Table of Contents

1 Foundation for Calculus: Functions and Limits
 1.1 Functions and Change
 1.2 Exponential Functions
 1.3 New Functions from Old
 1.4 Logarithmic Functions
 1.5 Trigonometric Functions
 1.6 Powers, Polynomials, and Rational Functions
 1.7 Introduction to Limits and Continuity
 1.8 Extending the Idea of a Limit
 1.9 Further Limit Calculations Using Algebra
 1.10 Optional Preview of the Formal Definition of a Limit

2 Key Concept: The Derivative
 2.1 How Do We Measure Speed?
 2.2 The Derivative at a Point
 2.3 The Derivative Function
 2.4 Interpretations of the Derivative
 2.5 The Second Derivative
 2.6 Differentiability

3 ShortCuts to Differentiation
 3.1 Powers and Polynomials
 3.2 The Exponential Function
 3.3 The Product and Quotient Rules
 3.4 The Chain Rule
 3.5 The Trigonometric Functions
 3.6 The Chain Rule and Inverse Functions
 3.7 Implicit Functions
 3.8 Hyperbolic Functions
 3.9 Linear Approximation and the Derivative
 3.10 Theorems about Differentiable Functions

4 Using the Derivative
 4.1 Using First and Second Derivatives
 4.2 Optimization
 4.3 Optimization and Modeling
 4.4 Families of Functions and Modeling

4.5 Applications to Marginality
 4.6 Rates and Related Rates
 4.7 L’Hopital’s Rule, Growth, and Dominance
 4.8 Parametric Equations

5 Key Concept: The Definite Integral
 5.1 How Do We Measure Distance Traveled?
 5.2 The Definite Integral
 5.3 The Fundamental Theorem and Interpretations
 5.4 Theorems about Definite Integrals

6 Constructing Antiderivatives
 6.1 Antiderivatives Graphically and Numerically
 6.2 Constructing Antiderivatives Analytically
 6.3 Differential Equations and Motion
 6.4 Second Fundamental Theorem of Calculus

7 Integration
 7.1 Integration by Substitution
 7.2 Integration by Parts
 7.3 Tables of Integrals

7.4 Algebraic Identities and Trigonometric Substitutions
 Integration by Partial Fractions with Linear Factors
 Integration by Partial Fractions with Nonrepeated Irreducible Quadratic Factors
 Integration by Partial Fractions with Quadratic Factors
 Integration by Partial Fractions of Improper Fractions
 Integration by Partial Fraction Decomposition
 Trigonometric Substitutions
 7.5 Numerical Methods for Definite Integrals
 7.6 Improper Integrals
 7.7 Comparison of Improper Integrals

8 Using the Definite Integral

8.1 Areas and Volumes
 8.2 Applications to Geometry
 8.3 Area and Arc Length in Polar Coordinates
 8.4 Density and Center of Mass
 8.5 Applications to Physics
 8.6 Applications to Economics
 8.7 Distribution Functions
 8.8 Probability, Mean, and Median

8.1 Areas and Volumes

9 Sequences and Series
 9.1 Sequences
 9.2 Geometric Series
 9.3 Convergence of Series
 9.4 Tests for Convergence
 9.5 Power Series and Interval of Convergence

10 Approximating Functions Using Series

10.1 Taylor Polynomials

10.2 Taylor Series
 10.3 Finding and Using Taylor Series
 10.4 The Error in Taylor Polynomial Approximations
 10.5 Fourier Series

10.1 Taylor Polynomials

11 Differential Equations
 11.1 What Is a Differential Equation?

11.2 Slope Fields

11.3 Euler’s Method
 11.4 Separation of Variables
 11.5 Growth and Decay
 11.6 Applications and Modeling
 11.7 The Logistic Model
 11.8 Systems of Differential Equations
 11.9 Analyzing the Phase Plane
 11.10 SecondOrder Differential Equations: Oscillations
 11.11 Linear SecondOrder Differential Equations

12 Functions of Several Variables
 12.1 Functions of Two Variables
 12.2 Graphs and Surfaces
 12.3 Contour Diagrams
 12.4 Linear Functions
 12.5 Functions of Three Variables
 12.6 Limits and Continuity

13 A Fundamental Tool: Vectors
 13.1 Displacement Vectors
 13.2 Vectors in General
 13.3 The Dot Product
 13.4 The Cross Product

14 Differentiating Functions of Several Variables
 14.1 The Partial Derivative
 14.2 Computing Partial Derivatives Algebraically
 14.3 Local Linearity and the Differential
 14.4 Gradients and Directional Derivatives in the Plane
 14.5 Gradients and Directional Derivatives in Space
 14.6 The Chain Rule
 14.7 SecondOrder Partial Derivatives
 14.8 Differentiability

15 Optimization: Local and Global Extrema
 15.1 Critical Points: Local Extrema and Saddle Points
 15.2 Optimization
 15.3 Constrained Optimization: Lagrange Multipliers

16 Integrating Functions of Several Variables
 16.1 The Definite Integral of a Function of Two Variables
 16.2 Iterated Integrals
 16.3 Triple Integrals
 16.4 Double Integrals in Polar Coordinates
 16.5 Integrals in Cylindrical and Spherical Coordinates
 16.6 Applications of Integration to Probability

17 Parameterization and Vector Fields
 17.1 Parameterized Curves
 17.2 Motion, Velocity, and Acceleration
 17.3 Vector Fields
 17.4 The Flow of a Vector Field

18 Line Integrals
 18.1 The Idea of a Line Integral
 18.2 Computing Line Integrals over Parameterized Curves
 18.3 Gradient Fields and PathIndependent Fields
 18.4 PathDependent Vector Fields and Green’s Theorem

19 Flux Integrals and Divergence
 19.1 The Idea of a Flux Integral
 19.2 Flux Integrals for Graphs, Cylinders, and Spheres
 19.3 The Divergence of a Vector Field
 19.4 The Divergence Theorem

20 The Curl and Stokes’ Theorem
 20.1 The Curl of a Vector Field
 20.2 Stokes’ Theorem
 20.3 The Three Fundamental Theorems

21 Parameters, Coordinates, and Integrals
 21.1 Coordinates and Parameterized Surfaces
 21.2 Change of Coordinates in a Multiple Integral
 21.3 Flux Integrals over Parameterized Surfaces