Activity 1.3.3.
A water balloon is tossed vertically in the air from a window. The balloon’s height in feet at time \(t\) in seconds after being launched is given by \(s(t) = -16t^2 + 16t + 32\text{.}\) Use this function to respond to each of the following questions.
(a)
Sketch an accurate, labeled graph of \(s\) on the axes provided. You should be able to do this without using computing technology.
(b)
Compute the average rate of change of \(s\) on the time interval \([1,2]\text{.}\) Include units on your answer and write one sentence to explain the meaning of the value you found.
(c)
Use the limit definition to compute the instantaneous rate of change of \(s\) with respect to time, \(t\text{,}\) at the instant \(a = 1\text{.}\) Show your work using proper notation, include units on your answer, and write one sentence to explain the meaning of the value you found.
(d)
On your graph in (a), sketch two lines: one whose slope represents the average rate of change of \(s\) on \([1,2]\text{,}\) the other whose slope represents the instantaneous rate of change of \(s\) at the instant \(a=1\text{.}\) Label each line clearly.
(e)
For what values of \(a\) do you expect \(s'(a)\) to be positive? Why? Answer the same questions when “positive” is replaced by “negative” and “zero.”
