### Chapter 1: Functions and Models

- Four Ways to Represent a Function
- Mathematical Models: A Catalog of Essential Functions
- New Functions from Old Functions
- Graphing Calculators and Computers
- Exponential Functions
- Inverse Functions and Logarithms
- Principles of Problem Solving

### Chapter 2: Limits and Derivatives

- The Tangent and Velocity Problems
- The Limit of a Function
- Calculating Limits Using the Limit Laws
- The Precise Definition of a Limit
- Continuity
- Limits at Infinity; Horizontal Asymptotes
- Derivatives and Rates of Change
- Writing Project: Early Methods for Finding Tangents
- The Derivative as a Function

### Chapter 3: Differentiation Rules

- Derivatives of Polynomials and Exponential Functions
- Applied Project: Building a Better Roller Coaster
- The Product and Quotient Rules
- Derivatives of Trigonometric Functions
- The Chain Rule
- Applied Project: Where Should a Pilot Start Descent?
- Implicit Differentiation
- Laboratory Project: Families of Implicit Curves
- Derivatives of Logarithmic Functions
- Rates of Change in the Natural and Social Sciences
- Exponential Growth and Decay
- Related Rates
- Linear Approximations and Differentials
- Laboratory Project: Taylor Polynomials
- Hyperbolic Functions

### Chapter 4: Applications of Differentiation

- Maximum and Minimum Values
- Applied Project: The Calculus of Rainbows
- The Mean Value Theorem
- How Derivatives Affect the Shape of a Graph
- Indeterminate Forms and L’Hospital’s Rule
- Writing Project: The Origins of L’Hospital’s Rule
- Summary of Curve Sketching
- Graphing with Calculus and Calculators
- Optimization Problems
- Applied Project: The Shape of a Can
- Newton’s Method
- Antiderivatives

### Chapter 5: Integrals

- Areas and Distances
- The Definite Integral
- Discovery Project: Area Functions
- The Fundamental Theorem of Calculus
- Indefinite Integrals and the Net Change Theorem
- Writing Project: Newton, Leibniz, and the Invention of Calculus
- The Substitution Rule

### Chapter 6: Applications of Integration

- Areas Between Curves
- Applied Project: The Gini Index
- Volumes
- Volumes by Cylindrical Shells
- Work
- Average Value of a Function
- Applied Project: Calculus and Baseball
- Applied Project: Where to Sit at the Movies

### Chapter 7: Techniques of Integration

- Integration by Parts
- Trigonometric Integrals
- Trigonometric Substitution
- Integration of Rational Functions by Partial Fractions
- Strategy for Integration
- Integration Using Tables and Computer Algebra Systems
- Discovery Project: Patterns in Integrals

### Chapter 8: Further Applications of Integration

- Arc Length
- Discovery Project: Arc Length Contest
- Area of a Surface of Revolution
- Discovery Project: Rotating on a Slant
- Applications to Physics and Engineering
- Discovery Project: Complementary Coffee Cups
- Applications to Economics and Biology
- Probability

### Chapter 9: Differential Equations

- Modeling with Differential Equations
- Direction Fields and Euler’s Method
- Separable Equations
- Applied Project: How Fast Does a Tank Drain?
- Applied Project: Which Is Faster, Going Up or Coming Down?
- Models for Population Growth
- Linear Equations
- Predator-Prey Systems

### Chapter 10: Parametric Equations and Polar Coordinates

- Curves Defined by Parametric Equations
- Laboratory Project: Running Circles around Circles
- Calculus with Parametric Curves
- Laboratory Project: Bézier Curves
- Polar Coordinates
- Laboratory Project: Families of Polar Curves
- Areas and Lengths in Polar Coordinates
- Conic Sections
- Conic Sections in Polar Coordinates

### Chapter 11: Infinite Sequences and Series

- Sequences
- Laboratory Project: Logistic Sequences
- Series
- The Integral Test and Estimates of Sums
- The Comparison Tests
- Alternating Series
- Absolute Convergence and the Ratio and Root Tests
- Strategy for Testing Series
- Power Series
- Representations of Functions as Power Series
- Taylor and Maclaurin Series
- Laboratory Project: An Elusive Limit
- Writing Project: How Newton Discovered the Binomial Series
- Applications of Taylor Polynomials
- Applied Project: Radiation from the Stars

### Chapter 12: Vectors and the Geometry of Space

- Three-Dimensional Coordinate Systems
- Vectors
- The Dot Product
- The Cross Product
- Discovery Project: The Geometry of a Tetrahedron
- Equations of Lines and Planes
- Laboratory Project: Putting 3D in Perspective
- Cylinders and Quadric Surfaces

### Chapter 13: Vector Functions

- Vector Functions and Space Curves
- Derivatives and Integrals of Vector Functions
- Arc Length and Curvature
- Motion in Space: Velocity and Acceleration
- Applied Project: Kepler’s Laws

### Chapter 14: Partial Derivatives

- Functions of Several Variables
- Limits and Continuity
- Partial Derivatives
- Tangent Planes and Linear Approximations
- The Chain Rule
- Directional Derivatives and the Gradient Vector
- Maximum and Minimum Values
- Applied Project: Designing a Dumpster
- Discovery Project: Quadratic Approximations and Critical Points

### Chapter 15: Multiple Integrals

- Double Integrals over Rectangles
- Iterated Integrals
- Double Integrals over General Regions
- Double Integrals in Polar Coordinates
- Applications of Double Integrals
- Surface Area
- Triple Integrals
- Discovery Project: Volumes of Hyperspheres
- Triple Integrals in Cylindrical Coordinates
- Discovery Project: The Intersection of Three Cylinders
- Triple Integrals in Spherical Coordinates
- Applied Project: Roller Derby
- Change of Variables in Multiple Integrals

### Chapter 16: Vector Calculus

- Vector Fields
- Line Integrals
- The Fundamental Theorem for Line Integrals
- Green’s Theorem
- Curl and Divergence
- Parametric Surfaces and Their Areas
- Surface Integrals
- Stokes’ Theorem
- Writing Project: Three Men and Two Theorems
- The Divergence Theorem
- Summary

### Chapter 17: Second-Order Differential Equations

- Second-Order Linear Equations
- Nonhomogeneous Linear Equations
- Applications of Second-Order Differential Equations
- Series Solutions