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Copyright, vi
Preface, xiii
To the Student, xxiii
Diagnostic Tests, xxiv
A Preview of Calculus, 3
1. Functions and Models, 11
1.1. Four Ways to Represent a Function, 12
1.2. Mathematical Models: A Catalog of Essential Functions, 25
1.3. New Functions from Old Functions, 37
1.4. Graphing Calculators and Computers, 46
1.5. Exponential Functions, 52
1.6. Inverse Functions and Logarithms, 61
1.7. Parametric Curves, 71
Laboratory Project: Running Circles Around Circles, 79
Review, 80
Principles of Problem Solving, 83
2. Limits and Derivatives, 89
2.1. The Tangent and Velocity Problems, 90
2.2. The Limit of a Function, 95
2.3. Calculating Limits Using the Limit Laws, 104
2.4. Continuity, 113
2.5. Limits Involving Infinity, 124
2.6. Derivatives and Rates of Change, 135
Writing Project: Early Methods for Finding Tangents, 145
2.7. The Derivative as a Function, 146
2.8. What Does f’ Say about f’ ?, 158
Review, 164
Focus on Problem Solving, 169
3. Differentiation Rules, 173
3.1. Derivatives of Polynomials and Exponential Functions, 174
Applied Project: Building a Better Roller Coaster, 183
3.2. The Product and Quotient Rules, 183
3.3. Derivatives of Trigonometric Functions, 190
3.4. The Chain Rule, 197
Laboratory Project: Bézier Curves, 208
Applied Project: Where Should a Pilot Start Descent?, 209
3.5. Implicit Differentiation, 209
3.6. Inverse Trigonometric Functions and Their Derivatives, 216
3.7. Derivatives of Logarithmic Functions, 221
Discovery Project: Hyperbolic Functions, 227
3.8. Rates of Change in the Natural and Social Sciences, 228
3.9. Linear Approximations and Differentials, 240
Laboratory Project: Taylor Polynomials, 247
Review, 248
Focus on Problem Solving, 251
4. Applications of Differentiation, 255
4.1. Related Rates, 256
4.2. Maximum and Minimum Values, 262
Applied Project: The Calculus of Rainbows, 270
4.3. Derivatives and the Shapes of Curves, 271
4.4. Graphing with Calculus and Calculators, 282
4.5. Indeterminate Forms and l’Hospital’s Rule, 290
Writing Project: The Origins of l’Hospital’s Rule, 299
4.6. Optimization Problems, 299
Applied Project: The Shape of a Can, 311
4.7. Newton’s Method, 312
4.8. Antiderivatives, 317
Review, 323
Focus on Problem Solving, 327
5. Integrals, 331
5.1. Areas and Distances, 332
5.2. The Definite Integral, 343
5.3. Evaluating Definite Integrals, 356
Discovery Project: Area Functions, 366
5.4. The Fundamental Theorem of Calculus, 367
Writing Project: Newton, Leibniz, and the Invention of Calculus, 374
5.5. The Substitution Rule, 375
5.6. Integration by Parts, 383
5.7. Additional Techniques of Integration, 389
5.8. Integration Using Tables and Computer Algebra Systems, 394
Discovery Project: Patterns in Integrals, 400
5.9. Approximate Integration, 401
5.10. Improper Integrals, 413
Review, 423
Focus on Problem Solving, 428
6. Applications of Integration, 431
6.1. More About Areas, 432
6.2. Volumes, 438
Discovery Project: Rotating on a Slant, 448
6.3. Volumes by Cylindrical Shells, 449
6.4. Arc Length, 455
Discovery Project: Arc Length Contest, 460
6.5. Average Value of a Function, 460
Applied Project: Where To Sit at the Movies, 464
6.6. Applications to Physics and Engineering, 464
Discovery Project: Complementary Coffee Cups, 475
6.7. Applications to Economics and Biology, 476
6.8. Probability, 480
Review, 487
Focus on Problem Solving, 491
7. Differential Equations, 493
7.1. Modeling with Differential Equations, 494
7.2. Direction Fields and Euler’s Method, 499
7.3. Separable Equations, 508
Applied Project: How Fast Does a Tank Drain?, 517
Applied Project: Which Is Faster, Going Up or Coming Down?, 518
7.4. Exponential Growth and Decay, 519
Applied Project: Calculus and Baseball, 529
7.5. The Logistic Equation, 530
7.6. Predator-Prey Systems, 540
Review, 547
Focus on Problem Solving, 551
8. Infinite Sequences and Series, 553
8.1. Sequences, 554
Laboratory Project: Logistic Sequences, 564
8.2. Series, 565
8.3. The Integral and Comparison Tests; Estimating Sums, 575
8.4. Other Convergence Tests, 585
8.5. Power Series, 592
8.6. Representations of Functions as Power Series, 598
8.7. Taylor and Maclaurin Series, 604
Laboratory Project: An Elusive Limit, 618
Writing Project: How Newton Discovered the Binomial Series, 618
8.8. Applications of Taylor Polynomials, 619
Applied Project: Radiation from the Stars, 627
Review, 628
Focus on Problem Solving, 631
Appendixes, A1
A. Intervals, Inequalities, and Absolute Values, A2
B. Coordinate Geometry, A7
C. Trigonometry, A17
D. Precise Definitions of Limits, A26
E. A Few Proofs, A35
F. Sigma Notation, A37
G. Integration of Rational Functions by Partial Fractions, A43
H. Polar Coordinates, A51
I. Complex Numbers, A67
J. Answers to Odd-Numbered Exercises, A76
Index, A115

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