

a

b3 2 a3 . 3

29–30 Express the integral as a limit of Riemann sums. Do not evaluate the limit. 5 1 3 30. y x 2 1 dx 29. y s4 1 x 2 dx 2 1 x

S

CAS

D

31–32 Express the integral as a limit of sums. Then evaluate, using a computer algebra system to find both the sum and the limit. 31.

y

0

32.

sin 5x dx

y

10

2

x 6 dx

33. The graph of f is shown. Evaluate each integral by interpreting it in terms of areas. (a)

y

2

(c)

y

7

0

5

f sxd dx

(b)

y

5

f sxd dx

(d)

y

9

0

0

f sxd dx f sxd dx

y

17–20 Express the limit as a definite integral on the given interval. n e xi 17. lim o Dx, f0, 1g n l ` i−1 1 1 x i

y=ƒ

2

0

2

4

6

8

x

n

18. lim

o xi s1 1 x i3

n l ` i−1

Dx, f2, 5g

n

o x i lns1 1 x i2 d Dx, n l ` i−1

19. lim

n

20. lim

o

n l ` i−1

f2, 6] 34. The graph of t consists of two straight lines and a semicircle. Use it to evaluate each integral.

cos x i Dx, f, 2g xi

(a) 21–25 Use the form of the definition of the integral given in Theorem 4 to evaluate the integral. 21.

y

5

y

0

y

1

21

23.

22

25.

0

s1 1 3xd dx

22.

sx 2 1 x d dx

24.

y

4

y

2

1

0

sx 2 1 2x 2 5 d dx

y

2

0

tsxd dx

(b)

y

6

2

tsxd dx

(c)

y

7

0

tsxd dx

y 4 2

s2x 2 x 3 d dx 0

4

7 x

sx 3 2 3x 2 d dx

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390

Chapter 5

Integrals

35–40 Evaluate the integral by interpreting it in terms of areas. 35.

y

4

2

37.

s1 2 xd dx

36.

9

0

y (1 1 s9 2 x ) dx

38.

y | 12 x | dx

40.

0

2

23

39.

y ( 13 x 2 2) dx y ( x 2 s25 2 x ) dx 5

52. If Fsxd − y2 f std dt, where f is the function whose graph is given, which of the following values is largest? (A) Fs0d (B) Fs1d (C) Fs2d (D) Fs3d (E) Fs4d x

2

y

25

3

24

y | 2x 2 1 | dx 1

y=f(t)

0

0

41. Evaluate y s1 1 x 4 dx. 1

1

2

3

t

4

1

42. Given that y sin4 x dx − 38 , what is y sin4  d?

0

0

43. In Example 5.1.2 we showed that y0 x 2 dx − 13. Use this fact 1

53. Each of the regions A, B, and C bounded by the graph of f and the x-axis has area 3. Find the value of

y

and the properties of integrals to evaluate y0 s5 2 6x 2 d dx. 1

2

f f sxd 1 2x 1 5g dx

24

y

44. Use the properties of integrals and the result of Example 3 to evaluate y13 s2e x 2 1d dx.

B

45. Use the result of Example 3 to evaluate y13 e x12 dx. 46. Use the result of Exercise 27 and the fact that y2 y0 cos x dx − 1 (from Exercise 5.1.31), together with the properties of integrals, to evaluate y0y2 s2 cos x 2 5xd dx. 47. Write as a single integral in the form

y

2

22

yab

f sxd dx 1 y f sxd dx 2 y 5

f sxd dx:

21

22

2

f sxd dx

48. If y28 f sxd dx − 7.3 and y24 f sxd dx − 5.9, find y48 f sxd dx. 49. If y09 f sxd dx − 37 and y09 tsxd dx − 16, find

y

9

0

f2 f sxd 1 3tsxdg dx

_4

0

2

C

x

54. Suppose f has absolute minimum value m and absolute maximum value M. Between what two values must y02 f sxd dx lie? Which property of integrals allows you to make your conclusion? 55–58 Use the properties of integrals to verify the inequality without evaluating the integrals. 55.

y

4

y

1

0

56.

0

sx 2 2 4x 1 4d dx > 0 s1 1 x 2 dx < y s1 1 x dx 1

0

57. 2 < y s1 1 x 2 dx < 2 s2 1

50. Find y0 f sxd dx if 5

f sxd −

H

21

3 x

for x , 3 for x > 3

58.

51. For the function f whose graph is shown, list the following quantities in increasing order, from smallest to largest, and explain your reasoning. (A) y08 f sxd dx

(B) y03 f sxd dx

8 (D) y4 f sxd dx

(E) f 9s1d

y3 s3  < y sin x dx < y6 12 12

59–64 Use Property 8 to estimate the value of the integral. 59.

y

1

y

2

y

2

x 3 dx

60.

61.

0

1 dx 1 1 x2

62.

63.

0

xe2x dx

64.

1 x 1 4 dx

y

3

y

2

y

2

0

0

(C) y38 f sxd dx

y

0

sx 3 2 3x 1 3d dx sx 2 2 sin xd dx

65–66 Use properties of integrals, together with Exercises 27 and 28, to prove the inequality.

2

0

_2

A

5

x

65.

y

3

1

sx 4 1 1 dx >

26 3

66.

y

y2

0

x sin x dx
21, let Asxd − y s1 1 t 2 d dt x

21

Asxd represents the area of a region. Sketch that region. (b) Use the result of Exercise 5.2.28 to find an expression for Asxd. (c) Find A9sxd. What do you notice? (d) If x > 21 and h is a small positive number, then Asx 1 hd 2 Asxd represents the area of a region. Describe and sketch the region. (e) Draw a rectangle that approximates the region in part (d). By comparing the areas of these two regions, show that Asx 1 hd 2 Asxd < 1 1 x2 h (f) Use part (e) to give an intuitive explanation for the result of part (c). 2 ; 3. (a) Draw the graph of the function f sxd − cossx d in the viewing rectangle f0, 2g by f21.25, 1.25g. (b) If we define a new function t by

tsxd − y cos st 2 d dt x

0

then tsxd is the area under the graph of f from 0 to x [until f sxd becomes negative, at which point tsxd becomes a difference of areas]. Use part (a) to determine the value

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392

Chapter 5

Integrals

of x at which tsxd starts to decrease. [Unlike the integral in Problem 2, it is impossible to evaluate the integral defining t to obtain an explicit expression for tsxd.] (c) Use the integration command on your calculator or computer to estimate ts0.2d, ts0.4d, ts0.6d, . . . , ts1.8d, ts2d. Then use these values to sketch a graph of t. (d) Use your graph of t from part (c) to sketch the graph of t9 using the interpretation of t9sxd as the slope of a tangent line. How does the graph of t9 compare with the graph of f ? 4. Suppose f is a continuous function on the interval fa, bg and we define a new function t by the equation x tsxd − y f std dt a

Based on your results in Problems 1–3, conjecture an expression for t9sxd.

The Fundamental Theorem of Calculus is appropriately named because it establishes a  connection between the two branches of calculus: differential calculus and integral calculus. Differential calculus arose from the tangent problem, whereas integral calculus arose from a seemingly unrelated problem, the area problem. Newton’s mentor at Cambridge, Isaac Barrow (1630 –1677), discovered that these two problems are actually closely related. In fact, he realized that differentiation and integration are inverse processes. The Fundamental Theorem of Calculus gives the precise inverse relationship between the derivative and the integral. It was Newton and Leibniz who exploited this relationship and used it to develop calculus into a systematic mathematical method. In particular, they saw that the Fundamental Theorem enabled them to compute areas and integrals very easily without having to compute them as limits of sums as we did in Sections 5.1 and 5.2. The first part of the Fundamental Theorem deals with functions defined by an equation of the form

y

y=f(t)

0

a

x

b

t

FIGURE 1 y 2

tsxd − y f std dt x

a

where f is a continuous function on fa, bg and x varies between a and b. Observe that t depends only on x, which appears as the variable upper limit in the integral. If x is a fixed number, then the integral yax f std dt is a definite number. If we then let x vary, the number yax f std dt also varies and defines a function of x denoted by tsxd. If f happens to be a positive function, then tsxd can be interpreted as the area under the graph of f from a to x, where x can vary from a to b. (Think of t as the “area so far” function; see Figure 1.)

y=f(t) 1 0

1

2

4

t

ExamplE 1 If f is the function whose graph is shown in Figure 2 and tsxd − y0x f std dt, find the values of ts0d, ts1d, ts2d, ts3d, ts4d, and ts5d. Then sketch a rough graph of t. SOLUtION First we notice that ts0d − y0 f std dt − 0. From Figure 3 we see that ts1d is 0

the area of a triangle: FIGURE 2

ts1d − y f std dt − 12 s1 ? 2d − 1 1

0

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393

SeCtION 5.3 The Fundamental Theorem of Calculus

To find ts2d we add to ts1d the area of a rectangle: ts2d − y f std dt − y f std dt 1 y f std dt − 1 1 s1 ? 2d − 3 2

1

0

2

0

1

We estimate that the area under f from 2 to 3 is about 1.3, so ts3d − ts2d 1 y f std dt < 3 1 1.3 − 4.3 3

2

y 2

y 2

y 2

y 2

y 2

1

1

1

1

1

0

t

1

0

1

g(1)=1

2

t

0

g(2)=3

1

2

3

t

0

1

2

4

t

0

1

2

4

t

g(3)Å4.3

g(4)Å3

g(5)Å1.7

FIGURE 3 y

For t . 3, f std is negative and so we start subtracting areas:

4

ts4d − ts3d 1 y f std dt < 4.3 1 s21.3d − 3.0

g

4

3

3

ts5d − ts4d 1 y f std dt < 3 1 s21.3d − 1.7

2

5

4

1 0

1

2

4

3

We use these values to sketch the graph of t in Figure 4. Notice that, because f std is positive for t , 3, we keep adding area for t , 3 and so t is increasing up to x − 3, where it attains a maximum value. For x . 3, t decreases because f std is negative. ■

5 x

FIGURE 4 tsxd − y f std dt x

If we take f std − t and a − 0, then, using Exercise 5.2.27, we have

0

tsxd − y t dt − x

0

Notice that t9sxd − x, that is, t9 − f . In other words, if t is defined as the integral of f by Equation 1, then t turns out to be an antiderivative of f , at least in this case. And if we sketch the derivative of the function t shown in Figure 4 by estimating slopes of tangents, we get a graph like that of f in Figure 2. So we suspect that t9 − f in Example 1 too. To see why this might be generally true we consider any continuous function f with x f sxd > 0. Then tsxd − ya f std dt can be interpreted as the area under the graph of f from a to x, as in Figure 1. In order to compute t9sxd from the definition of a derivative we first observe that, for h . 0, tsx 1 hd 2 tsxd is obtained by subtracting areas, so it is the area under the graph of f from x to x 1 h (the blue area in Figure 5). For small h you can see from the figure that this area is approximately equal to the area of the rectangle with height f sxd and width h:

y

h ƒ 0

a

FIGURE 5

x

x2 2

x+h

b

tsx 1 hd 2 tsxd < hf sxd

t

so

tsx 1 hd 2 tsxd < f sxd h

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394

Chapter 5

Integrals

Intuitively, we therefore expect that t9sxd − lim

hl0

tsx 1 hd 2 tsxd − f sxd h

The fact that this is true, even when f is not necessarily positive, is the first part of the Fundamental Theorem of Calculus.

We abbreviate the name of this theorem as FTC1. In words, it says that the derivative of a definite integral with respect to its upper limit is the integrand evaluated at the upper limit.

the Fundamental theorem of calculus, Part 1 If f is continuous on fa, bg, then the function t defined by tsxd − y f std dt x

a 0 when f > 0. The Fundamental Theorem of Calculus applies to continuous functions. It can’t be applied here because f sxd − 1yx 2 is not continuous on f21, 3g. In fact, f has an infinite discontinuity at x − 0, so

y

3

21

1 dx x2

does not exist.

differentiation and Integration as Inverse processes We end this section by bringing together the two parts of the Fundamental Theorem. the Fundamental theorem of calculus Suppose f is continuous on fa, bg. 1. If tsxd − ya f std dt, then t9sxd − f sxd. x

2. ya f sxd dx − Fsbd 2 Fsad, where F is any antiderivative of f , that is, F9− f. b

We noted that Part 1 can be rewritten as d dx

y

x

a

f std dt − f sxd

which says that if f is integrated and then the result is differentiated, we arrive back at the original function f. Since F9sxd − f sxd, Part 2 can be rewritten as

y

b

a

F9sxd dx − Fsbd 2 Fsad

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SeCtION 5.3 The Fundamental Theorem of Calculus

This version says that if we take a function F, first differentiate it, and then integrate the result, we arrive back at the original function F, but in the form Fsbd 2 Fsad. Taken together, the two parts of the Fundamental Theorem of Calculus say that differentiation and integration are inverse processes. Each undoes what the other does. The Fundamental Theorem of Calculus is unquestionably the most important theorem in calculus and, indeed, it ranks as one of the great accomplishments of the human mind. Before it was discovered, from the time of Eudoxus and Archimedes to the time of Galileo and Fermat, problems of finding areas, volumes, and lengths of curves were so difficult that only a genius could meet the challenge. But now, armed with the systematic method that Newton and Leibniz fashioned out of the Fundamental Theorem, we will see in the chapters to come that these challenging problems are accessible to all of us.

1. Explain exactly what is meant by the statement that “differentiation and integration are inverse processes.” x

2. Let tsxd − y0 f std dt, where f is the function whose graph is shown. (a) Evaluate tsxd for x − 0, 1, 2, 3, 4, 5, and 6. (b) Estimate ts7d. (c) Where does t have a maximum value? Where does it have a minimum value? (d) Sketch a rough graph of t.

(d) Where does t have a maximum value? (e) Sketch a rough graph of t. (f) Use the graph in part (e) to sketch the graph of t9sxd. Compare with the graph of f. y 2