Final answer:
The average value of the function f(x) = (x - 8)^2 on the interval [7, 10] is 1. The values of c where f(c) equals the average value fave are c = 7 and c = 9.
Step-by-step explanation:
The student has requested help in finding the average value fave of a function on a certain interval and then finding the value of c such that the average value equals f(c). The function is f(x) = (x - 8)^2, and the interval is [7, 10].
To find the average value of f on the provided interval, we use the formula for the average value of a function over an interval [a, b], which is:
- Calculate the integral of f over the interval [a, b].
- Divide the result by the length of the interval, which is (b - a).
The integral of f(x) = (x - 8)^2 from 7 to 10 is:
∫ f(x) dx = ∫ (x - 8)^2 dx
= [(x - 8)^3 / 3] from 7 to 10
= [2^3 / 3] - [(-1)^3 / 3]
= [8/3] - [-1/3]
= 9/3
= 3
The length of the interval [7, 10] is 3. Therefore, the average value fave is:
fave = Integral result / (b - a) = 3 / 3
= 1.
To find the value(s) of c such that f(c) = fave, we set f(c) equal to fave:
(c - 8)^2 = 1
Take the square root of both sides:
c - 8 = ±1
Solve for c:
c = 8 + 1 or c = 8 - 1
c = 9 or c = 7.
Both of these values fall within the original interval of [7, 10]. So the list of values for c is 7, 9.