Table of Contents

1 Functions and Models
 1.1 Four Ways to Represent a Function
 1.2 Mathematical Models: A Catalog of Essential Functions
 1.3 New Functions from Old Functions
 1.4 Exponential Functions
 1.5 Inverse Functions and Logarithms

2 Limits and Derivatives
 2.1 The Tangent and Velocity Problems
 2.2 The Limit of a Function
 2.3 Calculating Limits Using the Limit Laws
 2.4 The Precise Definition of a Limit
 2.5 Continuity
 2.6 Limits at Infinity; Horizontal Asymptotes
 2.7 Derivatives and Rates of Change
 2.8 The Derivative as a Function

3 Differentiation Rules
 3.1 Derivatives of Polynomials and Exponential Functions
 3.2 The Product and Quotient Rules
 3.3 Derivatives of Trigonometric Functions
 3.4 The Chain Rule
 3.5 Implicit Differentiation
 3.6 Derivatives of Logarithmic Functions

3.7 Rates of Change in the Natural and Social Sciences
 3.8 Exponential Growth and Decay
 3.9 Related Rates

3.10 Linear Approximations and Differentials
 3.11 Hyperbolic Functions

4 Applications of Differentiation
 4.1 Maximum and Minimum Values
 4.2 The Mean Value Theorem
 4.3 How Derivatives Affect the Shape of a Graph
 4.4 Indeterminate Forms and L’Hospital’s Rule
 4.5 Summary of Curve Sketching
 4.6 Graphing with Calculus and Calculators
 4.7 Optimization Problems
 4.8 Newton’s Method
 4.9 Antiderivatives

5 Integrals
 5.1 Areas and Distances
 5.2 The Definite Integral
 5.3 The Fundamental Theorem of Calculus
 5.4 Indefinite Integrals and the Net Change Theorem
 5.5 The Substitution Rule

6 Applications of Integration
 6.1 Areas Between Curves
 6.2 Volumes
 6.3 Volumes by Cylindrical Shells
 6.4 Work
 6.5 Average Value of a Function

7 Techniques of Integration
 7.1 Integration by Parts
 7.2 Trigonometric Integrals
 7.3 Trigonometric Substitution
 7.4 Integration of Rational Functions by Partial Fractions
 7.5 Strategy for Integration
 7.6 Integration Using Tables and Computer Algebra Systems
 7.7 Approximate Integration
 7.8 Improper Integrals

8 Further Application of Integration
 8.1 Arc Length

8.2 Area of a Surface of Revolution
 8.3 Applications to Physics and Engineering

8.4 Applications to Economics and Biology
 8.5 Probability

9 Differential Equations
 9.1 Modeling with Differential Equations

9.2 Direction Fields and Euler’s Method
 9.3 Separable Equations
 9.4 Models for Population Growth
 9.5 Linear Equations
 9.6 PredatorPrey Systems

10 Parametric Equations and Polar Coordinates
 10.1 Curves Defined by Parametric Equations
 10.2 Calculus with Parametric Curves
 10.3 Polar Coordinates
 10.4 Areas and Lengths in Polar Coordinates
 10.5 Conic Sections
 10.6 Conic Sections in Polar Coordinates

11 Infinite Sequences and Series
 11.1 Sequences
 11.2 Series
 11.3 The Integral Test and Estimates of Sums
 11.4 The Comparison Tests
 11.5 Alternating Series
 11.6 Absolute Convergence and the Ratio and Root Tests
 11.7 Strategy for Testing Series
 11.8 Power Series
 11.9 Representations of Functions as Power Series
 11.10 Taylor and Maclaurin Series
 11.11 Applications of Taylor Polynomials

12 Vectors and the Geometry of Space
 12.1 ThreeDimensional Coordinate Systems
 12.2 Vectors
 12.3 The Dot Product
 12.4 The Cross Product
 12.5 Equations of Lines and Planes

12.6 Cylinders and Quadric Surfaces

13 Vector Functions
 13.1 Vector Functions and Space Curves
 13.2 Derivatives and Integrals of Vector Functions

13.3 Arc Length and Curvature
 13.4 Motion in Space: Velocity and Acceleration

14 Partial Derivatives
 14.1 Functions of Several Variables
 14.2 Limits and Continuity
 14.3 Partial Derivatives
 14.4 Tangent Planes and Linear Approximations
 14.5 The Chain Rule
 14.6 Directional Derivatives and the Gradient Vector
 14.7 Maximum and Minimum Values

14.8 Lagrange Multipliers

15 Multiple Integrals
 15.1 Double Integrals over Rectangles
 15.2 Double Integrals over General Regions
 15.3 Double Integrals in Polar Coordinates
 15.4 Applications of Double Integrals
 15.5 Surface Area
 15.6 Triple Integrals
 15.7 Triple Integrals in Cylindrical Coordinates

15.8 Triple Integrals in Spherical Coordinates
 15.9 Change of Variables in Multiple Integrals

16 Vector Calculus
 16.1 Vector Fields
 16.2 Line Integrals
 16.3 The Fundamental Theorem for Line Integrals
 16.4 Green’s Theorem

16.5 Curl and Divergence
 16.6 Parametric Surfaces and Their Areas

16.7 Surface Integrals

16.8 Stokes’ Theorem

16.9 The Divergence Theorem
 16.10 Summary

17 SecondOrder Differential Equations
 17.1 SecondOrder Linear Equations
 17.2 Nonhomogeneous Linear Equations

17.3 Applications of SecondOrder Differential Equations
 17.4 Series Solutions

Appendixes
 A Numbers, Inequalities, and Absolute Values
 B Coordinate Geometry and Lines
 C Graphs of SecondDegree Equations
 D Trigonometry
 E Sigma Notation
 F Proofs of Theorems
 G The Logarithm Defined as an Integral
 H Complex Numbers