Single Variable Calculus: Early Transcendentals, 9th Edition
By James Stewart, Daniel K. Clegg and Saleem Watson
Contents:
Preface x
A Tribute to James Stewart xxii
About the Authors xxiii
Technology in the Ninth Edition xxiv
To the Student xxv
Diagnostic Tests xxvi
A Preview of Calculus 1
1 Functions and Models 7
1.1 Four Ways to Represent a Function 8
1.2 Mathematical Models: A Catalog of Essential Functions 21
1.3 New Functions from Old Functions 36
1.4 Exponential Functions 45
1.5 Inverse Functions and Logarithms 54
Review 67
Principles of Problem Solving 70
2 Limits and Derivatives 77
2.1 The Tangent and Velocity Problems 78
2.2 The Limit of a Function 83
2.3 Calculating Limits Using the Limit Laws 94
2.4 The Precise Definition of a Limit 105
2.5 Continuity 115
2.6 Limits at Infinity; Horizontal Asymptotes 127
2.7 Derivatives and Rates of Change 140
writing project • Early Methods for Finding Tangents 152
2.8 The Derivative as a Function 153
Review 166
Problems Plus 171
3 Differentiation Rules 173
3.1 Derivatives of Polynomials and Exponential Functions 174
applied project • Building a Better Roller Coaster 184
3.2 The Product and Quotient Rules 185
3.3 Derivatives of Trigonometric Functions 191
3.4 The Chain Rule 199
applied project • Where Should a Pilot Start Descent? 209
3.5 Implicit Differentiation 209
discovery project • Families of Implicit Curves 217
3.6 Derivatives of Logarithmic and Inverse Trigonometric Functions 217
3.7 Rates of Change in the Natural and Social Sciences 225
3.8 Exponential Growth and Decay 239
applied project • Controlling Red Blood Cell Loss During Surgery 247
3.9 Related Rates 247
3.10 Linear Approximations and Differentials 254
discovery project • Polynomial Approximations 260
3.11 Hyperbolic Functions 261
Review 269
Problems Plus 274
4 Applications of Differentiation 279
4.1 Maximum and Minimum Values 280
applied project • The Calculus of Rainbows 289
4.2 The Mean Value Theorem 290
4.3 What Derivatives Tell Us about the Shape of a Graph 296
4.4 Indeterminate Forms and l’Hospital’s Rule 309
writing project • The Origins of l’Hospital’s Rule 319
4.5 Summary of Curve Sketching 320
4.6 Graphing with Calculus and Technology 329
4.7 Optimization Problems 336
applied project • The Shape of a Can 349
applied project • Planes and Birds: Minimizing Energy 350
4.8 Newton’s Method 351
4.9 Antiderivatives 356
Review 364
Problems Plus 369
5 Integrals 371
5.1 The Area and Distance Problems 372
5.2 The Definite Integral 384
discovery project • Area Functions 398
5.3 The Fundamental Theorem of Calculus 399
5.4 Indefinite Integrals and the Net Change Theorem 409
writing project • Newton, Leibniz, and the Invention of Calculus 418
5.5 The Substitution Rule 419
Review 428
Problems Plus 432
6 Applications of Integration 435
6.1 Areas Between Curves 436
applied project • The Gini Index 445
6.2 Volumes 446
6.3 Volumes by Cylindrical Shells 460
6.4 Work 467
6.5 Average Value of a Function 473
applied project • Calculus and Baseball 476
applied project • Where to Sit at the Movies 478
Review 478
Problems Plus 481
7 Techniques of Integration 485
7.1 Integration by Parts 486
7.2 Trigonometric Integrals 493
7.3 Trigonometric Substitution 500
7.4 Integration of Rational Functions by Partial Fractions 507
7.5 Strategy for Integration 517
7.6 Integration Using Tables and Technology 523
discovery project • Patterns in Integrals 528
7.7 Approximate Integration 529
7.8 Improper Integrals 542
Review 552
Problems Plus 556
8 Further Applications of Integration 559
8.1 Arc Length 560
discovery project • Arc Length Contest 567
8.2 Area of a Surface of Revolution 567
discovery project • Rotating on a Slant 575
8.3 Applications to Physics and Engineering 576
discovery project • Complementary Coffee Cups 587
8.4 Applications to Economics and Biology 587
8.5 Probability 592
Review 600
Problems Plus 602
9 Differential Equations 605
9.1 Modeling with Differential Equations 606
9.2 Direction Fields and Euler’s Method 612
9.3 Separable Equations 621
applied project • How Fast Does a Tank Drain? 630
9.4 Models for Population Growth 631
9.5 Linear Equations 641
applied project • Which Is Faster, Going Up or Coming Down? 648
9.6 Predator-Prey Systems 649
Review 656
Problems Plus 659
10 Parametric Equations and Polar Coordinates 661
10.1 Curves Defined by Parametric Equations 662
discovery project • Running Circles Around Circles 672
10.2 Calculus with Parametric Curves 673
discovery project • Bézier Curves 684
10.3 Polar Coordinates 684
discovery project • Families of Polar Curves 694
10.4 Calculus in Polar Coordinates 694
10.5 Conic Sections 702
10.6 Conic Sections in Polar Coordinates 711
Review 719
Problems Plus 722
11 Sequences, Series, and Power Series 723
11.1 Sequences 724
discovery project • Logistic Sequences 738
11.2 Series 738
11.3 The Integral Test and Estimates of Sums 751
11.4 The Comparison Tests 760
11.5 Alternating Series and Absolute Convergence 765
11.6 The Ratio and Root Tests 774
11.7 Strategy for Testing Series 779
11.8 Power Series 781
11.9 Representations of Functions as Power Series 787
11.10 Taylor and Maclaurin Series 795
discovery project • An Elusive Limit 810
writing project • How Newton Discovered the Binomial Series 811
11.11 Applications of Taylor Polynomials 811
applied project • Radiation from the Stars 820
Review 821
Problems Plus 825
12 Vectors and the Geometry of Space 829
12.1 Three-Dimensional Coordinate Systems 830
12.2 Vectors 836
discovery project • The Shape of a Hanging Chain 846
12.3 The Dot Product 847
12.4 The Cross Product 855
discovery project • The Geometry of a Tetrahedron 864
12.5 Equations of Lines and Planes 864
discovery project • Putting 3D in Perspective 874
12.6 Cylinders and Quadric Surfaces 875
Review 883
Problems Plus 887
13 Vector Functions 889
13.1 Vector Functions and Space Curves 890
13.2 Derivatives and Integrals of Vector Functions 898
13.3 Arc Length and Curvature 904
13.4 Motion in Space: Velocity and Acceleration 916
applied project • Kepler’s Laws 925
Review 927
Problems Plus 930
14 Partial Derivatives 933
14.1 Functions of Several Variables 934
14.2 Limits and Continuity 951
14.3 Partial Derivatives 961
discovery project • Deriving the Cobb-Douglas Production Function 973
14.4 Tangent Planes and Linear Approximations 974
applied project • The Speedo LZR Racer 984
14.5 The Chain Rule 985
14.6 Directional Derivatives and the Gradient Vector 994
14.7 Maximum and Minimum Values 1008
discovery project • Quadratic Approximations and Critical Points 1019
14.8 Lagrange Multipliers 1020
applied project • Rocket Science 1028
applied project • Hydro-Turbine Optimization 1030
Review 1031
Problems Plus 1035
15 Multiple Integrals 1037
15.1 Double Integrals over Rectangles 1038
15.2 Double Integrals over General Regions 1051
15.3 Double Integrals in Polar Coordinates 1062
15.4 Applications of Double Integrals 1069
15.5 Surface Area 1079
15.6 Triple Integrals 1082
discovery project • Volumes of Hyperspheres 1095
15.7 Triple Integrals in Cylindrical Coordinates 1095
discovery project • The Intersection of Three Cylinders 1101
15.8 Triple Integrals in Spherical Coordinates 1102
applied project • Roller Derby 1108
15.9 Change of Variables in Multiple Integrals 1109
Review 1117
Problems Plus 1121
16 Vector Calculus 1123
16.1 Vector Fields 1124
16.2 Line Integrals 1131
16.3 The Fundamental Theorem for Line Integrals 1144
16.4 Green’s Theorem 1154
16.5 Curl and Divergence 1161
16.6 Parametric Surfaces and Their Areas 1170
16.7 Surface Integrals 1182
16.8 Stokes’ Theorem 1195
16.9 The Divergence Theorem 1201
16.10 Summary 1208
Review 1209
Problems Plus 1213
Appendixes A1
A Numbers, Inequalities, and Absolute Values A2
B Coordinate Geometry and Lines A10
C Graphs of Second-Degree Equations A16
D Trigonometry A24
E Sigma Notation A36
F Proofs of Theorems A41
G The Logarithm Defined as an Integral A53
H Answers to Odd-Numbered Exercises A61
Index A143
About the Authors xxiii
Technology in the Ninth Edition xxiv
To the Student xxv
Diagnostic Tests xxvi
A Preview of Calculus 1
1 Functions and Models 7
1.1 Four Ways to Represent a Function 8
1.2 Mathematical Models: A Catalog of Essential Functions 21
1.3 New Functions from Old Functions 36
1.4 Exponential Functions 45
1.5 Inverse Functions and Logarithms 54
Review 67
Principles of Problem Solving 70
2 Limits and Derivatives 77
2.1 The Tangent and Velocity Problems 78
2.2 The Limit of a Function 83
2.3 Calculating Limits Using the Limit Laws 94
2.4 The Precise Definition of a Limit 105
2.5 Continuity 115
2.6 Limits at Infinity; Horizontal Asymptotes 127
2.7 Derivatives and Rates of Change 140
writing project • Early Methods for Finding Tangents 152
2.8 The Derivative as a Function 153
Review 166
Problems Plus 171
3 Differentiation Rules 173
3.1 Derivatives of Polynomials and Exponential Functions 174
applied project • Building a Better Roller Coaster 184
3.2 The Product and Quotient Rules 185
3.3 Derivatives of Trigonometric Functions 191
3.4 The Chain Rule 199
applied project • Where Should a Pilot Start Descent? 209
3.5 Implicit Differentiation 209
discovery project • Families of Implicit Curves 217
3.6 Derivatives of Logarithmic and Inverse Trigonometric Functions 217
3.7 Rates of Change in the Natural and Social Sciences 225
3.8 Exponential Growth and Decay 239
applied project • Controlling Red Blood Cell Loss During Surgery 247
3.9 Related Rates 247
3.10 Linear Approximations and Differentials 254
discovery project • Polynomial Approximations 260
3.11 Hyperbolic Functions 261
Review 269
Problems Plus 274
4 Applications of Differentiation 279
4.1 Maximum and Minimum Values 280
applied project • The Calculus of Rainbows 289
4.2 The Mean Value Theorem 290
4.3 What Derivatives Tell Us about the Shape of a Graph 296
4.4 Indeterminate Forms and l’Hospital’s Rule 309
writing project • The Origins of l’Hospital’s Rule 319
4.5 Summary of Curve Sketching 320
4.6 Graphing with Calculus and Technology 329
4.7 Optimization Problems 336
applied project • The Shape of a Can 349
applied project • Planes and Birds: Minimizing Energy 350
4.8 Newton’s Method 351
4.9 Antiderivatives 356
Review 364
Problems Plus 369
5 Integrals 371
5.1 The Area and Distance Problems 372
5.2 The Definite Integral 384
discovery project • Area Functions 398
5.3 The Fundamental Theorem of Calculus 399
5.4 Indefinite Integrals and the Net Change Theorem 409
writing project • Newton, Leibniz, and the Invention of Calculus 418
5.5 The Substitution Rule 419
Review 428
Problems Plus 432
6 Applications of Integration 435
6.1 Areas Between Curves 436
applied project • The Gini Index 445
6.2 Volumes 446
6.3 Volumes by Cylindrical Shells 460
6.4 Work 467
6.5 Average Value of a Function 473
applied project • Calculus and Baseball 476
applied project • Where to Sit at the Movies 478
Review 478
Problems Plus 481
7 Techniques of Integration 485
7.1 Integration by Parts 486
7.2 Trigonometric Integrals 493
7.3 Trigonometric Substitution 500
7.4 Integration of Rational Functions by Partial Fractions 507
7.5 Strategy for Integration 517
7.6 Integration Using Tables and Technology 523
discovery project • Patterns in Integrals 528
7.7 Approximate Integration 529
7.8 Improper Integrals 542
Review 552
Problems Plus 556
8 Further Applications of Integration 559
8.1 Arc Length 560
discovery project • Arc Length Contest 567
8.2 Area of a Surface of Revolution 567
discovery project • Rotating on a Slant 575
8.3 Applications to Physics and Engineering 576
discovery project • Complementary Coffee Cups 587
8.4 Applications to Economics and Biology 587
8.5 Probability 592
Review 600
Problems Plus 602
9 Differential Equations 605
9.1 Modeling with Differential Equations 606
9.2 Direction Fields and Euler’s Method 612
9.3 Separable Equations 621
applied project • How Fast Does a Tank Drain? 630
9.4 Models for Population Growth 631
9.5 Linear Equations 641
applied project • Which Is Faster, Going Up or Coming Down? 648
9.6 Predator-Prey Systems 649
Review 656
Problems Plus 659
10 Parametric Equations and Polar Coordinates 661
10.1 Curves Defined by Parametric Equations 662
discovery project • Running Circles Around Circles 672
10.2 Calculus with Parametric Curves 673
discovery project • Bézier Curves 684
10.3 Polar Coordinates 684
discovery project • Families of Polar Curves 694
10.4 Calculus in Polar Coordinates 694
10.5 Conic Sections 702
10.6 Conic Sections in Polar Coordinates 711
Review 719
Problems Plus 722
11 Sequences, Series, and Power Series 723
11.1 Sequences 724
discovery project • Logistic Sequences 738
11.2 Series 738
11.3 The Integral Test and Estimates of Sums 751
11.4 The Comparison Tests 760
11.5 Alternating Series and Absolute Convergence 765
11.6 The Ratio and Root Tests 774
11.7 Strategy for Testing Series 779
11.8 Power Series 781
11.9 Representations of Functions as Power Series 787
11.10 Taylor and Maclaurin Series 795
discovery project • An Elusive Limit 810
writing project • How Newton Discovered the Binomial Series 811
11.11 Applications of Taylor Polynomials 811
applied project • Radiation from the Stars 820
Review 821
Problems Plus 825
12 Vectors and the Geometry of Space 829
12.1 Three-Dimensional Coordinate Systems 830
12.2 Vectors 836
discovery project • The Shape of a Hanging Chain 846
12.3 The Dot Product 847
12.4 The Cross Product 855
discovery project • The Geometry of a Tetrahedron 864
12.5 Equations of Lines and Planes 864
discovery project • Putting 3D in Perspective 874
12.6 Cylinders and Quadric Surfaces 875
Review 883
Problems Plus 887
13 Vector Functions 889
13.1 Vector Functions and Space Curves 890
13.2 Derivatives and Integrals of Vector Functions 898
13.3 Arc Length and Curvature 904
13.4 Motion in Space: Velocity and Acceleration 916
applied project • Kepler’s Laws 925
Review 927
Problems Plus 930
14 Partial Derivatives 933
14.1 Functions of Several Variables 934
14.2 Limits and Continuity 951
14.3 Partial Derivatives 961
discovery project • Deriving the Cobb-Douglas Production Function 973
14.4 Tangent Planes and Linear Approximations 974
applied project • The Speedo LZR Racer 984
14.5 The Chain Rule 985
14.6 Directional Derivatives and the Gradient Vector 994
14.7 Maximum and Minimum Values 1008
discovery project • Quadratic Approximations and Critical Points 1019
14.8 Lagrange Multipliers 1020
applied project • Rocket Science 1028
applied project • Hydro-Turbine Optimization 1030
Review 1031
Problems Plus 1035
15 Multiple Integrals 1037
15.1 Double Integrals over Rectangles 1038
15.2 Double Integrals over General Regions 1051
15.3 Double Integrals in Polar Coordinates 1062
15.4 Applications of Double Integrals 1069
15.5 Surface Area 1079
15.6 Triple Integrals 1082
discovery project • Volumes of Hyperspheres 1095
15.7 Triple Integrals in Cylindrical Coordinates 1095
discovery project • The Intersection of Three Cylinders 1101
15.8 Triple Integrals in Spherical Coordinates 1102
applied project • Roller Derby 1108
15.9 Change of Variables in Multiple Integrals 1109
Review 1117
Problems Plus 1121
16 Vector Calculus 1123
16.1 Vector Fields 1124
16.2 Line Integrals 1131
16.3 The Fundamental Theorem for Line Integrals 1144
16.4 Green’s Theorem 1154
16.5 Curl and Divergence 1161
16.6 Parametric Surfaces and Their Areas 1170
16.7 Surface Integrals 1182
16.8 Stokes’ Theorem 1195
16.9 The Divergence Theorem 1201
16.10 Summary 1208
Review 1209
Problems Plus 1213
Appendixes A1
A Numbers, Inequalities, and Absolute Values A2
B Coordinate Geometry and Lines A10
C Graphs of Second-Degree Equations A16
D Trigonometry A24
E Sigma Notation A36
F Proofs of Theorems A41
G The Logarithm Defined as an Integral A53
H Answers to Odd-Numbered Exercises A61
Index A143