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

The average value is

Here,

Therefore,

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To find the average value of ( f(x) = (x - 3)^2 ) as ( x ) varies between ( [2, 5] ), you use the following formula:

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

Substitute ( a = 2 ) and ( b = 5 ) into the formula and find the integral:

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

Now, integrate ( (x - 3)^2 ) with respect to ( x ):

[ \int_{2}^{5} (x - 3)^2 , dx = \left[ \frac{(x - 3)^3}{3} \right]_{2}^{5} ]

[ = \left[ \frac{(5 - 3)^3}{3} - \frac{(2 - 3)^3}{3} \right] ]

[ = \left[ \frac{2^3}{3} - \frac{-1^3}{3} \right] ]

[ = \left[ \frac{8}{3} + \frac{1}{3} \right] ]

[ = \frac{9}{3} ]

[ = 3 ]

Thus, the average value of ( f(x) = (x - 3)^2 ) as ( x ) varies between ( [2, 5] ) is ( 3 ).

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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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