Calculus Using Mathematica
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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