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20 May, 01:44

Find two values of c in ( - π / 4, π / 4) such that f (c) is equal to the average value of f (x) = 2 cos (2x) on ( - π / 4, π / 4). Round your answers to three decimal places.

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  1. 20 May, 02:29
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    c₁ = 1/2 cos⁻¹ (2/π) = 0.44

    c₂ = - 1/2 cos⁻¹ (2/π) = - 0.44

    Step-by-step explanation:

    the average value of f (x) = 2 cos (2x) on ( - π / 4, π / 4) is

    av f (x) = ∫[2*cos (2x) ] dx / (∫dx) between limits of integration - π / 4 and π / 4

    thus

    av f (x) = ∫[cos (2x) ] dx / (∫dx) = [sin (2 * π / 4) - sin (2 * ( - π / 4) ] / [ π / 4 - (-π / 4) ]

    = 2*sin (π/2) / (π/2) = 4/π

    then the average value of f (x) is 4/π. Thus the values of c such that f (c) = av f (x) are

    4/π = 2 cos (2c)

    2/π = cos (2c)

    c = 1/2 cos⁻¹ (2/π) = 0.44

    c = 0.44

    since the cosine function is symmetrical with respect to the y axis then also c = - 0.44 satisfy the equation

    thus

    c₁ = 1/2 cos⁻¹ (2/π) = 0.44

    c₂ = - 1/2 cos⁻¹ (2/π) = - 0.44
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