Solve the below:
Q: The temperature, in degrees Celsius, 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.
|
T (in days)
|
0
|
3
|
6
|
9
|
12
|
15
|
|
W(t)
|
20
|
31
|
28
|
24
|
22
|
21
|
a) Use the date 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 (greater than or equal to) t (greater than or equal to) 15 days by using a trapezoidal approximation with subintervals of length 3 days.
c) A student proposes the function P, given by P(t)=20+10te^(-t/3) as a model for the temperature 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 (greater than or equal to) t (greater than or equal to) 15 days.