Calculus Using Mathematica

Calculus Using Mathematica

1st Edition - January 1, 1993

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  • Author: K.D. Stroyan
  • eBook ISBN: 9781483267975

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Description

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


  • 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

Product details

  • No. of pages: 558
  • Language: English
  • Copyright: © Academic Press 1993
  • Published: January 1, 1993
  • Imprint: Academic Press
  • eBook ISBN: 9781483267975

About the Author

K.D. Stroyan

Affiliations and Expertise

Department of Mathematics, The University of Iowa Iowa City. Iowa

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