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A PREVIEW OF CALCULUS 1

1 Functions and Models 9

1.1 Four Ways to Represent a Function 10

1.2 Mathematical Models:A Catalog of Essential Functions 23

1.3 New Functions from Old Functions 36

1.4 Graphing Calculators and Computers 44

1.5 Exponential Functions 51

1.6 Inverse Functions and Logarithms 58

Review 72

Principles of Problem Solving 75

2 Limits and Derivatives 81

2.1 The Tangent and Velocity Problems 82

2.2 The Limit of a Function 87

2.3 Calculating Limits Using the Limit Laws 99

2.4 The Precise Definition of a Limit 108

2.5 Continuity 118

2.6 Limits at Infinity;Horizontal Asymptotes 130

2.7 Derivatives and Rates of Change 143

Writing Project Early Methods for Finding Tangents 153

2.8 The Derivative as a Function 154

Review 165

Problems Plus 170

3 Differentiation Rules 173

3.1 Derivatives of Polynomials and Exponential Functions 174

Applied Project Building a Better Roller Coaster 184

3.2 The Product and Quotient Rules 184

3.3 Derivatives of Trigonometric Functions 191

3.4 The Chain Rule 198

Applied Project Where Should a Pilot Start Descent? 208

3.5 Implicit Differentiation 209

Laboratory Project Families of Implicit Curves 217

3.6 Derivatives of Logarithmic Functions 218

3.7 Rates of Change in the Natural and Social Sciences 224

3.8 Exponential Growth and Decay 237

3.9 Related Rates 244

3.10 Linear Approximations and Differentials 250

Laboratory Project Taylor Polynomials 256

3.11 Hyperbolic Functions 257

Review 264

Problems Plus 268

4 Applications of Differentiation 273

4.1 Maximum and Minimum Values 274

Applied Project The Calculus of Rainbows 282

4.2 The Mean Value Theorem 284

4.3 How Derivatives Affect the Shape of a Graph 290

4.4 Indeterminate Forms and l'Hospital's Rule 301

Writing Project The Origins of I'Hospital's Rule 310

4.5 Summary of Curve Sketching 310

4.6 Graphing with Calculus and Calculators 318

4.7 Optimization Problems 325

Applied Project The Shape of a Can 337

4.8 Newton's Method 338

4.9 Antiderivatives 344

Review 351

Problems Plus 355

5 Integrals 359

5.1 Areas and Distances 360

5.2 The Definite Integral 371

Discovery Project Area Functions 385

5.3 The Fundamental Theorem of Calculus 386

5.4 Indefinite Integrals and the Net Change Theorem 397

Writing Project Newton,Leibniz,and the Invention of Calculus 406

5.5 The Substitution Rule 407

Review 415

Problems Plus 419

6 Applications of Integration 421

6.1 Areas Between Curves 422

Applied Project The Gini Index 429

6.2 Volumes 430

6.3 Volumes by Cylindrical Shells 441

6.4 Work 446

6.5 Average Value of a Function 451

Applied Project Calculus and Baseball 455

Applied Project Where to Sit at the Movies 456

Review 457

Problems Plus 459

7 Techniques of Integration 463

7.1 Integration by Parts 464

7.2 Trigonometric Integrals 471

7.3 Trigonometric Substitution 478

7.4 Integration of Rational Functions by Partial Fractions 484

7.5 Strategy for Integration 494

7.6 Integration Using Tables and Computer Algebra Systems 500

Discovery Project Patterns in Integrals 505

7.7 Approximate Integration 506

7.8 Improper Integrals 519

Review 529

Problems Plus 533

8 Further Applications of Integration 537

8.1 Arc Length 538

Discovery Project Arc Length Contest 545

8.2 Area of a Surface of Revolution 545

Discovery Project Rotating on a Slant 551

8.3 Applications to Physics and Engineering 552

Discovery Project Complementary Coffee Cups 562

8.4 Applications to Economics and Biology 563

8.5 Probability 568

Review 575

Problems Plus 577

9 Differential Equations 579

9.1 Modeling with Differential Equations 580

9.2 Direction Fields and Euler's Method 585

9.3 Separable Equations 594

Applied Project How Fast Does a Tank Drain? 603

Applied Project Which Is Faster,Going Up or Coming Down? 604

9.4 Models for Population Growth 605

9.5 Linear Equations 616

9.6 Predator-Prey Systems 622

Review 629

Problems Plus 633

10 Parametric Equations and Polar Coordinates 635

10.1 Curves Defined by Parametric Equations 636

Laboratory Project Running Circles around Circles 644

10.2 Calculus with Parametric Curves 645

Laboratory Project Bézier Curves 653

10.3 Polar Coordinates 654

Laboratory Project Families of Polar Curves 664

10.4 Areas and Lengths in Polar Coordinates 665

10.5 Conic Sections 670

10.6 Conic Sections in Polar Coordinates 678

Review 685

Problems Plus 688