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If the graph at right represents distance over time between a space shuttle and Earth, how would you determine the average velocity? Describe the method you would use.

Given a distance function, will average velocity be represented by 'average area under the curve' or 'average rate of change'?

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.

First quadrant, x axis labeled time, hours, y axis labeled distance, miles, decreasing curve starting at top of y axis, dropping about half way & running about a third, leveling out, running another sixth, changing from concave up to concave down, running about another sixth, then dropping to the x axis.