This chapter centers on the notion of ‘rate of change’, going from the average rate to the instantaneous one. This is done by examining different problems of growth. Our mathematical models lead us to geometric progressions, exponential functions, difference and differential equations.

We start with two different real-life situations: the struggle for life, and radioactive decay, both of which lead to the same mathematical model. Both units are written as texts for students.

In section 3.3 we show how to present geometric and arithmetic progressions in a meaningful way. Arithmetic progressions of a higher order are treated at the junior high school level.

Difference equations versus differential equations are considered in section 3.4. We consider the study of this dual approach to mathematical models a must for college-level mathematics. In our text we examine both points of view, the relationship between them, and their relationship with exponential, or logarithmic, functions. We also explain, at some length, how to approach exact computations, a topic which is hard to find treated in textbooks. Pointers are given on how to study a differential equation directly. The chapter ends with some more general remarks on mathematical models.

The mathematical theory of the struggle for life

Historical note

During World War 1, fishing in the Adriatic Sea (locate this on a map) was interrupted. After the war, when the Italians resumed fishing they were surprised to find fewer fish of the kind they had been catching than there had been before. They thought, of course, that since they had not been catching these fish for four years there would be many more of them.