The temperature, in degrees Celsius (${}^{\circ}\mathrm{C}$), of the water in a pond is a differentiable function $W$ of time $t$. The table below shows the water temperature as recorded every 3 days over a 15-day period.

\begin{center}
\begin{tabular}{ | c | c | }
\hline
\begin{tabular}{ c }
$t$ \\
(days) \\
\end{tabular} & \begin{tabular}{ c }
$W(t)$ \\
$\left({}^{\circ}\mathrm{C}\right)$ \\
\end{tabular} \\
\hline\hline
0 & 20 \\
3 & 31 \\
6 & 28 \\
9 & 24 \\
12 & 22 \\
15 & 21 \\
\hline
\end{tabular}
\end{center}

(a) Use data from the table to find an approximation for $W'(12)$. Show the computations that lead to your answer. Indicate units of measure.

(b) Approximate the average temperature, in degrees Celsius, of the water over the time interval $0 \leq t \leq 15$ days by using a trapezoidal approximation with subintervals of length $\Delta t = 3$ days.

(c) A student proposes the function $P$, given by $P(t) = 20 + 10te^{(-t/3)}$, as a model for the temperature of the water in the pond at time $t$, where $t$ is measured in days and $P(t)$ is measured in degrees Celsius. Find $P'(12)$. Using appropriate units, explain the meaning of your answer in terms of water temperature.

(d) Use the function $P$ defined in part (c) to find the average value, in degrees Celsius, of $P(t)$ over the time interval $0 \leq t \leq 15$ days.