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{tabular}{ c } $t$ |
| (days) |
&
| $W(t)$ |
| $\left({}^{\circ}\mathrm{C}\right)$ |
\hline\hline 0 & 20
3 & 31 6 & 28 9 & 24 12 & 22 15 & 21 \hline \end{tabular}
(a) Use data from the table to find an approximation for $W^{\prime}(12)$. 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 $P^{\prime}(12)$. 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.