0 BEFORE CALCULUS
0.1 Functions 1
0.2 New Functions from Old 15
0.3 Families of Functions 27
0.4 Inverse Functions; Inverse Trigonometric Functions 38
0.5 Exponential and Logarithmic Functions 52
1 LIMITS AND CONTINUITY
1.1 Limits (An Intuitive Approach) 67
1.2 Computing Limits 80
1.3 Limits at Infinity; End Behavior of a Function 89
1.4 Limits (Discussed More Rigorously) 100
1.5 Continuity 110
1.6 Continuity of Trigonometric, Exponential, and Inverse Functions 121
2 THE DERIVATIVE
2.1 Tangent Lines and Rates of Change 131
2.2 The Derivative Function 143
2.3 Introduction to Techniques of Differentiation 155
2.4 The Product and Quotient Rules 163
2.5 Derivatives of Trigonometric Functions 169
2.6 The Chain Rule 174
3 TOPICS IN DIFFERENTIATION
3.1 Implicit Differentiation 185
3.2 Derivatives of Logarithmic Functions 192
3.3 Derivatives of Exponential and Inverse Trigonometric Functions 197
3.4 Related Rates 204
3.5 Local Linear Approximation; Differentials 212
3.6 L'Hôpital?s Rule; Indeterminate Forms 219
4 THE DERIVATIVE IN GRAPHING AND APPLICATIONS
4.1 Analysis of Functions I: Increase, Decrease, and Concavity 232
4.2 Analysis of Functions II: Relative Extrema; Graphing Polynomials 244
4.3 Analysis of Functions III: Rational Functions, Cusps, and Vertical Tangents 254
4.4 Absolute Maxima and Minima 266
4.5 Applied Maximum and Minimum Problems 274
4.6 Rectilinear Motion 288
4.7 Newton?s Method 296
4.8 Rolle?s Theorem; Mean-Value Theorem 302
5 INTEGRATION
5.1 An Overview of the Area Problem 316
5.2 The Indefinite Integral 322
5.3 Integration by Substitution 332
5.4 The Definition of Area as a Limit; Sigma Notation 340
5.5 The Definite Integral 353
5.6 The Fundamental Theorem of Calculus 362
5.7 Rectilinear Motion Revisited Using Integration 376
5.8 Average Value of a Function and its Applications 385
5.9 Evaluating Definite Integrals by Substitution 390
5.10 Logarithmic and Other Functions Defined by Integrals 396
6 APPLICATIONS OF THE DEFINITE INTEGRAL IN GEOMETRY, SCIENCE, AND ENGINEERING
6.1 Area Between Two Curves 413
6.2 Volumes by Slicing; Disks and Washers 421
6.3 Volumes by Cylindrical Shells 432
6.4 Length of a Plane Curve 438
6.5 Area of a Surface of Revolution 444
6.6 Work 449
6.7 Moments, Centers of Gravity, and Centroids 458
6.8 Fluid Pressure and Force 467
6.9 Hyperbolic Functions and Hanging Cables 474
7 PRINCIPLES OF INTEGRAL EVALUATION
7.1 An Overview of Integration Methods 488
7.2 Integration by Parts 491
7.3 Integrating Trigonometric Functions 500
7.4 Trigonometric Substitutions 508
7.5 Integrating Rational Functions by Partial Fractions 514
7.6 Using Computer Algebra Systems and Tables of Integrals 523
7.7 Numerical Integration; Simpson?s Rule 533
7.8 Improper Integrals 547
8 MATHEMATICAL MODELING WITH DIFFERENTIAL EQUATIONS
8.1 Modeling with Differential Equations 561
8.2 Separation of Variables 568
8.3 Slope Fields; Euler?s Method 579
8.4 First-Order Differential Equations and Applications 586
9 INFINITE SERIES
9.1 Sequences 596
9.2 Monotone Sequences 607
9.3 Infinite Series 614
9.4 Convergence Tests 623
9.5 The Comparison, Ratio, and Root Tests 631
9.6 Alternating Series; Absolute and Conditional Convergence 638
9.7 Maclaurin and Taylor Polynomials 648
9.8 Maclaurin and Taylor Series; Power Series 659
9.9 Convergence of Taylor Series 668
9.10 Differentiating and Integrating Power Series; Modeling with Taylor Series 678
10 PARAMETRIC AND POLAR CURVES; CONIC SECTIONS
10.1 Parametric Equations; Tangent Lines and Arc Length for Parametric Curves 692
10.2 Polar Coordinates 705
10.3 Tangent Lines, Arc Length, and Area for Polar Curves 719
10.4 Conic Sections 730
10.5 Rotation of Axes; Second-Degree Equations 748
10.6 Conic Sections in Polar Coordinates 754
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