Calculus Using Mathematica (eBook)

K.D. Stroyan (Autor)

eBook Download: PDF

2014 | 1. Auflage
558 Seiten
Elsevier Science (Verlag)
9781483267975 (ISBN)

Infos

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.

Inhalt

Front Cover 1
Calculus Using Mathematica 4
Copyright Page 5
Table of Contents 6
Contents of Mathematica NoteBooks for Calculus Using Mathematica 16
Preface 24
Acknowledgments 26
CHAPTER 1. Introduction 28
1.1. A Mathematica Introduction with aMathcalntro.ma 30
1.2. Mathematica on a NeXT 31
1.3. Mathematica on a Macintosh 36
1.4. Mathematica on DOS Windows (IBM) 37
1.5. Free Advice 37
Part 1: Differentiation in One Variable 40
CHAPTER 2. Using Calculus to Model Epidemics 42
2.1. The First Model 43
2.2. Shortening the Time Steps 48
2.3. The Continuous Variable Model 49
2.4. Calculus and the S-I-R Differential Equations 53
2.5. The Big Picture 55
2.6. Projects 55
CHAPTER 3. Numerics, Symbolics and Graphics in Science 58
3.1. Functions from Formulas 59
3.2. Types of Explicit Functions 62
3.3. Logs and Exponentials 65
3.4. Chaining Variables or Composition of Functions 67
3.5. Graphics and Formulas 69
3.6. Graphs without Formulas 72
3.7. Parameters 72
3.8. Background on Functional Identities 75
CHAPTER 4. Linearity vs. Local Linearity 76
4.1. Linear Approximation of Oxbows 78
4.2. The Algebra of Microscopes 78
4.3. Mathematica Increments and Microscopes 83
4.4. Functions with Kinks and Jumps 83
4.5. The Cool Canary - Another Kind of Linearity 85
CHAPTER 5. Direct Computation of Increments 88
5.1. How Small is Small Enough? 89
5.2. Derivatives as Limits 90
5.3. Small, Medium and Large Numbers 90
5.4. Rigorous Technical Summary 95
5.5. Increment Computations 96
5.6. Derivatives of Sine and Cosine 101
5.7. Continuity and the Derivative 107
5.8. Instantaneous Rates of Change 108
5.9. Projects 110
CHAPTER 6. Symbolic Differentiation 112
6.1. Rules for Special Functions 113
6.2. The Superposition Rule 115
6.3. Symbolic Differentiation with Mathematica 118
6.4. The Product Rule 119
6.5. The Expanding House 124
6.6. The Chain Rule 124
6.7. Derivatives of Other Exponentials by the Chain Rule 126
6.8. Derivative of The Natural Logarithm 127
6.9. Combined Symbolic Rules 128
6.10. Test Your Differentiation Skills 130
CHAPTER 7. Basic Applications of Differentiation 132
7.1. Differentiation with Parameters and Other Variables 132
7.2. Linked Variables and Related Rates 133
7.3. Review - Inside the Microscope 143
7.4. Review - Numerical Increments 144
7.5. Differentials and The (x, y)-Equation of the Tangent Line 147
CHAPTER 8. The Natural Logarithm and Exponential 150
8.1. The Official Definition of the Natural Exponential 152
8.2. Properties Follow from The Official Definition 155
8.3. e As a "Natural" Base for Exponentials and Logs 156
8.4. Growth of Log and Exp Compared with Powers 158
8.5. Mathematica Limits 161
8.6. Projects 161
CHAPTER 9. Graphs and the Derivative 164
9.1. Planck's Radiation Law 164
9.2. Graphing and The First Derivative 171
9.3. The Theorems of Bolzano and Darboux 177
9.4. Graphing and the Second Derivative 177
9.5. Another Kind of Graphing from the Slope 185
9.6. Projects 186
CHAPTER 10. Velocity, Acceleration and Calculus 188
10.1. Velocity and the First Derivative 188
10.2. Acceleration and the Second Derivative 189
10.3. Galileo's Law of Gravity 192
10.4. Projects 194
CHAPTER 11. Maxima and Minima in One Variable 196
11.1. Critical Points 197
11.2. Max - min with Endpoints 200
11.3. Max - min without Endpoints 206
11.4. Supply and Demand in Economics 209
11.5. Geometric Max-min Problems 213
11.6. Max-min with Parameters 221
11.7. Max-min in S-I-R Epidemics 224
11.8. Projects 226
CHAPTER 12. Discrete Dynamical Systems 228
12.1. Two Models for Price Adjustment by Supply and Demand 230
12.2. Function Iteration, Equilibria and Cobwebs/indexequilibrium, discrete 235
12.3. The Linear System 239
12.4. Nonlinear Models 242
12.5. Local Stability - Calculus and Nonlinearity 243
12.6. Projects 247
Part 2: Integration in One Variable 248
CHAPTER 13. Basic Integration 250
13.1. Geometric Approximations by Sums of Slices 251
13.2. Extension of the Distance Formula, D = R.T 258
13.3. The Definition of the Definite Integral 261
13.4. Mathematica Summation 262
13.5. The Algebra of Summation 263
13.6. The Algebra of Infinite Summation 267
13.7. The Fundamental Theorem of Integral Calculus, Part 1 269
13.8. The Fundamental Theorem of Integral Calculus, Part 2 272
CHAPTER 14. Symbolic Integration 276
14.1. Indefinite Integrals 277
14.2. Specific Integral Formulas 278
14.3. Superposition of Antiderivatives 279
14.4. Change of Variables or 'Substitution' 280
14.5. Trig Substitutions (Optional) 284
14.6. Integration by Parts 288
14.7. Combined Integration 293
14.8. Impossible Integrals 294
CHAPTER 15. Applications of Integration 296
15.1. The Infinite Sum Theorem: Duhamel's Principle 296
15.2. A Project on Geometric Integrals 305
15.3. Other Projects 309
Part 3: Vector Geometry 310
CHAPTER 16. Basic Vector Geometry 312
16.1. Cartesian Coordinates 312
16.2. Position Vectors 317
16.3. Basic Geometry of Vectors 319
16.4. The Geometry of Vector Addition 327
16.5. The Geometry of Scalar Multiplication 334
16.6. Vector Difference and Oriented Displacements 342
CHAPTER 17. Analytical Vector Geometry 348
17.1. A Lexicon of Geometry and Algebra 348
17.2. The Vector Parametric Line 351
17.3. Radian Measure and Parametric Curves 354
17.4. Parametric Tangents and Velocity Vectors 360
17.5. The Implicit Equation of a Plane 363
17.6. Wrap-up Exercises 366
CHAPTER 18. Linear Functions and Graphs in Several Variables 370
18.1. Vertical Slices and Chickenwire Plots 373
18.2. Horizontal Slices and Contour Graphs 377
18.3. Mathematica Plots 378
18.4. Linear Functions and Gradient Vectors 379
18.5. Explicit, Implicit and Parametric Graphs 382
Part 4: Differentiation in Several Variables 386
CHAPTER 19. Differentiation of Functions of Several Variables 388
19.1. Definition of Partial and Total Derivatives 389
19.2. Geometric Interpretation of the Total Derivative 390
19.3. Partial differentiation examples 391
19.4. Applications of the Total Differential Approximation 401
19.5. The Meaning of the Gradient Vector 404
19.6. Review Exercises 410
CHAPTER 20. Maxima and Minima in Several Variables 416
20.1. Zero Gradients and Horizontal Tangent Planes 417
20.2. Implicit Differentiation (Again) 427
20.3. Extrema Over Noncompact Regions 429
20.4. Projects on Max - min 437
Part 5: Differential Equations 438
CHAPTER 21. Continuous Dynamical Systems 440
21.1. One Dimensional Continuous Initial Value Problems 440
21.2. Euler's Method 450
21.3. Some Theory 453
21.4. Separation of Variables 455
21.5. The Geometry of Autonomous Equations in Two Dimensions 459
21.6. Flow Analysis of S-I-R Epidemics 469
21.7. Projects 473
CHAPTER 22. Autonomous Linear Dynamical Systems 474
22.1. Autonomous Linear Constant Coefficient Equations 474
22.2. Symbolic Exponential Solutions 476
22.3. Rotation & Euler's Formula
22.4. Superposition and The General Solution 483
22.5. Specific Solutions 489
22.6. Symbolic Solution of Second Order Autonomous I.V.P.s 489
22.7. Your Car's Worn Shocks 493
22.8. Projects 494
CHAPTER 23. Equilibria of Continuous Dynamical Systems 496
23.1. Dynamic Equilibria in One Dimension 496
23.2. Linear Equilibria in Two Dimensions 498
23.3. Nonlinear Equilibria in Two Dimensions 503
23.4. Explicit Solutions, Phase Portraits & Invariants
23.5. Review: Looking at Equilibria 512
23.6. Projects 515
Part 6: Infinite Series 518
CHAPTER 24. Geometric Series 520
24.1. Geometric Series 522
24.2. Convergence by Comparison 531
CHAPTER 25. Power Series 534
25.1. Computation of Power Series 534
25.2. The Ratio Test for Convergence of Power Series 540
25.3. Integration of Series 542
25.4. Differentiation of Power Series 546
CHAPTER 26. The Edge of Convergence 550
26.1. Alternating Series 550
26.2. Telescoping Series 552
26.3. Comparison of Integrals and Series 553
26.4. Limit Comparisons 556
26.5. Fourier Series 556
Index 558

Produkt-Details

• Erscheint lt. Verlag: 10.5.2014
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Technik
• ISBN-13: 9781483267975 / 9781483267975