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During the semester, Arezo’s daily calculus grade at time, (in days), is given by . She will receive if she passes calculus, which requires an average grade of or better. If her semester grade (over weeks) is the average grade during that time period, should she celebrate?


Find the average value of , given .

Average (Mean) Values

To calculate the mean (average) value of a finite set of items, add up the values of items and divide by the number of items.

Integrals help us add over a continuous interval. Therefore, for any continuous function :

mean value of over

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.

Since , we can also calculate the average value of any function using its antiderivative . Its average slope gives the average rate of change of , which is the same as the average value of

mean rate of change of over

First quadrant, 2 tick marks on x axis, first at the origin labeled, A, second almost to the right end, labeled b, Increasing curve labeled, capital F of x, starting at the origin, changing from concave up to concave down, in center of quadrant, ending at point corresponding to, b, almost at the top, with dashed segment labeled, m = average, from origin to end point of curve.