1 Foundation For Calculus: Functions and Limits 1

1.1 Functions and Change 2

1.2 Exponential Functions 14

1.3 New Functions From Old 26

1.4 Logarithmic Functions 34

1.5 Trigonometric Functions 42

1.6 Powers, Polynomials, and Rational Functions 53

1.7 Introduction to Limits and Continuity 62

1.8 Extending The Idea of a Limit 71

1.9 Further Limit Calculations Using Algebra 80

1.10 Preview of The Formal Definition of a Limit Online

2 Key Concept: The Derivative 87

2.1 How Do We Measure Speed? 88

2.2 The Derivative At a Point 96

2.3 The Derivative Function 105

2.4 Interpretations of The Derivative 113

2.5 The Second Derivative 121

2.6 Differentiability 130

3 Short-Cuts to Differentiation 135

3.1 Powers and Polynomials 136

3.2 The Exponential Function 146

3.3 The Product and Quotient Rules 151

3.4 The Chain Rule 158

3.5 The Trigonometric Functions 165

3.6 The Chain Rule and Inverse Functions 171

3.7 Implicit Functions 178

3.8 Hyperbolic Functions 181

3.9 Linear Approximation and The Derivative 185

3.10 Theorems About Differentiable Functions 193

4 Using The Derivative 199

4.1 Using First and Second Derivatives 200

4.2 Optimization 211

4.3 Optimization and Modeling 220

4.4 Families of Functions and Modeling 234

4.5 Applications to Marginality 244

4.6 Rates and Related Rates 253

4.7 L’hopital’s Rule, Growth, and Dominance 264

4.8 Parametric Equations 271

5 Key Concept: The Definite Integral 285

5.1 How Do We Measure Distance Traveled? 286

5.2 The Definite Integral 298

5.3 The Fundamental Theorem and Interpretations 308

5.4 Theorems About Definite Integrals 319

6 Constructing Antiderivatives 333

6.1 Antiderivatives Graphically and Numerically 334

6.2 Constructing Antiderivatives Analytically 341

6.3 Differential Equations and Motion 348

6.4 Second Fundamental Theorem of Calculus 355

7 Integration 361

7.1 Integration By Substitution 362

7.2 Integration By Parts 373

7.3 Tables of Integrals 380

7.4 Algebraic Identities and Trigonometric Substitutions 386

7.5 Numerical Methods For Definite Integrals 398

7.6 Improper Integrals 408

7.7 Comparison of Improper Integrals 417

8 Using The Definite Integral 425

8.1 Areas and Volumes 426

8.2 Applications to Geometry 436

8.3 Area and Arc Length In Polar Coordinates 447

8.4 Density and Center of Mass 456

8.5 Applications to Physics 467

8.6 Applications to Economics 478

8.7 Distribution Functions 489

8.8 Probability, Mean, and Median 497

9 Sequences and Series 507

9.1 Sequences 508

9.2 Geometric Series 514

9.3 Convergence of Series 522

9.4 Tests For Convergence 529

9.5 Power Series and Interval of Convergence 539

10 Approximating Functions Using Series 549

10.1 Taylor Polynomials 550

10.2 Taylor Series 560

10.3 Finding and Using Taylor Series 567

10.4 The Error In Taylor Polynomial Approximations 577

10.5 Fourier Series 584

11 Differential Equations 599

11.1 What Is a Differential Equation? 600

11.2 Slope Fields 605

11.3 Euler’s Method 614

11.4 Separation of Variables 619

11.5 Growth and Decay 625

11.6 Applications and Modeling 637

11.7 The Logistic Model 647

11.8 Systems of Differential Equations 657

11.9 Analyzing The Phase Plane 667

11.10 Second-Order Differential Equations: Oscillations 674

11.11 Linear Second-Order Differential Equations 682

Appendices Online

A Roots, Accuracy, and Bounds Online

B Complex Numbers Online

C Newton’s Method Online

D Vectors In The Plane Online

Ready Reference 693

Answers to Odd Numbered Problems 705

Index 743

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