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Note: Each chapter concludes with Review Exercises and P.S. Problem Solving | |
Preparation for Calculus | |
Graphs and Models | |
Linear Models and Rates of Change | |
Functions and Their Graphs | |
Fitting Models to Data | |
Inverse Functions | |
Exponential and Logarithmic Functions | |
Limits and Their Properties | |
A Preview of Calculus | |
Finding Limits Graphically and Numerically | |
Evaluating Limits Analytically | |
Continuity and One–Sided Limits | |
Infinite Limits | |
Section Project: Graphs and Limits of Trigonometric Functions | |
Differentiation | |
The Derivative and the Tangent Line Problem | |
Basic Differentiation Rules and Rates of Change | |
The Product and Quotient Rules and Higher–Order Derivatives | |
The Chain Rule | |
Implicit Differentiation | |
Section Project: Optical Illusions | |
Derivatives of Inverse Functions | |
Related Rates | |
Newton's Method | |
Applications of Differentiation | |
Extrema on an Interval | |
Rolle's Theorem and the Mean Value Theorem | |
Increasing and Decreasing Functions and the First Derivative Test | |
Section Project: Rainbows | |
Concavity and the Second Derivative Test | |
Limits at Infinity | |
A Summary of Curve Sketching | |
Optimization Problems | |
Section Project: Connecticut River | |
Differentials | |
Integration | |
Antiderivatives and Indefinite Integration | |
Area | |
Riemann Sums and Definite Integrals | |
The Fundamental Theorem of Calculus | |
Section Project: Demonstrating the Fundamental Theorem | |
Integration by Substitution | |
Numerical Integration | |
The Natural Logarithmic Function: Integration | |
Inverse Trigonometric Functions: Integration | |
Hyperbolic Functions | |
Section Project: St. Louis Arch | |
Differential Equations | |
Slope Fields and Euler's Method | |
Differential Equations: Growth and Decay | |
Differential Equations: Separation of Variables | |
The Logistic Equation | |
First–Order Linear Differential Equations | |
Section Project: Weight Loss | |
Predator–Prey Differential Equations | |
Applications of Integration | |
Area of a Region Between Two Curves | |
Volume: The Disk Method | |
Volume: The Shell Method | |
Section Project: Saturn | |
Arc Length and Surfaces of Revolution | |
Work | |
Section Project: Tidal Energy | |
Moments, Centers of Mass, and Centroids | |
Fluid Pressure and Fluid Force | |
Integration Techniques, L'Hopital's Rule, and Improper Integrals | |
Basic Integration Rules | |
Integration by Parts | |
Trigonometric Integrals | |
Section Project: Power Lines | |
Trigonometric Substitution | |
Partial Fractions | |
Integration by Tables and Other Integration Techniques | |
Indeterminate Forms and L'Hopital's Rule | |
Improper Integrals | |
Infinite Series | |
Sequences | |
Series and Convergence | |
Section Project: Cantor's Disappearing Table | |
The Integral Test andp–Series | |
Section Project: The Harmonic Series | |
Comparisons of Series | |
Section Project: Solera Method | |
Alternating Series | |
The Ratio and Root Tests | |
Taylor Polynomials and Approximations | |
Power Series | |
Representation of Functions by Power Series | |
Taylor and Maclaurin Series | |
Conics, Parametric Equations, and Polar Coordinates | |
Conics and Calculus | |
Plane Curves and Parametric Equations | |
Section Project: Cycloids | |
Parametric Equations and Calculus | |
Polar Coordinates and Polar Graphs | |
Section Project: Anamorphic Art | |
Area and Arc Length in Polar Coordinates | |
Polar Equations of Conics and Kepler's Laws | |
Appendices | |
Appendix | |
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