Exponential Growth and Decay Help

Introduction to Exponential Growth and Decay

Many processes of nature and many mathematical applications involve logarithmic and exponential functions. For example, if we examine a population of bacteria, we notice that the rate at which the population grows is proportional to the number of bacteria present. To see that this makes good sense, suppose that a bacterium reproduces itself every 4 hours. If we begin with 5 thousand bacteria then

after 4 hours there are 10 thousand bacteria

after 8 hours there are 20 thousand bacteria

after 12 hours there are 40 thousand bacteria

after 16 hours there are 80 thousand bacteria ...

etc.

The point is that each new generation of bacteria also reproduces , and the older generations reproduce as well. A sketch (Fig. 6.12) of the bacteria population against time shows that the growth is certainly not linear—indeed the shape of the curve appears to be of exponential form.

Fig. 6.12

Notice that when the number of bacteria is large, then different generations of bacteria will be reproducing at different times. So, averaging out, it makes sense to hypothesize that the growth of the bacteria population varies continuously as in Fig. 6.13. Here we are using a standard device of mathematical analysis: even though the number of bacteria is always an integer, we represent the graph of the population of bacteria by a smooth curve. This enables us to apply the tools of calculus to the problem.

Fig. 6.13

A Differential Equation

If B ( t ) represents the number of bacteria present in a given population at time t , then the preceding discussion suggests that

where K is a constant of proportionality. This equation expresses quantitatively the assertion that the rate of change of B ( t ) (that is to say, the quantity dB/dt ) is proportional to B ( t ). To solve this equation, we rewrite it as

We integrate both sides with respect to the variable t :

The left side is

ln | B ( t )| + C

and the right side is

where C and are constants of integration. We thus obtain

ln | B ( t )| = Kt + D ,

where we have amalgamated the two constants into a single constant D . Exponentiating both sides gives

| B ( t )| = e ^{Kt + D}

or

Notice that we have omitted the absolute value signs since the number of bacteria is always positive. Also we have renamed the constant e ^D with the simpler symbol P .

Equation (*) will be our key to solving exponential growth and decay problems.

We motivated our calculation by discussing bacteria, but in fact the calculation applies to any function which grows at a rate proportional to the size of the function.

Next we turn to some examples.

Bacterial Growth Examples

Example 1

It is important to realize that population growth problems cannot be described using just arithmetic. Exponential growth is nonlinear, and advanced analytical ideas (such as calculus) must be used to understand it.

Example 2

You Try It : A petri dish has 5000 bacteria at 1:00 p.m. on a certain day and 8000 bacteria at 5:00 p.m. that same day. How many bacteria were there at noon?

Find practice problems and solutions for these concepts at: Transcendental Functions Practice Test.