10: Exponential Functions
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- 121132
"The mathematics of uncontrolled growth are frightening. A single cell of the bacterium E. coli would, under ideal circumstances, divide every twenty minutes. That is not particularly disturbing until you think about it, but the fact is that bacteria multiply geometrically: one becomes two, two become four, four become eight, and so on. In this way it can be shown that in a single day, one cell of \(E\). coli could produce a super-colony equal in size and weight to the entire planet Earth." - Michael Crichton, The Andromeda Strain, p. 247 [Crichton, 1969]
In this chapter, we introduce the exponential functions. We first describe the discrete process of population doubling, represented by powers of 2 , namely, \(2^{n}\), where \(n\) is some integer. We generalize to a continuous function \(2^{x}\) where \(x\) is any real number. We can then attach meaning to the notion of the derivative of an exponential function. In doing so, we encounter a specially convenient base denote \(e\), leading to the most useful member of this class of functions, \(y=e^{x}\). We discuss applications to unlimited growth in a population.