MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
Consider the function q(x)=∣sinx∣ on the interval [−π,π].
(i) Sketch one complete cycle of y=q(x) over [−π,π], labelling the points where the formula for q(x) changes.
(ii) Express q(x) without the absolute-value symbol by splitting [−π,π] into suitable sub-intervals, and then compute its average value on [−π,π]. State the exact value obtained.
(iii) Verify that the value found in (ii) satisfies the conclusion of the Mean Value Theorem for Integrals by finding at least one number c∈(−π,π) such that q(c) equals this average value.