Question

Find the average value of f(x) = 4 − x^2 on the interval [−2, 3].

Answer 1

The average value of the function f(x) = 4 - x^2 on the interval [−2, 3] is 2.733.

Step-by-step explanation:

To find the average value of the function f(x) = 4 - x2 on the interval [−2, 3], we use the formula for the average value of a function:

Average Value = (1/(b - a)) ∫ab f(x) dx

Here, a = -2 and b = 3, so the interval length is 3 - (-2) = 5.

We integrate f(x) from -2 to 3:

1. ∫ f(x) dx = ∫ (4 - x2) dx from -2 to 3
2. = [4x - (1/3)x3] from -2 to 3
3. = (4(3) - (1/3)(3)3) - (4(-2) - (1/3)(-2)3)
4. = (12 - 9) - (-8 - (-8/3))
5. = 3 + 8 + 8/3
6. = 3 + 24/3 + 8/3
7. = 3 + 32/3
8. = 3 + 10.666...
9. = 13.666...

Now we divide by the length of the interval (5) to find the average value:

Average Value = 13.666... / 5

Average Value = 2.733...