PREFACE vii

SUPPLEMENTS ix

ACKNOWLEDGMENTS xi

THE ROOTS OF CALCULUS xv

1 Limits and Continuity 1

1.1 Limits (An Intuitive Approach) 1

1.2 Computing Limits 13

1.3 Limits at Infinity; End Behavior of a Function 21

1.4 Limits (Discussed More Rigorously) 30

1.5 Continuity 39

1.6 Continuity of Trigonometric Functions 50

1.7 Inverse Trigonometric Functions 55

1.8 Exponential and Logarithmic Functions 62

2 The Derivative 77

2.1 Tangent Lines and Rates of Change 77

2.2 The Derivative Function 87

2.3 Introduction to Techniques of Differentiation 98

2.4 The Product and Quotient Rules 105

2.5 Derivatives of Trigonometric Functions 110

2.6 The Chain Rule 114

3 Topics in Differentiation 124

3.1 Implicit Differentiation 124

3.2 Derivatives of Logarithmic Functions 131

3.3 Derivatives of Exponential and Inverse Trigonometric Functions 136

3.4 Related Rates 142

3.5 Local Linear Approximation; Differentials 149

3.6 L’Hoˆ pital’s Rule; Indeterminate Forms 157

4 The Derivative in Graphing and Applications 169

4.1 Analysis of Functions I: Increase, Decrease, and Concavity 169

4.2 Analysis of Functions II: Relative Extrema; Graphing Polynomials 180

4.3 Analysis of Functions III: Rational Functions, Cusps, and Vertical Tangents 189

4.4 Absolute Maxima and Minima 200

4.5 Applied Maximum and Minimum Problems 208

4.6 Rectilinear Motion 222

4.7 Newton’s Method 230

4.8 Rolle’s Theorem; Mean-Value Theorem 235

5 Integration 249

5.1 An Overview of the Area Problem 249

5.2 The Indefinite Integral 254

5.3 Integration by Substitution 264

5.4 The Definition of Area as a Limit; Sigma Notation 271

5.5 The Definite Integral 281

5.6 The Fundamental Theorem of Calculus 290

5.7 Rectilinear Motion Revisited Using Integration 302

5.8 Average Value of a Function and its Applications 310

5.9 Evaluating Definite Integrals by Substitution 315

5.10 Logarithmic and Other Functions Defined by Integrals 320

6 Applications of the Definite Integral in Geometry, Science, and Engineering 336

6.1 Area Between Two Curves 336

6.2 Volumes by Slicing; Disks and Washers 344

6.3 Volumes by Cylindrical Shells 354

6.4 Length of a Plane Curve 360

6.5 Area of a Surface of Revolution 365

6.6 Work 370

6.7 Moments, Centers of Gravity, and Centroids 378

6.8 Fluid Pressure and Force 387

6.9 Hyperbolic Functions and Hanging Cables 392

7 Principles of Integral Evaluation 406

7.1 An Overview of Integration Methods 406

7.2 Integration by Parts 409

7.3 Integrating Trigonometric Functions 417

7.4 Trigonometric Substitutions 424

7.5 Integrating Rational Functions by Partial Fractions 430

7.6 Using Computer Algebra Systems and Tables of Integrals 437

7.7 Numerical Integration; Simpson’s Rule 446

7.8 Improper Integrals 458

8 Mathematical Modeling with Differential Equations 471

8.1 Modeling with Differential Equations 471

8.2 Separation of Variables 477

8.3 Slope Fields; Euler’s Method 488

8.4 First-Order Differential Equations and Applications 494

9 Infinite Series 504

9.1 Sequences 504

9.2 Monotone Sequences 513

9.3 Infinite Series 520

9.4 Convergence Tests 528

9.5 The Comparison, Ratio, and Root Tests 534

9.6 Alternating Series; Absolute and Conditional Convergence 539

9.7 Maclaurin and Taylor Polynomials 549

9.8 Maclaurin and Taylor Series; Power Series 559

9.9 Convergence of Taylor Series 567

9.10 Differentiating and Integrating Power Series; Modeling with Taylor Series 575

10 Parametric and Polar Curves; Conic Sections 588

10.1 Parametric Equations; Tangent Lines and Arc Length for Parametric Curves 588

10.2 Polar Coordinates 600

10.3 Tangent Lines, Arc Length, and Area for Polar Curves 613

10.4 Conic Sections 622

10.5 Rotation of Axes; Second-Degree Equations 639

10.6 Conic Sections in Polar Coordinates 644

11 Three-dimensional Space; Vector

11.1 Rectangular Coordinates in 3-Space; Spheres; Cylindrical Surfaces 657

11.2 Vectors 663

11.3 Dot Product; Projections 673

11.4 Cross Product 682

11.5 Parametric Equations of Lines 692

11.6 Planes in 3-Space 698

11.7 Quadric Surfaces 705

11.7 Cylindrical and Spherical Coordinates 715

12 Vector-Valued Functions 723

12.1 Introduction to Vector-Valued Functions 723

12.2 Calculus of Vector-Valued Functions 729

12.3 Change of Parameter; Arc Length 738

12.4 Unit Tangent, Normal, and Binormal Vectors 746

12.5 Curvature 751

12.6 Motion Along a Curve 759

12.7 Kepler’s Laws of Planetary Motion 771

13 Partial Derivatives 781

13.1 Functions of Two or More Variables 781

13.2 Limits and Continuity 791

13.3 Partial Derivatives 800

13.4 Differentiability, Differentials, and Local Linearity 812

13.5 The Chain Rule 820

13.6 Directional Derivatives and Gradients 830

13.7 Tangent Planes and Normal Vectors 840

13.8 Maxima and Minima of Functions of Two Variables 845

13.9 Lagrange Multipliers 856

14 Multiple Integrals 925

14.1 Double Integrals 925

14.2 Double Integrals Over Nonrectangular Regions 932

14.3 Double Integrals in Polar Coordinates 941

14.4 Surface Area; Parametric Surfaces 948

14.5 Triple Integrals 961

14.6 Triple Integrals in Cylindrical and Spherical Coordinates 968

14.7 Change of Variables in Multiple Integrals; Jacobians 977

14.8 Centers of Gravity Using Multiple Integrals 989

15 Topics in Vector Calculus 1001

15.1 Vector Fields 1001

15.2 Line Integrals 1010

15.3 Independence of Path; Conservative Vector Fields 1025

15.4 Green’s Theorem 1035

15.5 Surface Integrals 1042

15.6 Applications of Surface Integrals; Flux 1049

15.7 The Divergence Theorem 1058

15.8 Stokes’ Theorem 1067

APPENDIX A A1

APPENDIX B 00

APPENDIX C 00

APPENDIX D 00

APPENDIX E 00

ANSWERS 00

INDEX I1

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