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Solution Manual for A First Course in Mathematical Modeling, 5th Edition

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    ’s Manual with Solutions n • Ko in • A First Course in Mathematical Modeling FIFTH EDITION Frank R. Giordano Naval Postgraduate School William P. Fox Naval Postgraduate School Steven B. Horton United States Military Academy SOLUTIONS GUIDE

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    CONTENTS PART I: ORGANIZING A MODELING COURSE 1 PART II: TEACHING SUGGESTIONS 7 1 Modeling Change 7 2 The Modeling Process, Proportionality, and Geometric Similarity 16 3 Model Fitting 20 4 Experimental Modeling 25 5 Simulation Modeling 29 6 Discrete Probabilistic Modeling 38 7 Optimization of Discrete Models 39 8 Modeling Using Graph Theory 41 9 Modeling with Decision Theory 43 10 Game Theory 44 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 III: SAMPLE PROBLEM SOLUTIONS 62 1 Modeling Change 62 2 The Modeling Process 70 3 Model Fitting 4 Experimental Modeling 103 5 Simulation Modeling 111 6 Discrete Probabilistic Modeling 121 7 Optimization of Discrete Models 124 Proportionality, and Geometric Similarity , 88

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    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 8 Modeling sing Graph Theory 143 9 Modeling with Decision Theory 147 10 Game Theory 158 U 46

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    PART I ORGANIZING A MODELING COURSE ISSUES Organizing a modeling course is a significant, educational challenge. A number of resource materials must be gathered: an appropriate text, supplemental references both for students as well as the instructor, sources and scenarios for student projects, and, possibly, computer software. Furthermore, there are crucial pedagogical issues that must be resolved in designing the course, including the following: 1. Course objectives 2. Course prerequisites 3. Course content 4. Number and type of student projects 5. Individual versus group projects 6. The role of computation 7. Grading considerations 8. Opportunities for follow-on modeling courses. We believe that a textbook can only serve as a base for a modeling course, which must then be tailored to meet the specific needs of students, as well as overall objectives in the curriculum. Moreover, a modeling course needs to be flexible and dynamic to allow for each individual instructor to take advantage of his or her particular mathematical expertise, experiences, and modeling preferences. OBJECTIVES The overall goal of our course is to provide a thorough introduction to the entire modeling process while affording students the opportunity to practice: 1. Creative and Empirical Model Construction: Given a real-world scenario, the student must identify a problem, make assumptions and collect data, propose a model, test the assumptions, refine the model as necessary, fit the model to data if appropriate, and analyze the underlying mathematical structure of the model in order to appraise the sensitivity of the conclusions in relation to the assumptions. Furthermore, the student should be able to generalize the construction to related scenarios. 2. Model Analysis Given a model, the student must work backward to uncover the implicit underlying assumptions, assess critically how well the assumptions reflect the scenario at hand, and estimate the sensitivity of the conclusions when the assumptions are not precisely met. 3. Model Research: The student investigates an area of interest to gain knowledge, understanding, and an ability to use what already has been created or discovered. Model research pr nin ‘ art’ : 1

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    can be organized in support of a modeling course. COURSE PREREQUISITES There are strong arguments to require such courses as advanced calculus, linear algebra, differential equations, probability, numerical analysis, and optimization as prerequisites to an introductory modeling course. Certainly the level and sophistication of the mathematics that students are capable of using increases significantly as more advanced courses are added to their programs. However, our desire is to gain the modeling experience as early as possible in the student’s career. Although some unfamiliar mathematical ideas are taught as part of the modeling process, the emphasis is on using mathematics already known to the student after completing high school. As outlined in the preface to the text, a course can be constructed requiring only high school mathematics as a prerequisite. Some sections do require an introductory calculus course as a prerequisite or corequisite, as detailed in the preface to the text. In our modeling courses, we emphasize teaching students how to use mathematics they already know in a context of significant applications with which they can readily identify. This approach stimulates student interest in mathematics and motivates them to study more advanced topics such as those mentioned above. Moreover, our students are eager to see meaningful applications of the mathematics they have learned. To accomplish our goals, we provide students with a diversity of scenarios for practicing all three of these facets of modeling. In addition, normal and routine exercises are assigned to test the student’s understanding of the instructional material we present. Thus, our textbook provides expository material and a framework around which a diversity of additional materials COURSE CONTENT Many modeling courses select from an inventory of specific model types which can be adapted to a variety of situations. Certainly model selection is a valid step in the problem-solving process and it is important that students learn to use what already has been created. However, our experience is that undergraduate students seldom comprehend the assumptions inherent in type models. We want our students to realize the necessity of making assumptions, the need to determine the appropriateness of the assumptions, and the importance of investigating how sensitive the conclusions are to the assumptions. Consequently, while we do discuss how to fit type models in the text, we have chosen to emphasize model construction, leaving the study of type models for more advanced courses. We feel that model construction promotes student creativity, demonstrates the artistic nature of model building, and develops an appreciation for how mathematics can be used effectively in various settings. Since the student needs practice in the first several steps of the problem-solving process–identifying the problem, making assumptions, determining interrelationships between the variables and submodels–it is tempting to compose an entire course of creative model construction. However, there are serious difficulties with such a course. Typically, students are very anxious, at least initially, because they don’t know how to begin the modeling process. After all, they have probably never before attacked an open-ended semester is dedicated to creative model construction. Moreover, a course consisting entirely of problem. When they are successful in constructing a model, they usually find the procedure enjoyable and exhilarating. Nevertheless, they tend to “burn out” if an entire quarter or 2

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    creative model construction cannot address other important aspects of modeling, like experimentation and simulation. Furthermore, there are difficulties with such a course for the instructor. Preparation of the course requires enormous effort in researching and generating scenarios to be modeled. Grading is difficult and tends to be subjective because each student approaches each project in a different way, yet students need constant feedback on their work. These difficulties then contribute to the anxieties of the student who is overly concerned about being graded in class under a time constraint in an area perceived to be relatively subjective. Under these conditions, success of the course becomes highly instructor dependent and circumstantial. For the above reasons, we have chosen to design a course consisting of a mixture of both creative and empirical modeling projects, along with projects in model analysis and model research. We begin by interactively constructing graphical models in class to engage students immediately in model analysis, which is relatively familiar to them. The transition to creative model construction commences with the students learning to make assumptions about a real-world behavior and by providing them with data to check their assumptions (initially with simple proportionality arguments). We then apply the modeling process to construct interactively in class relatively simple submodels and models in a variety of settings covering many disciplines. Students begin to see that situations arise where it may be very difficult or impossible to construct an analytic model, yet predictive capabilities are highly desirable. This perception motivates the study of empirical modeling. Students find empirical model construction more procedural and reproducible than creative model construction, and we find they welcome the mixture. In the text, some scenarios are modeled several ways creatively, and several ways empirically, so students can experience the alternatives that may be available. STUDENT PROJECTS We have developed our modeling courses to promote progressive development in the student’s modeling capabilities. To achieve our objectives, we require each student to complete at least 6 significant problem assignments or projects to be handed-in for a grade. Each student is assigned a mixture of problems/projects in creative and empirical model construction, model analysis, and model research. We purposely select problems/projects which address scenarios for which there are no unique solutions. Several of the projects include real data that the student is either given or can readily collect. If the course is taught early in the student’s program we recommend a combination of individual and group projects. Individual work is essential if the student is to develop adequate modeling skills. By way of contrast, group projects are exhilarating and allow for the experience of the synergistic effect that takes place in a ‘brainstorming’ session. For example, during the instruction of Chapter 2 on proportionality, we give a choice of 5 or 6 different scenarios for which students individually create a model. During the instruction of the next chapter on model fitting, teams are organized to test, refine, and fit these models to data. The resulting group model is typically substantially superior to any of the individual models. A similar procedure is followed during the instruction of the simulation chapter where students are required to develop models individually followed by a team effort to refine the models and implement them on the computer. 3

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    Finally, we choose projects from a wide variety of disciplines including biology, economics and the physical sciences. We make certain that these projects require little overhead to be paid by the student in order to understand, and successfully attack, the problem. We also allow students to choose a project that can be turned in at the end of the course. Students are free to choose from a diversity of possibilities such as completing UMAP modules, researching a model studied earlier, developing a model in a scenario of interest to them, or analyzing a model presented in another subject they are studying. This chosen project gives the student an opportunity to pursue a subject in some detail. Typically, our students choose a project of interest from one of the Project sections in the text. We require that students complete one of the following for their selected projects: 1. Model Construction Develop a scenario of interest to the student and develop a model to address the situation posed. For example, the student may pose a relevant economic question and construct a graphical model to explain the behavior being studied. 2. Model Analysis: Analyze a model of interest by identifying the underlying assumptions and discussing their applicability. Determine the mathematical conclusions of the model and the sensitivity of the conclusions to the assumptions made. Finally, interpret the mathematical conclusions for real-world scenarios. 3. Model Research Research a model studied in class to determine the ‘state of the art’ in that area. For example, many UMAP modules are available that treat scenarios discussed in class in greater detail. The student may also study a section of the text not covered in class and complete the problems from the corresponding problem set. THE ROLE OF COMPUTATION We emphasize that computing and programming capabilities are not required for our modeling courses. However, beginning with Chapter One, Modeling Change, computation does play a role of increasing importance. The use of computers can significantly enhance a modeling course: in a demonstrational mode to facilitate student understanding of a concept (such as a graphical solution of a linear program) or in a computational mode to reduce the tediousness in carrying out certain numerical procedures. We have found the integration of a computer in a supportive role adds significantly to student interest and to the realism of our modeling course. We provide our students with packaged software for making scatterplots, fitting models according to various criteria, solving linear programs by the Simplex method, constructing divided difference tables and cubic spline models, solving initial value problems by various methods, and graphing functions. A TYPICAL COURSE In this edition, we have provided several options for organizing a course. We have detailed the prerequisites in Figure 2 of the Preface. We incorporate lecture/discussion lessons, hour exams, and computer workshop sessions as part of our modeling courses. Additionally, time is provided for students to complete modeling projects. Detailed discussions of the various lessons and suggestions on how many lessons to devote to each section appear in the Teaching Suggestions portion of this Manual, arranged according to the chapters of our text. : : 4

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    GRADING CONSIDERATIONS The difficulty in grading a modeling course has dissuaded many instructors from teaching modeling, and discouraged many students from taking the course as well. Given the objectives of our course, both creative and technical skills are to be acquired by the student. We have found it best to have the creative requirements accomplished outside the classroom, without the added pressure of a fixed, short time constraint. We use the classroom exams strictly for testing techniques we expect the student to master. These classroom exams test the understanding of basic concepts or straightforward applications of particular techniques, such as fitting a cubic spline (see the section on Sample Tests in this Manual). The more interpretive applications of the modeling techniques are treated in problems and projects assigned for work outside the classroom. The classroom exams are similar to those in other mathematics courses and present no added difficulty in grading. For most of the student projects we are able to establish a grading scale that assigns weights to various parts of each problem. Homework is collected often, principally for purposes of feedback but also for a grade. (Some homework is collected but not graded, to assist students in allowing their minds to roam without fear of being ‘wrong’ while at the same time providing some guidance and feedback.) THE USE OF UMAP MODULES We have found material provided by The Consortium For Mathematics and Its Applications (COMAP) to be outstanding and particularly well suited to the course we propose. COMAP started under a grant from the National Science Foundation and has as its goal the production of instructional materials to introduce applications of mathematics into the undergraduate curriculum. The individual modules may be used in a variety of ways. First, they may be used as instructional material to support several lessons. (We have incorporated several modules in the text in precisely this manner.) In this mode a student completes the self-study module by working through its exercises (the detailed solutions provided with the module can be removed conveniently before it is issued). Another option is to put together a block of instruction covering, for example, linear programming or difference equations, using as instructional material one or more UMAP modules suggested in the projects sections of the text. The modules also provide excellent sources for ‘model research’ since they cover a wide variety of applications of mathematics in many fields. In this mode, a student is given an appropriate module to be researched and is asked to complete and report on the module. Finally, the modules are excellent resources for scenarios for which students can practice model construction. In this mode the teacher writes a scenario for a student project based on an application addressed in a particular module and uses the module as background material, perhaps having the student complete the module at a later date. Information may be obtained by contacting COMAP Inc., Suite 3B, 175 Middlesex Turnpike, Bedford, MA 01730, 800-722-6627. 5

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    FOLLOW-ON MODELING COURSES Many of our students express an interest in taking an additional modeling course. While we have tailored many courses to the individual needs of the students’ programs, a very successful and popular course is one which combines studies from various UMAP modules selected to fit a student’s particular interests, coupled with significant advanced student projects. The proportion of time devoted to the modules versus the student projects varies, depending on the nature of the projects that are available. We are fortunate to have real-world projects readily available at our schools and appropriate to assign for advanced student work. There are, however, several other excellent sources which can be used as background material for student projects The comprehensive four-volume series Modules in Applied Mathematics , edited by William F. Lucas and published by Springer-Verlag, provides important and realistic applications of mathematics appropriate for undergraduates. The volumes treat differential equations models, political and related models, discrete and system models, and life science models. Another book, Case Studies in Mathematical Modelling, edited by D. J. James and J. J. McDonald and published by Halsted Press, provides case studies explicitly designed to facilitate the development of mathematical models. Finally, scenarios with an industrial flavor are contained in the excellent modular series edited by J. L. Agnew and M. S. Keener of Oklahoma State University. Frank R. Giordano William P. Fox Steven B. Horton Maurice D. Weir 6

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    PART II TEACHING SUGGESTIONS CHAPTER ONE Modeling Change SUGGESTED SYLLABUS Class Hours Topic Text 1 Modeling Change with Difference Equations Sect 1.1 1 Approximating Change with Difference Equations Sect 1.2 1 Solutions to Dynamical Systems Sect 1.3 1 Systems of Difference Equations Sect 1.4 OBJECTIVES The major objectives of this chapter are: 1. To build and solve models involving change that takes place in discrete intervals. 2. To prepare the students for modeling change taking place continuously in Chapters 11 and 12. 3. To introduce numerical solutions by iterating difference equations. 4. 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 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 intro sit and 7

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    A spreadsheet is of great use to iterate and graph the models the students build. We have found that even if the students have not used a spreadsheet before, they learn the few commands they need for this chapter quickly. They do not have difficulties extending their knowledge of spreadsheets to handle systems of difference equations in Section 1.4. We added a new modeling example and problems on SIR models. We add a few additional examples for nonlinear DDS here for the instructors use. We build nonlinear discrete dynamical systems to describe the change in behavior of the quantities we study. We also will study systems of DDS to describe the changes in various systems that act together in some way or ways. We define a nonlinear DDS– If the function of a n involves powers of a n (like a n 2  , or a functional relationship (like a n a n − 1 ), we will say that the discrete dynamical system is nonlinear. A sequence is a function whose domain is the set of non-negative integers ( n  0, 1, 2, . . . ). We will restrict our model solution to the numerical and graphical solutions. Analytical solutions may be studied in more advanced mathematics courses. Example 1. Growth of a Yeast Culture We often model population growth by assuming that the change in population is directly proportional to the current size of the given population. This produces a simple, first order DDS similar to those seen earlier. It might appear reasonable at first examination, but the long-term behavior of growth without bound is disturbing. Why would growth without bound of a yeast culture in a jar (or controlled space) be alarming? There are certain factors that affect population growth. Things include resources (food, oxygen, space, etc.) These resources can support some maximum population. As this number is approached, the change (or growth rate) should decrease and the population should never exceed its resource supported amount. Problem Identification Predict the growth of yeast in a controlled environment as a function of the resources available and the current population. Assumptions and Variables We assume that the population size is best described by the weight of the biomass of the culture. We define y n as the population size of the yeast culture after period n . There exists a maximum carrying capacity, M , that is sustainable by the resources available. The yeast culture is growing under the conditions established. Model: y n  1  y n  k  y n   M − y n  where y n is the population size after period n n is the time period measured in hours k is the growth (or decay) constant of proportionality M is the carrying capacity of our system In our experiment, we first plot y n versus n and find a stable equilibrium value of approximately 665. Next, we plot y n  1 − y n versus y n  665 − y n  to find the slope, k ; it is approximately 0. 00082 , With k  0. 00082 and the carrying capacity in biomass is 665. This analysis is currently in the textbook. The final model is y n  1  y n  0. 00082  y n   665 − y n  : : 8

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    Again, this is nonlinear because of the y n 2 term obtained in the expansion. The solution is iterated via technology (there is no closed form analytical solution for this equation) from an initial condition, biomass, of 9. 6: Plot of DDS from growth of a yeast culture. The model shows stability in that the population (biomass) of the yeast culture approaches 665 as n gets large. Thus, the population is eventually stable at approximately 665 units. Example 2. Spread of a Contagious Disease There are one thousand students in a college dormitory, and some students have been diagnosed with meningitis, a highly contagious disease. The health center wants to build a model to determine how fast the disease will spread. Problem Identification: Predict the number of students affected with meningitis as a function of time. Assumptions and Variables: Let m  n  be the number of students affected with meningitis after n days. We assume all students are susceptible to the disease. The possible interactions of infected and susceptible students are proportional to their product (as an interaction term). The model is, m n  1  m n  k  m n  1000 − m n  9

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    Two students returned from spring break with meningitis. The rate of spreading per day is characterized by k  0. 0090. It is assumed that a vaccine can be in place and students vaccinated within 1-2 weeks. m n  1  m n  0. 00090  m n  1000 m n  We iterated with technology: We see that without treatment everyone will eventually get the disease. Plot of DDS for the spread of a disease. − Interpretation: The results show that most students will be affected within 2 weeks. Since only about 10% will be affected within one week, every effort must be made to get the vaccination at the school and get the students vaccinated within one week. We also add an additional example for systems here for the instructor’s use as needed. This is a military model concerning insurgencies. 10

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    Example Modeling Military Interactions and Insurgencies Insurgent forces have a strong foothold in the city of Urbania, a major metropolis in the center of the country of Ibestan. Intelligence estimates they currently have a force of about 1000 fighters. The local police force has approximately 1300 officers, many of which have had no formal training in law enforcement methods or modern tactics for addressing insurgent activity. Based on data collected over the past year, approximately 8% of insurgents switch sides and join the police each week, whereas about 11% of police switch sides and join the insurgents. Intelligence also estimates that around 120 new insurgents arrive from the neighboring country of Moronka each week. Recruiting efforts in Ibestan yield about 85 new police recruits each week as well. In armed conflict with insurgent forces, the local police are able to capture or kill approximately 10% of the insurgent force each week on average while losing about 3% of their force. Problem Statement: Build a mathematical model of this insurgency. Determine the equilibrium state (if it exists) for this DDS. We define the variables P n  the number of police in the system after time period n. I n  the number of insurgents in the system after time period n. n  0, 1, 2, 3, … weeks Model: P n  1  P n − 0. 03 P n − 0. 11 P n  0. 08 I n  85, P 0  1300 I n  1  I n  0. 11 P n − 0. 08 I n − 0. 01 I n  120, I 0  1000 P n I n 1300 1000 1283 1083 1275.02 1149.19 1273.452 1202.588 1276.376 1246.202 1282.38 1282.287 1290.429 1312.537 1299.772 1338.228 1309.862 1360.322 1320.307 1379.549 1330.828 1396.464 1341.229 1411.491 1351.377 1424.958 1361.18 1437.117 1370.585 1448.166 1379.556 1458.26 1388.079 1467.525 1396.15 1476.059 1403.774 1483.945 1410.961 1491.25 1417.726 1498.031 3. 11

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    1424.087 1504.335 1430.062 1510.204 1435.669 1515.674 1440.93 1520.777 1445.862 1525.539 Plot of Insurgents I n over time. 12

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