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?

Answer 1

See the explanation below.

The average value of #f(x) = 7 + 10x − 9x^2# on the interval #[0, b]# is
#1/(b-0)int_0^b (7 + 10x − 9x^2) dx#

which can be assessed in order to,

#1/b[7x+5x^2-3x^3]_0^b = (7b+5b^2-3b^3)/b#
# = 7+5b-3b^2#
We want the average value to be #8#, so we need to solve
# 7+5b-3b^2 = 8#.

To obtain, use the formula or finish the square.

two solutions: #b=(5-sqrt13)/6# and #b=(5+sqrt13)/6#.
Since both are positive, the interval #[0,b]# exists for either of these.
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Answer 2

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:

  1. Compute the definite integral of ( f(x) ) over the interval ([0, b]).
  2. Divide the result by the length of the interval ([0, b]).
  3. 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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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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