PART I: TEACHING SUGGESTIONS 7

Instructor’s Manual with Solutions A First Course in Mathematical Modeling

FIFTH EDITION

Frank R. Giordano

William P. Fox

Steven B. Horton 1 / 4

CONTENTS

PART I: TEACHING SUGGESTIONS 7

• Modeling Change7
• The Modeling Process,

Proportionality, and Geometric Similarity16

• Model Fitting20
• Experimental Modeling25
• Simulation Modeling29
• Discrete Probabilistic Modeling 38
• Optimization of Discrete Models 39
• Modeling Using Graph Theory41
• Modeling with Decision Theory 43

10 Game Theory44 11 Modeling with a Differential Equation 46 12 Modeling with Systems of Differential Equations 49 13 Optimization of Continuous Models 52 14 Dimensional Analysis and Similitude 54 15 Graphs as Functions as Model 59

PART II: SAMPLE PROBLEM SOLUTIONS 62

• Modeling Change62
• The Modeling Process

70

• Model Fitting4 Experimental Modeling103
• Simulation Modeling111
• Discrete Probabilistic Modeling 121
• Optimization of Discrete Models 124

Proportionality, and Geometric Similarity , 88

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© 2014 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.11 Modeling with a Differential Equation 167 12 Modeling with Systems of Differential Equations 189 13 Optimization of Continuous Modeling 209 14 Dimensional Analysis and Similitude 226 15 Graphs of Functions as Models 2 iv

• Modeling sing Graph Theory143
• Modeling with Decision Theory 147

10 Game Theory158 U 46 Not For Sale © Cengage Learning. All rights reserved. No distribution allowed without express authorization. 3 / 4

PART I

TEACHING SUGGESTIONS

CH

APTER ONE

Modeling Change

SUGGESTED SYLLABUS

Class Hours TopicText

• Modeling Change with Difference Equations Sect 1.1
• Approximating Change with Difference Equations Sect 1.2
• Solutions to Dynamical SystemsSect 1.3
• Systems of Difference EquationsSect 1.4

OBJECTIVES

The major objectives of this chapter are:

• To build and solve models involving change that takes place in discrete intervals.
• To prepare the students for modeling change taking place continuously in Chapters 11

and 12.

• To introduce numerical solutions by iterating difference equations.
• To extend the modeling process by modeling interactive systems early.

DISCUSSION

We have found that freshman students model dynamical systems quite naturally. Since they can iterate the systems they build given an initial value, they gain intuition by graphing their results and analyzing the long-term behavior. Further, by experimenting with different initial values, they begin to appreciate the sensitivity of the model’s conclusions to the initial conditions. Finally, the experience of this chapter prepares them for modeling with differential equations and systems of differential equations, which students generally find more difficult.We strongly suggest beginning with behavior that can be modeled exactly, such as the accumulation of money in a savings account. Moving to annuities or mortgages retains student interest while introducing more sophisticated models. Since the students can readily enumerate these sequences even before building the model, they gain confidence in their work. Once the students are confident with exactly modeling behavior, we move to approximating change with discrete systems. Only an elementary notion of proportionality is needed for the scenarios in this chapter. (The concepts of proportionality and geometric similarity are studied in more detail in Chapter Two). A powerful advantage of studying discrete systems early is that the © 2014 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.

resulting models can be iterated if starting values are known. We have found that students have no difficulty iterating the systems. Finally, introducing systems of difference equations permits the students to model interesting systems with rather elementary mathematics. Many of the scenarios introduced here are revisited in Chapters 11 and 12.7 Not For Sale © Cengage Learning. All rights reserved. No distribution allowed without express authorization.

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