Elementary Mathematical Models Order Aplenty and a Glimpse of Chaos

362 Pages·1998·20.497 MB·English

by  Dan Kalman

Elementary Mathematical Models
Order Aplenty and a Glimpse of Chaos
Inspired by the work of M. C. Escher, the cover design exhibits order aplenty, and provides a
glimpseofchaos. Intertwinedwiththefishareimagesofabifurcationdiagramthatarisesinthe
discussionofchaosinChapter14. Andjustasastudyoffishpopulationsistransformedintoan
introductiontochaos,sothefishinthecoverdesignaretransformedintobifurcationdiagramsas
theyswimacrossthepage.
(cid:13)c 1997by
TheMathematicalAssociationofAmerica(Incorporated)
LibraryofCongressCatalogCardNumber97-74331
PrintISBN978-0-88385-707-6
ElectronicISBN978-1-61444-601-9
PrintedintheUnitedStatesofAmerica
CurrentPrinting(lastdigit):
109876543
Elementary Mathematical Models
Order Aplenty and a Glimpse of Chaos
Dan Kalman
American University
Published and distributedby
The Mathematical Association of America
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101Careersin Mathematics,2ndEdition editedbyAndrew Sterrett
Archimedes: WhatDid HeDoBesidesCryEureka?,ShermanStein
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ACoursein MathematicalModeling,DouglasMooneyandRandallSwift
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Preface
College professors could learn something from comedian Don Novello. In his Father
GuidoSarduccipersonna,Novellodescribedarevolutionaryapproachtohighereducation:
the five minute university. This is an accelerated course of study in which the students
take five minutes to learn what the average college graduate remembers five years after
graduation. For example, the five minute university course in economics is Supply and
Demand. That is not the title of the course, it is the course. Similarly, the five minute
universitycourse in English literatureis To Be, or Not To Be.
There is insight in Novello’s satire, and a pointed question for educators. What is
it that we really want our students to take away from their college courses? After the
cramming and exams are done, afterthe studentsleave school and go about thebusiness
of living, what is it that we want them to retain?
TheyDon’tTeachCollegeAlgebraattheFiveMinuteUniversity. Asacaseinpoint,
consider a typical course in college algebra. The goal of this course is to train students
in techniques of algebraic manipulation. There is a heavy dose of algebraic operations
thatstudentswillseldomsee again, ifever, evenin moreadvanced mathematicscourses:
the simplification of complicated rational functions, compound radical expressions with
several different radices, and logarithmic equations with variable bases, to name a few.
Students are expected to master the rules that apply to each kind of symbolic form, the
assumed precedence for the order of operations, and the proper use of parentheses, as
wellas avarietyofstandardforms, andalgorithmsforobtainingtheseforms. How much
of all this does the average student remember five years after graduation? My guess is
zero. That is why the five minuteuniversitydoesn’t have a college algebra course.
Prompted by this insight, I set out to create a new kind of course. I ended up writing
thebook you now hold. Its developmentwas influencedby boththe immediateneeds of
students to hone mathematical skills for other courses, as well as longer term goals that
mightwarrant a mentionat the five minuteuniversity.
In this Preface I willdescribe the studentsfor whom the book is intended,and what I
hope they will get out of the course. There will also be a discussion of the philosophy
followed in developing the course, followed by teaching suggestions. These will pertain
v
vi ELEMENTARYMATHEMATICALMODELS
bothtomethodsof presentation,as wellas tothepace andcoverage appropriatefora 14
week semester course. To simplifythe exposition, EMM willbe used as an abbreviation
for Elementary MathematicalModels.
Students Served. The EMM course seeks to serve the same students as traditional
collegealgebraandliberalartsmathematicscourses. Thesearestudentswhohavestudied
at least a year of algebra in highschool, but who are notheaded for calculus. Theymay
need to take a mathematics course for a general education requirement. They may also
needfamiliaritywiththe mainideas of college algebra forquantitativegeneral education
courses in areas outside of mathematics, such as science, economics, and business. For
most of these studentsthis will be the only mathematics course completedin college.
Development Philosophy. The developmentof EMM reflects two importantconcerns.
First, I wanted the course to possess an internal integrity. My aim was to make the
material intrinsically interesting and worthwhile, to present a coherent story throughout
the semester, and to convey something of the utility,power, and methodof mathematics.
Among other things, this demands genuineand understandableapplications.
In formulatingthese goals, I was strongly influencedby the ground-breakingstatistics
text by Freedman, Pisani, and Purves1. The authors’ account2 of the development of
this book is a model of clarity and insight. It describes a succession of attempts to
refinean introductorystatisticscourse. Ultimately,the authorswere forced to focus on a
single fundamental question: What are the main ideas the field of statistics has to offer
the intelligentoutsider? Their answer to this question became the foundation for a new
course in statistics. In the same way, I have triedto capture some of the main ideas that
arise in the applications of mathematics, and to present these ideas in an interestingand
intelligiblecontext. I am pleased to acknowledge theinfluenceof Freedman, Pisani, and
Purves on my thinking.
Course developmentwas also guided by a second fundamental concern, to emphasize
the topics that students are most likely to meet in mathematical applications in other
disciplines. I am thinkinghere of the most elementaryapplications, formulatedin terms
of arithmeticand simplealgebraic operations: linear, quadratic, polynomial,and rational
functions; square roots; exponential and logarithmic functions. These functions are the
building blocks for the simple models that appear in first courses in the physical, life,
and social sciences. Many studentsinmycourse have learnedabout some or allof these
functionsinpriorclasses. Thechallengeisto makethematerialfreshand interestingfor
these students, and at the same time, accessible as a first exposure. I tried to do this by
creepinguponeachmathematicaltopicinthecontextofanapplication. Theapplications
areallanalyzedusingacommonmethodology,involvingdifferenceequations. Thispoint
will be discussed in greater detail presently. It suffices here to indicate that difference
equation methods are applicable to a number of problems that (I hope) have obvious
significanceandrelevance. ThemathematicaltopicsIwanttocovercomeupinanatural
wayas differenceequationsareusedtostudytheapplications. Thealgebraicemphasisis
1 DavidFreedman,RobertPisani,andRogerPurves. Statistics. W.W.Norton,NewYork,1978.
2 Seethefirst6pagesoftheintroductiontotheinstructorsmanualforthetext.
Preface vii
restricted to what is really required for working with these simple models. More arcane
aspects of algebra, like those described earlier, have been banished from my course.
ThatisnottosaythatIhaveeliminatedallalgebra. Farfromit. Algebraisanimportant
andpowerfultool, andstudentsshouldappreciateandrememberthatfact. Thiscan even
bea partofthefiveminuteuniversitycurriculum. Algebraisanimportantandpowerful
tool. ButIwantthestudentstomakethatobservationthemselves,fromseeingalgebrain
action. Thealgebrathatdoesappearinthebookispresentedinaway thatmakesitclear
why algebra is needed, and what it contributesto formulatingand analyzing models.
I believe EMM is a hybrid of liberal arts mathematics courses and traditional college
algebracourses. Thegoalsofinstruction,particularlyasregardsthefiveminuteuniversity
ideas, arelikethoseofa liberalartsmathematicscourse. Thecontentemphasesare more
like those of a college algebra course. I have set out to infuse quadratic equations and
logarithms with the same kind of enthusiasm and vitality that authors like Jacobs3 have
broughttotheirworks. Mineis a terminalmathematicscourse thatdoesn’tforedoomthe
student to terminating the study of mathematics. He or she will be well prepared to go
ontoprecalculusandcalculus. Itisanentrylevelmathematicscoursethatstudentsdon’t
have to slog through motivated only by vague promises that the material will be needed
in future studies. They can see as they take the course what this material is good for
and howitcan beused. Hershand Davis4 have describedmathematicsas thepartof the
humanitiesthat has the qualitiesof science. In an analogous way, my course is the entry
level college algebra course that has the qualitiesof liberal arts mathematics.
Based on the concerns discussed above, my development of EMM was guided by the
followingprinciples:
• Introduce each new mathematicaloperation in the context of a believableproblem;
• Weave all of the topics into an integratedwhole;
• Provide as a theme a methodology that can be used in a large number of problem
contexts;
• Emphasize conceptual understandingof how the mathematics contributes to solving
problemsover technical mastery of each mathematical topic for its own sake.
The second and third principles are reflected in the choice of topics for the course. A
brief discussion of the organization of topics willbe presentednext.
CourseOutline. Inoverview,EMMconcernsdiscreteandcontinuousmodelsofgrowth.
Each applicationstartswitha discretemodelbuiltaroundasequence{an}. Thereisalso
asimplehypothesisdescribingthewaysuccessivetermsdependonprecedingterms. One
example of such a hypothesis is that each new term an+1 can be obtained by adding a
fixed constant to the preceding term, an. The algebraic expression of thishypothesis,
an+1 =an+a constant
is a difference equation. Throughout the course, a succession of simple hypotheses are
considered: the terms of the sequence increase by a constant amount; the terms increase
3 HaroldR.Jacobs. Mathematics,AHumanEndeavor,SecondEdition. W.H.Freeman, NewYork,1982.
4 ReubenHershandPhillipJ.Davis. TheMathematicalExperience. Birkhauser,Boston,MA,1980.
viii ELEMENTARYMATHEMATICALMODELS
by varying amounts which themselves increase by a constant amount; theterms increase
by a constant percentage; and so on. Each of these hypotheses finds expression as a
particular kind of difference equation. Each new class of difference equation is studied
using a common methodology. In outline, the unifyingmethodology is
• Formulate a family of difference equations;
• Develop solutionsto the difference equations;
• Study the behavior of the solutions.
Whilethemodelsareinitiallyformulatedintermsofdiscretevariables,thesolutionsofthe
difference equationscan generallybe interpretedin the context of continuous variables.
The course starts with arithmetic growth and builds up through successively more
complicated models: quadratic growth, geometric growth, mixtures of arithmetic and
geometricgrowth,andfinallylogisticgrowth. EMMclimaxeswithastudyofthechaotic
behavior that can occur in logistic models. Interspersed among the discussions of the
various growth models are units on the families of functions that appear as solutions
to difference equations. Thus, the study of arithmetic growth leads into a unit on
linearfunctions;quadratic growthmodelsprovidea settingfor studyingthepropertiesof
quadratic functions, and so on.
Unifying Themes. Throughout the course, several themes are touched on repeatedly.
Obviously, the formulation of a model in terms of a difference equation, and the
developmentofa solutiontothatdifferenceequationhas aconstantpresence. Theuseof
systematicinvestigationandpatternstoidentifythesolutionsofthedifferenceequationsis
another common theme. Numerical and graphical methods are used systematicallyin all
thetopics. Theideaofexpressingthesolutiontothedifferenceequationasafunction,and
ofevaluatingorinvertingthatfunctiontoanswerquestionsaboutthemodelisalsostressed
repeatedly. Theprogressionoftopicsdemonstratestheincrementalnatureofmodeling,as
earlier models are refined or modifiedto obtain later, more realistic models. Finally, the
mostfundamentalthemeofEMMistheapplicabilityofthemathematicaltopics. Students
recognize this from the start, because the entire course evolves out of the investigation
of real problems. By the end of the course, they are studying sophisticated models with
unexpectedandnontrivialbehaviors. Thepowerofalgebratoanswerimportantquestions
about these models is always on display.
Teaching EMM
Scope and Sequence. There is more in the book than I have been able to cover in a
semester. Here are a few suggestions for a subset thatwillmake up a reasonable course.
Thecore materialinthe course consists of chapters 1{6, 9, 10, 12. Thiswillcover most
of the same functions as the standard college algebra course, though in less algebraic
detail. When I teach the course, I cover this core material, plus chapters 7, 13, and 14.
That is everythingin the book except chapters 8 and 11. If that pace turns out to be too
fast, you can eliminatethe sections of chapter 13 that involve harvesting, and part or all
of chapter 7. The resultwillstillbe a coherent course witha definitestory to tell,and a
climactic finish.
Preface ix
You mayprefertocover somethingsthatIleave out. Chapter8uses whathas already
beendevelopedabout quadratic functionsto presenttheideas behindleast squares linear
regression. I put it in the book as a significant application of quadratic functions, and
one which connects to an importanttool usedthroughoutthebook: fittinga lineto data.
Chapter11 gives greater depth about logarithmicfunctions, particularlytheideas behind
logarithmicscalesandlog-linearregression. Ifyouwanttocoverthismaterial,youmight
elect to skip chaos (chapter 14) and possiblylogistic growth (chapter 13), as well.
I go quickly through the first two chapters. The first chapter presents an overview of
the course that willnot make a great deal of sense at thestart. It is providedto give the
students a feel for what is coming. It might be productive to have the students reread it
after finishingchapter 6. Chapter 2 introduces difference equations in a general setting.
Somestudentshavedifficultywiththesubscriptnotationandtheideaofdifferenceversus
functional equations. My approach is to go over it quickly, indicating that the students
will have plenty of opportunityfor reviewing the same ideas in later chapters, and then
providetime inchapters 3 and 5 to again work on thetroublesomeideas. An alternative
is to slow down in chapter 2, and then go faster through chapters 3 and 5.
Even if you don’t do chapter 14, take a look at the first section. It provides a
retrospective review of most of the difference equation models used in the course. You
may wish to assign some or all of this material as review at the end of the course.
A wordof warning. Youwillnotbeabletopresentinclass everythingineach chapter
that you cover. I wrote this book so that I could say on paper just about everything I
wanted to present to the students. I let them know early that they needed to read the
material before class, and that I would not lecture on every topic. Instead, I used a
combination of students working in groups, lab activities, and presenting solutions to
problems to augment what the students learned by studying outside class. If you are
not comfortable withthat approach, you may have to reduce the numberof chapters you
cover.
The schedule below shows how I organize my semester course.
Week Topics
1 Chapters 1 and 2
2 Chapter 3
3 Chapter 4
4 Chapter 5
5 Exam, review
6 Chapter 6
7 Chapter 7
8 Chapter 9
9 Chapter 10
10 Exam, review
11 Chapter 12
12 Chapter 13
13 Chapter 14
14 Review