Activity 1.3.3.
Let’s consider two different functions and see how different computations of their average rate of change tells us about their respective behavior. Plots of \(q\) and \(h\) are shown in the figures in part (c).
(a)
Consider the function \(q(x) = 4-(x-2)^2\text{.}\) Compute \(AV_{[0,1]}\text{,}\) \(AV_{[1,2]}\text{,}\) \(AV_{[2,3]}\text{,}\) and \(AV_{[3,4]}\text{.}\) What do your last two computations tell you about the behavior of the function \(q\) on \([2,4]\text{?}\)
(b)
Consider the function \(h(t) = 3 - 2(0.5)^t\text{.}\) Compute \(AV_{[-1,1]}\text{,}\) \(AV_{[1,3]}\text{,}\) and \(AV_{[3,5]}\text{.}\) What do your computations tell you about the behavior of the function \(h\) on \([-1,5]\text{?}\)
(c)
On the graphs that follow (\(q\) at left, \(h\) at right), plot the line segments whose respective slopes are the average rates of change you computed in (a) and (b).
(d)
True or false: Since \(AV_{[0,3]} = 1\text{,}\) the function \(q\) is increasing on the interval \((0,3)\text{.}\) Justify your decision.
(e)
Give an example of a function that has the same average rate of change no matter what interval you choose. You can provide your example through a table, a graph, or a formula; regardless of your choice, write a sentence to explain.
