Calculus [Eight edition] 9781285740621, 9781305271760, 1285740629, 1305271769

Table of contents :
Contents......Page 5
Preface......Page 13
To the Student......Page 25
Calculators, Computers, and Other Graphing Devices......Page 26
Diagnostic Tests......Page 28
A Preview of Calculus......Page 33
Ch 1: Functions and Limits......Page 41
1.1: Four Ways to Represent a Function......Page 42
1.2: Mathematical Models: A Catalog of Essential Functions......Page 55
1.3: New Functions from Old Functions......Page 68
1.4: The Tangent and Velocity Problems......Page 77
1.5: The Limit of a Function......Page 82
1.6: Calculating Limits Using the Limit Laws......Page 94
1.7: The Precise Definition of a Limit......Page 104
1.8: Continuity......Page 114
1: Review......Page 126
Principles of Problem Solving......Page 130
Ch 2: Derivatives......Page 137
2.1: Derivatives and Rates of Change......Page 138
2.2: The Derivative as a Function......Page 149
2.3: Differentiation Formulas......Page 162
2.4: Derivatives of Trigonometric Functions......Page 176
2.5: The Chain Rule......Page 184
2.6: Implicit Differentiation......Page 193
2.7: Rates of Change in the Natural and Social Sciences......Page 201
2.8: Related Rates......Page 213
2.9: Linear Approximations and Differentials......Page 220
2: Review......Page 227
Problems Plus......Page 232
Ch 3: Applications of Differentiation......Page 235
3.1: Maximum and Minimum Values......Page 236
3.2: The Mean Value Theorem......Page 247
3.3: How Derivatives Affect the Shape of a Graph......Page 253
3.4: Limits at Infinity; Horizontal Asymptotes......Page 263
3.5: Summary of Curve Sketching......Page 276
3.6: Graphing with Calculus and Calculators......Page 283
3.7: Optimization Problems......Page 290
3.8: Newton's Method......Page 304
3.9: Antiderivatives......Page 310
3: Review......Page 317
Problems Plus......Page 321
Ch 4: Integrals......Page 325
4.1: Areas and Distances......Page 326
4.2: The Definite Integral......Page 338
4.3: The Fundamental Theorem of Calculus......Page 352
4.4: Indefinite Integrals and the Net Change Theorem......Page 362
4.5: The Substitution Rule......Page 372
4: Review......Page 380
Problems Plus......Page 384
Ch 5: Applications of Integration......Page 387
5.1: Areas between Curves......Page 388
5.2: Volumes......Page 398
5.3: Volumes by Cylindrical Shells......Page 409
5.4: Work......Page 415
5.5: Average Value of a Function......Page 421
5: Review......Page 425
Problems Plus......Page 427
Ch 6: Inverse Functions......Page 431
6.1: Inverse Functions......Page 432
6.2: Exponential Functions and Their Derivatives......Page 440
6.3: Logarithmic Functions......Page 453
6.4: Derivatives of Logarithmic Functions......Page 460
6.5: Exponential Growth and Decay......Page 498
6.6: Inverse Trigonometric Functions......Page 506
6.7: Hyperbolic Functions......Page 516
6.8: Indeterminate Forms and l'Hospital's Rule......Page 523
6: Review......Page 535
Problems Plus......Page 540
Ch 7: Techniques of Integration......Page 543
7.1: Integration by Parts......Page 544
7.2: Trigonometric Integrals......Page 551
7.3: Trigonometric Substitution......Page 558
7.4: Integration of Rational Functions by Partial Fractions......Page 565
7.5: Strategy for Integration......Page 575
7.6: Integration Using Tables and Computer Algebra Systems......Page 580
7.7: Approximate Integration......Page 586
7.8: Improper Integrals......Page 599
7: Review......Page 609
Problems Plus......Page 612
Ch 8: Further Applications of Integration......Page 615
8.1: Arc Length......Page 616
8.2: Area of a Surface of Revolution......Page 623
8.3: Applications to Physics and Engineering......Page 630
8.4: Applications to Economics and Biology......Page 641
8.5: Probability......Page 645
8: Review......Page 653
Problems Plus......Page 655
Ch 9: Differential Equations......Page 657
9.1: Modeling with Differential Equations......Page 658
9.2: Direction Fields and Euler's Method......Page 663
9.3: Separable Equations......Page 671
9.4: Models for Population Growth......Page 682
9.5: Linear Equations......Page 692
9.6: Predator-Prey Systems......Page 699
9: Review......Page 706
Problems Plus......Page 709
Ch 10: Parametric Equations and Polar Coordinates......Page 711
10.1: Curves Defined by Parametric Equations......Page 712
10.2: Calculus with Parametric Curves......Page 721
10.3: Polar Coordinates......Page 730
10.4: Areas and Lengths in Polar Coordinates......Page 741
10.5: Conic Sections......Page 746
10.6: Conic Sections in Polar Coordinates......Page 754
10: Review......Page 761
Problems Plus......Page 764
Ch 11: Infinite Sequences and Series......Page 765
11.1: Sequences......Page 766
11.2: Series......Page 779
11.3: The Integral Test and Estimates of Sums......Page 791
11.4: The Comparison Tests......Page 799
11.5: Alternating Series......Page 804
11.6: Absolute Convergence and the Ratio and Root Tests......Page 809
11.7: Strategy for Testing Series......Page 816
11.8: Power Series......Page 818
11.9: Representations of Functions as Power Series......Page 824
11.10: Taylor and Maclaurin Series......Page 831
11.11: Applications of Taylor Polynomials......Page 846
11: Review......Page 856
Problems Plus......Page 859
Ch 12: Vectors and the Geometry of Space......Page 863
12.1: Three-Dimensional Coordinate Systems......Page 864
12.2: Vectors......Page 870
12.3: The Dot Product......Page 879
12.4: The Cross Product......Page 886
12.5: Equations of Lines and Planes......Page 895
12.6: Cylinders and Quadric Surfaces......Page 906
12: Review......Page 913
Problems Plus......Page 916
Ch 13: Vector Functions......Page 919
13.1: Vector Functions and Space Curves......Page 920
13.2: Derivatives and Integrals of Vector Functions......Page 927
13.3: Arc Length and Curvature......Page 933
13.4: Motion in Space: Velocity and Acceleration......Page 942
13: Review......Page 953
Problems Plus......Page 956
Ch 14: Partial Derivatives......Page 959
14.1: Functions of Several Variables......Page 960
14.2: Limits and Continuity......Page 975
14.3: Partial Derivatives......Page 983
14.4: Tangent Planes and Linear Approximations......Page 999
14.5: The Chain Rule......Page 1009
14.6: Directional Derivatives and the Gradient Vector......Page 1018
14.7: Maximum and Minimum Values......Page 1031
14.8: Lagrange Multipliers......Page 1043
14: Review......Page 1053
Problems Plus......Page 1057
Ch 15: Multiple Integrals......Page 1059
15.1: Double Integrals over Rectangles......Page 1060
15.2: Double Integrals over General Regions......Page 1073
15.3: Double Integrals in Polar Coordinates......Page 1082
15.4: Applications of Double Integrals......Page 1088
15.5: Surface Area......Page 1098
15.6: Triple Integrals......Page 1101
15.7: Triple Integrals in Cylindrical Coordinates......Page 1112
15.8: Triple Integrals in Spherical Coordinates......Page 1117
15.9: Change of Variables in Multiple Integrals......Page 1124
15: Review......Page 1133
Problems Plus......Page 1137
Ch 16: Vector Calculus......Page 1139
16.1: Vector Fields......Page 1140
16.2: Line Integrals......Page 1147
16.3: The Fundamental Theorem for Line Integrals......Page 1159
16.4: Green's Theorem......Page 1168
16.5: Curl and Divergence......Page 1175
16.6: Parametric Surfaces and Their Areas......Page 1183
16.7: Surface Integrals......Page 1194
16.8: Stokes' Theorem......Page 1206
16.9: The Divergence Theorem......Page 1213
16.10: Summary......Page 1219
16: Review......Page 1220
Problems Plus......Page 1223
Ch 17: Second-Order Differential Equations......Page 1225
17.1: Second-Order Linear Equations......Page 1226
17.2: Nonhomogeneous Linear Equations......Page 1232
17.3: Applications of Second-Order Differential Equations......Page 1240
17.4: Series Solutions......Page 1248
17: Review......Page 1253
Appendixes......Page 1255
A: Numbers, Inequalities, and Absolute Values......Page 1256
B: Coordinate Geometry and Lines......Page 1264
C: Graphs of Second-Degree Equations......Page 1270
D: Trigonometry......Page 1278
E: Sigma Notation......Page 1288
F: Proofs of Theorems......Page 1293
G: Complex Numbers......Page 1302
H: Answers to Odd-Numbered Exercises......Page 1311
Index......Page 1385
Reference......Page 1407
Concept Check Answers......Page 1417

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