Exercises

Although it is not necessary to focus on a single base, for the purposes of this discussion let us continue to work with the function . In each of the following questions where you are asked to describe the behavior of a function f, describe its behavior in the region of your plot and as x approaches both and .

1. 1. Let . Using what we know about , plot g and describe its behavior. Explain the behavior of g in terms of the behavior of f.
   2. Is g a function that can be written in the form where k is a constant? Explain why or why not?
   3. How would your plot and description change if 1 were replaced by a constant b?
2. 1. Let . Using what we know about , plot g and describe its behavior. Explain the behavior of g in terms of the behavior of f.
   2. How would your plot and description change for the function ? Explain your answer.
   3. How would your plot and description from (a) change for the function ? Explain your answer.
3. 1. Let . Plot g and describe its behavior based upon your knowledge of .
   2. How would your answer change if instead we defined g by ?
   3. How would your answer change if instead we defined g by ?
4. In the first three problems, we have worked our way up to thinking about functions that take the form , where P, A, and k are fixed positive numbers. Now let us apply this to a particular example.

   The United States experienced a polio epidemic in 1949. At the end of January, 494 cases of polio had been reported. By the end of December, a total of 42,375 cases had been reported. Let us assume that the maximum number that was eventually reported was 43,000.

   1. Explain how would you use this information to construct a function of the form to model the spread of the polio epidemic. That is, how would you go about finding particular P, A, and k to fit this information. (Assume that the time t is measured in months.)
   2. Carry out the calculations to find P, A, and k.
   3. Plot your function f for the values you have from (b) and describe the course of the epidemic. In particular, when was the epidemic spreading the fastest, when was it spreading the slowest?