A student asserts: “For any continuous even function defined on with the average…

MATHEMATICS · Single correct · AP · Calculus AB · Applications of Integration · Applications of Integration

Problem

A student asserts: “For any continuous even function gg defined on [−L,L][-L,L] with L>0,L>0, the average value of gg on [−L,L][-L,L] equals 1L∫0Lg(x) dx.\dfrac{1}{L}\displaystyle \int_{0}^{L} g(x)\,dx.” Which of the following best evaluates the student’s assertion? (a) The assertion is invalid because the average value formula always divides by 2L2L, even for an even function. (b) The assertion is valid because for an even function ∫−LLg(x) dx=2∫0Lg(x) dx,\displaystyle\int_{-L}^{L} g(x)\,dx = 2\int_{0}^{L} g(x)\,dx, so 12L∫−LLg(x) dx=1L∫0Lg(x) dx.\dfrac{1}{2L}\int_{-L}^{L} g(x)\,dx = \dfrac{1}{L}\int_{0}^{L} g(x)\,dx. (c) The assertion is invalid because symmetry forces ∫−LLg(x) dx=0\displaystyle\int_{-L}^{L} g(x)\,dx=0, so the average would have to be 00. (d) The assertion is valid only if g(0)=0g(0)=0; otherwise the lack of symmetry about the yy-axis changes the average value.

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