How do you find the average value of the function for #f(x)=x^3, 0<=x<=2#?

Answer 1

#2#

The average value is basically the area under the curve divided by the length of the base, i.e.:

#1/(color(blue)(2)-color(blue)(0)) int_color(blue)(0)^color(blue)(2) \ x^3 \ dx = 1/2 [ 1/4 x^4 ]_color(blue)(0)^color(blue)(2) = 1/8((color(blue)(2))^4 - (color(blue)(0))^4) = 1/8(16-0) = 2#
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Answer 2

To find the average value of the function ( f(x) = x^3 ) over the interval ([0, 2]), you can use the formula:

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

In this case, ( a = 0 ) and ( b = 2 ). Substituting these values and the function ( f(x) = x^3 ) into the formula, you can evaluate the integral. Then, divide the result by the length of the interval, which is ( b - a = 2 - 0 = 2 ). This will give you the average value of the function over the interval ([0, 2]).

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Answer from HIX Tutor

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.

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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