Chapter 1. Precalculus Review.1
1.1 What is Calculus? 1
1.2 Review of Elementary Mathematics.3
1.3 Review of Inequalities.11
1.4 Coordinate Plane; Analytic Geometry.17
1.5 Functions.24
1.6 The Elementary Functions.32
1.7 Combinations of Functions.41
1.8 A Note on Mathematical Proof; Mathematical Induction.47
Chapter 2. Limits and Continuity.53
2.1 The Limit Process (An Intuitive Introduction).53
2.2 Definition of Limit.64
2.3 Some Limit Theorems.73
2.4 Continuity.82
2.5 The Pinching Theorem; Trigonometric Limits.91
2.6 Two Basic Theorems.97
Project 2.6 The Bisection Method for Finding the Roots of f (x) = 0 102
Chapter 3. The Derivative; The Process of Differentiation.105
3.1 The Derivative.105
3.2 Some Differentiation Formulas.115
3.3 The d/dx Notation; Derivatives of Higher Order.124
3.4 The Derivative as a Rate of Change.130
3.5 The Chain Rule.133
3.6 Differentiating the Trigonometric Functions.142
3.7 Implicit Differentiation; Rational Powers.147
Chapter 4. The Mean-Value Theorem; Applications of the First and Second Derivatives.154
4.1 The Mean-Value Theorem.154
4.2 Increasing and Decreasing Functions.160
4.3 Local Extreme Values.167
4.4 Endpoint Extreme Values; Absolute Extreme Values.174
4.5 Some Max-Min Problems.182
Project 4.5 Flight Paths of Birds 190
4.6 Concavity and Points of Inflection.190
4.7 Vertical and Horizontal Asymptotes; Vertical Tangents and Cusps.195
4.8 Some Curve Sketching.201
4.9 Velocity and Acceleration; Speed.209
Project 4.9A Angular Velocity; Uniform Circular Motion 217
Project 4.9B Energy of a Falling Body (Near the Surface of the Earth) 217
4.10 Related Rates of Change Per Unit Time.218
4.11 Differentials.223
Project 4.11 Marginal Cost, Marginal Revenue, Marginal Profit 228
4.12 Newton-Raphson Approximations.229
Chapter 5. Integration.234
5.1 An Area Problem; A Speed-Distance Problem.234
5.2 The Definite Integral of a Continuous Function.234
5.3 The Function f(x) = Integral from a to x of f(t) dt.246
5.4The Fundamental Theorem of Integral Calculus.254
5.5 Some Area Problems.260
Project 5.5 Integrability; Integrating Discontinuous Functions 266
5.6 Indefinite Integrals.268
5.7 Working Back from the Chain Rule; the u-Substitution.274
5.8 Additional Properties of the Definite Integral.281
5.9 Mean-Value Theorems for Integrals; Average Value of a Function.285
Chapter 6. Some Applications of the Integral.292
6.1 More on Area.292
6.2 Volume by Parallel Cross-Sections; Discs and Washers.296
6.3 Volume by the Shell Method.306
6.4 The Centroid of a Region; Pappus's Theorem on Volumes.312
Project 6.4 Centroid of a Solid of Revolution 319
6.5 The Notion of Work.319
6.6 Fluid Force.327
Chapter 7. The Transcendental Functions.333
7.1 One-to-One Functions; Inverse Functions.333
7.2 The Logarithm Function, Part I.342
7.3 The Logarithm Function, Part II.347
7.4 The Exponential Function.356
Project 7.4 Some Rational Bounds for the Number e 364
7.5 Arbitrary Powers; Other Bases.364
7.6 Exponential Growth and Decay.370
7.7 The Inverse Trigonometric Functions.378
Project 7.7 Refraction 387
7.8 The Hyperbolic Sine and Cosine.388
7.9 The Other Hyperbolic Functions.392
Chapter 8. Techniques of Integration.398
8.1 Integral Tables and Review.398
8.2 Integration by Parts.402
Project 8.2 Sine Waves y = sin nx and Cosine Waves y = cos nx 410
8.3 Powers and Products of Trigonometric Functions.411
8.4 Integrals Featuring Square Root of (a^2 - x^2), Square Root of (a^2 + x^2), and Square Root of (x^2 - a^2).417
8.5 Rational Functions; Partial Functions.422
8.6 Some Rationalizing Substitutions.430
8.7 Numerical Integration.433
Chapter 9. Some Differential Equations.443
9.1 First-Order Linear Equations.444
9.2 Integral Curves; Separable Equations.451
Project 9.2 Orthogonal Trajectories 458
9.3 The Equation y¿¿ + ay¿+ by = 0.459
Chapter 10. The Conic Sections; Polar Coordinates; Parametric Equations.469
10.1 Geometry of Parabola, Ellipse, Hyperbola.469
10.2 Polar Coordinates.478
10.3 Graphing in Polar Coordinates.484
Project 10.3 Parabola, Ellipse, Hyperbola in Polar Coordinates 491
10.4 Area in Polar Coordinates.492
10.5 Curves Given Parametrically.496
Project 10.5 Parabolic Trajectories 503
10.6 Tangents to Curves Given Parametrically.503
10.7 Arc Length and Speed.509
10.8 The Area of a Surface of Revolution; Pappus's Theorem on Surface. Area 517
Project 10.8 The Cycloid 525
Chapter 11. Sequences; Indeterminate Forms; Improper Integrals.528
11.1 The Least Upper Bound Axiom.528
11.2 Sequences of Real Numbers.532
11.3 The Limit of a Sequence.538
Project 11.3 Sequences and the Newton-Raphson Method 547
11.4 Some Important Limits.550
11.5 The Indeterminate Forms (0/0).554
11.6 The Indeterminate Form (¿/¿); Other Indeterminate Forms.560
11.7 Improper Integrals.565
Chapter 12. Infinite Series.575
12.1 Sigma Notation 575
12.2 Infinite Series 577
12.3 The Integral Test; Basic Comparison, Limit Comparison 585
12.4 The Root Test; the Ratio Test 593
12.5 Absolute Convergence and Conditional Convergence; Alternating Series 597
12.6 Taylor Polynomials in x; Taylor Series in x 602
12.7 Taylor Polynomials and Taylor Series in x ¿ a 613
12.8 Power Series 616
12.9 Differentiation and Integration of Power Series 623
Project 12.9A The Binomial Series 633
Project 12.9B Estimating ¿ 634
Appendix. A. Some Additional Topics. A-1
A.1 Rotation of Axes; Eliminating the xy-Term A-1
A.2 Determinants A-3
Appendix B. Some Additional Proofs. A-8
B.1 The Intermediate-Value Theorem A-8
B.2 Boundedness; Extreme-Value Theorem A-9
B.3 Inverses A-10
B.4 The Integrability of Continuous Functions A-11
B.5 The Integral as the Limit of Riemann Sums A-14
Answers to Odd-Numbered Exercises A-15
Index I-1
Table of Integrals Inside Covers
