Calculus Using Mathematica

Calculus Using Mathematica is intended for college students taking a course in calculus. It teaches the basic skills of differentiation and integration and how to use Mathematica, a scientific software language, to perform very elaborate symbolic and numerical computations.

This is a set composed of the core text, science and math projects, and computing software for symbolic manipulation and graphics generation. Topics covered in the core text include an introduction on how to get started with the program, the ideas of independent and dependent variables and parameters in the context of some down-to-earth applications, formulation of the main approximation of differential calculus, and discrete dynamical systems. The fundamental theory of integration, analytical vector geometry, and two dimensional linear dynamical systems are elaborated as well.

This publication is intended for beginning college students.

Table of Contents to NoteBooks Diskette

Preface

Acknowledgments

Chapter 1. Introduction

1.1. A Mathematica Introduction with aMathcaIntro.ma

1.2. Mathematica on a NeXT

1.3. Mathematica on a Macintosh

1.4. Mathematica on DOS Windows (IBM)

1.5. Free Advice

Part 1 Differentiation in One Variable

Chapter 2. Using Calculus to Model Epidemics

2.1. The First Model

2.2. Shortening the Time Steps

2.3. The Continuous Variable Model

2.4. Calculus and the S-I-R Differential Equations

2.5. The Big Picture

2.6. Projects

Chapter 3. Numerics, Symbolics and Graphics in Science

3.1. Functions from Formulas

3.2. Types of Explicit Functions

3.3. Logs and Exponentials

3.4. Chaining Variables or Composition of Functions

3.5. Graphics and Formulas

3.6. Graphs without Formulas

3.7. Parameters

3.8. Background on Functional Identities

Chapter 4. Linearity vs. Local Linearity

4.1. Linear Approximation of Oxbows

4.2. The Algebra of Microscopes

4.3. Mathematica Increments and Microscopes

4.4. Functions with Kinks and Jumps

4.5. The Cool Canary - Another Kind of Linearity

Chapter 5. Direct Computation of Increments

5.1. How Small is Small Enough?

5.2. Derivatives as Limits

5.3. Small, Medium and Large Numbers

5.4. Rigorous Technical Summary

5.5. Increment Computations

5.6. Derivatives of Sine and Cosine

5.7. Continuity and the Derivative

5.8. Instantaneous Rates of Change

5.9. Projects

Chapter 6. Symbolic Differentiation

6.1. Rules for Special Functions

6.2. The Superposition Rule

6.3. Symbolic Differentiation with Mathematica

6.4. The Product Rule

6.5. The Expanding House

6.6. The Chain Rule

6.7. Derivatives of Other Exponentials by the Chain Rule

6.8. Derivative of The Natural Logarithm

6.9. Combined Symbolic Rules

6.10. Test Your Differentiation Skills

Chapter 7. Basic Applications of Differentiation

7.1. Differentiation with Parameters and Other Variables

7.2. Linked Variables and Related Rates

7.3. Review - Inside the Microscope

7.4. Review - Numerical Increments

7.5. Differentials and The (x,y)-Equation of the Tangent Line

Chapter 8. The Natural Logarithm and Exponential

8.1. The Official Definition of the Natural Exponential

8.2. Properties Follow from The Official Definition

8.3. e As a "Natural" Base for Exponentials and Logs

8.4. Growth of Log and Exp Compared with Powers

8.5. Mathematica Limits

8.6. Projects

Chapter 9. Graphs and the Derivative

9.1. Planck's Radiation Law

9.2. Graphing and The First Derivative

9.3. The Theorems of Bolzano and Darboux

9.4. Graphing and the Second Derivative

9.5. Another Kind of Graphing from the Slope

9.6. Projects

Chapter 10. Velocity, Acceleration and Calculus

10.1. Velocity and the First Derivative

10.2. Acceleration and the Second Derivative

10.3. Galileo's Law of Gravity

10.4. Projects

Chapter 11. Maxima and Minima in One Variable

11.1. Critical Points

11.2. Max - min with Endpoints

11.3. Max - min without Endpoints

11.4. Supply and Demand in Economics

11.5. Geometric Max-min Problems

11.6. Max-min with Parameters

11.7. Max-min in S-I-R Epidemics

11.8. Projects

Chapter 12. Discrete Dynamical Systems

12.1. Two Models for Price Adjustment by Supply and Demand

12.2. Function Iteration, Equilibria and Cobwebs/indexequilibrium, Discrete

12.3. The Linear System

12.4. Nonlinear Models

12.5. Local Stability - Calculus and Nonlinearity

12.6. Projects

Part 2 Integration in One Variable

Chapter 13. Basic Integration

13.1. Geometric Approximations by Sums of Slices

13.2. Extension of the Distance Formula, D = R·T

13.3. The Definition of the Definite Integral

13.4. Mathematica Summation

13.5. The Algebra of Summation

13.6. The Algebra of Infinite Summation

13.7. The Fundamental Theorem of Integral Calculus, Part 1

13.8. The Fundamental Theorem of Integral Calculus, Part 2

Chapter 14. Symbolic Integration

14.1. Indefinite Integrals

14.2. Specific Integral Formulas

14.3. Superposition of Antiderivatives

14.4. Change of Variables or 'Substitution'

14.5. Trig Substitutions (Optional)

14.6. Integration by Parts

14.7. Combined Integration

14.8. Impossible Integrals

Chapter 15. Applications of Integration

15.1. The Infinite Sum Theorem: Duhamel's Principle

15.2. A Project on Geometric Integrals

15.3. Other Projects

Part 3 Vector Geometry

Chapter 16. Basic Vector Geometry

16.1. Cartesian Coordinates

16.2. Position Vectors

16.3. Basic Geometry of Vectors

16.4. The Geometry of Vector Addition

16.5. The Geometry of Scalar Multiplication

16.6. Vector Difference and Oriented Displacements

Chapter 17. Analytical Vector Geometry

17.1. A Lexicon of Geometry and Algebra

17.2. The Vector Parametric Line

17.3. Radian Measure and Parametric Curves

17.4. Parametric Tangents and Velocity Vectors

17.5. The Implicit Equation of a Plane

17.6. Wrap-up Exercises

Chapter 18. Linear Functions and Graphs in Several Variables

18.1. Vertical Slices and Chickenwire Plots

18.2. Horizontal Slices and Contour Graphs

18.3. Mathematica Plots

18.4. Linear Functions and Gradient Vectors

18.5. Explicit, Implicit and Parametric Graphs

Part 4 Differentiation in Several Variables

Chapter 19. Differentiation of Functions of Several Variables

19.1. Definition of Partial and Total Derivatives

19.2. Geometric Interpretation of the Total Derivative

19.3. Partial differentiation Examples

19.4. Applications of the Total Differential Approximation

19.5. The Meaning of the Gradient Vector

19.6. Review Exercises

Chapter 20. Maxima and Minima in Several Variables

20.1. Zero Gradients and Horizontal Tangent Planes

20.2. Implicit Differentiation (Again)

20.3. Extrema Over Noncompact Regions

20.4. Projects on Max - min

Part 5 Differential Equations

Chapter 21. Continuous Dynamical Systems

21.1. One Dimensional Continuous Initial Value Problems

21.2. Euler's Method

21.3. Some Theory

21.4. Separation of Variables

21.5. The Geometry of Autonomous Equations in Two Dimensions

21.6. Flow Analysis of S-I-R Epidemics

21.7. Projects

Chapter 22. Autonomous Linear Dynamical Systems

22.1. Autonomous Linear Constant Coefficient Equations

22.2. Symbolic Exponential Solutions

22.3. Rotation & Euler's Formula

22.4. Superposition and The General Solution

22.5. Specific Solutions

22.6. Symbolic Solution of Second Order Autonomous I.V.P.s

22.7. Your Car's Worn Shocks

22.8. Projects

Chapter 23. Equilibria of Continuous Dynamical Systems

23.1. Dynamic Equilibria in One Dimension

23.2. Linear Equilibria in Two Dimensions

23.3. Nonlinear Equilibria in Two Dimensions

23.4. Explicit Solutions, Phase Portraits & Invariants

23.5. Review: Looking at Equilibria

23.6. Projects

Part 6 Infinite Series

Chapter 24. Geometric Series

24.1. Geometric Series

24.2. Convergence by Comparison

Chapter 25. Power Series

25.1. Computation of Power Series

25.2. The Ratio Test for Convergence of Power Series

25.3. Integration of Series

25.4. Differentiation of Power Series

Chapter 26. The Edge of Convergence

26.1. Alternating Series

26.2. Telescoping Series

26.3. Comparison of Integrals and Series

26.4. Limit Comparisons

26.5. Fourier Series

Index