Calculus for The Life Sciences

738 Pages·2008·13.274 MB·English

by  Sebastian J. Schreiber, Karl Smith, Wayne Getz

Calculus for the Life Sciences
Sebastian J. Schreiber, Karl J. Smith, and Wayne M. Getz
April 7, 2008
2
About the authors:
Sebastian J. Schreiber receivedhis B.A. in mathematics fromBostonUniversity in1989andhis Ph.D. inmathe-
maticsfromthe UniversityofCalifornia,Berkeleyin1995. He iscurrentlyProfessorofEcologyandEvolutionatthe
University of California, Davis. Previously, he was an associate professor of mathematics at the College of William
and Mary, where he was the 2005 recipient of the Simon Prize for Excellence in the Teaching of Mathematics, and
Western Washington University. Professor Schreiber’s research on stochastic processes, nonlinear dynamics, and
applications to ecology, evolution, and epidemiology has been supported by grants from the U.S. National Science
Foundation and the U.S. National Oceanic and Atmospheric Administration. He is the author or co-author of over
40scientificpapersinpeer-reviewedmathematicsandbiologyjournals. Severalofthese papersareco-authoredwith
undergraduatestudentsthatweresupportedbytheNationalScienceFoundation. ProfessorSchreiberiscurrentlyon
the editorial boards of the research journals: Mathematical Medicine and Biology, Journal of Biological Dynamics,
and Theoretical Ecology.
Karl J. Smith received his B.A. and M.A. (in 1967) degrees in mathematics form UCLA. He moved to northern
Californiain1968toteachatSantaRosaJuniorCollege,wherehetaughtuntilhisretirementin1993. Alongtheway,
heservedasdepartmentchair,andhereceivedaPh.D.in1979inmathematicseducationatSoutheasternUniversity.
A past president of the American Mathematical Association of Two-Year Colleges, Professor Smith is very active
nationallyinmathematics education. He wasfounding editorofWesternAMATYC News, achairpersonofthe com-
mittee on Mathematics Excellence, and a NSF grant reviewer. He was a recipient in 1979 of an Outstanding Young
MenofAmericaAward,in1980ofanOutstandingEducatorAward,andin1989ofanOutstandingTeacherAward.
Professor Smith is the author of over 60 successful textbooks. Over two million students have learned mathematics
from his textbooks.
Wayne M. Getz received his B.Sc , B.Sc. Hons, and Ph.D. in applied mathematics from the University of the
Witwatersrand, South Africa, in 1971, 1972, and 1976 respectively. He was a research scientist at the National
ResearchInstitute forMathematicalSciences inSouthAfrica until he movedto take upa faculty positionin1979at
the University of California,Berkeley. He is currently Professorof Environmental Science and Chair of the Division
of Environmental Biology at UC Berkeley. Professor Getz also has a D.Sc. from the University of Cape Town
and is an Extraordinary Professor at the University of Pretoria, both in South Africa. Recognition for his research
in biomathematics and its application to various areas of physiology, behavior, ecology, and evolution include an
Alexander von Humbold US Senior Scientist Research Award in 1992, election to the American Association for the
Advancement of Science (1995), the California Academy of Sciences (2000), and the Royal Society of South Africa
(2003). He was appointed as a Chancellor’s Professor at Berkeley from 1998-2001. Professor Getz has served as a
consultanttotheUSandCanadianGovernmentsandaUSDistrictJudgeonmatterspertainingto themanagement
of Fisheries, as a member of two National Academy of Sciences review panels, and is a founder and Trustee of the
South African Centre for Epidemiological modeling and Analysis. His research over the past 25 years has been
supported by the U. S. National Science Foundation, the National Institutes of Health, California Department of
FoodandAgriculture,CaliforniaSeaGrant,theA.P.SloanFoundation,theWhitehallFoundation,DARPA,andthe
Ellison Medical Foundation. Recently he received a prestigious James S. McDonnell 21st Century Science Initiative
Award. Professor Getz has published a book entitled Population Harvesting in the Princeton Monographs in
PopulationBiologyseries,edited other books andvolumes,and is anauthor or coauthoronmore than 150scientific
papers in over 50 different peer-reviewed applied mathematics and biology journals.
©2008 Schreiber, Smith & Getz
Preface
If the 20th century belonged to physics, the 21st century may well belong to biology. Just 50 years after
the discovery of DNA’s chemical structure and the invention of the computer experiment, a revolution is
occurring in biology, driven by mathematical and computational science.
JimAustin,USEditorofScience,andCarlosCastillo-Chavez,ProfessorofBiomathematics, Science,February6,2004
Calculus was invented in the second half of the seventeenth century by Isaac Newton and Gottfried Leibniz
to solve problems in physics and geometry. Calculus heralded in the age-of-physics with many of the advances in
mathematics over the past 300 years going hand-in-hand with the development of various fields of physics, such as
mechanics, thermodynamics, fluid dynamics, electromagnetism, and quantum mechanics. Today, physics and some
branchesof mathematics areobligate mutualists: unable to exist withoutone another. This history ofthe growthof
this obligate association is evident in the types of problems that pervade modern calculus textbooks and contribute
to the canonical lower division mathematics curricula offered at educational institutions around the world.
The age-of-biology is most readily identified with two seminal events: the publication of Charles Darwin’s, On
The Origin of Species, in 1859; and, almost 100 years later, Francis Crick and James Watson’s discovery in 1953 of
the genetic code. About mathematics, Darwin stated
I have deeply regretted that I did not proceed far enough at least to understand something of the great
leading principles of mathematics; for men thus endowed seem to have an extra sense.
Despite Darwin’s assertion, mathematics was not as important in the initial growth of biology as it was in physics.
However, in the past decades, dramatic advances in biological understanding and experimental techniques have
unveiledcomplexnetworksofinteractingcomponentsandhaveyieldedvastdatasets. Toextractmeaningfulpatterns
from these complexities, mathematical methods applied to the study of such patterns is going to be crucial to the
maturationofmanyfieldsofbiology. Itsrole,however,willbemorecomputationalthananalytical. Mathematicswill
function as a tool to dissect out the complexities inherent in biologicalsystems rather than be used to encapsulated
physical theories through elegant mathematical equations.
The reason that mathematics will ultimately play a different type of role in the age-of-biology than it did in
the age-of-physics is largely due to the units of analysis in biology being extraordinarily more complex than those
of physics. The difference between an ideal billiard ball and a real billiard ball or an ideal beam and a real beam
completely pales in comparison with the difference between an ideal and a real salmonella bacterium, let alone an
ideal and a real elephant. Biology, unlike physics, has no axiomatic laws that provide a precise and coherent theory
upon which to build powerful predictive models. The closest biology comes to this ideal is in the theory of enzyme
kinetics associated with the simplest cellular processes and the theory of population genetics that only works for a
smallhandfulofdiscrete,environmentallyinsensitive,individualtraitsdeterminedbytheparticularallelesoccupying
discrete identifiable genetic loci. Eye color in humans provides one such example.
This complexity in biology means that accurate theories are much more detailed than in physics, and precise
predictions,ifpossibleatall,aremuchmorecomputationallydemandingthancomparableprecisioninphysics. Only
with the advent of extremely powerful computers can we begin to aspire to solve the problems of how a string of
peptidesfoldsintoanenzymewithpredictedcatalyticproperties,tounderstandhowaneuropilstructureinthebrain
of some animalrecognizesa sound, a smell, or the shape of anobject, or to predict how the species compositionofa
lakewillchangewithaninfluxofheat,pesticides,orfertilizer. Ontheotherhand,predictionsregardingtheresponse
oflargersystemsconsistingofcommunitiesofindividualsorwholeecosystemstoexternalperturbationsoftencannot
be tested without irreversibly damaging an irreplaceable or unique system. Hence, mathematical models provide a
powerful tool to explore the potential effects of these perturbations.
©2008 Schreiber, Smith & Getz 3
4
Itiscriticalthatallbiologistsinvolvedinmodelingareproperlytrainedtounderstandthemeaningofoutputfrom
models and to have a proper perspective on the limitations of the models themselves. Just as we would not allow a
butcher with a fine setofscalpels to performexploratorysurgeryfor cancerin a humanbeing, so we shouldbe wary
of allowingbiologists poorly trainedin the mathematical sciences to use powerfulsimulationsoftwareto analyze the
behaviorofbiologicalsystems. If,forexample,anenvironmentalimpactanalysisisdramaticallywronginpredicting
how a lake will respond to an influx of heat coming from a power plant to be located on its shores, then the flora
andfauna inthe lakeandonits surroundingshorescouldendup being degradedto the pointwhere the recreational
value of the lake is destroyed. Consequently, the time has come for all biologists, who are interested in more than
just the natural history of their subject, to obtain a sufficiently rigorous grounding in mathematics and modeling
so that they can appropriately interpret models with an awarenessof their meaning and limitations. Reflecting this
view, in a news release of the National Institute of General Medical Sciences (NIGMS), Dr. Judith H. Greenberg,
acting director of NIGMS states: “Advances in biomedical research in the 21st century will be critically dependent
on collaboration between biologists and scientists in other disciplines, such as mathematics.” And NIGMS, along
∗
with the National Science Foundation (NSF), intends to “put their money where their mouth is” because these
organizationsanticipatespendingmorethan$24millionto“encouragetheuseofmathematicaltoolsandapproaches
to study biology.”
About this Book
In training biologists to be scientists, it is no longer adequate for them to study either an engineering calculus or
a “watered-down” version of the calculus. The application of mathematics to biology has progressed sufficiently
far in the last two decades and mathematical modeling is sufficiently ubiquitous in biology to justify an overhaul
of how mathematics is taught to students in the life sciences. In a recent article “Math and Biology: Careers at
the Interface,” the authors state, “Today a biology department or research medical school without ‘theoreticians’
∗
is almost unthinkable. Biology departments at research universities and medical schools routinely carry out inter-
disciplinary projects that involve computer scientists, mathematicians, physicists, statisticians, and computational
scientists. And mathematics departments frequently engage professors whose main expertise is in the analysis of
biological problems.”In other words, mathematics and biology departments at universities and colleges around the
world can no longer afford to build separate educational empires, but instead need to provide coordinated training
for students wishing to experience and ultimately contribute to the explosion of quantitatively rigorous research in
ecology, epidemiology, genetics, immunology, physiology, and molecular and cellular biology. To meet this need,
interdisciplinary courses are becoming more common at both large and small universities and colleges.
In this text, we present material to cover one year of calculus, which, when combined with a statistics course,
will make students conversant in the use of mathematics in the natural sciences and to inspire them to take further
coursesinmathematics. Inparticular,the book canbe viewedasa gatewayto the exciting interfaceofmathematics
and biology. As a calculus based introduction to this interface, the main goals of this book are
• to provide students with a thorough grounding in calculus’ concepts and applications, analytical techniques,
and numerical methods.
• to have students understand how, when, and why calculus can be used to model biological phenomena.
To achieve these goals, the book has several important features.
Features
First,andforemost,everytopicismotivatedbyasignificantbiologicalapplicationseveralofwhichappearinnoother
texts. These topics include CO build-up at the Mauna Loa observatory in Hawaii, scaling of metabolic rates with
2
body size, enzyme activity in response to temperature, optimal harvesting in patchy environments, developmental
rates and degree days, sudden population disappearances, stooping peregrine falcons, drug infusion, measuring
cardiac output, in vivo HIV dynamics, and mechanisms of memory formation. Many of these examples involve real
world data and whenever possible, we use these examples to motivate and develop formal definitions, procedures,
and theorems. Since students learn by doing, every section ends with a set of applied problems that expose them to
∗Pressrelease,oftheNational Institutes ofHealth,AlisaZappMachalek,August22,2002.
∗“MathandBiology: CareersattheInterface,”JimAustinandCarlosCastillo-Chavez,Science,February6,2004.
©2008 Schreiber, Smith & Getz
5
additional applications as well as further developing applications presentedwithin the text. These applied problems
are always preceded by a set of drill problems designed to provide students with the practice they need to master
the methods and concepts that underlie many of the applied problems.
Second, for more in depth applications, each chapter will include at least two projects which can be used for
individual or group work. These projects will be diverse in scope ranging from a study of enzyme kinetics to the
heart rates in mammals to disease outbreaks.
Third,sequences,differenceequations,andtheirapplicationsareinterwovenatthesectionallevelinthefirstfour
chapters. We include sequences in the first half of the book for three reasons. The first reason is that difference
equations are a fundamental tool in modeling and give rise to a variety of exciting applications (e.g. population
genetics),mathematicalphenomena(e.g. chaos)andnumericalmethods(i.e. Newton’smethodandEuler’smethod).
Hence, students get exposed to discrete dynamical models in the first half of the book and continuous dynamical
models in the second half of the book. The second reason is that two of the most important concepts, limits and
derivatives, provide fundamental ways to explore the behavior of difference equations (e.g., using limits to explore
asymptotic behavior and derivatives to linearize equilibria). The third reason is that integrals are defined as limits
of sequences. Consequently, it only makes sense to present sequences before one discusses integrals. The material
on sequences is placed in clearly marked sections so that instructors wishing to teach this topic during the second
semester can do so easily.
Fourth, we introduce two topics, bifurcation diagrams and life history tables, that are not covered by other
calculusbooks. Bifurcationdiagramsforunivariatedifferentialequationsareaconceptuallyrichyetaccessibletopic.
Theyprovideanopportunitytoillustratethatsmallparameterchangescanhavelargedynamicaleffects. Lifehistory
tables provide students with an introduction to age structured populations and the net reproductive number R of
0
a population or a disease.
Fifth, throughout the text are problems described as Historical Quest. These problems are not just historical
notes to help one see mathematics and biology as living and breathing disciplines, but are designed to involve the
studentinthequestofpursingsomegreatideasinthehistoryofscience. Yes,theywillgivesomeinterestinghistory,
but then lead one on a quest which should be interesting for those willing to pursue the challenge they offer.
Sixth, throughout the book, concepts are presented visually, numerically, algebraically, and verbally. By pre-
senting these different perspectives, we hope to enhance as well as reinforce the students understanding of and
appreciation for the main ideas.
Seventh, we include well-developed review sections at the end of each chapter that contain lists of definitions,
important ideas, important applications, as well as review questions.
Content
Chapter 1: This chapter begins with a brief overview of the role of modeling in the life sciences. It then focuses
on reviewing fundamental concepts from precalculus and probability. While many of the precalculus concepts are
familiar,theemphasisonmodelingandverbal,numericalandvisualrepresentationsofconceptswillbenewtomany
students. Basicprobabilityconceptsareintroducedbecausethey playafundamentalroleinmanybiologicalmodels.
This chapter also includes an introduction to sequences through an emphasis on elementary difference equations.
Chapter 2: In this chapter, the concepts of limits, continuity, and asymptotic behavior at infinity are first
discussed. The notion of a derivative at a point is defined and its interpretation as a tangent line to a function is
discussed. The idea of differentiability of functions and the realization of the derivative as a function itself are then
explored. Examples and problems focus on investigating the meaning of a derivative in a variety of contexts.
Chapter3: Inthis chapter,the basic rulesof differentiationarefirstdevelopedfor polynomials andexponentials.
Theproductandquotientrulesarethencovered,followedbythechainruleandtheconceptofimplicitdifferentiation.
Derivatives for the trigonometric functions are explored and biological examples are developed throughout. The
chapterconcludeswithsectionsonlinearapproximation(includingsensitivityanalysis),higherorderderivativesand
l’Hˆopital’s rule.
Chapter 4: In chapter 4, we complete our introduction to differential calculus by demonstrating its application
to curve sketching, optimization, and analysis of the stability of dynamic processes described through the use of
derivatives. Applications include canonical problems in physiology, behavior, ecology, and resource economics.
Chapter 5: This chapter begins by motivating integration as the inverse of differentiation and in the process
introduces the concept of differential equations and their solution through the construction of slope fields. The
conceptoftheintegralasan“areaunderacurve”andnetchangeisthendiscussedandmotivatesthedefinitionofan
©2008 Schreiber, Smith & Getz
6
integralas the limit of Riemann sums. The concept of the definite integralis developed as a precursor to presenting
TheFundamentalTheoremofCalculus. Integrationbysubstitution,byparts,andthroughtheuseofpartialfractions
are discussed with a particular focus on biological applications. The chapter concludes with a section on numerical
integration and a final section on additional applications including estimation of cardiac output, survival-renewal
processes, and work as measured by energy output.
Chapter6: Inthis chapterweprovideacomprehensiveintroductiontounivariatedifferentialequations. Qualita-
tive,numerical,andanalyticapproachesarecoveredandamodellingthemeunitesallsections. Studentsareexposed
via phase line diagrams, classification of equilibria, and bifurcation diagrams to the modern approach of studying
differential equations. Applications to in vivo HIV dynamics, population collapse, evolutionary games, continuous
drug infusion, and memory formation are presented.
Chapter 7: In this chapter we introduce applications of integration to probability. Probability density functions
are motivated by approximating histograms of real world data sets. Improper integration is presented and used as
a tool to computes expectations and variances. Distributions covered in the context of describing real world data
include the uniform, Pareto,exponential,logistic,normal,andlognormaldistributions. The chapterconcludes with
a section on life history tables and the net reproductive number of an age-structured population.
Supplemetary Material
To be added later.
Acknowledgements
To be added later.
©2008 Schreiber, Smith & Getz
Contents
1 Modeling with Functions 3
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Real Numbers and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.3 Data fitting with Linear and Periodic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
1.4 Power Functions and Scaling Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
1.5 Exponentials and Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
1.6 Function Building . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
1.7 Sequences and Difference Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
1.8 Summary and Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
1.9 Group Projects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
2 Limits and Derivatives 141
2.1 Rates of Change and Tangent Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
2.2 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
2.3 Limit Laws and Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
2.4 To Infinity and Beyond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
2.5 Sequential Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
2.6 The Derivative at a Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
2.7 Derivatives as Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
2.8 Summary and Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
2.9 Group Projects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
3 Derivative Rules and Tools 261
3.1 Derivatives of Polynomials and Exponentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
3.2 Product and Quotient Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
3.3 Chain Rule and Implicit Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
3.4 Trigonometric Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304
3.5 Linear Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312
3.6 Higher-Order Derivatives and Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
3.7 l’Hˆopital’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
3.8 Summary and Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346
3.9 Group Projects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350
4 Applications of Differentiation 353
4.1 Graphing with Gusto . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
4.2 Getting Extreme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366
4.3 Optimization in Biology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379
4.4 Applications to Optimal Behavior. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393
4.5 Linearization and Difference Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408
4.6 Summary and Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424
4.7 Group Projects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429
©2008 Schreiber, Smith & Getz 1
2 CONTENTS
5 Integration 433
5.1 Antiderivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435
5.2 Accumulated Change and Area under a Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449
5.3 The Definite Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464
5.4 The Fundamental Theorem of Calculus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477
5.5 Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487
5.6 Integration by Parts and Partial Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496
5.7 Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508
5.8 Applications of Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524
5.9 Summary and Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539
5.10 Group Projects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542
6 Differential Equations 547
6.1 A Modeling Introduction to Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549
6.2 Separable Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562
6.3 Linear Models in Biology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572
6.4 Slope Fields and Euler’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584
6.5 Phase Lines and Classifying Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601
6.6 Bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616
6.7 Summary and Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 628
6.8 Group Projects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633
7 Probabilistic Applications of Integration 637
7.1 Histograms, PDFs and CDFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639
7.2 Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 659
7.3 Mean and Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674
7.4 Bell-shaped distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 691
7.5 Life tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 712
7.6 Summary and Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 726
7.7 Group Research Projects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 729
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 731
©2008 Schreiber, Smith & Getz
Chapter 1
Modeling with Functions
1.1 Introduction, p. 5
1.2 Real Numbers and Functions, p. 17
1.3 Data Fitting with Linear and Periodic Functions, p. 40
1.4 Power Functions and Scaling Laws, p. 57
1.5 Exponentials and Logarithms, p. 71
1.6 Function Building, p. 87
1.7 Sequences and Difference Equations, p. 107
1.9 Summary and Review, p. 128
Figure 1.1: The humpback whale (Megaptera novaeangliae) is found in all the world’s oceans. They are known for
the complex “songs” which last 10-20 minutes. (See Problem 29, Section 1.5)
PREVIEW
The interface between mathematics and biology presents challenges and opportunities for both mathematicians and biol-
ogists. Unique opportunities forresearch have surfacedwithin the last ten to twenty years, both because of the explosion
ofbiologicaldata withtheadvent ofnew technologies andbecauseoftheavailabilityofadvanced andpowerfulcomputers
that can organize the plethora of data. For biology, the possibilities range from the level of the cell and molecule to the
©2008 Schreiber, Smith & Getz 3
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biosphere. Formathematics, thepotential isgreatintraditionalappliedareassuchasstatisticsanddifferentialequations,
aswellasinsuchnon-traditional areasasknottheory.
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Thesechallenges: aggregation ofcomponents toelucidate thebehavior ofensembles, integration across scales, andinverse
problems,arebasictoallsciences, andavarietyof techniques existtodealwiththem andtobegintosolvethebiological
problemsthat generate them. However,theuniqueness ofbiological systems,shapedbyevolutionary forces,willposenew
difficulties, mandate new perspectives, and led to the development of new mathematics. The excitement of this area of
scienceisalreadyevident,andissuretogrowintheyearstocome. -Executive Summary from a NSF-Sponsored Workshop
Led by Simon Levin (1990)
The above quotation is as true today as when it was written. It provides a hint of the exciting opportunities
that exists at the interface of mathematics and biology. The goal of this course is to provide you with a strong
grounding in calculus while, at the same time introducing you to various research areas of mathematical biology
and inspiring you to take more courses at this interdisciplinary interface. In this chapter, we will set the tone for
the entire book and will provide you with some of the skills you will need to work at this interface. As the title of
the chapter suggests, we introduce you to modeling with mathematical functions. In the first section, the idea of
mathematical modeling is introduced. In the next five sections, we remind you of the mathematical concepts that
will be important to you as you make your journey through this book. Throughout the book you will find real life
problems that can be solved using mathematics. For example, the decline of whales is a serious problem that we
inherited fromthe whaling activities of the past two centuries. The InternationalWhaling Commissionin 1966gave
thehumpbackwhaleworldwideprotectionstatus,buttheirpopulationtodayisonlyabout30-35%oftheirestimated
originalpopulationlevels. Inthe lastproblemin this chapter,we use a modelto explorethe densities we canexpect
a whale population to recover to after harvesting individuals in the population has ceased.
©2008 Schreiber, Smith & Getz