Find the average value of the function
f(x) 8x^2-9x+8
on the interval [3,5]. Find the value of the x-coordinate at which the function assumes it's average value
The average value is ?
the x-coordinate is ?
Please help and showing work is much appreciated. Thank you
To find the average value of a function on the interval [a, b], we need to calculate the definite integral of the function over that interval and then divide it by the length of the interval (b - a).
Given the function f(x) = 8x^2 - 9x + 8 and the interval [3, 5], we can proceed as follows:
Step 1: Calculate the definite integral of f(x) over the interval [3, 5].
∫[3,5] (8x^2 - 9x + 8) dx
To simplify the integral, we can use the power rule of integration:
= [8/3 * x^3 - 9/2 * x^2 + 8x] evaluated from 3 to 5
= (8/3 * 5^3 - 9/2 * 5^2 + 8 * 5) - (8/3 * 3^3 - 9/2 * 3^2 + 8 * 3)
= (8/3 * 125 - 9/2 * 25 + 40) - (8/3 * 27 - 9/2 * 9 + 24)
= (1000/3 - 225/2 + 40) - (216/3 - 81/2 + 24)
= (1000/3 - 1125/6 + 240/6) - (216/3 - 162/6 + 144/6)
= (2000/6 - 1125/6 + 240/6) - (720/6 - 162/6 + 144/6)
= (1115/6) - (702/6)
= 413/6
Step 2: Calculate the length of the interval [3, 5].
length = 5 - 3 = 2
Step 3: Divide the result from Step 1 by the length from Step 2 to find the average value of f(x).
Average Value = (413/6) / 2
= (413/6) * (1/2)
= 413/12
≈ 34.42
Therefore, the average value of the function f(x) = 8x^2 - 9x + 8 on the interval [3, 5] is approximately 34.42.
To find the x-coordinate at which the function assumes its average value, we need to solve the equation f(x) = average value. In this case, we need to solve the equation:
8x^2 - 9x + 8 = 34.42
To solve this quadratic equation, we can rearrange it into standard form:
8x^2 - 9x + 8 - 34.42 = 0
8x^2 - 9x - 26.42 = 0
We can now use the quadratic formula to find the x-coordinate:
x = (-b ± √(b^2 - 4ac)) / (2a)
For the equation 8x^2 - 9x - 26.42 = 0, we have:
a = 8, b = -9, c = -26.42
Plugging these values into the quadratic formula, we get:
x = (-(-9) ± √((-9)^2 - 4 * 8 * (-26.42))) / (2 * 8)
x = (9 ± √(81 + 845.12)) / 16
x = (9 ± √926.12) / 16
The solutions for x will be the x-coordinates at which the function assumes its average value.
Therefore, the x-coordinates at which the function f(x) = 8x^2 - 9x + 8 assumes its average value are approximately ((9 + √926.12)/16) and ((9 - √926.12)/16).