Activity 4.2.4.
Suppose that an object moving along a straight line path has its velocity \(v\) (in feet per second) at time \(t\) (in seconds) given by
\begin{equation*}
v(t) = \frac{1}{2}t^2 - 3t + \frac{7}{2}\text{.}
\end{equation*}
(a)
Compute \(M_5\text{,}\) the middle Riemann sum, for \(v\) on the time interval \([1,5]\text{.}\) Be sure to clearly identify the value of \(\Delta t\) as well as the locations of \(t_0\text{,}\) \(t_1\text{,}\) \(\cdots\text{,}\) \(t_5\text{.}\) In addition, provide a careful sketch of the function and the corresponding rectangles that are being used in the sum.
(b)
Building on your work in (a), estimate the total change in position of the object on the interval \([1,5]\text{.}\)
(c)
Re-interpret your work in (a) and (b) to estimate the total distance traveled by the object on \([1,5]\text{.}\)
(d)
Use appropriate computing technology such as this applet to compute \(M_{10}\) and \(M_{20}\text{.}\) What exact value do you think the middle sum eventually approaches as \(n\) increases without bound? What does that number represent in the physical context of the overall problem?
