\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | }
\hline
$t$ (hours) & 0 & 2 & 4 & 6 & 8 & 10 & 12 \\
\hline
$R ( t )$ (vehicles per hour) & 2935 & 3653 & 3442 & 3010 & 3604 & 1986 & 2201 \\
\hline
\end{tabular}
\end{center}

On a certain weekday, the rate at which vehicles cross a bridge is modeled by the differentiable function $R$ for $0 \leq t \leq 12$, where $R ( t )$ is measured in vehicles per hour and $t$ is the number of hours since 7:00 A.M. $(t = 0)$. Values of $R ( t )$ for selected values of $t$ are given in the table above.\\
(a) Use the data in the table to approximate $R ^ { \prime } ( 5 )$. Show the computations that lead to your answer. Using correct units, explain the meaning of $R ^ { \prime } ( 5 )$ in the context of the problem.\\
(b) Use a midpoint sum with three subintervals of equal length indicated by the data in the table to approximate the value of $\int _ { 0 } ^ { 12 } R ( t ) \, d t$. Indicate units of measure.\\
(c) On a certain weekend day, the rate at which vehicles cross the bridge is modeled by the function $H$ defined by $H ( t ) = - t ^ { 3 } - 3 t ^ { 2 } + 288 t + 1300$ for $0 \leq t \leq 17$, where $H ( t )$ is measured in vehicles per hour and $t$ is the number of hours since 7:00 A.M. $(t = 0)$. According to this model, what is the average number of vehicles crossing the bridge per hour on the weekend day for $0 \leq t \leq 12$?\\
(d) For $12 < t < 17$, $L ( t )$, the local linear approximation to the function $H$ given in part (c) at $t = 12$, is a better model for the rate at which vehicles cross the bridge on the weekend day. Use $L ( t )$ to find the time $t$, for $12 < t < 17$, at which the rate of vehicles crossing the bridge is 2000 vehicles per hour. Show the work that leads to your answer.