Preface

Introduction to this book

Calculus arose as a tool for solving practical scientific problems through the centuries. However, it is often taught as a technical subject with rules and formulas (and occasionally theorems), devoid of its connection to applications. In this course, the applications form an important focal point, with emphasis on life sciences. This places the techniques and concepts into practical context, as well as motivating quantitative approaches to biology taught to undergraduates. While many of the examples have a biological flavor, the level of biology needed to understand those examples is kept at a minimum. The problems are motivated with enough detail to follow the assumptions, but are simplified for the purpose of pedagogy.

The mathematical philosophy is as follows:

We start with basic observations about functions and graphs, with an emphasis on power functions and polynomials. We use elementary properties of a function to sketch its graph and to understand its shape, even before discussing derivatives; later we refine such graph-sketching skills. We consider useful ideas with biological implications even in this basic context. In fact, we discuss several examples in which two processes (such as growth and predation or nutrient uptake and consumption) are at balance. We show how setting up the relevant algebraic problem revels when such a balance can Power functions Polynominals, Rational functions

exist.

We introduce the derivative in three complementary ways: (1) As a rate of change, (2) as the slope we see when we zoom into the graph of a function, and (3) as a computational quantity that can be approximated by a finite difference. We discuss (1) by first defining an average rate of change over a finite time interval. We use actual data to do so, but then by refining the time interval, we show how this average rate of change approaches the instantaneous rate, i.e. the derivative. This helps to make the idea of the limit more intuitive, and not simply a formal calculation. We illustrate (2) using a sequence of graphs or interactive graphs with increasing magnification. We illustrate (3) using simple computation that can be carried out on a spreadsheet. The actual formal definition of the derivative (while presented and used) takes a back-seat to this discussion.

The next philosophical aspect of the course is that we develop all the ideas and applications of calculus using simple functions (power and polynomials) first, before introducing the more elaborate technical calculations. The aim is to highlight the usefulness of derivatives for understanding functions (sketching and interpreting their behavior), and for optimization problems, before having to grapple with the chain rule and more intricate computation of derivatives. This helps to illustrate what calculus can achieve, and decrease the focus on rote mechanical calculations.

Once this entire "tour" of calculus is complete, we introduce the chain rule and its applications, and then the transcendental functions (exponentials and trigonometric). Both are used to illustrate biological phenomena (population growth and decay, then, later on, cyclic processes). Both allow a repeated exposure to the basic ideas of calculus - curve sketching, optimization, and applications to related rates. This means that the important concepts picked up earlier in the context of simpler functions can be reinforced again. At this point, it is time to practice and apply the chain rule, and to compute more technically involved derivatives. But, even more than that, both these topics allow us to informally introduce a powerful new idea, that of a differential equation.

By making the link between the exponential function and the differential equation \(d y / d x=k y\), we open the door to a host of biological applications where we seek to understand how a system changes: how a population size grow? how does the mass of a cell change as nutrients are taken up and consumed? By revisiting our initial discussions, we identify the "balance points" as steady states, and we develop arguments to predict what changes with time would be observed. The idea of a deviation away from steady state also leads us to find the behavior of solutions to the differential equation \(d y / d t=a-b y\). This leads to many useful applications. including the temperature of a cooling object, the level of drug in the bloodstream, simple chemical reactions, and many more.

Ultimately, a first semester calculus course is all about the applications of a derivative. We use this fact to explore nonlinear differential equations of the first order, using qualitative sketches of the direction field and the state space of the equation. Even though some of the (integration) methods for solving a differential equation are developed only in a second semester calculus course, we provide here the background to understanding what such equations are saying, and what they imply. These simple yet powerful qualitative methods allow us to get intuition to the behavior of more realistic biological models, including density-dependent (logistic) growth and even the spread of disease. Many of the ideas here are geometric, and we return to interpreting the meaning of graphs and slopes yet again in this context.

The idea of a computational approach is introduced and practiced in several places, as appropriate. We use simple examples to motivate linear approximation and Newton’s method for finding zeros of a function. Later, we use Euler’s method to solve a simple differential equation computationally. All these methods are based on the derivative, and most introduce the idea of an iterated (repeated) process that is ideally handled by computer or calculator. The exposure to these computational methods, while novel and sometimes daunting, provides an important set of examples of how properly understanding the mathematics can be used directly for effective design of computational algorithms.