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Calculus for Life Sciences: a Modeling Approach

J. L. Cornette and R. A. Ackerman
Cornette and Ackerman
Publication Date: 
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Electronic Book
electronic book
[Reviewed by
Steven Deckelman and George Jennings
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Traditional calculus courses were developed with majors in physics and engineering in mind. Students in the life sciences, however, probably need a curriculum better attuned to life sciences applications, focusing in particular on mathematical modeling as a means of understanding biological systems. The Cornette-Ackermann text is jointly written by a mathematician and a biologist (ecologist) and is premised on three principles. First, that life sciences students are motivated by and respond well to actual data related to real life sciences problems; second, that the ultimate goal of a life sciences approach to calculus is primarily to model living systems with difference and differential equations; and third, that the level of rigor in such an approach should be comparable to that of traditional physics/engineering calculus.

After an introduction to the general process of mathematical modeling in the context of difference equations, functions are introduced as descriptions of biological patterns. The first year course covers the derivative and integral (including logs and exponentials) up though integration by substitution, integration by parts and applications including solids of revolution. Trigonometric functions are introduced in a context of traditional spring-mass oscillations à la Hooke’s Law along with life sciences analogs such as oscillations in predator-prey models. Perhaps the most advanced topic covered is functions of two variables and the diffusion equation. The diffusion equation is both derived and its application to molecular diffusion discussed.

A strength of the text lies in the many diverse examples from the life and biomedical sciences. These are typically based on real world scientific studies and data, and include references to the literature. Often examples are presented first, so the reader has a concrete question to think about, and then the theory is developed to explain the examples. For example a discrete model for a population growing with constant relative growth rate is worked out before the derivative is introduced, to help the reader think about measuring rates of change and using them to make predictions about long-term behavior. The same mathematical model is used to explain the decrease in the intensity of sunlight with increasing depth below the surface of a lake.

The examples are interesting and varied. The sudden drop in the density of trees at the tree line illustrates a discontinuous function. The section on maxima and minima estimates the optimal dimensions of a spider web. The section on integration begins by asking for the area of an oak leaf then goes on to calculate cardiac output. The method of slices in multiple integration is introduced by asking for the volume of a potato, then asking the reader to estimate the volume of a human brain based on images of slices taken 1 cm apart. This is not a dry book.

Volume II treats systems of difference equations, equilibrium points and their stability, cobweb diagrams, compartment models, logistic models of population, the SIR model for the spread of infectious disease, and examples from population dynamics. Armed with this experience the reader is prepared to tackle the final chapter which introduces systems of ordinary differential equations, direction fields, phase planes, etc.

Of course this book does not cover some topics that one finds in a traditional calculus book. There is no chapter on special techniques of integration, or on sequences and series. Exponential functions dominate, trigonometric functions play a minor part. Partial derivatives, maxima and minima of multivariable functions, and least squares regression are treated, but there is little on analytic geometry or vector calculus. The theory of limits is presented rigorously at the beginning, with a section on epsilons and deltas and a convergence theory grounded in Dedekind cuts, but later on when it comes to integration the treatment is considerably less formal.

There are many interesting exercises, and solutions to selected exercises are provided. Listings for both MATLAB and TI-86 calculator programs are included. There are many figures, tables and graphs. One may download both volumes for free at the authors’ website and copy them under the Creative Commons license. Source code is not provided. The free online edition has beautiful color graphics. Black and white print editions are also available, and very inexpensive. One may use the book as the textbook for a course, or download individual chapters and use them separately. One may even want to download and read it oneself just because it is so interesting.

Online edition at

Print edition are available at CreateSpace: volume I, $9.94; volume II, $8.27.

Steven Deckelman ( is a Professor of Mathematics at the University of Wisconsin-Stout. His interests include complex analysis, bioinformatics and mathematical neuroscience. George Jennings ( is a Professor of Mathematics at California State University, Dominguez Hills. His interests include differential and algebraic geometry, applied math, and education.

Volume I

Chapter 1.   Mathematical Models of Biological Processes
Chapter 2.   Functions
Chapter 3.   The Derivative
Chapter 4.   Continuity and the Power Chain Rule
Chapter 5.   Derivatives of the Exponential and Logarithm Functions
Chapter 6.   Derivatives of Products, Quotients, and Compositions
Chapter 7.   Derivatives of Trigonometric Functions
Chapter 8.   Applications of Derivatives
Chapter 9.   The Mean Value Theorem and Taylor's Polynomials
Chapter 10.   Partial Derivatives and the Diffusion Equation
Chapter 11.   The Intgegral
Chapter 12.   The Fundamental Theorem of Calculus
Chapter 13.   Applications of the Fundamental Theorem of Calculus

Volume II

Chapter 1.   Mathematical Models of Biological Processes
Chapter D   Dynamical Equations from Volume I
Chapter 14   First Order Difference Equations
Chapter 15   Discrete Dynamical Systems
Chapter 16   Nonlinear Dynamical Systems
Chapter 17   First Order Differential Equations
Chapter 18   Second Order Differential Equations, Systems of two Equations