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 Find the average value of cos t on the intervals [0, pi],[0,(pi/2)], [0,(pi/4)],[0,...

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cspanutius | Student, Undergraduate | (Level 1) Salutatorian

Posted August 3, 2012 at 9:33 PM via web

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 Find the average value of cos t on the intervals [0, pi],[0,(pi/2)], [0,(pi/4)],[0, 0.01].

please answer # A in the following question:

(a) Find the average value of cos t on the intervals [0, π],[0,(π/2)], [0,(π/4)],[0, 0.01].

(b) Determine the general formula for f bar (the letter f with a bar over it) (subscript [0, x] the average of  cos t  over the interval [0, x].

(c) Calculate lim(x->0) f bar (the letter f with a bar over it)  (subscript [0, x])

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lfryerda | High School Teacher | (Level 2) Educator

Posted August 3, 2012 at 10:11 PM (Answer #1)

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To find the average value of a function, use the formula `1/{b-a}int_a^b f(t)dt`.

This means that the average of `cos t` in an interval is:

`1/{b-a}int_a^b cos t dt`

`=1/{b-a} (sin b-sin a)`

`={sin b-sin a}/{b-a}`

To answer Part A of the question, we use that result for the intervals:

`[0,pi]` has average `1/pi(sin pi-sin 0) = 0` .

`[0,pi/2]` has average `1/{pi/2}(sin (pi/2)-sin 0) = 2/pi` .

`[0,pi/4]` has average `1/{pi/4}(sin (pi/4)-sin0)=4/{pi sqrt 2}` .

`[0,0.01]` has average `1/0.01(sin 0.01-sin0)=0.99998` .

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