# How do you find the average value of #f(x)=x^5-4x^3+2x-1# as x varies between #[-2,2]#?

The average value of a function is found using the following equation:

Therefore:

This is a basic integral.

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Morgan has given a fine answer. I want to mention a fact that can simplify the integration needed for this question.

In this question the first four terms of the polynomial form an odd function that is integrable on any closed interval, so

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To find the average value of the function (f(x) = x^5 - 4x^3 + 2x - 1) over the interval ([-2, 2]), you use the formula for the average value of a function:

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

Substituting the given function and interval into this formula:

[ \text{Average value} = \frac{1}{2 - (-2)} \int_{-2}^{2} (x^5 - 4x^3 + 2x - 1) , dx ]

You then evaluate this integral to find the average value of the function over the given interval.

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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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