# What is the average value of the function #f(x)=cos(16x) # on the interval #[0,pi/2]#?

This question asks for average value, but is posted under the topic "average rate of change" Here are solutions to both.

Average Change Rate

The inquiry seeks Average Value.

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To find the average value of the function ( f(x) = \cos(16x) ) on the interval ([0, \frac{\pi}{2}]), we need to compute the definite integral of ( f(x) ) over that interval and then divide it by the length of the interval.

[ \text{Average value} = \frac{1}{\frac{\pi}{2} - 0} \int_0^{\frac{\pi}{2}} \cos(16x) , dx ]

[ = \frac{2}{\pi} \int_0^{\frac{\pi}{2}} \cos(16x) , dx ]

[ = \frac{2}{\pi} \left[\frac{\sin(16x)}{16}\right]_0^{\frac{\pi}{2}} ]

[ = \frac{2}{\pi} \left[\frac{\sin(8\pi) - \sin(0)}{16}\right] ]

[ = \frac{2}{\pi} \left[\frac{0 - 0}{16}\right] ]

[ = \frac{2}{\pi} \times 0 ]

[ = 0 ]

Therefore, the average value of the function ( f(x) = \cos(16x) ) on the interval ([0, \frac{\pi}{2}]) is ( 0 ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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