PREFACE

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 branches of mathematics are obligate mutualists: unable to exist without one another. The history of the growth of 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. Over the past three decades, however, dramatic advances in biological understanding and experimental techniques have unveiled complex networks of interacting components and have yielded vast sets of data about the structure of genomes and the variation and distribution of organisms in space and time. To extract meaningful patterns from these complexities, mathematical methods applied to the study of such patterns is crucial to the maturation of many fields of biology. Mathematics will function as a tool to dissect out the complexities inherent in biological systems rather than be used to encapsulate physical theories through elegant mathematical equations.

Mathematics will ultimately play a different type of role in biology than in physics because the units of analysis in biology are 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 dramatically 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 small handful of discrete, environmentally insensitive, individual traits determined by the particular alleles occupying 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 those in physics, and precise predictions, if possible at all, are much more computationally demanding than comparable precision in physics. Only with the advent of extremely powerful computers can we aspire to use mathematical models to solve the problems of how a string of peptides folds into an enzyme with predicted catalytic properties, how a neuropil structure recognizes and categorizes an object, or how the species composition of a lake changes with an influx of heat, pesticides, or fertilizer.

It is critical that all biologists involved in modeling are properly trained to understand the meaning of output from mathematical models and to have a proper perspective on the limitations of the models themselves to address real problems. Just as we would not allow a butcher with a fine set of scalpels to perform exploratory surgery for cancer in a human being, so we should be wary of allowing biologists poorly trained in the mathematical sciences to use powerful simulation software to analyze the behavior of biological systems. 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 awareness of their meaning and limitations.

About This Book

It is no longer adequate for biologists 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 past three 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 titled 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 interdisciplinary 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; instead, they 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 the basic canons of first-year calculus—but motivated through real biological problems. When combined with a course in statistics, students will have the quantitative foundation needed for research in the biological sciences and the background to take further courses in mathematics. In particular, this book can be viewed as a gateway to the exciting interface of mathematics and biology. As a calculus-based introduction to this interface, the main goals of this book are twofold:

• To provide students with a thorough grounding in concepts and applications, analytical techniques, and numerical methods of calculus
• 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

Concepts Motivated through Applications

First, and foremost, topics are motivated where possible by significant biological applications, several of which appear in no other introductory calculus texts. Significant applications include CO₂ buildup at the Mauna Loa observatory in Hawaii, scaling of metabolic rates with body size, optimal exploitation of resources in patchy environments, insect developmental rates and degree days, rapid decline of populations, velocities of stooping peregrine falcons, drug infusion and accumulation rates, measurement of cardiac output, in vivo HIV dynamics, mechanisms of memory formation, and spread of disease in human populations. 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 we learn by doing, every section ends with a set of applied problems that expose students to additional applications, as well as recurring applications that are further developed as more knowledge is gained. 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.

Chapter Projects

Second, for more in-depth applications, each chapter includes one or more projects that can be used for individual or group work. These projects are diverse in scope, ranging from a study of enzyme kinetics to heart rates in mammals to disease outbreaks.

Early Use of Sequences and Difference Equations

Third, sequences, difference equations, and their applications are interwoven at the sectional level in the first four 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), mathematical phenomena (e.g. chaos) and numerical methods (i.e. Newton's method and Euler's method). Hence, students are exposed to discrete dynamical models in the first half of the book and continuous dynamical models in the second half. The second reason for including sequences 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 discussing 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.

Inclusion of Bifurcation Diagrams and Life History Tables

Fourth, we introduce two topics, bifurcation diagrams and life history tables that are often not covered in other calculus books. Bifurcation diagrams for univariate differential equations are a conceptually rich yet accessible topic. They provide an opportunity to illustrate that small parameter changes can have large dynamical effects. Life history tables provide students with an introduction to age structured populations and the net reproductive number R₀ of a population or a disease.

Historical Quests

Fifth, throughout the text there are problems labeled as . These problems are not just historical notes to help students see mathematics and biology as living and breathing disciplines; rather, they are designed to involve students in the quest of pursuing great ideas in the history of science. Yes, they provide some interesting history, but they also lead students on a quest that should be rewarding for those willing to pursue the challenges they offer.

Multiple Representations of Topics

Sixth, throughout the book, concepts are presented visually, numerically, algebraically, and verbally. By using these different perspectives, we hope to enhance as well as reinforce understanding of and appreciation for the main ideas.

Review Sections

Seventh, we include review questions at the end of each chapter that cover concepts from each section in that chapter.

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, including power functions, the exponential function, inverse functions and logarithms. While most of these concepts are familiar, the emphasis on modeling and verbal, numerical and visual representations of concepts will be new to many students. The chapter includes a strong emphasis on working with real data including fitting linear and periodic functions to data. 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.

Chapter 3: In this chapter, the basic rules of differentiation are first developed for polynomials and exponentials. The product and quotient rules are then covered, followed by the chain rule and the concept of implicit differentiation. Derivatives for the trigonometric functions are explored and biological examples are developed throughout. The chapter concludes with sections on linear approximation (including sensitivity analysis), higher order derivatives and l'Hôpital's rule.

Chapter 4: In this chapter, 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 concept of the integral as an “area under a curve” and net change is then discussed and motivates the definition of an integral as the limit of Riemann sums. The concept of the definite integral is developed as a precursor to presenting the fundamental theorem of calculus. Integration by substitution, by parts, and through the use of partial fractions is discussed with a particular focus on biological applications. The chapter concludes with sections on numerical integration and additional applications, including estimation of cardiac output, survival-renewal processes, and work as measured by energy output.

Chapter 6: In this chapter, we provide a comprehensive introduction to univariate differential equations. Qualitative, numerical, and analytical approaches are covered, and a modeling theme unites all sections. Students are exposed 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 compute expectations and variances. Distributions covered in the context of describing real-world data include the uniform, Pareto, exponential, logistic, normal, and lognormal distributions. The chapter concludes with a section on life history tables and the net reproductive number of an age-structured population.

Chapter 8: In this chapter, we introduce functions of two variables, particularly in the context of the representation of surfaces in three-dimensional space, which has general relevance as well as biological modeling relevance. We then provide an introduction to 2-by-2 matrices, 2-D vectors, and related eigenvalues and eigenvectors—purely in terms of their relevance to modeling 2-D systems using linear differential equations and finding their equilibria. The chapter concludes with a section on phase-plane methods used to explore the behavior of nonlinear 2-D differential equation models, with examples drawn from pharmacology, cell biology, ecology, and epidemiology.

Supplementary Materials for Students and Instructors

Instructor's Solutions Manual This supplement, written by Tamas Wiandt, provides worked-out solutions to most exercises in the text (ISBN 9781118645567).

Student Solutions Manual This supplement, written by Tamas Wiandt, provides detailed solutions to most odd-numbered exercises (ISBN 9781118645598).

Instructor's Manual This supplement, written by Eli Goldwyn, contains teaching tips, additional examples, and sample assignments (ISBN 9781118676981).

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Acknowledgments

Two of us (Getz and Smith) owe a debt of gratitude to Ben Becker, Michael Westphal, and George Lobell: without their efforts more than fifteen years ago, the seeds of the project that culminated in the production of this book would never have been sown. The three of us are even more deeply indebted to our families for the tolerance shown while we scrambled to meet deadlines; they patiently waited for a seemingly endless project to reach completion. We thank the editors at Wiley for their belief in our book, particularly our acquisitions editors David Dietz and Shannon Corliss, and project editor, Ellen Keohane, who picked up the text at a time when the publishing industry as a whole was going through a recession. Through Ellen's tireless efforts and her orchestration of a highly professional group of designers, illustrators, and typesetters, we were able to turn our LaTeX manuscript into a beautifully designed and produced book. We reserve special thanks to Celeste Hernandez and Marie Vanisko for scrutinizing our text and checking for errors in our formulations and solutions. We also thank Tamas Wiandt, Erin Roberts, Ben Weingartner, Sarah Day, David Brown, and Omar Shairzay for their corrections and suggestions. All remaining errors are our responsibility, and we apologize to readers for any of these that may have given them cause for pause. Toward the end of the project, Ellen moved on and was replaced by Anne Scanlan-Rohrer. Anne has our admiration for the professional way she kept our project on track and saw it through to completion. Finally, we thank all our colleagues who patiently reviewed different parts of the text at various stages of writing and provided invaluable feedback that has greatly improved the final form of this book.

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