Answer – The average value of the function f(x) = 15 − x on the interval [4,9] can be found using the formula $f_{avg} = \frac{1}{b - a}{\int }_a^{b}f\left(x\right)dx$ as 17/2.

Explanation:

Before getting into the calculations for this problem, it might be useful to first visualize the graph of a function f(x) that needs to be averaged on an interval a to b. The below graph illustrates such a function:

Remember that we arrive at the area under the curve when we calculate the value of the integral of the function from a to b.

Now, let us assume that the rectangle in the graph is a container holding water, which settles at a particular level once the water is steady. The height of this steady level is what we can consider to be a representation of the average value of the function f(x) on the interval [a,b], indicated by f_avg .

f_avg can now be defined so that the area of the rectangle equals the area under the curve (which is the highlighted area).

$Area of the rec\tan gle = Area under the curve$

$Height of the rec\tan gle x Width of the rec\tan gle ={\int }_a^{b}f\left(x\right)dx$

$f_{avg} (b - a) ={\int }_a^{b}f\left(x\right)dx$

$f_{avg} = \frac{1}{b–a}{\int }_a^{b}f\left(x\right)dx$

This is the formula to find the average value of a function f(x) on the interval [a, b].

If we are unable to remember the formula at any point, we can easily deduce it by following the above steps.

Now, coming back to the given question, we have f(x) = 15 − x, b = 9, and  a = 4.

On substituting these values in the equation for the average value of a function, we get:

$f_{avg} = \frac{1}{9–4}{\int }_4^9(15 - x)dx$

$f_{avg} = \frac{1}{5} (15x - \frac{x²}{2})$, which must be evaluated for the interval 4 to 9

$f_{avg} = \frac{1}{5} \left[\right(15\left(9\right) - \frac{9^2}{2}) - (15\left(4\right) - \frac{4^2}{2}\left)\right]$

$= \frac{1}{5} \left[\right(135 - \frac{81}{2}) - (60 - \frac{16}{2}\left)\right]$

$= \frac{1}{5}\left[\right(\frac{189}{2}) - 52]$

$= \frac{1}{5} \left[\frac{85}{2}\right]$

$= \frac{17}{2}$

Thus, the average value of the function f(x) = 15 − x on the interval [4,9] is 17/2.

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