Question

Which of the functions given below has an average (mean) value of zero on the interval -a=x=a, a>0 ? a.abs(x) b.cos(x) c. e^x d. sin(x) e. x^2+x^3

Answer 1

Odd functions have the property that the average on the interval [-a,a] is zero, because of this:

The definition of the average of a differentiable function is:

Average = { ∫ f(x)dx from - a to a } / [ a - (-a)]

And for an odd fuction ∫f(x)dx from - a to a is zero => Average = 0

The only odd function in the list is cos(x), then the answer is b. cos(x).