Calculus, Ninth Edition
By James Stewart, Daniel 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 Limits 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 The Tangent and Velocity Problems 45
1.5 The Limit of a Function 51
1.6 Calculating Limits Using the Limit Laws 62
1.7 The Precise Definition of a Limit 73
1.8 Continuity 83
Review 95
Principles of Problem Solving 99
2 Derivatives 107
2.1 Derivatives and Rates of Change 108
writing project • Early Methods for Finding Tangents 120
2.2 The Derivative as a Function 120
2.3 Differentiation Formulas 133
applied project • Building a Better Roller Coaster 147
2.4 Derivatives of Trigonometric Functions 148
2.5 The Chain Rule 156
applied project • Where Should a Pilot Start Descent? 164
2.6 Implicit Differentiation 164
discovery project • Families of Implicit Curves 172
2.7 Rates of Change in the Natural and Social Sciences 172
2.8 Related Rates 185
2.9 Linear Approximations and Differentials 192
discovery project • Polynomial Approximations 198
Review 199
Problems Plus 204
3 Applications of Differentiation 209
3.1 Maximum and Minimum Values 210
applied project • The Calculus of Rainbows 219
3.2 The Mean Value Theorem 220
3.3 What Derivatives Tell Us about the Shape of a Graph 226
3.4 Limits at Infinity; Horizontal Asymptotes 237
3.5 Summary of Curve Sketching 250
3.6 Graphing with Calculus and Technology 258
3.7 Optimization Problems 265
applied project • The Shape of a Can 278
applied project • Planes and Birds: Minimizing Energy 279
3.8 Newton’s Method 280
3.9 Antiderivatives 285
Review 292
Problems Plus 297
4 Integrals 301
4.1 The Area and Distance Problems 302
4.2 The Definite Integral 314
discovery project • Area Functions 328
4.3 The Fundamental Theorem of Calculus 329
4.4 Indefinite Integrals and the Net Change Theorem 339
writing project • Newton, Leibniz, and the Invention of Calculus 348
4.5 The Substitution Rule 349
Review 357
Problems Plus 361
5 Applications of Integration 363
5.1 Areas Between Curves 364
applied project • The Gini Index 373
5.2 Volumes 374
5.3 Volumes by Cylindrical Shells 388
5.4 Work 395
5.5 Average Value of a Function 401
applied project • Calculus and Baseball 404
Review 405
Problems Plus 408
6 Inverse Functions: 411
Exponential, Logarithmic, and Inverse Trigonometric Functions
6.1 Inverse Functions and Their Derivatives 412
Instructors may cover either Sections 6.2–6.4 or Sections 6.2*–6.4*. See the Preface.
6.2 Exponential Functions and
Their Derivatives 420
6.2* The Natural Logarithmic
Function 451
6.3 Logarithmic
Functions 433
6.3* The Natural Exponential
Function 460
6.4 Derivatives of Logarithmic
Functions 440
6.4* General Logarithmic and
Exponential Functions 468
6.5 Exponential Growth and Decay 478
applied project • Controlling Red Blood Cell Loss During Surgery 486
6.6 Inverse Trigonometric Functions 486
applied project • Where to Sit at the Movies 495
6.7 Hyperbolic Functions 495
6.8 Indeterminate Forms and l’Hospital’s Rule 503
writing project • The Origins of l’Hospital’s Rule 515
Review 516
Problems Plus 520
7 Techniques of Integration 523
7.1 Integration by Parts 524
7.2 Trigonometric Integrals 531
7.3 Trigonometric Substitution 538
7.4 Integration of Rational Functions by Partial Fractions 545
7.5 Strategy for Integration 555
7.6 Integration Using Tables and Technology 561
discovery project • Patterns in Integrals 566
7.7 Approximate Integration 567
7.8 Improper Integrals 580
Review 590
Problems Plus 594
8 Further Applications of Integration 597
8.1 Arc Length 598
discovery project • Arc Length Contest 605
8.2 Area of a Surface of Revolution 605
discovery project • Rotating on a Slant 613
8.3 Applications to Physics and Engineering 614
discovery project • Complementary Coffee Cups 625
8.4 Applications to Economics and Biology 625
8.5 Probability 630
Review 638
Problems Plus 640
9 Differential Equations 643
9.1 Modeling with Differential Equations 644
9.2 Direction Fields and Euler’s Method 650
9.3 Separable Equations 659
applied project • How Fast Does a Tank Drain? 668
9.4 Models for Population Growth 669
9.5 Linear Equations 679
applied project • Which Is Faster, Going Up or Coming Down? 686
9.6 Predator-Prey Systems 687
Review 694
Problems Plus 697
10 Parametric Equations and Polar Coordinates 699
10.1 Curves Defined by Parametric Equations 700
discovery project • Running Circles Around Circles 710
10.2 Calculus with Parametric Curves 711
discovery project • Bézier Curves 722
10.3 Polar Coordinates 722
discovery project • Families of Polar Curves 732
10.4 Calculus in Polar Coordinates 732
10.5 Conic Sections 740
10.6 Conic Sections in Polar Coordinates 749
Review 757
Problems Plus 760
11 Sequences, Series, and Power Series 761
11.1 Sequences 762
discovery project • Logistic Sequences 776
11.2 Series 776
11.3 The Integral Test and Estimates of Sums 789
11.4 The Comparison Tests 798
11.5 Alternating Series and Absolute Convergence 803
11.6 The Ratio and Root Tests 812
11.7 Strategy for Testing Series 817
11.8 Power Series 819
11.9 Representations of Functions as Power Series 825
11.10 Taylor and Maclaurin Series 833
discovery project • An Elusive Limit 848
writing project • How Newton Discovered the Binomial Series 849
11.11 Applications of Taylor Polynomials 849
applied project • Radiation from the Stars 858
Review 859
Problems Plus 863
12 Vectors and the Geometry of Space 867
12.1 Three-Dimensional Coordinate Systems 868
12.2 Vectors 874
discovery project • The Shape of a Hanging Chain 884
12.3 The Dot Product 885
12.4 The Cross Product 893
discovery project • The Geometry of a Tetrahedron 902
12.5 Equations of Lines and Planes 902
discovery project • Putting 3D in Perspective 912
12.6 Cylinders and Quadric Surfaces 913
Review 921
Problems Plus 925
13 Vector Functions 927
13.1 Vector Functions and Space Curves 928
13.2 Derivatives and Integrals of Vector Functions 936
13.3 Arc Length and Curvature 942
13.4 Motion in Space: Velocity and Acceleration 954
applied project • Kepler’s Laws 963
Review 965
Problems Plus 968
14 Partial Derivatives 971
14.1 Functions of Several Variables 972
14.2 Limits and Continuity 989
14.3 Partial Derivatives 999
discovery project • Deriving the Cobb-Douglas Production Function 1011
14.4 Tangent Planes and Linear Approximations 1012
applied project • The Speedo LZR Racer 1022
14.5 The Chain Rule 1023
14.6 Directional Derivatives and the Gradient Vector 1032
14.7 Maximum and Minimum Values 1046
discovery project • Quadratic Approximations and Critical Points 1057
14.8 Lagrange Multipliers 1058
applied project • Rocket Science 1066
applied project • Hydro-Turbine Optimization 1068
Review 1069
Problems Plus 1073
15 Multiple Integrals 1075
15.1 Double Integrals over Rectangles 1076
15.2 Double Integrals over General Regions 1089
15.3 Double Integrals in Polar Coordinates 1100
15.4 Applications of Double Integrals 1107
15.5 Surface Area 1117
15.6 Triple Integrals 1120
discovery project • Volumes of Hyperspheres 1133
15.7 Triple Integrals in Cylindrical Coordinates 1133
discovery project • The Intersection of Three Cylinders 1139
15.8 Triple Integrals in Spherical Coordinates 1140
applied project • Roller Derby 1146
15.9 Change of Variables in Multiple Integrals 1147
Review 1155
Problems Plus 1159
16 Vector Calculus 1161
16.1 Vector Fields 1162
16.2 Line Integrals 1169
16.3 The Fundamental Theorem for Line Integrals 1182
16.4 Green’s Theorem 1192
16.5 Curl and Divergence 1199
16.6 Parametric Surfaces and Their Areas 1208
16.7 Surface Integrals 1220
16.8 Stokes’ Theorem 1233
16.9 The Divergence Theorem 1239
16.10 Summary 1246
Review 1247
Problems Plus 1251
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 Answers to Odd-Numbered Exercises A51
Index A135