Activity 1.1.2.
The following questions concern the position function given by \(s(t) = 64 - 16(t-1)^2\text{,}\) considered in Preview Activity 1.1.1, and plotted in the given figure.
(a)
Compute the average velocity of the ball on each of the following time intervals: \([0.4,0.8]\text{,}\) \([0.7,0.8]\text{,}\) \([0.79, 0.8]\text{,}\) \([0.799,0.8]\text{,}\) \([0.8,1.2]\text{,}\) \([0.8,0.9]\text{,}\) \([0.8,0.81]\text{,}\) \([0.8,0.801]\text{.}\) Include units for each value.
(b)
On the graph provided in (a), sketch the line that passes through the points \(A=(0.4, s(0.4))\) and \(B=(0.8, s(0.8))\text{.}\) What is the meaning of the slope of this line? In light of this meaning, what is a geometric way to interpret each of the values computed in the preceding question?
(c)
Use a graphing utility to plot the graph of \(s(t) = 64 - 16(t-1)^2\) on an interval containing the value \(t = 0.8\text{.}\) Then, zoom in repeatedly on the point \((0.8, s(0.8))\text{.}\) What do you observe about how the graph appears as you view it more and more closely?
(d)
What do you conjecture is the velocity of the ball at the instant \(t = 0.8\text{?}\) Why?
