Calculus

1 Functions and Graphs
1.1 What Is Calculus?
1.2 Preliminaries
1.3 Lines in the Plane; Parametric Equations
1.4 Functions and Graphs
1.5 Inverse Functions; Inverse Trigonometric Functions
Chapter 1 Review
Book Report: Ethnomathematics by Marcia Ascher
Chapter 1 Group Research Project

2 Limits and Continuity
2.1 The Limit of a Function
2.2 Algebraic Computation of Limits
2.3 Continuity
2.4 Exponential and Logarithmic Functions
Chapter 2 Review
Chapter 2 Group Research Project

3 Differentiation
3.1 An Introduction to the Derivative: Tangents
3.2 Techniques of Differentiation
3.3 Derivatives of Trigonometric, Exponential, and Logarithmic Functions
3.4 Rates of Change: Modeling Rectilinear Motion
3.5 The Chain Rule
3.6 Implicit Differentiation
3.7 Related Rates and Applications
3.8 Linear Approximation and Differentials
Chapter 3 Review
Book Report: Fermat's Enigma by Simon Singh
Chapter 3 Group Research Project

4 Additional Applications of the Derivative
4.1 Extreme Values of a Continuous Function
4.2 The Mean Value Theorem
4.3 Using Derivatives to Sketch the Graph of a Function
4.4 Curve Sketching with Asymptotes: Limits Involving Infinity
4.5 Hopital's Rule
4.6 Optimization in the Physical Sciences and Engineering
4.7 Optimization in Business, Economics, and the Life Sciences
Chapter 4 Review
Chapter 4 Group Research Project

5 Integration
5.1 Antidifferentiation
5.2 Area as the Limit of a Sum
5.3 Riemann Sums and the Definite Integral
5.4 The Fundamental Theorems of Calculus
5.5 Integration by Substitution
5.6 Introduction to Differential Equations
5.7 The Mean Value Theorem for Integrals; Average Value
5.8 Numerical Integration: The Trapezoidal Rule and Simpson's Rule
5.9 An Alternative Approach: The Logarithm as an Integral
Chapter 5 Review
Chapter 5 Group Research Project

Cumulative Review Problems - Chapters 1–5

6 Additional Applications of the Integral
6.1 Area Between Two Curves
6.2 Volume
6.3 Polar Forms and Area
6.4 Arc Length and Surface Area
6.5 Physical Applications: Work, Liquid Force, and Centroids
6.6 Applications to Business, Economics, and Life Sciences
Chapter 6 Review
Book Report: To Infinity and Beyond, A Cultural History of the Infinite, by Eli Maor
Chapter 6 Group Research Project

7 Methods of Integration
7.1 Review of Substitution and Integration by Table
7.2 Integration By Parts
7.3 Trigonometric Methods
7.4 Method of Partial Fractions
7.5 Summary of Integration Techniques
7.6 First-Order Differential Equations
7.7 Improper Integrals
7.8 Hyperbolic and Inverse Hyperbolic Functions
Chapter 7 Review
Chapter 7 Group Research Project

8 Infinite Series
8.1 Sequences and Their Limits
8.2 Introduction to Infinite Series; Geometric Series
8.3 The Integral Test; p-series
8.4 Comparison Tests
8.5 The Ratio Test and the Root Test
8.6 Alternating Series; Absolute and Conditional Convergence
8.7 Power Series
8.8 Taylor and Maclaurin Series
Chapter 8 Review
Chapter 8 Group Research Project

Cumulative Review Problems - Chapters 6–8

9 Vectors in the Plane and in Space
9.1 Vectors in R2
9.2 Coordinates and Vectors in R3
9.3 The Dot Product
9.4 The Cross Product
9.5 Lines in R3
9.6 Planes in R3
9.7 Quadric Surfaces
Chapter 9 Review
Chapter 9 Group Research Project

10 Vector-Valued Functions
10.1 Introduction to Vector Functions
10.2 Differentiation and Integration of Vector Functions
10.3 Modeling Ballistics and Planetary Motion
10.4 Unit Tangent and Principal Unit Normal Vectors; Curvature
10.5 Tangential and Normal Components of Acceleration
Chapter 10 Review
Chapter 10 Group Research Project

Cumulative Review Problems - Chapters 1–10

11 Partial Differentiation
11.1 Functions of Several Variables
11.2 Limits and Continuity
11.3 Partial Derivatives
11.4 Tangent Planes, Approximations, and Differentiability
11.5 Chain Rules
11.6 Directional Derivatives and the Gradient
11.7 Extrema of Functions of Two Variables
11.8 Lagrange Multipliers
Chapter 11 Review
Book Report: Hypatia's Heritage by Margaret Alic
Chapter 11 Group Research Project

12 Multiple Integration
12.1 Double Integration over Rectangular Regions
12.2 Double Integration over Nonrectangular Regions
12.3 Double Integrals in Polar Coordinates
12.4 Surface Area
12.5 Triple Integrals
12.6 Mass, Moments, and Probability Density Functions
12.7 Cylindrical and Spherical Coordinates
12.8 Jacobians: Change of Variables
Chapter 12 Review
Chapter 12 Group Research Project

13 Vector Analysis
13.1 Properties of a Vector Field: Divergence and Curl
13.2 Line Integrals
13.3 The Fundamental Theorem and Path Independence
13.4 Green's Theorem
13.5 Surface Integrals
13.6 Stokes' Theorem and Applications
13.7 Divergence Theorem and Applications
Chapter 13 Review
Chapter 13 Group Research Project

Cumulative Review Problems - Chapters 11–13

14 Introduction to Differential Equations
14.1 First-Order Differential Equations
14.2 Second-Order Homogeneous Linear Differential Equations
14.3 Second-Order Nonhomogeneous Linear Differential Equations
Chapter 14 Review
Book Report: Mathematical Experience by Philip J. Davis and Reuben Hersh
Chapter 14 Group Research Project

Appendices
A: Introduction to the Theory of Limits
B: Selected Proofs
C: Significant Digits
D: Short Table of Integrals
E: Trigonometry
F: Parabolas
G: Ellipses
H: Hyperbolas
I: Determinants
J: Answers to Selected Problems