Preview Activity 1.3.1.
Let the height function for a ball tossed vertically be given by \(s(t) = 64 - 16(t-1)^2\text{,}\) where \(t\) is measured in seconds and \(s\) is measured in feet above the ground.
(a)
Compute the value of \(AV_{[1.5,2.5]}\text{.}\)
(b)
What are the units on the quantity \(AV_{[1.5,2.5]}\text{?}\) What is the meaning of this number in the context of the rising/falling ball?
(c)
In Desmos, plot the function \(s(t) = 64 - 16(t-1)^2\) along with the points \((1.5,s(1.5))\) and \((2.5, s(2.5))\text{.}\) Make a copy of your plot on the axes in the figure provided, labeling key points as well as the scale on your axes. What is the domain of the model? The range? Why?
(d)
Work by hand to find the equation of the line through the points \((1.5,s(1.5))\) and \((2.5, s(2.5))\text{.}\) Write the line in the form \(y = mt + b\) and plot the line in Desmos, as well as on the axes above.
(e)
What is a geometric interpretation of the value \(AV_{[1.5,2.5]}\) in light of your work in the preceding questions?
(f)
How do your answers in the preceding questions change if we instead consider the interval \([0.25, 0.75]\text{?}\) \([0.5, 1.5]\text{?}\) \([1,3]\text{?}\)
