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In order to calculate the average value of a function, sometimes it makes sense to integrate while other times the slope of a secant line is determined. When do you need to use each strategy?

When trying to find the average value of , the strategy depends on what function is given: or .

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

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