1. Functions, Graphs, and Models

Functions and Mathematical Modeling
Graphs of Equations and Functions
Polynomials and Algebraic Functions
Transcendental Functions
Preview: What Is Calculus?

2. Prelude to Calculus

Tangent Lines and Slope Predictors
The Limit Concept
More about Limits
The Concept of Continuity

3. The Derivative

The Derivative and Rates of Change
Basic Differentiation Rules
The Chain Rule
Derivatives of Algebraic Functions
Maxima and Minima of Functions on Closed Intervals
Applied Optimization Problems
Derivatives of Trigonometric Functions
Successive Approximations and Newton's Method

4. Additional Applications of the Derivative

Implicit Differentiation and Related Rates
Increments, Differentials, and Linear Approximation
Increasing and Decreasing Functions and the Mean Value Theorem
The First Derivative Test and Applications
Simple Curve Sketching
Higher Derivatives and Concavity
Curve Sketching and Asymptotes

5. The Integral

Introduction
Antiderivatives and Initial Value Problems
Elementary Area Computations
Riemann Sums and the Integral
Evaluation of Integrals
The Fundamental Theorem of Calculus
Integration by Substitution
Areas of Plane Regions
Numerical Integration

6. Applications of the Integral

Riemann Sum Approximations
Volumes by the Method of Cross Sections
Volumes by the Method of Cylindrical Shells
Arc Length and Surface Area of Revolution
Force and Work
Centroids of Plane Regions and Curves

7. Calculus of Transcendental Functions

Exponential and Logarithmic Functions
Indeterminate Forms and L'Hopîtal's Rule
More Indeterminate Forms
The Logarithm as an Integral
Inverse Trigonometric Functions
Hyperbolic Functions

8. Techniques of Integration

Introduction
Integral Tables and Simple Substitutions
Integration by Parts
Trigonometric Integrals
Rational Functions and Partial Fractions
Trigonometric Substitutions
Integrals Involving Quadratic Polynomials
Improper Integrals

9. Differential Equations

Simple Equations and Models
Slope Fields and Euler's Method
Separable Equations and Applications
Linear Equations and Applications
Population Models
Linear Second-Order Equations
Mechanical Vibrations

10. Polar Coordinates and Parametric Curves

Analytic Geometry and the Conic Sections
Polar Coordinates
Area Computations in Polar Coordinates
Parametric Curves
Integral Computations with Parametric Curves
Conic Sections and Applications

11. Infinite Series

Introduction
Infinite Sequences
Infinite Series and Convergence
Taylor Series and Taylor Polynomials
The Integral Test
Comparison Tests for Positive-Term Series
Alternating Series and Absolute Convergence
Power Series
Power Series Computations
Series Solutions of Differential Equations

12. Vectors, Curves, and Surfaces in Space

Vectors in the Plane
Rectangular Coordinates and Three-Dimensional Vectors
The Cross Product of Vectors
Lines and Planes in Space
Curves and Motions in Space
Curvature and Acceleration
Cylinders and Quadric Surfaces
Cylindrical and Spherical Coordinates

13. Partial Differentiation

Introduction
Functions of Several Variables
Limits and Continuity
Partial Derivatives
Multivariable Maxima and Minima
Increments and Linear Approximation
The Multivariable Chain Rule
Directional Derivatives and Gradient Vectors
Lagrange Multipliers and Constrained Optimization
Critical Points of Multivariable Functions

14. Multiple Integrals

Double Integrals
Double Integrals over More General Regions
Area and Volume by Double Integration
Double Integrals in Polar Coordinates
Applications of Double Integrals
Triple Integrals
Integration in Cylindrical and Spherical Coordinates
Surface Area
Change of Variables in Multiple Integrals

15. Vector Calculus

Vector Fields
Line Integrals
The Fundamental Theorem and Independence of Path
Green's Theorem
Surface Integrals
The Divergence Theorem
Stokes' Theorem

Appendices
Answers
Index