# How do you find the numbers b such that the average value of #f(x) = 7 + 10x − 9x^2# on the interval [0, b] is equal to 8?

See the explanation below.

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To obtain, use the formula or finish the square.

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To find the number ( b ) such that the average value of ( f(x) = 7 + 10x - 9x^2 ) on the interval ([0, b]) is equal to 8, follow these steps:

- Compute the definite integral of ( f(x) ) over the interval ([0, b]).
- Divide the result by the length of the interval ([0, b]).
- Set the quotient equal to 8 and solve for ( b ).

Mathematically, this process can be represented as:

[ \frac{\int_{0}^{b} (7 + 10x - 9x^2) , dx}{b - 0} = 8 ]

Solve this equation for ( b ) to find the numbers that satisfy the given condition.

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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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