Table of Contents

Preface

Introduction

Review Chapter: Functions and Their Graphs

0.0 Introduction
0.1 Ways to define a function
0.2 Polynomials and rational functions
0.3 Transcendental functions
0.4 Plotting equations
0.5 Parametric curves
0.6 Shifting, scaling, and combining functions

Chapter 1: Tangents and Derivatives

1.0 Introduction
1.2 Local linearity
1.3 Limits
1.4 Continuity
1.5 The derivative at a point
1.6 Derivatives as functions
1.7 Asymptotic behavior

Chapter 2: Differentiation Rules and Properties

2.0 Introduction
2.1 Product and quotient rules
2.2 Chain rule
2.3 Derivatives of trigonometric functions
2.4 Differentiating implicit functions
2.5 Higher derivatives
2.6 Differentials
2.7 Parametric curves

Chapter 3: Applications of Derivatives

3.0 Introduction
3.1 Extreme values
3.2 Mean value theorem
3.3 Shape of a graph
3.4 Optimization
3.5 Newton's method
3.6 Indeterminate forms
3.7 Related rates

Chapter 4: Integrals

4.0 Introduction
4.1 Area function
4.2 Antiderivatives
4.3 Definite integrals
4.4 Fundamental theorem of calculus
4.5 Change of variable

Chapter 5: Applications of Integration

5.0 Introduction
5.1 Velocity and acceleration
5.2 Area between curves
5.3 Volume by cross sections
5.4 Volume by cylindrical shells
5.5 Average value of a function

Chapter 6: Calculus of Transcendental Functions

6.0 Introduction
6.1 Inverse functions
6.2 Natural logarithm
6.3 Exponential functions
6.4 Inverse trigonometric functions
6.5 Hyperbolic functions

Chapter 7: Techniques of Integration

7.0 Introduction
7.1 Integration by parts
7.2 Trigonometric functions
7.3 Trigonometric substitution
7.4 Partial fractions
7.5 Tables of integrals and further substitutions
7.6 Improper integrals

Chapter 8: Further Applications of Integration

8.0 Introduction
8.1 Polar coordinates
8.2 Arc length
8.3 Surface of revolution
8.4 Exponential growth and decay
8.5 Moments and center of mass

Chapter 9: Function Approximations

9.0 Introduction
9.1 Taylor polynomials
9.2 Polynomial interpolation
9.3 Splines
9.4 Bézier curves
9.5 Rational functions
9.6 Trigonometric functions

Chapter 10: Infinite Series

10.0 Introduction
10.1 Sequences
10.2 Series
10.3 Convergence tests
10.4 Power series
10.5 Maclaurin and Taylor series
10.6 Complex functions

Chapter 11: Numerical Integration

11.0 Introduction
11.1 Riemann sums
11.2 Simpson's rule
11.3 Taylor polynomials
11.4 Other numerical integration methods
11.5 Euler's method

Chapter 12: Vectors in Two and Three Dimensions

12.0 Introduction
12.1 Vectors in the plane
12.2 Vectors in space
12.3 Inner products and projections
12.4 Cross product
12.5 Lines and planes
12.6 Cylindrical and spherical coordinate systems
12.7 Surfaces

Chapter 13: Partial Derivatives

13.0 Introduction
13.1 Functions of several variables
13.2 Partial derivatives
13.3 Rules for partial derivatives
13.4 Local linearity
13.5 Directional derivatives and the gradient
13.6 Normals and the tangent plane
13.7 Extrema
13.8 Lagrange multipliers

Chapter 14: Multiple Integrals

14.0 Introduction
14.1 Double integrals
14.2 Iterated integrals
14.3 Double integrals in polar coordinates
14.4 Surface area
14.5 Triple integrals
14.6 Cylindrical and spherical coordinates

Chapter 15: Vector-Valued Functions

15.0 Introduction
15.1 Space curves
15.2 Derivatives and integrals
15.3 Arc length and curvature
15.4 Velocity and acceleration

Chapter 16: Vector Calculus

16.0 Introduction
16.1 Vector fields
16.2 Line integrals
16.3 Green's theorem
16.4 Surface integrals
16.6 Divergence theorem

Chapter 17: Differential Equations

17.0 Introduction
17.1 Solutions to differential equations
17.2 Differential equations with separable variables
17.3 Homogeneous differential equations
17.4 Exact differential equations
17.5 Exactness from integrating factors

Appendix A: Animations

A0: Introduction
A1: 2D Animations
A2: 3D Plots
A3: 3D Animations

Appendix B: Business Examples

B.1 Marginal analysis
B.2 Interest
B.3 Consumer and producer surplus
B.4 Probability
B.5 Expected value

Appendix C: Complex Numbers

C.1 Complex numbers

Appendix D: Matrices and Determinants

D0: Introduction
D1: Matrices and vectors
D2: Determinants
D3: Geometric transformations in two dimensions
D4: Geometric transformations in three dimensions

Appendix E: Engineering Examples

E.1 Work
E.2 Pressure
E.3 Moments of inertia

Index

Examples

Explorations