Activity 4.2.3.
Suppose that an object moving along a straight line path has its velocity in feet per second at time \(t\) in seconds given by \(v(t) = \frac{2}{9}(t-3)^2 + 2\text{.}\)
(a)
Carefully sketch the region whose exact area will tell you the value of the distance the object traveled on the time interval \(2 \le t \le 5\text{.}\)
(b)
Estimate the distance traveled on \([2,5]\) by computing \(L_4\text{,}\) \(R_4\text{,}\) and \(M_4\text{.}\)
(c)
Does averaging \(L_4\) and \(R_4\) result in the same value as \(M_4\text{?}\) If not, what do you think the average of \(L_4\) and \(R_4\) measures?
(d)
For this question, think about an arbitrary function \(f\text{,}\) rather than the particular function \(v\) given above. If \(f\) is positive and increasing on \([a,b]\text{,}\) will \(L_n\) over-estimate or under-estimate the exact area under \(f\) on \([a,b]\text{?}\) Will \(R_n\) over- or under-estimate the exact area under \(f\) on \([a,b]\text{?}\) Explain.
