Single variable calculus : concepts and contexts

Front Cover......Page 1
Title Page......Page 8
Copyright......Page 9
Contents......Page 12
Preface......Page 16
To the Student......Page 26
Diagnostic Tests......Page 27
A Preview of Calculus......Page 34
1. Functions and Models......Page 42
1.1 Four Ways to Represent a Function......Page 43
1.2 Mathematical Models: A Catalog of Essential Functions......Page 56
1.3 New Functions from Old Functions......Page 68
1.4 Graphing Calculators and Computers......Page 77
1.5 Exponential Functions......Page 83
1.6 Inverse Functions and Logarithms......Page 92
1.7 Parametric Curves......Page 102
Laboratory Project:Running Circles Around Circles......Page 110
Review......Page 111
Principles of Problem Solving......Page 114
2. Limits and Derivatives......Page 120
2.1 The Tangent and Velocity Problems......Page 121
2.2 The Limit of a Function......Page 126
2.3 Calculating Limits Using the Limit Laws......Page 135
2.4 Continuity......Page 144
2.5 Limits Involving Infinity......Page 154
2.6 Derivatives and Rates of Change......Page 166
Writing Project:Early Methods for Finding Tangents......Page 176
2.7 The Derivative as a Function......Page 177
2.8 What Does f' Say about f?......Page 189
Review......Page 195
Focus on Problem Solving......Page 200
3. Differentiation Rules......Page 204
3.1 Derivatives of Polynomials and Exponential Functions......Page 205
3.2 The Product and Quotient Rules......Page 214
3.3 Derivatives of Trigonometric Functions......Page 221
3.4 The Chain Rule......Page 228
Laboratory Project:Bézier Curves......Page 239
3.5 Implicit Differentiation......Page 240
3.6 Inverse Trigonometric Functions and Their Derivatives......Page 247
3.7 Derivatives of Logarithmic Functions......Page 252
Discovery Project:Hyperbolic Functions......Page 258
3.8 Rates of Change in the Natural and Social Sciences......Page 259
3.9 Linear Approximations and Differentials......Page 271
Laboratory Project:Taylor Polynomials......Page 278
Review......Page 279
Focus on Problem Solving......Page 282
4. Applications of Differentiation......Page 286
4.1 Related Rates......Page 287
4.2 Maximum and Minimum Values......Page 293
Applied Project:The Calculus of Rainbows......Page 301
4.3 Derivatives and the Shapes of Curves......Page 302
4.4 Graphing with Calculus and Calculators......Page 313
4.5 Indeterminate Forms and l’Hospital’s Rule......Page 321
4.6 Optimization Problems......Page 330
Applied Project ■ The Shape of a Can......Page 342
4.7 Newton’s Method......Page 343
4.8 Antiderivatives......Page 348
Review......Page 354
Focus on Problem Solving......Page 358
5. Integrals......Page 362
5.1 Areas and Distances......Page 363
5.2 The Definite Integral......Page 374
5.3 Evaluating Definite Integrals......Page 387
Discovery Project:Area Functions......Page 397
5.4 The Fundamental Theorem of Calculus......Page 398
Writing Project:Newton, Leibniz, and the Invention of Calculus......Page 405
5.5 The Substitution Rule......Page 406
5.6 Integration by Parts......Page 414
5.7 Additional Techniques of Integration......Page 420
5.8 Integration Using Tables and Computer Algebra Systems......Page 425
Discovery Project:Patterns in Integrals......Page 431
5.9 Approximate Integration......Page 432
5.10 Improper Integrals......Page 444
Review......Page 454
Focus on Problem Solving......Page 459
6. Applications of Integration......Page 462
6.1 More About Areas......Page 463
6.2 Volumes......Page 469
Discovery Project:Rotating on a Slant......Page 479
6.3 Volumes by Cylindrical Shells......Page 480
6.4 Arc Length......Page 486
6.5 Average Value of a Function......Page 491
6.6 Applications to Physics and Engineering......Page 495
Discovery Project:Complementary Coffee Cups......Page 506
6.7 Applications to Economics and Biology......Page 507
6.8 Probability......Page 511
Review......Page 518
Focus on Problem Solving......Page 522
7. Differential Equations......Page 524
7.1 Modeling with Differential Equations......Page 525
7.2 Direction Fields and Euler’s Method......Page 530
7.3 Separable Equations......Page 539
Applied Project:How Fast Does a Tank Drain?......Page 548
Applied Project:Which Is Faster, Going Up or Coming Down?......Page 549
7.4 Exponential Growth and Decay......Page 550
Applied Project:Calculus and Baseball......Page 560
7.5 The Logistic Equation......Page 561
7.6 Predator-Prey Systems......Page 571
Review......Page 578
Focus on Problem Solving......Page 582
8. Infinite Sequences and Series......Page 584
8.1 Sequences......Page 585
Laboratory Project:Logistic Sequences......Page 595
8.2 Series......Page 596
8.3 The Integral and Comparison Tests; Estimating Sums......Page 606
8.4 Other Convergence Tests......Page 616
8.5 Power Series......Page 623
8.6 Representations of Functions as Power Series......Page 629
8.7 Taylor and Maclaurin Series......Page 635
Writing Project:How Newton Discovered the Binomial Series......Page 649
8.8 Applications of Taylor Polynomials......Page 650
Applied Project:Radiation from the Stars......Page 658
Review......Page 659
Focus on Problem Solving......Page 662
Appendixes......Page 664
A: Intervals, Inequalities, and Absolute Values......Page 665
B: Coordinate Geometry......Page 670
C: Trigonometry......Page 680
D: Precise Definitions of Limits......Page 689
E: A Few Proofs......Page 698
F: Sigma Notation......Page 700
G: Integration of Rational Functions by Partial Fractions......Page 706
H: Polar Coordinates......Page 714
I: Complex Numbers......Page 730
J: Answers to Odd-Numbered Exercises......Page 739
Index......Page 778