| |

| |

| |

The Cartesian Plane and Functions | |

| |

| |

| |

Real Numbers and the Real Line | |

| |

| |

| |

The Cartesian Plane | |

| |

| |

| |

Graphs of Equations | |

| |

| |

| |

Lines in the Plane | |

| |

| |

| |

Functions | |

| |

| |

| |

Limits and Their Properties | |

| |

| |

| |

An Introduction to Limits | |

| |

| |

| |

Techniques for Evaluating Limits | |

| |

| |

| |

Continuity | |

| |

| |

| |

Infinite Limits | |

| |

| |

| |

e-d Definition of Limits | |

| |

| |

| |

Differentiation | |

| |

| |

| |

The Derivative and the Tangent Line Problem | |

| |

| |

| |

Velocity, Acceleration, and Other Rates of Change | |

| |

| |

| |

Differentiation Rules for Powers, Constant Multiples, and Sums | |

| |

| |

| |

Differentiation Rules for Products and Quotients | |

| |

| |

| |

The Chain Rule | |

| |

| |

| |

Implicit Differentiation | |

| |

| |

| |

Related Rates | |

| |

| |

| |

Applications of Differentiation | |

| |

| |

| |

Extrema on an Interval | |

| |

| |

| |

Rolle's Theorem and the Mean Value Theorem | |

| |

| |

| |

Increasing and Decreasing Functions and the First Derivative Test | |

| |

| |

| |

Concavity and the Second Derivative Test | |

| |

| |

| |

Limits at Infinity | |

| |

| |

| |

A Summary of Curve Sketching | |

| |

| |

| |

Optimization Problems | |

| |

| |

| |

Newton's Method | |

| |

| |

| |

Differentials | |

| |

| |

| |

Business and Economics Applications | |

| |

| |

| |

Integration | |

| |

| |

| |

Antiderivatives and Indefinite Integration | |

| |

| |

| |

Area | |

| |

| |

| |

Riemann Sums and the Definite Integral | |

| |

| |

| |

The Fundamental Theorem of Calculus | |

| |

| |

| |

Integration by Substitution | |

| |

| |

| |

Numerical Integration | |

| |

| |

| |

Applications of Integration | |

| |

| |

| |

Area of a Region Between Two Curves | |

| |

| |

| |

Volume: The Disc Method | |

| |

| |

| |

Volume: The Shell Method | |

| |

| |

| |

Arc Length and Surfaces of Revolution | |

| |

| |

| |

Work | |

| |

| |

| |

Fluid Pressure and Fluid Force | |

| |

| |

| |

Moments, Centers of Mass, and Centroids | |

| |

| |

| |

Exponential and Logarithmic Functions | |

| |

| |

| |

Exponential Functions | |

| |

| |

| |

Differentiation and Integration of Exponential Functions | |

| |

| |

| |

Inverse Functions | |

| |

| |

| |

Logarithmic Functions | |

| |

| |

| |

Logarithmic Functions and Differentiation | |

| |

| |

| |

Logarithmic Functions and Integration | |

| |

| |

| |

Growth and Decay | |

| |

| |

| |

Indeterminate Forms and L'Hopital's Rule | |

| |

| |

| |

Trigonometric Functions and Inverse Trigonometric Functions | |

| |

| |

| |

Review of Trigonometric Functions | |

| |

| |

| |

Graphs and Limits of Trigonometric Functions | |

| |

| |

| |

Derivatives of Trigonometric Functions | |

| |

| |

| |

Integrals of Trigonometric Functions | |

| |

| |

| |

Inverse Trigonometric Functions and Differentiation | |

| |

| |

| |

Inverse Trigonometric Functions: Integration and Completing the Square | |

| |

| |

| |

Hyperbolic Functions | |

| |

| |

| |

Integration Techniques and Improper Integrals | |

| |

| |

| |

Basic Integration Formulas | |

| |

| |

| |

Integration by Parts | |

| |

| |

| |

Trigonometric Integrals | |

| |

| |

| |

Trigonometric Substitution | |

| |

| |

| |

Partial Fractions | |

| |

| |

| |

Integration by Tables and Other Integration Techniques | |

| |

| |

| |

Improper Integrals | |

| |

| |

| |

Infinite Series | |

| |

| |

| |

Sequences | |

| |

| |

| |

Series and Convergence | |

| |

| |

| |

The Integral Test and p-Series | |

| |

| |

| |

Comparisons of Series | |

| |

| |

| |

Alternating Series | |

| |

| |

| |

The Ratio and Root Tests | |

| |

| |

| |

Taylor Polynomials and Approximations | |

| |

| |

| |

Power Series | |

| |

| |

| |

Representation of Functions by Power Series | |

| |

| |

| |

Taylor and Maclaurin Series | |

| |

| |

| |

Conic Sections | |

| |

| |

| |

Parabolas | |

| |

| |

| |

Ellipses | |

| |

| |

| |

Hyperbolas | |

| |

| |

| |

Rotation and the General Second-Degree Equation | |

| |

| |

| |

Plane Curves, Parametric Equations, and Polar Coordinates | |

| |

| |

| |

Plane Curves and Parametric Equations | |

| |

| |

| |

Parametric Equations and Calculus | |

| |

| |

| |

Polar Coordinates and Polar Graphs | |

| |

| |

| |

Tangent Lines and Curve Sketching in Polar Coordinates | |

| |

| |

| |

Area and Arc Length in Polar Coordinates | |

| |

| |

| |

Polar Equations for Conics and Kepler's Laws | |

| |

| |

| |

Vectors and Curves in the Plane | |

| |

| |

| |

Vectors in the Plane | |

| |

| |

| |

The Dot Product of Two Vectors | |

| |

| |

| |

Vector-Valued Functions | |

| |

| |

| |

Velocity and Acceleration | |

| |

| |

| |

Tangent Vectors and Normal Vectors | |

| |

| |

| |

Arc Length and Curvature | |

| |

| |

| |

Solid Analytic Geometry and Vectors in Space | |

| |

| |

| |

Space Coordinates and Vectors in Space | |

| |

| |

| |

The Cross Product of Two Vectors in Space | |

| |

| |

| |

Lines and Planes in Space | |

| |

| |

| |

Surfaces in Space | |

| |

| |

| |

Curves and Vector-Valued Functions in Space | |

| |

| |

| |

Tangent Vectors, Normal Vectors, and Curvature in Space | |

| |

| |

| |

Functions of Several Variables | |

| |

| |

| |

Introduction to Functions of Several Variables | |

| |

| |

| |

Limits and Continuity | |

| |

| |

| |

Partial Derivatives | |

| |

| |

| |

Differentials | |

| |

| |

| |

Chain Rules for Functions of Several Variables | |

| |

| |

| |

Directional Derivatives and Gradients | |

| |

| |

| |

Tangent Planes and Normal Lines | |

| |

| |

| |

Extrema of Functions of Two Variables | |

| |

| |

| |

Applications of Extrema of Functions of Two Variables | |

| |

| |

| |

Lagrange Multipliers | |

| |

| |

| |

Multiple Integration | |

| |

| |

| |

Iterated Integrals and Area in the Plane | |

| |

| |

| |

Double Integrals and Volume | |

| |

| |

| |

Change of Variables: Polar Coordinates | |

| |

| |

| |

Center of Mass and Moments of Inertia | |

| |

| |

| |

Surface Area | |

| |

| |

| |

Triple Integrals and Applications | |

| |

| |

| |

Cylindrical and Spherical Coordinates | |

| |

| |

| |

Triple Integrals in Cylindrical and Spherical Coordinates | |

| |

| |

| |

Change of Variables: Jacobians | |

| |

| |

| |

Vector Analysis | |

| |

| |

| |

Vector Fields | |

| |

| |

| |

Line Integrals | |

| |

| |

| |

Conservative Vector Fields and Independence of Path | |

| |

| |

| |

Green's Theorem | |

| |

| |

| |

Parametric Surfaces | |

| |

| |

| |

Surface Integrals | |

| |

| |

| |

Divergence Theorem | |

| |

| |

| |

Stokes's Theorem | |

| |

| |

| |

Differential Equations | |

| |

| |

| |

Definitions and Basic Concepts | |

| |

| |

| |

Separation of Variables in First-Order Equations | |

| |

| |

| |

Exact First-Order Equations | |

| |

| |

| |

First-Order Linear Differential Equations | |

| |

| |

| |

Second-Order Homogeneous Linear Equations | |

| |

| |

| |

Second-Order Nonhomogeneous Linear Equations | |

| |

| |

| |

Series Solutions of Differential Equations | |

| |

| |

| |

Proofs of Selected Theorems | |

| |

| |

| |

Basic Differentiation Rules for Elementary Functions | |

| |

| |

| |

Integration Tables Answers to Odd-Numbered Exercises | |