3.2.1 Decreasing Exponential Functions

Did you miss the Section 3.2 introductory page with details of the murder problem?

A key feature of the solution of the murder problem will be the role of decreasing exponential functions. As preparation for finding the time of death, we examine these functions first.

Activity 1

1. Use a computer algebra tool or a graphing calculator to graph all three of the following exponential functions in a single graphing window:

   ${\displaystyle y=e^{-1.7t}}$

   ${\displaystyle y=e^{-2.5t}}$

   ${\displaystyle y=e^{-3.7t}}$

2. What point do the three graphs have in common?

3. What happens to each of the three graphs as t becomes large?

4. What can you say in general about functions of the form $y=e^{-kt}$ where k is a positive number?

5. What connection do you find between the size of `k` and the rate of decrease of $y=e^{-kt}$?

Comment on Activity 1

We already know how to differentiate exponential functions

for any constant `r`. In particular,

for all constants `k`.

Checkpoint 1