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 Applications 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

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
 I Answers to OddNumbered Exercises