Question

Find the average value of f(x) = (π/2)cos(x) on the interval [0, π/2]

a) π/4
b) π/2
c) 1
d) 0

Answer 1

The average value of f(x) = (π/2)cos(x) on the interval [0, π/2] is 1/π ≈ 0.318.

Step-by-step explanation:

The average value of a function over an interval is given by the formula:

average value = (1/(b-a)) * integral from a to b of f(x) dx

In this case, the given function is f(x) = (π/2)cos(x) and the interval is [0, π/2].

So, the average value of f(x) on the interval [0, π/2] can be found by evaluating the integral:

average value = (1/(π/2 - 0)) * integral from 0 to π/2 of (π/2)cos(x) dx

Simplifying the integral, we get:

average value = (1/π) * integral from 0 to π/2 of cos(x) dx

The integral of cos(x) from 0 to π/2 is sin(π/2) - sin(0) = 1 - 0 = 1.

So, the average value of f(x) on the interval [0, π/2] is:

average value = (1/π) * 1 = 1/π ≈ 0.318