Question

Let f(x)=cos(π/2x).What is the average value of f(x) on the interval [0,3]?

a) 0
b) 1
c) 2/π
d) 1/2

Answer 1

The average value of f(x) = cos(π/2x) over [0,3] is undefined, as the function is not integrable over the entire interval due to it not being defined at x = 0.

Step-by-step explanation:

The average value of a function f(x) on an interval [a, b] can be found by integrating the function from a to b and then dividing by the length of the interval. For the function f(x) = cos(π/2x) over the interval [0,3], we calculate the average value using the formula:

Average value = (1/(b-a)) ∫ from a to b of f(x) dx

Here, a = 0, b = 3, so the length of the interval is 3 - 0 = 3. We will integrate f(x) from 0 to 3 and then divide by 3. Performing this integration may be complex or undefined due to the nature of the cosine function and its period. In this case, the average is undefined because the function is not integrable over the entire interval [0, 3] (the function is not defined at x = 0).