Preview Activity 4.2.1.
A person walking along a straight path has her velocity in miles per hour at time \(t\) given by the function \(v(t) = 0.25t^3-1.5t^2+3t+0.25\text{,}\) for times in the interval \(0 \le t \le 2\text{.}\) The graph of this function is also given in each of the three diagrams in the following figure.
Note that in each diagram, we use four rectangles to estimate the area under \(y = v(t)\) on the interval \([0,2]\text{,}\) but the method by which the four rectangles’ respective heights are decided varies among the three individual graphs.
(a)
How are the heights of rectangles in the left-most diagram being chosen? Explain, and hence determine the value of
\begin{equation*}
S = A_1 + A_2 + A_3 + A_4
\end{equation*}
by evaluating the function \(y = v(t)\) at appropriately chosen values and observing the width of each rectangle. Note, for example, that
\begin{equation*}
A_3 = v(1) \cdot \frac{1}{2} = 2 \cdot \frac{1}{2} = 1\text{.}
\end{equation*}
(b)
Explain how the heights of rectangles are being chosen in the middle diagram and find the value of
\begin{equation*}
T = B_1 + B_2 + B_3 + B_4\text{.}
\end{equation*}
(c)
Likewise, determine the pattern of how heights of rectangles are chosen in the right-most diagram and determine
\begin{equation*}
U = C_1 + C_2 + C_3 + C_4\text{.}
\end{equation*}
(d)
Of the estimates \(S\text{,}\) \(T\text{,}\) and \(U\text{,}\) which do you think is the best approximation of \(D\text{,}\) the total distance the person traveled on \([0,2]\text{?}\) Why?
