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Tables of Contents for Calculus

Chapter/Section Title

Page #

Page Count

Preface

xi

Acknowledgments

xvi

Ancillaries

xx

For the Student

xxiii

Precalculus Review

2

2

Algebra

4

16

Functions and Their Graphs

20

16

Trigonometry

36

17

Exponentials and Logarithms

53

10

Conic Sections

63

19

Limits and Continuity

82

60

Introduction to Limits

84

13

Definition of Limit

97

9

Techniques for Finding Limits

106

10

Limits Involving Infinity

116

11

Continuous Functions

127

15

Chapter 1 Review Exercises

139

1

Extended Problems and Group Projects

140

2

The Derivative

142

104

Tangent Lines and Rates of Change

144

13

Definition of Derivative

157

16

Techniques of Differentiation

173

11

Derivatives of the Trigonometric Functions

184

11

The Chain Rule

195

11

Implicit Differentiation

206

8

Related Rates

214

10

Linear Approximations and Differentials

224

13

Newton's Method

237

9

Chapter 2 Review Exercises

242

3

Extended Problems and Group Projects

245

1

Applications of the Derivative

246

96

Extrema of Functions

248

13

The Mean Value Theorem

261

8

The First Derivative Test

269

10

Concavity and the Second Derivative Test

279

12

Summary of Graphical Methods

291

11

Optimization Problems

302

16

Velocity and Acceleration

318

10

Applications to Economics, Social Sciences, and Life Sciences

328

14

Chapter 3 Review Exercises

337

3

Extended Problems and Group Projects

340

2

Integrals

342

88

Antiderivatives, Indefinite Integrals, and Simple Differential Equations

344

13

Change of Variables in Indefinite Integrals

357

8

Summation Notation and Area

365

12

The Definite Integral

377

11

Properties of the Definite Integral

388

8

The Fundamental Theorem of Calculus

396

13

Numerical Integration

409

21

Chapter 4 Review Exercises

427

2

Extended Problems and Group Projects

429

1

Applications of the Definite Integral

430

84

Area

432

13

Solids of Revolution

445

12

Volumes by Cylindrical Shells

457

6

Volumes by Cross Sections

463

5

Arc Length and Surfaces of Revolution

468

12

Work

480

8

Moments and Centers of Mass

488

9

Other Applications

497

17

Chapter 5 Review Exercises

511

2

Extended Problems and Group Projects

513

1

Transcendental Functions

514

114

The Derivative of the Inverse Function

516

11

The Natural Logarithm Function

527

12

The Exponential Function

539

9

Integration Using Natural Logarithm and Exponential Functions

548

9

General Exponential and Logarithmic Functions

557

11

Separable Differential Equations and Laws of Growth and Decay

568

8

Inverse Trigonometric Functions

576

15

Hyperbolic and Inverse Hyperbolic Functions

591

19

Indeterminate Forms and I'Hopital's Rule

610

18

Chapter 6 Review Exercises

623

3

Extended Problems and Group Projects

626

2

Techniques of Integration

628

64

Integration by Parts

630

9

Trigonometric Integrals

639

5

Trigonometric Substitutions

644

6

Integrals of Rational Functions

650

10

Quadratic Expressions and Miscellaneous Substitutions

660

8

Tables of Integrals and Computer Algebra Systems

668

6

Improper Integrals

674

18

Chapter 7 Review Exercises

689

2

Extended Problems and Group Projects

691

1

Infinite Series

692

92

Sequences

694

15

Convergent or Divergent Series

709

14

Positive-term Series

723

11

The Ratio and Root Tests

734

5

Alternating Series and Absolute Convergence

739

8

Power Series

747

8

Power Series Representations of Functions

755

9

Maclaurin and Taylor Series

764

12

Applications of Taylor Polynomials

776

8

Chapter 8 Review Exercises

782

1

Extended Problems and Group Projects

783

1

Parametric Equations and Polar Coordinates

784

60

Parametric Equations

786

17

Arc Length and Surface Area

803

8

Polar Coordinates

811

11

Integrals in Polar Coordinates

822

10

Translation and Rotation of Axes

832

12

Chapter 9 Review Exercises

840

1

Extended Problems and Group Projects

841

3

Vectors and Surfaces

844

76

Vectors in Two Dimensions

846

11

Vectors in Three Dimensions

857

9

The Dot Product

866

10

The Vector Product

876

9

Lines and Planes

885

13

Surfaces

898

22

Chapter 10 Review Exercises

918

1

Extended Problems and Group Projects

919

1

Vector-Valued Functions

920

52

Vector-Valued Functions and Space Curves

922

7

Limits, Derivatives, and Integrals

929

8

Curvilinear Motion

937

10

Curvature

947

12

Tangential and Normal Components of Acceleration

959

6

Kepler's Laws

965

7

Chapter 11 Review Exercises

970

1

Extended Problems and Group Projects

971

1

Partial Differentiation

972

104

Functions of Several Variables

974

12

Limits and Continuity

986

12

Partial Derivatives

998

9

Increments and Differentials

1007

14

Chain Rules

1021

9

Directional Derivatives

1030

12

Tangent Planes and Normal Lines

1042

8

Extrema of Functions of Several Variables

1050

12

Lagrange Multipliers

1062

14

Chapter 12 Review Exercises

1071

3

Extended Problems and Group Projects

1074

2

Multiple Integrals

1076

90

Double Integrals

1078

12

Area and Volume

1090

13

Double Integrals in Polar Coordinates

1103

8

Surface Area

1111

4

Triple Integrals

1115

12

Moments and Center of Mass

1127

9

Cylindrical Coordinates

1136

8

Spherical Coordinates

1144

8

Change of Variables and Jacobians

1152

14

Chapter 13 Review Exercises

1163

2

Extended Problems and Group Projects

1165

1

Vector Calculus

1166

68

Vector Fields

1168

8

Line Integrals

1176

12

Independence of Path

1188

11

Green's Theorem

1199

7

Surface Integrals

1206

10

The Divergence Theorem

1216

7

Stokes's Theorem

1223

11

Chapter 14 Review Exercises

1232

1

Extended Problems and Group Projects

1233

1

Differential Equations

1234

Separable Differential Equations

1236

First-Order Linear Differential Equations

1245

Second-Order Linear Differential Equations

1257

Nonhomogeneous Linear Differential Equations

1262

Vibrations

1268

Chapter 15 Review Exercises

1277

Extended Problems and Group Projects

1278

Appendices

I Theorems on Limits and Integrals

A1

II Table of Integrals

A14

III The Binomial Series

A19

Answers to Selected Exercises

A24

Index

A97