# How do you find the average value of #f(x)=sqrtx# as x varies between #[0,4]#?

The average value of

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To find the average value of ( f(x) = \sqrt{x} ) as x varies between [0,4], we use the formula for the average value of a function on a closed interval [a, b]:

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

In this case, a = 0, b = 4, and ( f(x) = \sqrt{x} ). Substituting these values into the formula:

[ \text{Average Value} = \frac{1}{4 - 0} \int_{0}^{4} \sqrt{x} , dx ]

[ = \frac{1}{4} \int_{0}^{4} x^{\frac{1}{2}} , dx ]

[ = \frac{1}{4} \left[ \frac{2}{3}x^{\frac{3}{2}} \right]_{0}^{4} ]

[ = \frac{1}{4} \left( \frac{2}{3}(4^{\frac{3}{2}}) - \frac{2}{3}(0^{\frac{3}{2}}) \right) ]

[ = \frac{1}{4} \left( \frac{2}{3}(8) - \frac{2}{3}(0) \right) ]

[ = \frac{1}{4} \left( \frac{16}{3} \right) ]

[ = \frac{4}{3} ]

Therefore, the average value of ( f(x) = \sqrt{x} ) as x varies between [0,4] is ( \frac{4}{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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