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CHAPTER 0 PRELIMINARIES 1

0.1 The Real Numbers and the Cartesian Plane 2

0.2 Lines and Functions 11

0.3 Graphing Calculators and Computer Algebra Systems 24

0.4 Solving Equations 34

0.5 Trigonometric Functions 40

0.6 Exponential and Logarithmic Functions 50

0.7 Transformations of Functions 63

0.8 Preview of Calculus 72

CHAPTER 1 LIMITS AND CONTINUITY 81

1.1 The Concept of Limit 82

1.2 Computation of Limits 91

1.3 Continuity and Its Consequences 102

1.4 Limits Involving Infinity 114

1.5 Formal Definition of the Limit 124

1.6 Limits and Loss-of-Significance Errors 137

CHAPTER 2 DIFFERENTIATION:ALGEBRAIC,TRIGONOMETRIC,EXPONENTIAL AND LOGARITHMIC FUNCTIONS 149

2.1 Tangent Lines and Velocity 150

2.2 The Derivative 164

2.3 Computation of Derivatives:The Power Rule 176

2.4 The Product and Quotient Rules 187

2.5 Derivatives of Trigonometric Functions 196

2.6 Derivatives of Exponential and Logarithmic Functions 205

2.7 The Chain Rule 213

2.8 Implicit Differentitiion and Related Rates 220

2.9 The Mean Value Theorem 229

CHAPTER 3 APPLICATIONS OF DIFFERENTIATION 241

3.1 Linear Approximations and L'H pital's Rule 242

3.2 Newton's Method 251

3.3 Maximum and Minimum Values 258

3.4 Increasing and Decreasing Functions 269

3.5 Concavity 278

3.6 Overview of Curve Sketching 286

3.7 Optimization 298

3.8 Rates of Change in Applications 310

CHAPTER 4 INTEGRATION 321

4.1 Antiderivatives 322

4.2 Sums and Sigma Notation 334

4.3 Area 342

4.4 The Definite Integral 350

4.5 The Fundamental Theorem of Calculus 364

4.6 Integration by Substitution 374

4.7 Numerical Integration 384

CHAPTER 5 APPLICATIONS OF THE DEFINITE INTEGRAL 401

5.1 Area between Curves 402

5.2 Volume 411

5.3 Volumes by Cylindrical Shells 425

5.4 Arc Length and Surface Area 434

5.5 Projectile Motion 442

5.6 Work,Moments and Hydrostatic Force 453

5.7 Probability 465

CHAPTER 6 EXPONENTIALS,LOGARITHMS AND OTHER TRANSCENDENTAL FUNCTIONS 479

6.1 The Natural Logarithm Revisited 480

6.2 Inverse Functions 487

6.3 The Exponential Function Revisited 495

6.4 Growth and Decay Problems 503

6.5 Separable Differential Equations 512

6.6 Euler's Method 521

6.7 The Inverse Trigonometric Functions 530

6.8 The Calculus of the Inverse Trigonometric Functions 536

6.9 The Hyperbolic Functions 543

CHAPTER 7 INTEGRATION TECHNIQUES 555

7.1 Review of Formulas and Techniques 556

7.2 Integration by Parts 560

7.3 Trigonometric Techniques of Integration 568

7.4 Integration of Rational Functions Using Partial Fractions 578

7.5 Integration Tables and Computer Algebra Systems 586

7.6 Indeterminate Forms and L'H？pital's Rule 596

7.7 Improper Integrals 604

CHAPTER 8 INFINITE SERIES 621

8.1 Sequences of Real Numbers 622

8.2 Infinite Series 636

8.3 The Integral Test and Comparison Tests 647

8.4 Alternating Series 658

8.5 Absolute Convergence and the Ratio Test 666

8.6 Power Series 674

8.7 Taylor Series 682

8.8 Applications of Taylor Series 695

8.9 Fourier Series 703

CHAPTER 9 PARAMETRIC EQUATIONS AND POLAR COORDINATES 721

9.1 Plane Curves and Parametric Equations 722

9.2 Calculus and Parametric Equations 732

9.3 Arc Length and Surface Area in Parametric Equations 739

9.4 Polar Coordinates 746

9.5 Calculus and Polar Coordinates 760

9.6 Conic Sections 769

9.7 Conic Sections in Polar Coordinates 779

CHAPTER 10 VECTORS AND THE GEOMETRY OF SPACE 787

10.1 Vectors in the Plane 788

10.2 Vectors in Space 798

10.3 The Dot Product 805

10.4 The Cross Product 814

10.5 Lines and Planes in Space 827

10.6 Surfaces in Space. 836

CHAPTER 11 VECTOR-VALUED FUNCTIONS 851

11.1 Vector-Valued Functions 852

11.2 The Calculus of Vector-Valued Functions 861

11.3 Motion in Space 872

11.4 Curvature 882

11.5 Tangent and Normal Vectors 890

CHAPTER 12 FUNCTIONS OF SEVERAL VARIABLES AND PARTIAL DIFFERENTIATION 907

12.1 Functions of Several Variables 908

12.2 Limits and Continuity 924

12.3 Partial Derivatives 936

12.4 Tangent Planes and Linear Approximations 948

12.5 The Chain Rule 960

12.6 The Gradient and Directional Derivatives 967

12.7 Extrema of Functions of Several Variables 979

12.8 Constrained Optimization and Lagrange Multipliers 994

CHAPTER 13 MULTIPLE INTEGRALS 1011

13.1 Double Integrals 1012

13.2 Area,Volume and Center of Mass 1028

13.3 Double Integrals in Polar Coordinates 1039

13.4 Surface Area 1046

13.5 Triple Integrals 1052

13.6 Cylindrical Coordinates 1064

13.7 Spherical Coordinates 1071

13.8 Change of Variables in Multiple Integrals 1079

CHAPTER 14 VECTOR CALCULUS 1095

14.1 Vector Fields 1096

14.2 Line Integrals 1108

14.3 Independence of Path and Conservative Vector Fields 1123

14.4 Green's Theorem 1134

14.5 Curl and Divergence 1143

14.6 Surface Integrals 1153

14.7 The Divergence Theorem 1167

14.8 Stokes'Theorem 1175

APPENDIX A PROOFS OF SELECT THEOREMS 1188

APPENDIX B ANSWERS TO ODD-NUMBERED EXERCISES 1199

BIBLIOGRAPHY 1251

CREDITS 1261

INDEX 1262