A First Course in Mathematical Modeling

by Giordano, Frank R.; Weir, Maurice D.; Fox, William P.

• ISBN13: 9780534384289
• ISBN10: 0534384285
• Edition: 3rd
• Format: Hardcover
• Copyright: 2002-07-29
• Publisher: Brooks Cole

Summary

This text provides an introduction to the entire modeling process. Throughout the book, students practice key facets of modeling, including creative and empirical model construction, model analysis, and model research. The authors apply a proven six-step problem solving process to enhance a student's problem solving capabilities. Rather than simply emphasizing the calculation step, the authors first ensure that students learn how to identify problems, construct or select models, and figure out what data needs to be collected. By involving students in the mathematical process as early as possible, beginning with short projects, the book facilitates their progressive development and confidence in mathematics and modeling.

Table of Contents

Modeling Change	1	(51)
Introduction	1	(1)
Testing for Proportionality	2	(2)
Modeling Change with Difference Equations	4	(5)
A Savings Certificate	5	(1)
Mortgaging a Home	6	(3)
Approximating Change with Difference Equations	9	(9)
Growth of a Yeast Culture	9	(1)
Growth of a Yeast Culture Revisited	10	(2)
Spread of a Contagious Disease	12	(1)
Decay of Digoxin in the Bloodstream	13	(1)
Heating of a Cooled Object	14	(4)
Solutions to Dynamical Systems	18	(17)
A Savings Certificate Revisited	18	(3)
Sewage Treatment	21	(4)
Prescription for Digoxin	25	(1)
An Investment Annuity	26	(2)
A Checking Account	28	(2)
An Investment Annuity Revisited	30	(5)
Systems of Difference Equations	35	(17)
A Car Rental Company	35	(3)
The Battle of Trafalgar	38	(3)
Competitive Hunter Model-Spotted Owls and Hawks	41	(3)
Using Tendencies of the Political Parties	44	(8)
The Modeling Process, Proportionality, and Geometric Similarity	52	(45)
Introduction	52	(2)
Mathematical Models	54	(11)
Vehicular Stopping Distance	59	(6)
Modeling Using Proportionality	65	(10)
Kepler's Third Law	67	(8)
Modeling Using Geometric Similarity	75	(13)
Raindrops from a Motionless Cloud	77	(2)
Modeling a Bass Fishing Derby	79	(9)
Automobile Gasoline Mileage	88	(3)
Body Weight and Height, Strength and Agility	91	(6)
Model Fitting	97	(29)
Introduction	97	(4)
Fitting Models to Data Graphically	101	(6)
Analytic Methods of Model Fitting	107	(7)
Applying the Least-Squares Criterion	114	(5)
Choosing a Best Model	119	(7)
Vehicular Stopping Distance	122	(4)
Experimental Modeling	126	(49)
Introduction	126	(1)
Harvesting in the Chesapeake Bay and Other One-Term Models	127	(11)
Harvesting Bluefish	130	(1)
Harvesting Blue Crabs	131	(7)
High-Order Polynomial Models	138	(8)
Elapsed Time of a Tape Recorder	140	(6)
Smoothing: Low-Order Polynomial Models	146	(13)
Elapsed Time of a Tape Recorder Revisited	147	(4)
Elapsed Time of a Tape Recorder Revisited Again	151	(2)
vehicle Stopping Distance	153	(2)
Growth of a Yeast Culture	155	(4)
Cubic Spline Models	159	(16)
Vehicle Stopping Distance Revisited	167	(8)
Simulation Modeling	175	(42)
Introduction	175	(2)
Simulating Deterministic Behavior: Area Under a Curve	177	(5)
Generating Random Numbers	182	(4)
Simulating Probabilistic Behavior	186	(8)
Inventory Model: Gasoline and Consumer Demand	194	(11)
Queuing Models	205	(12)
A Harbor System	205	(8)
Morning Rush Hour	213	(4)
Discrete Probabilistic Modeling	217	(19)
Introduction	217	(1)
Probabilistic Modeling with Discrete Systems	217	(6)
Rental Car Company Revisited	217	(2)
Voting Tendencies	219	(4)
Modeling Component and System Reliability	223	(4)
Series Systems	224	(1)
Parallel Systems	224	(1)
Series and Parallel Combinations	224	(3)
Linear Regression	227	(9)
Ponderosa Pines	229	(2)
The Bass Fishing Derby Revisited	231	(5)
Discrete Optimization Modeling	236	(56)
An Overview of Discrete Optimization Modeling	237	(13)
Determining a Production Schedule	237	(3)
Space Shuttle Cargo	240	(1)
Approximation by a Piecewise Linear Function	241	(9)
Linear Programming I: Geometric Solutions	250	(9)
The Carpenter's Problem	251	(1)
A Data-Fitting Problem	252	(7)
Linear Programming II: Algebraic Solutions	259	(4)
Solving the Carpenter's Problem Algebraically	261	(2)
Linear Programming III: The Simplex Method	263	(10)
The Carpenter's Problem Revisited	268	(3)
Using the Tableau Format	271	(2)
Linear Programming IV: Sensitivity Analysis	273	(6)
Numerical Search Methods	279	(13)
Using the Dichotomous Search Method	282	(3)
Using the Golden Section Search Method	285	(2)
Model-Fitting Criterion Revisited	287	(1)
Optimizing Industrial Flow	288	(4)
Dimensional Analysis and Similitude	292	(44)
Introduction	292	(3)
Dimensions as Products	295	(9)
A Simple Pendulum	298	(3)
Wind Force on a Van	301	(3)
The Process of Dimensional Analysis	304	(9)
Terminal Velocity of a Raindrop	309	(2)
Automobile Gas Mileage Revisited	311	(2)
A Damped Pendulum	313	(6)
Examples Illustrating Dimensional Analysis	319	(11)
Explosion Analysis	319	(5)
How Long Should You Roast a Turkey?	324	(6)
Similitude	330	(6)
Drag Force on a Submarine	331	(5)
Graphs of Functions as Models	336	(32)
An Arms Race	336	(14)
Civil Defense	345	(1)
Mobile Launching Pads	346	(1)
Multiple Warheads	347	(1)
MIRVs Revisited: Counting Warheads	348	(2)
Modeling an Arms Race in Stages	350	(5)
Managing Nonrenewable Resources: The Energy Crisis	355	(4)
Effects of Taxation on the Energy Crisis	359	(5)
A Gasoline Shortage and Taxation	364	(4)
Modeling with a Differential Equation	368	(44)
Introduction	368	(3)
Population Growth	371	(11)
Prescribing Drug Dosage	382	(9)
Braking Distance Revisited	391	(4)
Graphical Solutions of Autonomous Differential Equations	395	(9)
Drawing a Phase Line and Sketching Solution Curves	396	(3)
Cooling Soup	399	(1)
Logistic Growth Revisited	400	(4)
Numerical Approximation Methods	404	(8)
Using Euler's Method	406	(1)
A Savings Certificate Revisited	407	(5)
Modeling with Systems of Differential Equations	412	(46)
Introduction	412	(1)
Graphical Solutions of Autonomous Systems of First-Order Differential Equations	413	(6)
A Linear Autonomous System	414	(1)
A Nonlinear Autonomous System	415	(4)
A Competitive Hunter Model	419	(8)
A Predator-Prey Model	427	(10)
Two Military Examples	437	(13)
Lanchester Combat Models	437	(7)
Economic Aspects of an Arms Race	444	(6)
Euler's Method for Systems of Differential Equations	450	(8)
Using Euler's Method for Systems	451	(1)
A Trajectory and Solution Curves	452	(6)
Continuous Optimization Modeling	458	(32)
Introduction	458	(1)
An Inventory Problem: Minimizing the Cost of Delivery and Storage	459	(9)
A Manufacturing Problem: Maximizing Profit in Producing Competing Products	468	(6)
Constrained Continuous Optimization	474	(6)
An Oil Transfer Company	474	(2)
A Space Shuttle Water Container	476	(4)
Managing Renewable Resources: The Fishing Industry	480	(10)
Appendix A Problems from the Mathematics Contest in Modeling, 1985-2002	490	(33)
Appendix B An Elevator Simulation Algorithm	523	(9)
Appendix C The Revised Simplex Method	532	(3)
Index	535