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What if the graph at right represents the velocity in miles per hour of the shuttle? In this case, how would you determine the average velocity? Describe the method you would use.

Given a velocity 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.