Question

Find the average value of f(x)=25−x2 on the interval [0,4]

Answer 1

The average value of the function f(x) = 25 - x² on the interval [0,4] is found to be 59/3 or approximately 19.67.

Step-by-step explanation:

The average value of a function f(x) on a given interval [a, b] can be found using the formula for the average value of a continuous function:

Average value = (1 / (b - a)) ∧ f(x) dx over [a, b]

To find the average value of the function f(x) = 25 - x² on the interval [0,4], we apply the formula:

1. Substitute a = 0 and b = 4 into the formula.

2. Calculate the integral: ∧ (25 - x²) dx from x = 0 to x = 4.

3. Divide the result by (4 - 0).

Performing the calculation:

• Integral of 25 - x² from 0 to 4:
  [25x - x³/3] from 0 to 4 = (100 - 64/3) - (0 - 0) = 100 - 64/3.
  

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Therefore, the average value of f(x) on the interval [0, 4] is 59/3 or approximately 19.67.