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1. Find the average value have of the function h on the given interval.

h(x) = 2 cos4(x) sin(x), [0, π]
2. Consider the given function and the given interval.
f(x) = 6 sin(x) − 3 sin(2x), [0, π]
(a) Find the average value fave of f on the given interval.
(b) Find c such that fave = f(c). (Round your answers to three decimal places.)
3. Find the numbers b such that the average value of
f(x) = 3 + 10x − 9x2
on the interval [0, b] is equal to 4.
4. In a certain city the temperature (in °F) t hours after 9 AM was modeled by the function
T(t) = 48 + 19 sin

πt
12

.
Find the average temperature Tave during the period from 9 AM to 9 PM. (Round your answer to the nearest whole number.)

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1 answer

  1. Recall that the average value of f(x) on the interval [a,b] is
    ∫[a,b] f(x) dx
    --------------------
    b-a

    So,
    #1.
    (∫[0,π] 2 cos4x sinx dx)/π = -4 / 15π

    #2. f(x) = 6 sin(x) − 3 sin(2x), [0, π]
    avg value is 12/π
    So, where is f(x) = 12/π ?

    #3. f(x) = 3 + 10x - 9x^2
    You want
    (∫[0,b] f(x) dx)/4 = 4
    (3x + 5x^2 - 3x^3)[0,b] = 4*4
    3b + 5b^2 - 3b^3 = 16
    b = -1.4724

    #4. Fix your formatting so we can read the actual function.
    Then apply the rules followed above.

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    oobleck

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