Table of Contents

1 Rates of Change, Limits, and the Derivative

1.1 Functions

1.2 Compositions of Functions

1.3 Slope as a Rate of Change

1.4 Calculating Rates of Change

1.5 Limits

1.6 More Work with Limits

1.7 The Derivative

• Review of Key Concepts
• Chapter Review Exercises
• Student Project: Exploring Chaos

2 Finding the Derivative

2.1 Derivatives of Polynomials

2.2 Derivatives of Products and Quotients

2.3 Differentiating Compositions

2.4 Implicit Differentiation

2.5 Trigonometric Functions

2.6 Exponential Functions

2.7 Logarithms

2.8 Inverse Functions

2.9 Inverse Trigonometric Functions

2.10 Modeling: Translating the World into Mathematics

• Review of Key Concepts
• Chapter Review Exercises
• Student Project: Lamborghini Diablo VT 6.0 Test Results

3 Motion, Vectors, and Parametric Equations 3.1 Motion Along a Line

3.2 Vectors

3.3 Parametric Equations

3.4 Velocity and Tangent Vectors

3.5 Dot Product

3.6 Newton's Laws

• Review of Key Concepts
• Chapter Review Exercises
• Student Project: Timing a Rifle Bullet
• Student Project: The Quarterback's Problem

4 Applications of the Derivative

4.1 The Tangent Line Approximation

4.2 Newton's Method

4.3 Increasing/Decreasing Functions; Concavity

4.4 Horizontal and Vertical Asymptotes

4.5 Tools for Optimization

4.6 Modeling Optimization Problems

4.7 Related Rates

4.8 Indeterminate Forms and l'Hôpital's Rules

4.9 Euler's Method

• Review of Key Concepts
• Chapter Review Exercises
• Student Project: Retrograde Motion of Mars
• Student Project: Faster Than Light?
• Student Project: Optimal Box Problem

5 The Integral

5.1 Summation Notation

5.2 The Definite Integral

5.3 The Fundamental Theorem of Calculus

5.4 The Indefinite Integral

5.5 Integration by Substitution

5.6 Areas between Curves

5.7 Integration by Parts

5.8 Integration by Partial Fractions

5.9 Solving Simple Differential Equations

5.10 Numerical Integration

• Review of Key Concepts
• Chapter Review Exercises
• Student Project: Small-Rocket Data
• Student Project: Improved Calculation of "Hyperbola-Areas"
• Student Project: How to Discover an Integration Algorithm

6 Applications of the Integral

6.1 Volumes by Cross Section

6.2 Volumes by Shells

6.3 Polar Coordinates and Parametric Equations

6.4 Arc Length and Unit Tangent Vectors

6.5 Areas of Regions Described by Polar Equations

6.6 Work

6.7 Center of Mass

6.8 Curvature, Acceleration, and Kepler's Second Law

6.9 Improper Integrals

• Review of Key Concepts
• Chapter Review Exercises
• Student Project: Tsar-Kolokol (Tsar Bell)
• Student Project: Optimal Sprayer

7 Infinite Series, Sequences, and Approximations

7.1 Taylor Polynomials

7.2 Approximations and Error

7.3 Sequences

7.4 Infinite Series

7.5 Tests for Convergence

7.6 Power Series and Taylor Series

7.7 Working with Power Series

• Review of Key Concepts
• Chapter Review Exercises
• Student Project: Signal Processing and Fourier Series

8 Vectors and Linear Functions

8.1 Vectors in Three Dimensions

8.2 Matrices and Determinants

8.3 The Cross Product

8.4 Linear Functions

8.5 The Geometry of Linear Functions

8.6 Planes

8.7 Motion in Three Dimensions

• Review of Key Concepts
• Chapter Review Exercises
• Student Project: The Cross Ratio and Photo Analysis

9 Functions of Several Variables

9.1 Conic Sections

9.2 Real-World Functions

9.3 Graphing: Surfaces and Level Curves

9.4 Graphing: Parametric Representations of Surfaces

9.5 Cylindrical and Spherical Coordinates

9.6 Limits

9.7 Derivatives

• Review of Key Concepts
• Chapter Review Exercises
• Student Project: Ruled Surfaces

10 Differentiable Functions of Several Variables

10.1 Differentiability

10.2 The Chain Rule

10.3 Applications of the Chain Rule

10.4 Further Applications of the Chain Rule

10.5 Optimization

10.6 Second Derivatives Test

10.7 Optimization with Constraints

• Review of Key Concepts
• Chapter Review Exercises
• Student Project: Cauchy's Method of Steepest Descent
• Student Project: Wave Equation
• Student Project: Rhumb Lines on Mercator Maps

11 Multiple Integrals

11.1 The Double Integral on Rectangles

11.2 Extending the Double Integral and Applications

11.3 Surface Area

11.4 Change-of-Variables Formula for Double Integrals

11.5 Triple Integrals

11.6 Change-of-Variables Formula for Triple Integrals

• Review of Key Concepts
• Chapter 11 Review Exercises
• Student Project: On the Attraction of Spheroids of Revolution

12 Line and Surface Integrals

12.1 The Line Integral

12.2 Vector Fields, Work, and Flows

12.3 The Fundamental Theorem of Line Integrals

12.4 Green's Theorem

12.5 Divergence and Curl

12.6 Surface Integrals

12.7 The Divergence Theorem

12.8 Stokes' Theorem

• Review of Key Concepts
• Chapter Review Exercises
• Student Project: Solid Angles
• Student Project: The Polar Planimeter

Appendix: Review Material

A.1 Algebra

A.2 Trigonometry

A.3 Polar Coordinates

A.4 Mathematical Induction

Answers to Selected Exercises

Index