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- anonymous

AP Calc help?
The graph of f ' (the derivative of the function f) is shown for the interval 0 to 10. The areas of the regions between the graph of f ' and the x-axis are 20, 6, and 4 respectively. If f(0) = 2. what is the maximum value of f on the closed interval 0 to 10?

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- anonymous

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- anonymous

can you post the graph?

- anonymous

http://www.siprep.org/faculty/kquattrin/documents/BC10DxApps.pdf
It's question 3 on this random pdf that google found for me.

- anonymous

An answer is circled but how did they get that?

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- anonymous

I did this to get 22. When you integrate f'(x) you get f(x) So that large area of 20 is the largest definite integral from 0 to any value b. Go farther and you have that negative area which the 4 does not make up for.
The lower limit of integration we are considering is always 0 to get the most area. Integral from 0 to b of f'(x)= f(b)-f(0). So the largest area will produce the largest f(b). Largest area is 20. f(b)-f(0)=20. f(b)=22

- anonymous

I'm sorry could you explain why/how you did that? I'm not seeing the connection between them asking for the maximum value on f and finding the largest definite integral. Or any further connections for that matter. I appreciate your help.

- anonymous

They give you the derivative f'(x) and give you the area under the derivative curve. These areas under are the definite integrals of the derivative in their respective intervals. You know that the definite integral of f'(x) from 0, the left endpoint to some value b is f(b)-f(0) = f(b)-2. So this is the formula that gives you the area under the derivative curve from 0 to anywhere you pick. To maximize the area is to find the maximum value of f(b) because they are related: Area = f(b)-2.

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