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Derivatives, Differential Equations and Rate of Change Word Problem : Trapezoidal Approximations

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T W(t)
(days) degrees C
0 20
3 31
6 28
9 24
12 22
The temperature. in degrees Celsius (°C). of the water in a pond is a differentiable function W of time r. The table above shows the water temperature as recorded every 3 days over a 15-day period.
(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 inten'aI 0 ≤ t ≤ 15 days by using a trapezoidal approximation with subintervals of length At = 3 days.
(c) A student proposes the function P, given by P(t) =.... as a model r the temperature. of the water in the pond at time t, where t is measured in days and !-(t) is ineasured in degrees Celsius. Find P'(12). Using appropriate units, explain the meaning of 'iour answer in terms of water temperature.
(d) Use the function P defined iii part (c) to find the average value. in degrees Celsius, of P(r) over the time a interval 0 ≤ t ≤ 15 days.

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A Rate of Change Word Problem is solved with Trapezoidal Approximations.

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