1. Functions
1.1 Functions and Their Graphs 1
1.2 Combining Functions; Shifting and Scaling Graphs 14
1.3 Trigonometric Functions 22
1.4 Graphing with Calculators and Computers 30
1.5 Exponential Functions 40
1.6 Inverse Functions and Logarithms 52
2. Limits and Derivatives
2.1 Rates of Change and Tangents to Curves 58
2.2 Limit of a Function and Limit Laws 65
2.3 Precise Definition of a Limit 76
2.4 One-Sided Limits 85
2.5 Continuity 92
2.6 Limits Involving Infinity, Asymptotes of Graphs 103
3. Differentiation
3.1 Tangents and the Derivative at a Point 122
3.2 The Derivative as a Function 126
3.3 Rules for Polynomials, Exponentials, Products, and Quotients 135
3.4 The Derivative as a Rate of Change 145
3.5 Derivatives of Trigonometric Functions 155
3.6 The Chain Rule 162
3.7 Implicit Differentiation 170
3.8 Derivatives of Inverse Functions and Logarithms 176
3.9 Inverse Trigonometric Functions 186
3.10 Related Rates 192
3.11 Linearization and Differentials 201
4. Applications of Derivatives
4.1 Extreme Values of Functions 222
4.2 The Mean Value Theorem 230
4.3 Monotonic Functions and the First Derivative Test 238
4.4 Concavity and Curve Sketching 243
4.5 Indeterminate Forms and L'Hopital's Rule 254
4.6 Applied Optimization 263
4.7 Newton's Method 274
4.8 Antiderivatives 279
5. Integration
5.1 Area and Estimating with Finite Sums 297
5.2 Sigma Notation and Limits of Finite Sums 307
5.3 The Definite Integral 313
5.4 The Fundamental Theorem of Calculus 325
5.5 Indefinite Integrals and the Substitution Rule 336
5.6 Substitution and Area Between Curves 344
6. Applications of Definite Integrals
6.1 Volumes Using Cross-Sections 363
6.2 Volumes Using Cylindrical Shells 374
6.3 Arc Length 382
6.4 Areas of Surfaces of Revolution 388
6.5 Work and Fluid Forces 393
6.6 Moments and Centers of Mass 402
7. Integrals and Transcendental Functions
7.1 The Logarithm Defined as an Integral 417
7.2 Exponential Change and Separable Differential Equations 427
7.3 Hyperbolic Functions 436
7.4 Relative Rates of Growth 444
8. Techniques of Integration
8.1 Integration by Parts 454
8.2 Trigonometric Integrals 462
8.3 Trigonometric Substitutions 467
8.4 Integration of Rational Functions by Partial Fractions 471
8.5 Integral Tables and Computer Algebra Systems 481
8.6 Numerical Integration 486
8.7 Improper Integrals 496
9. First-Order Differential Equations
9.1 Solutions, Slope Fields, and Euler's Method 514
9.2 First-Order Linear Equations 522
9.3 Applications 528
9.4 Graphical Solutions of Autonomous Equations 534
9.5 Systems of Equations and Phase Planes 541
10. Infinite Sequences and Series
10.1 Sequences 550
10.2 Infinite Series 562
10.3 The Integral Test 571
10.4 Comparison Tests 576
10.5 The Ratio and Root Tests 581
10.6 Alternating Series, Absolute and Conditional Convergence 586
10.7 Power Series 593
10.8 Taylor and Maclaurin Series 602
10.9 Convergence of Taylor Series 607
10.10 The Binomial Series and Applications of Taylor Series 614
11. Parametric Equations and Polar Coordinates
11.1 Parametrizations of Plane Curves 628
11.2 Calculus with Parametric Curves 636
11.3 Polar Coordinates 645
11.4 Graphing in Polar Coordinates 649
11.5 Areas and Lengths in Polar Coordinates 653
11.6 Conic Sections 657
11.7 Conics in Polar Coordinates 666
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