Hot water is dripping through a coffeemaker, filling a large cup with coffee. The amount of coffee in the cup at time $t$, $0 \leq t \leq 6$, is given by a differentiable function $C$, where $t$ is measured in minutes. Selected values of $C ( t )$, measured in ounces, are given in the table below.

\begin{center}
\begin{tabular}{ | c | | c | c | c | c | c | c | c | }
\hline
\begin{tabular}{ c }
$t$ \\
(minutes) \\
\end{tabular} & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\
\hline
\begin{tabular}{ c }
$C ( t )$ \\
(ounces) \\
\end{tabular} & 0 & 5.3 & 8.8 & 11.2 & 12.8 & 13.8 & 14.5 \\
\hline
\end{tabular}
\end{center}

(a) Use the data in the table to approximate $C ^ { \prime } ( 3.5 )$. Show the computations that lead to your answer, and indicate units of measure.

(b) Is there a time $t$, $2 \leq t \leq 4$, at which $C ^ { \prime } ( t ) = 2$? Justify your answer.

(c) Use a midpoint sum with three subintervals of equal length indicated by the data in the table to approximate the value of $\frac { 1 } { 6 } \int _ { 0 } ^ { 6 } C ( t ) \, dt$. Using correct units, explain the meaning of $\frac { 1 } { 6 } \int _ { 0 } ^ { 6 } C ( t ) \, dt$ in the context of the problem.

(d) The amount of coffee in the cup, in ounces, is modeled by $B ( t ) = 16 - 16 e ^ { - 0.4 t }$. Using this model, find the rate at which the amount of coffee in the cup is changing when $t = 5$.