Single Variable Calculus

Front Cover......Page 1
Title Page......Page 2
Copyright......Page 3
Contents......Page 6
Preface......Page 12
To the Student......Page 23
Diagnostic Tests......Page 25
A PREVIEW OF CALCULUS......Page 31
1 FUNCTIONS AND MODELS......Page 39
1.1 Four Ways to Represent a Function......Page 40
1.2 Mathematical Models: A Catalog of Essential Functions......Page 53
1.3 New Functions from Old Functions......Page 66
1.4 Graphing Calculators and Computers......Page 75
Review......Page 81
Principles of Problem Solving......Page 83
2 LIMITS......Page 89
2.1 The Tangent and Velocity Problems......Page 90
2.2 The Limit of a Function......Page 95
2.3 Calculating Limits Using the Limit Laws......Page 106
2.4 The Precise Definition of a Limit......Page 116
2.5 Continuity......Page 126
Review......Page 137
Problems Plus......Page 139
3 DERIVATIVES......Page 141
3.1 Derivatives and Rates of Change......Page 142
3.2 The Derivative as a Function......Page 152
3.3 Differentiation Formulas......Page 164
3.4 Derivatives of Trigonometric Functions......Page 177
3.5 The Chain Rule......Page 184
3.6 Implicit Differentiation......Page 193
3.7 Rates of Change in the Natural and Social Sciences......Page 199
3.8 Related Rates......Page 211
3.9 Linear Approximations and Differentials......Page 218
Laboratory Project: Taylor Polynomials......Page 224
Review......Page 225
Problems Plus......Page 229
4 APPLICATIONS OF DIFFERENTIATION......Page 233
4.1 Maximum and Minimum Values......Page 234
Applied Project: The Calculus of Rainbows......Page 242
4.2 The Mean Value Theorem......Page 243
4.3 How Derivatives Affect the Shape of a Graph......Page 249
4.4 Limits at Infinity; Horizontal Asymptotes......Page 259
4.5 Summary of Curve Sketching......Page 272
4.6 Graphing with Calculus and Calculators......Page 279
4.7 Optimization Problems......Page 285
Applied Project: The Shape of a Can......Page 297
4.8 Newton's Method......Page 298
4.9 Antiderivatives......Page 303
Review......Page 310
Problems Plus......Page 314
5 INTEGRALS......Page 317
5.1 Areas and Distances......Page 318
5.2 The Definite Integral......Page 329
Discovery Project: Area Functions......Page 341
5.3 The Fundamental Theorem of Calculus......Page 342
5.4 Indefinite Integrals and the Net Change Theorem......Page 353
Writing Project: Newton, Leibniz, and the Invention of Calculus......Page 361
5.5 The Substitution Rule......Page 362
Review......Page 369
Problems Plus......Page 373
6 APPLICATIONS OF INTEGRATION......Page 375
6.1 Areas Between Curves......Page 376
6.2 Volumes......Page 383
6.3 Volumes by Cylindrical Shells......Page 394
6.4 Work......Page 399
6.5 Average Value of a Function......Page 403
Review......Page 407
Problems Plus......Page 409
7 INVERSE FUNCTIONS: EXPONENTIAL, LOGARITHMIC, AND INVERSE TRIGONOMETRIC FUNCTIONS......Page 413
7.1 Inverse Functions......Page 414
7.2 Exponential Functions and Their Derivatives......Page 421
7.3 Logarithmic Functions......Page 434
7.4 Derivatives of Logarithmic Functions......Page 440
7.2* The Natural Logarithmic Function......Page 450
7.3* The Natural Exponential Function......Page 459
7.4* General Logarithmic and Exponential Functions......Page 467
7.5 Exponential Growth and Decay......Page 476
7.6 Inverse Trigonometric Functions......Page 483
7.7 Hyperbolic Functions......Page 492
7.8 Indeterminate Forms and L'Hospital's Rule......Page 499
Writing Project: The Origins of L'Hospital's Rule......Page 510
Review......Page 511
Problems Plus......Page 515
8 TECHNIQUES OF INTEGRATION......Page 517
8.1 Integration by Parts......Page 518
8.2 Trigonometric Integrals......Page 525
8.3 Trigonometric Substitution......Page 532
8.4 Integration of Rational Functions by Partial Fractions......Page 538
8.5 Strategy for Integration......Page 548
8.6 Integration Using Tables and Computer Algebra Systems......Page 554
Discovery Project: Patterns in Integrals......Page 559
8.7 Approximate Integration......Page 560
8.8 Improper Integrals......Page 573
Review......Page 583
Problems Plus......Page 586
9 FURTHER APPLICATIONS OF INTEGRATION......Page 589
9.1 Arc Length......Page 590
9.2 Area of a Surface of Revolution......Page 597
Discovery Project: Rotating on a Slant......Page 603
9.3 Applications to Physics and Engineering......Page 604
9.4 Applications to Economics and Biology......Page 615
9.5 Probability......Page 620
Review......Page 627
Problems Plus......Page 629
10 DIFFERENTIAL EQUATIONS......Page 631
10.1 Modeling with Differential Equations......Page 632
10.2 Direction Fields and Euler's Method......Page 637
10.3 Separable Equations......Page 645
Applied Project: How Fast Does a Tank Drain?......Page 653
Applied Project: Which Is Faster, Going Up or Coming Down?......Page 655
10.4 Models for Population Growth......Page 656
Applied Project: Calculus and Baseball......Page 666
10.5 Linear Equations......Page 667
10.6 Predator-Prey Systems......Page 673
Review......Page 679
Problems Plus......Page 683
11 PARAMETRIC EQUATIONS AND POLAR COORDINATES......Page 685
11.1 Curves Defined by Parametric Equations......Page 686
Laboratory Project: Running Circles Around Circles......Page 694
11.2 Calculus with Parametric Curves......Page 695
11.3 Polar Coordinates......Page 704
11.4 Areas and Lengths in Polar Coordinates......Page 715
11.5 Conic Sections......Page 719
11.6 Conic Sections in Polar Coordinates......Page 727
Review......Page 734
Problems Plus......Page 737
12 INFINITE SEQUENCES AND SERIES......Page 739
12.1 Sequences......Page 740
12.2 Series......Page 752
12.3 The Integral Test and Estimates of Sums......Page 762
12.4 The Comparison Tests......Page 770
12.5 Alternating Series......Page 775
12.6 Absolute Convergence and the Ratio and Root Tests......Page 779
12.7 Strategy for Testing Series......Page 786
12.8 Power Series......Page 788
12.9 Representations of Functions as Power Series......Page 793
12.10 Taylor and Maclaurin Series......Page 799
Writing Project: How Newton Discovered the Binomial Series......Page 813
12.11 Applications of Taylor Polynomials......Page 814
Applied Project: Radiation from the Stars......Page 822
Review......Page 823
Problems Plus......Page 826
APPENDIXES......Page 830
A: Numbers, Inequalities, and Absolute Values......Page 831
B: Coordinate Geometry and Lines......Page 839
C: Graphs of Second-Degree Equations......Page 845
D: Trigonometry......Page 853
E: Sigma Notation......Page 863
F: Proofs of Theorems......Page 868
G: Complex Numbers......Page 875
H: Answers to Odd-Numbered Exercises......Page 884
INDEX......Page 936