Calculus Using Mathematica

Table of Contents to NoteBooks DiskettePrefaceAcknowledgments 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 AdvicePart 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. ProjectsPart 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 ProjectsPart 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 GraphsPart 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 - minPart 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. ProjectsPart 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 SeriesIndex