Exercise 11

According to the U.S. census, the population of the city of Grand Rapids, MI, was 181,843 in 1980; 189,126 in 1990; and 197,800 in 2000.

1. Between 1980 and 2000, by how many people did the population of Grand Rapids grow?
2. In an average year between 1980 and 2000, by how many people did the population of Grand Rapids grow?
3. Just like we can find the average velocity of a moving body by computing change in position over change in time, we can compute the average rate of change of any function $f$. In particular, the average rate of change of a function $f$ over an interval $[a,b]$ is the quotient $$\begin{gathered} \frac{f(b)-f(a)}{b-a} \end{gathered}$$. What does the quantity $\frac{f(b)-f(a)}{b-a}$ measure on the graph of $y = f(x)$ over the interval $[a,b]$?
4. Let $P(t)$ represent the population of Grand Rapids at time $t$, where $t$ is measured in years from January 1, 1980. What is the average rate of change of $P$ on the interval $t = 0$ to $t = 20$? What are the units on this quantity?
5. If we assume the population of Grand Rapids is growing at a rate of approximately 4% per decade, we can model the population function with the formula $$\begin{gathered} P(t) = 181843 (1.04)^{t/10} \end{gathered}$$. Use this formula to compute the average rate of change of the population on the intervals $[5,10]$, $[5,9]$, $[5,8]$, $[5,7]$, and $[5,6]$.
6. How fast do you think the population of Grand Rapids was changing on January 1, 1985? Said differently, at what rate do you think people were being added to the population of Grand Rapids as of January 1, 1985? How many additional people should the city have expected in the following year? Why?