## Table of contents

**0 BEFORE CALCULUS 1**

0.1 Functions 1

0.2 New Functions from Old 15

0.3 Families of Functions 27

0.4 Inverse Functions 38

**1 LIMITS AND CONTINUITY 49**

1.1 Limits (An Intuitive Approach) 49

1.2 Computing Limits 62

1.3 Limits at Infinity; End Behavior of a Function 71

1.4 Limits (Discussed More Rigorously) 81

1.5 Continuity 90

1.6 Continuity of Trigonometric Functions 101

**2 THE DERIVATIVE 110**

2.1 Tangent Lines and Rates of Change 110

2.2 The Derivative Function 122

2.3 Introduction to Techniques of Differentiation 134

2.4 The Product and Quotient Rules 142

2.5 Derivatives of Trigonometric Functions 148

2.6 The Chain Rule 153

2.7 Implicit Differentiation 161

2.8 Related Rates 168

2.9 Local Linear Approximation; Differentials 175

**3 THE DERIVATIVE IN GRAPHING AND APPLICATIONS 187**

3.1 Analysis of Functions I: Increase, Decrease, and Concavity 187

3.2 Analysis of Functions II: Relative Extrema; Graphing Polynomials 197

3.3 Analysis of Functions III: Rational Functions, Cusps, and Vertical Tangents 207

3.4 Absolute Maxima and Minima 216

3.5 Applied Maximum and Minimum Problems 224

3.6 Rectilinear Motion 238

3.7 Newton’s Method 246

3.8 Rolle’s Theorem; Mean-Value Theorem 252

**4 INTEGRATION 265**

4.1 An Overview of the Area Problem 265

4.2 The Indefinite Integral 271

4.3 Integration by Substitution 281

4.4 The Definition of Area as a Limit; Sigma Notation 287

4.5 The Definite Integral 300

4.6 The Fundamental Theorem of Calculus 309

4.7 Rectilinear Motion Revisited Using Integration 322

4.8 Average Value of a Function and its Applications 332

4.9 Evaluating Definite Integrals by Substitution 337

**5 APPLICATIONS OF THE DEFINITE INTEGRAL IN GEOMETRY, SCIENCE, AND ENGINEERING 347**

5.1 Area Between Two Curves 347

5.2 Volumes by Slicing; Disks and Washers 355

5.3 Volumes by Cylindrical Shells 365

5.4 Length of a Plane Curve 371

5.5 Area of a Surface of Revolution 377

5.6 Work 382

5.7 Moments, Centers of Gravity, and Centroids 391

5.8 Fluid Pressure and Force 400

**6 EXPONENTIAL, LOGARITHMIC, AND INVERSE TRIGONOMETRIC FUNCTIONS 409**

6.1 Exponential and Logarithmic Functions 409

6.2 Derivatives and Integrals Involving Logarithmic Functions 420

6.3 Derivatives of Inverse Functions; Derivatives and Integrals Involving Exponential Functions 427

6.4 Graphs and Applications Involving Logarithmic and Exponential Functions 434

6.5 L’Hôpital’s Rule; Indeterminate Forms 441

6.6 Logarithmic and Other Functions Defined by Integrals 450

6.7 Derivatives and Integrals Involving Inverse Trigonometric Functions 462

6.8 Hyperbolic Functions and Hanging Cables 472

**7 PRINCIPLES OF INTEGRAL EVALUATION 488**

7.1 An Overview of Integration Methods 488

7.2 Integration by Parts 491

7.3 Integrating Trigonometric Functions 500

7.4 Trigonometric Substitutions 508

7.5 Integrating Rational Functions by Partial Fractions 514

7.6 Using Computer Algebra Systems and Tables of Integrals 523

7.7 Numerical Integration; Simpson’s Rule 533

7.8 Improper Integrals 547

**8 MATHEMATICAL MODELING WITH DIFFERENTIAL EQUATIONS 561**

8.1 Modeling with Differential Equations 561

8.2 Separation of Variables 568

8.3 Slope Fields; Euler’s Method 579

8.4 First-Order Differential Equations and Applications 586

**9 INFINITE SERIES 596**

9.1 Sequences 596

9.2 Monotone Sequences 607

9.3 Infinite Series 614

9.4 Convergence Tests 623

9.5 The Comparison, Ratio, and Root Tests 631

9.6 Alternating Series; Absolute and Conditional Convergence 638

9.7 Maclaurin and Taylor Polynomials 648

9.8 Maclaurin and Taylor Series; Power Series 659

9.9 Convergence of Taylor Series 668

9.10 Differentiating and Integrating Power Series; Modeling with Taylor Series 678

**10 PARAMETRIC AND POLAR CURVES; CONIC SECTIONS 692**

10.1 Parametric Equations; Tangent Lines and Arc Length for Parametric Curves 692

10.2 Polar Coordinates 705

10.3 Tangent Lines, Arc Length, and Area for Polar Curves 719

10.4 Conic Sections 730

10.5 Rotation of Axes; Second-Degree Equations 748

10.6 Conic Sections in Polar Coordinates 754

**11 THREE-DIMENSIONAL SPACE; VECTORS 767**

11.1 Rectangular Coordinates in 3-Space; Spheres; Cylindrical Surfaces 767

11.2 Vectors 773

11.3 Dot Product; Projections 785

11.4 Cross Product 795

11.5 Parametric Equations of Lines 805

11.6 Planes in 3-Space 813

11.7 Quadric Surfaces 821

11.8 Cylindrical and Spherical Coordinates 832

**12 VECTOR-VALUED FUNCTIONS 841**

12.1 Introduction to Vector-Valued Functions 841

12.2 Calculus of Vector-Valued Functions 848

12.3 Change of Parameter; Arc Length 858

12.4 Unit Tangent, Normal, and Binormal Vectors 868

12.5 Curvature 873

12.6 Motion Along a Curve 882

12.7 Kepler’s Laws of Planetary Motion 895

**13 PARTIAL DERIVATIVES 906**

13.1 Functions of Two or More Variables 906

13.2 Limits and Continuity 917

13.3 Partial Derivatives 927

13.4 Differentiability, Differentials, and Local Linearity 940

13.5 The Chain Rule 949

13.6 Directional Derivatives and Gradients 960

13.7 Tangent Planes and Normal Vectors 971

13.8 Maxima and Minima of Functions of Two Variables 977

13.9 Lagrange Multipliers 989

**14 MULTIPLE INTEGRALS 1000**

14.1 Double Integrals 1000

14.2 Double Integrals over Nonrectangular Regions 1009

14.3 Double Integrals in Polar Coordinates 1018

14.4 Surface Area; Parametric Surfaces 1026

14.5 Triple Integrals 1039

14.6 Triple Integrals in Cylindrical and Spherical Coordinates 1048

14.7 Change of Variables in Multiple Integrals; Jacobians 1058

14.8 Centers of Gravity Using Multiple Integrals 1071

**15 TOPICS IN VECTOR CALCULUS 1084**

15.1 Vector Fields 1084

15.2 Line Integrals 1094

15.3 Independence of Path; Conservative Vector Fields 1111

15.4 Green’s Theorem 1122

15.5 Surface Integrals 1130

15.6 Applications of Surface Integrals; Flux 1138

15.7 The Divergence Theorem 1148

15.8 Stokes’ Theorem 1158

A APPENDICES

A GRAPHING FUNCTIONS USING CALCULATORS AND COMPUTER ALGEBRA SYSTEMS A1

B TRIGONOMETRY REVIEW A13

C SOLVING POLYNOMIAL EQUATIONS A27

D SELECTED PROOFS A34

ANSWERS TO ODD-NUMBERED EXERCISES A45

INDEX I-1

WEB APPENDICES (online only)

Available for download at www.wiley.com/go/global/anton and in WileyPLUS.

E REAL NUMBERS, INTERVALS, AND INEQUALITIES

F ABSOLUTE VALUE

G COORDINATE PLANES, LINES, AND LINEAR FUNCTIONS

H DISTANCE, CIRCLES, AND QUADRATIC EQUATIONS

I EARLY PARAMETRIC EQUATIONS OPTION

J MATHEMATICAL MODELS

K THE DISCRIMINANT

L SECOND-ORDER LINEAR HOMOGENEOUS DIFFERENTIAL EQUATIONS

WEB PROJECTS: Expanding the Calculus Horizon (online only)

Available for download at www.wiley.com/go/global/anton and in WileyPLUS.

BLAMMO THE HUMAN CANNONBALL

COMET COLLISION

HURRICANE MODELING

ITERATION AND DYNAMICAL SYSTEMS

RAILROAD DESIGN

ROBOTICS