(Practice Exercises, Additional Exercises, and Questions to Guide Your Review appear at the end of each chapter.) | |
Preliminaries Real Numbers and the Real Line Lines, Circles, and Parabolas | |
Functions and Their Graphs | |
Identifying Functions; Mathematical Models | |
Combining Functions; Shifting and Scaling Graphs | |
Trigonometric Functions | |
Graphing with Calculators and Computers | |
Limits and Derivatives | |
Rates of Change and Limits | |
Calculating Limits Using the Limit Laws | |
Precise Definition of a Limit | |
One-Sided Limits and Limits at Infinity | |
Infinite Limits and Vertical Asymptotes | |
Continuity | |
Tangents and Derivatives | |
Differentiation | |
The Derivative as a Function | |
Differentiation Rules | |
The Derivative as a Rate of Change | |
Derivatives of Trigonometric Functions | |
The Chain Rule and Parametric Equations | |
Implicit Differentiation | |
Related Rates | |
Linearization and Differentials | |
Applications of Derivatives | |
Extreme Values of Functions | |
The Mean Value Theorem | |
Monotonic Functions and the First Derivative Test | |
Concavity and Curve Sketching | |
Applied Optimization Problems | |
Indeterminate Forms and L'Hopital's Rule | |
Newton's Method | |
Antiderivatives | |
Integration | |
Estimating with Finite Sums | |
Sigma Notation and Limits of Finite Sums | |
The Definite Integral | |
The Fundamental Theorem of Calculus | |
Indefinite Integrals and the Substitution Rule | |
Substitution and Area Between Curves | |
Applications of Definite Integrals | |
Volumes by Slicing and Rotation About an Axis | |
Volumes by Cylindrical Shells | |
Lengths of Plane Curves | |
Moments and Centers of Mass | |
Areas of Surfaces of Revolution and The Theorems of Pappus | |
WorkFluid Pressures and Forces | |
Transcendental Functions | |
Inverse Functions and their Derivatives | |
Natural Logarithms | |
The Exponential Functionax and loga xExponential Growth and Decay | |
Relative Rates of Growth | |
Inverse Trigonometric Functions | |
Hyperbolic Functions | |
Techniques of Integration | |
Basic Integration Formulas | |
Integration by Parts | |
Integration of Rational Functions by Partial Fractions | |
Trigonometric Integrals | |
Trigonometric Substitutions | |
Integral Tables and Computer Algebra Systems | |
Numerical Integration | |
Improper Integrals | |
Further Applications of Integration | |
Slope Fields and Separable Differential Equations | |
First-Order Linear Differential Equations | |
Euler's Method | |
Graphical Solutions of Autonomous Equations | |
Applications of First-Order Differential Equations | |
Conic Sections and Polar Coordinates | |
Conic Sections and Quadratic Equations | |
Classifying Conic Sections by Eccentricity | |
Quadratic Equations and Rotations | |
Conics and Parametric Equations; The Cycloid | |
Polar Coordinates | |
Graphing in Polar Coordinates | |
Area and Lengths in Polar Coordinates | |
Conic Sections in Polar Coordinates | |
Infinite Sequences and Series | |
Sequences Infinite Series | |
The Integral Test | |
Comparison Tests | |
The Ratio and Root Tests | |
Alternating Series, Absolute and Conditional Convergence | |
Power Series | |
Taylor and Maclaurin Series | |
Convergence of Taylor Series; Error Estimates | |
Applications of Power Series | |
Fourier Series | |
Appendices | |
Mathematical Induction | |
Proofs of Limit Theorems | |
Commonly Occurring Limits | |
Theory of the Real Numbers | |
Complex Numbers | |
The Distributive Law for Vector Cross Products | |
Determinants and Cramer's Rule | |
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