# What is the average value of the function #f(x)=cos(8x)# on the interval #[0,pi/2]#?

Thus, in this instance

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To find the average value of the function f(x) = cos(8x) on the interval [0, π/2], we use the formula for the average value of a function over an interval:

Average value = (1 / (b - a)) * ∫(from a to b) f(x) dx.

Substitute the given function and interval:

Average value = (1 / (π/2 - 0)) * ∫(from 0 to π/2) cos(8x) dx.

Now, integrate the function:

∫cos(8x) dx = (1/8) * sin(8x) + C.

Evaluate the integral at the upper and lower limits:

(1/8) * sin(8 * (π/2)) - (1/8) * sin(8 * 0).

Simplify:

(1/8) * sin(4π) - (1/8) * sin(0).

sin(4π) = 0, and sin(0) = 0.

So, the average value of the function f(x) = cos(8x) on the interval [0, π/2] is 0.

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