Question

Find the average value of f(x) = 25 - x² on the interval [0, 1].

Find a value c in the interval [0, 1] such that f(c) is equal to the average value.

Answer 1

The average value of the function f(x) = 25 - x² on the interval [0, 1] is 24 2/3, and the value c in this interval that yields this average is approximately 0.577.

Step-by-step explanation:

To find the average value of the function f(x) = 25 - x² on the interval [0, 1], we integrate the function over the interval and then divide by the length of this interval. The formula for the average value of a function f(x) over the interval [a, b] is given by:

Average value of f(x) = ± ∫ f(x) dx / (b - a)

Now let's carry out the calculation:

1. First, integrate the function f(x) = 25 - x² from 0 to 1: ∫ (25 - x²) dx = [25x - (x³/3)] from 0 to 1.
2. Substitute the bounds into the antiderivative to evaluate the integral: [25(1) - (1³/3)] - [25(0) - (0³/3)] = 25 - 1/3 - 0 = 24 2/3.
3. Divide this result by the length of the interval (1 - 0 = 1): 24 2/3 / 1 = 24 2/3.

The average value of the function on [0, 1] is therefore 24 2/3.

To find a value c in [0, 1] such that f(c) equals the average value, we solve the equation f(c) = 24 2/3:

25 - c² = 24 2/3

c² = 1/3

c = ±sqrt(1/3)

Since c must be in the interval [0, 1], only the positive root makes sense. Therefore, c = sqrt(1/3), which approximately equals 0.577.

So the value c in the interval [0, 1] that gives us the average value of the function is approximately 0.577.