Table of Contents

Chapter 1 Precalculus Review
 1.1 Real Numbers, Functions, and Graphs
 1.2 Linear and Quadratic Functions
 1.3 The Basic Classes of Functions
 1.4 Trigonometric Functions
 1.5 Technology: Calculators and Computers

Chapter 2 Limits
 2.1 Limits, Rates of Change, and Tangent Lines
 2.2 Limits: A Numerical and Graphical Approach
 2.3 Basic Limit Laws
 2.4 Limits and Continuity
 2.5 Evaluating Limits Algebraically
 2.6 Trigonometric Limits
 2.7 Limits at Infinity
 2.8 Intermediate Value Theorem
 2.9 The Formal Definition of a Limit

Chapter 3 Differentiation
 3.1 Definition of the Derivative
 3.2 The Derivative as a Function
 3.3 Product and Quotient Rules
 3.4 Rates of Change
 3.5 Higher Derivatives
 3.6 Trigonometric Functions
 3.7 The Chain Rule
 3.8 Implicit Differentiation
 3.9 Related Rates

Chapter 4 Applications of the Derivative
 4.1 Linear Approximation and Applications
 4.2 Extreme Values
 4.3 The Mean Value Theorem and Monotonicity
 4.4 The Shape of a Graph
 4.5 Graph Sketching and Asymptotes
 4.6 Applied Optimization
 4.7 Newton’s Method

Chapter 5 The Integral
 5.1 Approximating and Computing Area
 5.2 The Definite Integral
 5.3 The Indefinite Integral
 5.4 The Fundamental Theorem of Calculus, Part I
 5.5 The Fundamental Theorem of Calculus, Part II
 5.6 Net Change as the Integral of a Rate of Change
 5.7 Substitution Method

Chapter 6 Applications of the Integral
 6.1 Area between Two Curves
 6.2 Setting up Integrals: Volume, Density, Average Value
 6.3 Volumes of Revolution
 6.4 The Method of Cylindrical Shells
 6.5 Work and Energy

Chapter 7 Exponential Functions
 7.1 Derivative of 𝑓(𝑥) = 𝑏^𝑥 and the Number 𝑒
 7.2 Inverse Functions
 7.3 Logarithms and Their Derivatives
 7.4 Exponential Growth and Decay
 7.5 Compound Interest and Present Value
 7.6 Models Involving 𝑦′ = 𝑘(𝑦 − 𝑏)
 7.7 L’Hôpital’s Rule
 7.8 Inverse Trigonometric Functions
 7.9 Hyperbolic Functions

Chapter 8 Techniques of Integration
 8.1 Integration by Parts
 8.2 Trigonometric Integrals
 8.3 Trigonometric Substitution
 8.4 Integrals Involving Hyperbolic and Inverse Hyperbolic Functions
 8.5 The Method of Partial Fractions
 8.6 Strategies for Integration
 8.7 Improper Integrals
 8.8 Probability and Integration
 8.9 Numerical Integration

Chapter 9 Further Applications of the Integral and Taylor Polynomials
 9.1 Arc Length and Surface Area
 9.2 Fluid Pressure and Force
 9.3 Center of Mass
 9.4 Taylor Polynomials

Chapter 10 Introduction to Differential Equations
 10.1 Solving Differential Equations
 10.2 Graphical and Numerical Methods
 10.3 The Logistic Equation
 10.4 FirstOrder Linear Equations

Chapter 11 Infinite Series

11.1 Sequences
 11.2 Summing an Infinite Series
 11.3 Convergence of Series with Positive Terms
 11.4 Absolute and Conditional Convergence
 11.5 The Ratio and Root Tests and Strategies for Choosing Tests
 11.6 Power Series
 11.7 Taylor Series

11.1 Sequences

Chapter 12 Parametric Equations, Polar Coordinates, and Conic Sections
 12.1 Parametric Equations
 12.2 Arc Length and Speed
 12.3 Polar Coordinates
 12.4 Area and Arc Length in Polar Coordinates
 12.5 Conic Sections

Chapter 13 Vector Geometry
 13.1 Vectors in the Plane
 13.2 Vectors in Three Dimensions
 13.3 Dot Product and the Angle between Two Vectors
 13.4 The Cross Product
 13.5 Planes in 3Space
 13.6 A Survey of Quadric Surfaces
 13.7 Cylindrical and Spherical Coordinates

Chapter 14 Calculus of VectorValued Functions
 14.1 VectorValued Functions
 14.2 Calculus of VectorValued Functions
 14.3 Arc Length and Speed

14.4 Curvature
 14.5 Motion in 3Space
 14.6 Planetary Motion According to Kepler and Newton

Chapter 15 Differentiation in Several Variables
 15.1 Functions of Two or More Variables
 15.2 Limits and Continuity in Several Variables
 15.3 Partial Derivatives
 15.4 Differentiability and Tangent Planes
 15.5 The Gradient and Directional Derivatives
 15.6 The Chain Rule
 15.7 Optimization in Several Variables

15.8 Lagrange Multipliers: Optimizing with a Constraint

Chapter 16 Multiple Integration
 16.1 Integration in Two Variables
 16.2 Double Integrals over More General Regions
 16.3 Triple Integrals

16.4 Integration in Polar, Cylindrical, and Spherical Coordinates
 16.5 Applications of Multiple Integrals
 16.6 Change of Variables

Chapter 17 Line and Surface Integrals
 17.1 Vector Fields
 17.2 Line Integrals
 17.3 Conservative Vector Fields
 17.4 Parametrized Surfaces and Surface Integrals
 17.5 Surface Integrals of Vector Fields

Chapter 18 Fundamental Theorems of Vector Analysis
 18.1 Green’s Theorem
 18.2 Stokes’ Theorem
 18.3 Divergence Theorem

Appendices
 A The Language of Mathematics
 B Properties of Real Numbers
 C Induction and the Binomial Theorem
 D Additional Proofs

Additional Proofs
 L’Hôpital’s Rule
 Error Bounds for Numerical Integration
 Comparison Test for Improper Integrals

Additional Content
 Second Order Differential Equations
 Complex Numbers