# BIOCALCULUS

## Lectures on calculus for life science majors

by

 Mike Mesterton-GibbonsProfessorDepartment of MathematicsFlorida State UniversityTallahassee, Florida 32306-4510Phone: (850) 644-2580Email: mesterto@math.fsu.edu

## Introduction

Biocalculus introduces the fundamental ideas of calculus from the perspective of a biologist, i.e., it uses biological data to motivate and elucidate concepts that are essential for constructive use of Mathematica (or any other mathematical software) in solving biological problems. The approach is heuristic, but systematic. One can develop a great deal of mathematical maturity with remarkably little exposure to mathematical rigor, and in these lectures I encourage students to develop as much as possible of the first with as little exposure as possible to the second.

This is not, however, a no-brainer calculus for those who have drifted into biology under the illusion that it offers an escape from mathematics. On the contrary, calculus plays an increasingly central role in biology, and so its concepts must be mastered, despite -- or, rather, because of -- the widespread availability of powerful mathematical software. Every application of mathematics involves formulation (of a relevant problem), calculation (of requisite quantities) and interpretation (of results); and insofar as concepts and methods can be separated, formulation and interpretation require understanding of concepts, whereas calculation requires knowledge of method. Mathematica, in essence, is a magical black box for performing calculations. Although its graphical output may facilitate interpretation, Mathematica neither formulates nor interprets, and to that extent there is a greater need than ever before for biology majors to know the concepts thoroughly. This course addresses that need.

On the other hand, concepts and method are not so readily separated. Because all black boxes -- even magical black boxes -- are fallible, one cannot in general use mathematical software wisely unless one develops reliable instincts about whether it has truly yielded an answer to the problem one gave it (or meant to give it). Although using the software may refine such instincts, the only way to acquire them in the first instance is through extensive experience of solving problems without the software's help. In other words, there is still a need to enter the black box, which this course addresses as well.

An exception to the rule of fallibility, however, is that mathematical software is extraordinarily reliable for plotting simple graphs, in particular those of polynomials. I exploit that reliability by assuming at the outset that plotting graphs is a computer task, so that early exercises can introduce the software by requiring graphs to be drawn. In each set of exercises, an asterisked number indicates that a relevant Mathematica program can be downloaded from this site, and a number in bold refers to answers or hints at the end. All other problems can be solved by modifying an existing program or solution.

In sum, this course straddles the contentious divide between reform and tradition in calculus. Its goal is to mould biology majors into better scientists by enabling them to use Mathematica (or similar software) wisely, but its approach embodies a firm conviction that skill in using high technology for complex procedures requires skill in using low technology (e.g., pencil and paper) for simple procedures. So its outlook is thoroughly modern. But its style is deliberately old-fashioned.

## Contents

 Ordinary functions : a graphical perspective Ordinary functions: an algebraic perspective. Smoothness and concavity: a graphical perspective Quotients, inverses and limits. Modelling photosynthesis Ordinary sequences. Fibonacci's rapid rabbits Discrete probability distributions. Sums of powers of integers Function sequences. Compositions. The exponential and logarithm Index functions. Area and signed area From index function to ordinary function: ventricular recharge Area as limit of a function sequence. D'Arcy Thompson's mini minnows Arterial discharge: the area under a polynomial From ventricular inflow to volume: integration From ventricular volume to inflow: the derivative as growth rate Smoothness and concavity: an algebraic perspective Making joins smooth: the derivative of a piecewise-smooth function Differential notation. The derivative of a sum or multiple Sex allocation and the product rule How flat must a flatworm be, not to have a heart? The fundamental theorem Continuous probability distributions: the fundamental theorem again Derivatives of compositions: the chain rule Variation in rat pupil area. Implied distributions and integration by substitution Properties of exponential and logarithm. The empirical basis of allometry Periodic functions: models of rhythms in nature Bivariate functions and their extrema: a graphical approach More on bivariate functions: partial derivatives and integrals The mean and median of a distribution The variance. More on improper integrals Symmetric distributions Differential equations. The conceptual basis of allometry Trigonometric function properties The method of maximum likelihood