Question

On a certain day the temperature t hours past midnight was T(t)=60+5sin((pi/6)(t-10)). What was the average temperature between noon and mignight of the same day? I know that you would use the mean value theorem for integrals to solve this which I know how, but why?

Answer 1

Wouldn't you just use the mean value thoerem for derivatives since you're trying to find the average temperature? Not time*temperature as the area under the curve of this function suggests?

Answer 2

you're right we are looking for average temperature and we are looking at area under curve but not of T but T'
_>\[=1/12\int\limits_{0}^{12}T 'dt\]
which really is (T(12)-T(0))/12

Answer 3

think of it as Temp/time * time