You have a function of temperature, T(t), which is a linear function of time t.

Using an integral, show that the average temperature (Tavg) is equal to (1/2)(Tmax + Tmin)

Question

You have a function of temperature, T(t), which is a linear function of time t.

Using an integral, show that the average temperature (Tavg) is equal to (1/2)(Tmax + Tmin)

anonymous · Answer

Oh, and show in two cases: when the slope is positive and when the slope is negative.

unklerhaukus · Answer

linear?, how is this possible?

anonymous · Answer

Oh, I completely forgot! 
t itself is the # of hours past 9am. You are finding the average temperature between the time interval of 9am to 9pm (so from 0 to 12).

I know that, if the slope is positive, Tmax would occur at T(0), while the min would occur at T(12), and vice versa if the slope is negative, but that's about it.

unklerhaukus · Answer

oh linear with time,

anonymous · Answer

yes. Sorry if my wording is off/confused you.

anonymous · Answer

$$(1/12)\int\limits_{0}^{12}(mx+b)  dx = (1/12)*(1/2)*(12)*(f(0) + f(12))$$
The average value of the function is equal to 1/(b-a) times the area of the trapezoid created.

Sounds about right? Anyone?

unklerhaukus · Answer

ah, that kind of integral

anonymous · Answer

but is the reasoning behind it okay?

unklerhaukus · Answer

ah, that does make sense

anonymous · Answer

ok. thanks for the help!