# 3.1.5 AP Calculus Exam average temperature

• MHB
• karush
In summary, the question is asking for an estimate of the average temperature of water in a tub over a 20-minute period, given data points for the first 20 minutes. The estimate can be found by taking the average of the initial and final data points, which gives a temperature of approximately $63.0^o F$. Additionally, part (a) asks for an estimate of the derivative of the function at $t=12$ minutes, which can be found using the slope formula and the data points at $t=9$ and $t=15$. The approximation for the average temperature may either overestimate or underestimate the actual average, depending on the shape of the curve and the values of $W(t)$ between the given
karush
Gold Member
MHB
309 average temperature
$$\begin{array}{|c|c|c|c|c|c|c|} \hline t\,(minutes)&0&4&9&15&20\\ \hline W(t)\,(degrees Farrenheit)&55.0&57.1&61.8&67.9&71.0\\ \hline \end{array}$$
The temperature of water in a tub at time t is modeled by a strictly increasing, twice-differentiable function W. where W(t) is measured in degrees Fahrenheit and t is measured in minutes.
At time $t=0$. the temperature of the water is $55^o F$.
The water is heated for 30 minutes, beginning at time $t=0$,
Values of $W(t)$ at selected times t for the first 20 minutes are given in the table above.

a. Use the data in the table to estimate W'(12).
c. For $0\le t \le 20$ the average temperature of the water in the tub is
$\dfrac{1}{20}\int_0^{20} W(t)\, dt$
Does this approximation overoptimistic or underestimate the average temperature of the water over these 20 minutes? Explain.

ok not sure just what kind of equation we need for this since the points are close to a straight line

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Re: 309 AP Dalculus Exam average temperature

(a) $W'(12) \approx \dfrac{W(15)-W(9)}{15-9} \, \dfrac{^oF}{min}$

(b) ... is this part a complete citation of the given question? Does part (c) depend on a complete citation of the question in part (b) ?

Re: 309 AP Dalculus Exam average temperature

that was exactly the way it was given
so I presume first they just want average between the first and last point

then they want a addition of the sums of the intervals which would be close.

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