# What is the average value of the function #f(x)=sin4x# on the interval #[-pi,pi]#?

It is

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To find the average value of the function (f(x) = \sin(4x)) on the interval ([- \pi, \pi]), we use the formula:

[ \text{Average value} = \frac{1}{b - a} \int_{a}^{b} f(x) , dx ]

In this case, (a = -\pi) and (b = \pi). So, we have:

[ \text{Average value} = \frac{1}{\pi - (-\pi)} \int_{-\pi}^{\pi} \sin(4x) , dx ]

[ = \frac{1}{2\pi} \left[ -\frac{\cos(4x)}{4} \right]_{-\pi}^{\pi} ]

[ = \frac{1}{2\pi} \left[ -\frac{\cos(4\pi)}{4} + \frac{\cos(-4\pi)}{4} \right] ]

[ = \frac{1}{2\pi} \left[ -\frac{1}{4} + \frac{1}{4} \right] ]

[ = \frac{1}{2\pi} \times 0 ]

[ = 0 ]

Therefore, the average value of the function (f(x) = \sin(4x)) on the interval ([- \pi, \pi]) 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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