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7-18.

It just so happens that Hustling Harry’s grade in math class at any week during the semester is calculated by . At what point during the ‑week semester is Harry’s grade equal to his average grade for the semester?

This is an application of the Mean Value Theorem.

Show setup and steps.  weeks

The Mean Value Theorem

The Mean Value Theorem for Integrals

If is continuous on , then there exists at least one point in such that .

First quadrant, bell curve labeled, f of x, left end point on the y axis, labeled, a, right end point labeled, b, dashed horizontal segment, about 1 fourth up from x axis to peak, labeled average, & shaded rectangle between, A & b, segment & x axis, & circled points where the curve intersects the dashed segment, corresponding to tick marks on x axis, labeled, c subscript 1 & c subscript 2.

The Mean Value Theorem for Derivatives

If is continuous on and differentiable on , then there exists at least one point in such that .

Increasing curve labeled, capital F of x, starting at the origin, labeled a on the x axis, changing from concave up to concave down, in center of quadrant, ending at point corresponding to tick mark on x axis labeled, b, with dashed segment labeled, m = A R O C, from origin to end point, & 2 dashed parallel lines, one tangent before point of inflection, corresponding to tick mark on x axis labeled, c subscript 1, & one tangent after point of inflection, corresponding to tick mark on x axis labeled, c subscript 2.