**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 22

1.4 Limits (Discussed More Rigorously) 31

1.5 Continuity 40

1.6 Continuity of Trigonometric Functions 51

1.7 Inverse Trigonometric Functions 56

1.8 Exponential and Logarithmic Functions 63

**2 The Derivative 79**

2.1 Tangent Lines and Rates of Change 79

2.2 The Derivative Function 89

2.3 Introduction to Techniques of Differentiation 100

2.4 The Product and Quotient Rules 108

2.5 Derivatives of Trigonometric Functions 113

2.6 The Chain Rule 118

**3 Topics in Differentiation 129**

3.1 Implicit Differentiation 129

3.2 Derivatives of Logarithmic Functions 135

3.3 Derivatives of Exponential and Inverse Trigonometric Functions 140

3.4 Related Rates 146

3.5 Local Linear Approximation; Differentials 153

3.6 L’Hôpital’s Rule; Indeterminate Forms 160

**4 The Derivative in Graphing and Applications ****172**

4.1 Analysis of Functions I: Increase, Decrease, and Concavity 172

4.2 Analysis of Functions II: Relative Extrema; Graphing Polynomials 183

4.3 Analysis of Functions III: Rational Functions, Cusps, and Vertical Tangents 192

4.4 Absolute Maxima and Minima 204

4.5 Applied Maximum and Minimum Problems 212

4.6 Rectilinear Motion 225

4.7 Newton’s Method 233

4.8 Rolle’s Theorem; Mean-Value Theorem 239

**5 Integration 253**

5.1 An Overview of the Area Problem 253

5.2 The Indefinite Integral 258

5.3 Integration by Substitution 268

5.4 The Definition of Area as a Limit; Sigma Notation 275

5.5 The Definite Integral 285

5.6 The Fundamental Theorem of Calculus 294

5.7 Rectilinear Motion Revisited Using Integration 306

5.8 Average Value of a Function and its Applications 314

5.9 Evaluating Definite Integrals by Substitution 319

5.10 Logarithmic and Other Functions Defined by Integrals 325

**6 Applications of the Definite Integral in Geometry, Science, and Engineering 341**

6.1 Area Between Two Curves 341

6.2 Volumes by Slicing; Disks and Washers 349

6.3 Volumes by Cylindrical Shells 358

6.4 Length of a Plane Curve 364

6.5 Area of a Surface of Revolution 370

6.6 Work 375

6.7 Moments, Centers of Gravity, and Centroids 383

6.8 Fluid Pressure and Force 392

6.9 Hyperbolic Functions and Hanging Cables 398

**7 Principles of Integral Evaluation 412**

7.1 An Overview of Integration Methods 412

7.2 Integration by Parts 415

7.3 Integrating Trigonometric Functions 423

7.4 Trigonometric Substitutions 431

7.5 Integrating Rational Functions by Partial Fractions 437

7.6 Using Computer Algebra Systems and Tables of Integrals 445

7.7 Numerical Integration; Simpson’s Rule 454

7.8 Improper Integrals 467

**8 Mathematical Modeling with Differential Equations 481**

8.1 Modeling with Differential Equations 481

8.2 Separation of Variables 487

8.3 Slope Fields; Euler’s Method 498

8.4 First-Order Differential Equations and Applications 504

**9 Infinite Series 514**

9.1 Sequences 514

9.2 Monotone Sequences 524

9.3 Infinite Series 531

9.4 Convergence Tests 539

9.5 The Comparison, Ratio, and Root Tests 547

9.6 Alternating Series; Absolute and Conditional Convergence 553

9.7 Maclaurin and Taylor Polynomials 563

9.8 Maclaurin and Taylor Series; Power Series 573

9.9 Convergence of Taylor Series 582

9.10 Differentiating and Integrating Power Series; Modeling with Taylor Series 591

**10 Parametric and Polar Curves; Conic Sections 605**

10.1 Parametric Equations; Tangent Lines and Arc Length for Parametric Curves 605

10.2 Polar Coordinates 617

10.3 Tangent Lines, Arc Length, and Area for Polar Curves 630

10.4 Conic Sections 639

10.5 Rotation of Axes; Second-Degree Equations 656

10.6 Conic Sections in Polar Coordinates 661

**11 Three-Dimensional Space; Vectors 674**

11.1 Rectangular Coordinates in 3-Space; Spheres; Cylindrical Surfaces 674

11.2 Vectors 680

11.3 Dot Product; Projections 691

11.4 Cross Product 700

11.5 Parametric Equations of Lines 710

11.6 Planes in 3-Space 717

11.7 Quadric Surfaces 725

11.8 Cylindrical and Spherical Coordinates 735

**12 Vector-Valued Functions 744**

12.1 Introduction to Vector-Valued Functions 744

12.2 Calculus of Vector-Valued Functions 750

12.3 Change of Parameter; Arc Length 759

12.4 Unit Tangent, Normal, and Binormal Vectors 768

12.5 Curvature 773

12.6 Motion Along a Curve 781

12.7 Kepler’s Laws of Planetary Motion 794

**13 Partial Derivatives 805**

13.1 Functions of Two or More Variables 805

13.2 Limits and Continuity 815

13.3 Partial Derivatives 824

13.4 Differentiability, Differentials, and Local Linearity 837

13.5 The Chain Rule 845

13.6 Directional Derivatives and Gradients 855

13.7 Tangent Planes and Normal Vectors 866

13.8 Maxima and Minima of Functions of Two Variables 872

13.9 Lagrange Multipliers 883

**14 Multiple Integrals 894**

14.1 Double Integrals 894

14.2 Double Integrals over Nonrectangular Regions 902

14.3 Double Integrals in Polar Coordinates 910

14.4 Surface Area; Parametric Surfaces 918

14.5 Triple Integrals 930

14.6 Triple Integrals in Cylindrical and Spherical Coordinates 938

14.7 Change of Variables in Multiple Integrals; Jacobians 947

14.8 Centers of Gravity Using Multiple Integrals 959

**15 Topics in Vector Calculus 971**

15.1 Vector Fields 971

15.2 Line Integrals 980

15.3 Independence of Path; Conservative Vector Fields 995

15.4 Green’s Theorem 1005

15.5 Surface Integrals 1013

15.6 Applications of Surface Integrals; Flux 1021

15.7 The Divergence Theorem 1030

15.8 Stokes’ Theorem 1039

**A Appendices**

A Trigonometry Review (Summary) A1

B Functions (Summary) A8

C New Functions from Old (Summary) A11

D Families of Functions (Summary) A16

E Inverse Functions (Summary) A23

Answers to Odd-Numbered Exercises A28

Index I-1