1.1 Modeling Change with Difference Equations 4
Example I: A Savings Certificate 5
Example 2: Mortgaging a Home 6
1.2 Approximating Change with Difference Equations
Example 1: Growth of a Yeast Culture 9
Example 2: Growth of a Yeast Culture Revisited 10
Example 3: Spread of a Contagious Disease 12
Example 4: Decay of Digoxin in the Bloodstream 13
Example 5: Heating of a Cooled Object 14
1.3 Solutions to Dynamical Systems 18
Example 1: A Savings Certificate Revisited 18
Example 2: Sewage Treatment 21
Example 3: Prescription for Digoxin 25
Example 4: An Investment Annuity 26
Example 5: A Checking Account 28
Example 6: An Investment Annuity Revisited 30
1.4 Systems of Difference Equations 35
Example I: A Car Rental Company 35
Example 2: The Battle of Trafalgar 38
Example 3: Competitive Hunter Model--Spotted Owls and Hawks
Example 4: Voting Tendencies of the Political Parties 44
2 The Modeling Process, Proportionality, and Geometric Similarity Introduction 52
2.1 Mathematical Models 54
Example 1: Vehicular Stopping Distance 59
2.2 Modeling Using Proportionality 65
Example 1: Kepler's Third Law 67
2.3 Modeling Using Geometric Similarity 75
Example 1: Raindrops from a Motionless Cloud 77
Example 2: Modeling a Bass Fishing Derby 79
2.4 Automobile Gasoline Mileage 88
2.5 Body Weight and Height, Strength and Agility 91
Model Fitting 97
Introduction 97
3.1 Fitting Models to Data Graphically 101
3.2 Analytic Methods of Model Fitting 107
3.3 Applying the Least-Squares Criterion 114
3.4 Choosing a Best Model 119
Example 1: Vehicular Stopping Distance 122
4 Experimental Modeling 126
Introduction 126
4.1 Harvesting in the Chesapeake Bay and Other
One-Term Models 127
Example 1: Harvesting Bluefish 130
Example 2: Harvesting Blue Crabs 131
4.2 High-Order Polynomial Models 138
Example 1: Elapsed Time of a Tape Recorder 140
4.3 Smoothing: Low-Order Polynomial Models 146
Example 1: Elapsed Time of a Tape Recorder Revisited 147
Example 2: Elapsed Time of a Tape Recorder Revisited Again
Example 3: Vehicle Stopping Distance 153
Example 4: Growth of a Yeast Culture 155
4.4 Cubic Spline Models 159
Example 1: Vehicle Stopping Distance Revisited 167
5 Simulation Modeling 175
Introduction 175
5.1 Simulating Deterministic Behavior:Area Under a Curve
5.2 Generating Random Numbers 182
5.3 Simulating Probabilistic Behavior '186
5.4 Inventory Model: Gasoline and Consumer Demand 194
5.5 Queuing Models 205
Example 1: A Harbor System 205
Example 2: Morning Rush Hour 213
6 Discrete Probabilistic Modeling 217
Introduction 217
6.1 Probabilistic Modeling with Discrete Systems 217
Example 1: Rental Car Company Revisited 217
Example 2: Voting Tendencies 219
6.2 Modeling Component and System Reliability 223
Example 1: Series Systems 224
Example 2: Parallel Systems 224
Example 3: Series and Parallel Combinations 224
6.3 Linear Regression 227
Example 1: Ponderosa Pines 229
Example 2: The Bass Fishing Derby Revisited 231
7 Discrete Optimization Modeling 236
7.1 An Overview of Discrete Optimization Modeling 237
Example 1: Determining a Production Schedule 237
Example 2: Space Shuttle Cargo 240
Example 3: Approximation by a Piecewise Linear Function 241
7.2 Linear Programming h Geometric Solutions 250
Example 1: The Carpenter's Problem 251
Example 2: A Data-Fitting Problem 252
7.3 Linear Programming Ih Algebraic Solutions 259
Example 1: Solving the Carpenter's Problem Algebraically 261
7.4 Linear Programming III: The Simplex Method 263
Example 1: The Carpenter's Problem Revisited 268
Example 2: Using the Tableau Format 271
7.5 Linear Programming IV: Sensitivity Analysis 273
7.6 Numerical Search Methods 279
Example 1: Using the Dichotomous Search Method 282
Example 2: Using the Golden Section Search Method 285
Example 3: Model-Fitting Criterion Revisited 287
Example 4: Optimizing Industrial Flow 288
8 Dimensional Analysis and Similitude 292
Introduction 292
8.1 Dimensions as Products 295
Example 1: A Simple Pendulum 298
Example 2: Wind Force on a Van 301
8.2 The Process of Dimensional Analysis 304
Example l: Terminal Velocity of a Raindrop 309
Example 2: Automobile Gas Mileage Revisited 311
8.3 A Damped Pendulum 313
8.4 Examples Illustrating Dimensional Analysis 319
Example I : Explosion Analysis 319
Example 2: How Long Should You Roast a Turkey? 324
8.5 Similitude 330
Example I: Drag Force on a Submarine 331
9 Graphs of Functions as Models 336
9.1 An Arms Race 336
Example 1: Civil Defense 345
Example 2: Mobile Launching Pads 346
Example 3: Multiple Warheads 347
Example 4: MIRVs Revisited: Counting Warheads 348
9.2 Modeling an Arms Race in Stages 350
9.3 Managing Nonrenewable Resources: The Energy Crisis
9.4 Effects of Taxation on the Energy Crisis 359
9.5 A Gasoline Shortage and Taxation 364
10 Modeling with a Differential Equation 368
Introduction 368
10.1 Population Growth 371
10.2 Prescribing Drug Dosage
10.3 Braking Distance Revisited 391
10.4 Graphical Solutions of Autonomous Differential Equations
Example 1: Drawing a Phase Line and Sketching
Solution Curves 396
Example 2: Cooling Soup 399
Example 3: Logistic Growth Revisited 400
10.5 Numerical Approximation Methods 404
Example 1: Using Euler's Method 406
Example 2: A Savings Certificate Revisited 407
11 Modeling with Systems of
Differential Equations 412
Introduction 412
11.1 Graphical Solutions of Autonomous Systems of First-Order
Differential Equations 413
Example 1: A Linear Autonomous System 414
Example 2: A Nonlinear Autonomous System 415
11.2 A Competitive Hunter Model 419
11.3 A Predator-Prey Model 427
11.4 Two Military Examples 437
Example I: Lanchester Combat Models 437
Example 2: Economic Aspects of an Arms Race 444
11.5 Euler's Method for Systems of Differential Equations 450
Example 1: Using Euler' s Method for Systems 451
Example 2: A Trajectory and Solution Curves 452
12 Continuous Optimization Modeling 458
Introduction 458
12.1 An Inventory Problem: Minimizing the Cost of Delivery and
Storage 459
12.2 A Manufacturing Problem: Maximizing Profit in Producing
Competing Products 468
12.3 Constrained Continuous Optimization 474
Example l: An Oil Transfer Company 474
Example 2: A Space Shuttle Water Container 476
12.4 Managing Renewable Resources: The Fishing Industry 480
A Problems from the Mathematics Contest in
Modeling. 1985-2002 490
B An Elevator Simulation Algorithm s23
C The Revised Simplex Method
Index 535
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发表于 2003-7-25 20:15:21