(Mathematica^®) for Rogawski's Calculus

2nd Edition 2010
Based on Mathematica Version 7

Abdul Hassen, Gary Itzkowitz, Hieu D. Nguyen, Jay Schiffman

W. H. Freeman and Company
New York

© Copyright 2010

Table of Contents

Chapter  1	Introduction

1.1 	Getting Started

1.1.1 	First-Time Users of Mathematica 7

1.1.2 	Entering and Evaluating Input Commands

1.1.3 	Documentation Center (Help Menu)

1.2	Mathematica's Conventions for Inputting Commands

1.2.1 	Naming

1.2.2 	Parentheses, Brackets, and Braces

1.2.3 	Lists

1.2.4 	Equal Signs

1.2.5 	Referring to Previous Results

1.2.6 	Commenting

1.3 	Basic Calculator Operations

1.4 	Functions

1.5 	Palettes

1.6	Solving Equations

Chapter  2	Graphs, Limits, and Continuity of Functions

2.1 	Plotting Graphs

2.1.1 	Basic Plot

2.1.2 	Plot Options

2.2	Limits

2.2.1 	Evaluating Limits

2.2.2 	Limits Involving Trigonometric Functions

2.2.3 	Limits Involving Infinity

2.3	Continuity

Chapter 3.	Differentiation

3.1	The Derivative

3.1.1 	Slope of Tangent

3.1.2 	Derivative as a Function

3.2	Higher-Order Derivatives

3.3	Chain Rule and Implicit Differentiation

3.4	Derivatives of Inverse, Exponential and Logarithmic Functions

3.4.1 	Inverse of a Function

3.4.2	Exponential and Logarithmic Functions

Chapter  4	Applications of the Derivative

4.1 	Related Rates

4.2	Extrema

4.3	Optimization

4.3.1 	Traffic Flow

4.3.2 	Minimum Cost

4.3.3 	Packaging (Minimum Surface Area)

4.3.4 	Maximum Revenue

4.4	Newton's Method

4.4.1 	Programming Newton's Method

4.4.2 	Divergence

4.4.3 	Slow Convergence

Chapter  5	Integration

5.1 	Antiderivatives (Indefinite Integral)

5.2	Riemann Sums and the Definite Integral

5.2.1 	Riemann Sums Using Left Endpoints

5.2.2 	Riemann Sums Using Right Endpoints

5.2.3 	Riemann Sums Using Midpoints

5.3	The Fundamental Theorem of Calculus

5.4	Integrals Involving Trigonometric, Exponential, and Logarithmic Functions

Chapter  6	Applications of the Integral

6.1 	Area Between Curves

6.2 	Average Value

6.3	Volumes of Solids of Revolution

6.3.1 	The Method of Discs

6.3.2 	The Method of Washers

6.3.3 	The Method of Cylindrical Shells

Chapter  7	Techniques of Integration

7.1 	Numerical Integration

7.1.1 	Trapezoidal Rule

7.1.2 	Simpson's Rule

7.1.3 	Midpoint Rule

7.2 	Techniques of Integration

7.2.1 	Substitution

7.2.2 	Trigonometric Substitution

7.2.3 	Method of Partial Fractions

7.3 	Improper Integrals

7.4	Hyperbolic Functions

7.4.1 	Hyperbolic Functions

7.4.2 	Identities Involving Hyperbolic Functions

7.4.3 	Derivatives of Hyperbolic Functions

7.4.4 	Inverse Hyperbolic Functions

Chapter  8	Further Applications of Integration

8.1 	Arc Length and Surface Area

8.1.1 	Arc Length

8.1.2 	Surface Area

8.2 	Center of Mass

Chapter  9	Introduction to Differential Equations

9.1 	Solving Differential Equations

9.2 	Models of the Form y'=k(y-b)

9.2.1 	Bacteria Growth

9.2.2 	Radioactive Decay

9.2.3 	Annuity

9.2.4 	Newton's Law of Cooling

9.3	Numerical Methods Using Slope Fields

9.3.1 	Slope Fields

9.3.2 	Euler's Method

9.4	The Logistic Equation

Chapter  10	Infinite Series

10.1	Sequences

10.2	Infinite Series

10.2.1 	Finite Sums

10.2.2 	Partial Sums and Convergence

10.3	Tests for Convergence

10.3.1 	Comparison and Limit Comparison Tests

10.3.2 	The Integral Test

10.3.3 	Absolute and Conditional Convergence

10.3.4 	Ratio Test

10.3.5 	Root Test

10.4	Power Series

10.4.1 	Taylor Polynomials

10.4.2 	Convergence of Power Series

10.4.3 	Taylor Series

Chapter  11	Parametric Equations, Polar Coordinates, and Conic Sections

11.1 	Parametric Equations

11.1.1 	Plotting Parametric Equations

11.1.2 	Parametric Derivatives

11.1.3 	Arc Length and Speed

11.2	Polar Coordinates and Curves

11.2.1 	Conversion Formulas

11.2.2 	Polar Curves

11.2.3 	Calculus of Polar Curves

11.3	Conic Sections

Chapter  12	Vector Geometry

12.1 	Vectors

12.2 	Matrices and the Cross Product

12.3 	Planes in 3-Space

12.4	A Survey of Quadric Surfaces

12.4.1 	Ellipsoids

12.4.2 	Hyperboloids

12.4.3 	Paraboloids

12.4.4 	Quadratic Cylinders

12.5	Cylindrical and Spherical Coordinates

12.5.1 	Cylindrical Coordinates

12.5.2 	Spherical Coordinates

Chapter  13	Calculus of Vector-Valued Functions

13.1 	Vector-Valued Functions

13.2 	Calculus of Vector-Valued Functions

13.3 	Arc Length

13.4	Curvature

13.5	Motion in Space

Chapter  14	Differentiation in Several Variables

14.1 	Functions of Two or More Variables

14.1.1 	Graphs of Functions of Two Variables

14.1.2 	ParametricPlot3D and ContourPlot3D

14.2 	Limits and Continuity

14.2.1 	Limits

14.2.2 	Continuity

14.3 	Partial Derivatives

14.4	Tangent Planes

14.5	Gradient and Directional Derivatives

14.6	Chain Rule

14.7	Optimization

14.8	Lagrange Multipliers

Chapter  15	Multiple Integration

15.1 	Double Integration Over a Rectangle

15.1.1 	Double Integrals and Riemann Sums

15.1.2 	Double Integrals and Iterated Integrals in Mathematica

15.2 	Double Integrals Over More General Regions

15.3 	Triple Integrals

15.4	Integration in Polar, Cylindrical and Spherical Coordinates

15.4.1 	Double Integrals in Polar Coordinates

15.4.2 	Triple Integrals in Cylindrical Coordinates

15.4.3 	Triple Integrals in Spherical Coordinates

15.5	Applications of Multiple Integrals

15.6	Change of Variables

Chapter  16	Line and Surface Integrals

16.1 	Vector Fields

16.2 	Line Integrals

16.3 	Conservative Vector Fields

16.4	Parametrized Surfaces and Surface Integrals

16.5	Surface Integrals of Vector Fields

Chapter  17	Fundamental Theorems of Vector Analysis

17.1 	Green's Theorem

17.2 	Stokes' Theorem

17.3 	Divergence Theorem

Appendices - Quick Reference Guides

A.	Common Mathematical Operations - Traditional Notation versus Mathematica Notation

B. 	Useful Command for Plotting, Solving, and Manipulating Mathematical Expressions

C. 	Useful Editing and Programming Commands

D. 	Formatting Cells in a Notebook

E. 	Saving and Printing a Notebook

References