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\noindent {\bf Problem statement} Suppose that the outdoor temperature (in ${}^{\circ}$ F) on a
particular day was approximated by the function
$$ T(t) = 50 + 14\sin\!\left({{\pi t}\over{12}}\right)\!,$$
where $t$ is time (in hours) after 9 AM.

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\noindent a) Find the {\it maximum} temperature $T_{\rm max}$, {\it minimum}
temperature $T_{\rm min}$, and {\it average} temperature
$$ T_{\rm aver} = {1\over{12}} \int_0^{12} T(t) \, dt $$ on that day
 during the period between 9 AM and 9 PM.

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\noindent b) Show that $T_{\rm aver} \neq {1\over 2}(T_{\rm min} + T_{\rm
max})$. (This is the definition that the weather bureau uses for
``average temperature''.)

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\noindent c) Show that if $T$ is not given by the above formula, but
rather $T(t)$ is a {\it linear} function of $t$, then $T_{\rm aver} =
{1\over 2}(T_{\rm min} + T_{\rm max})$. (Use either geometric
reasoning or an integral.)

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