OpenStudy (anonymous):

what is the exact difference between mean value theorem and average value when it comes to integration?

7 years ago

The average value of a derivative is the mean value of its underlying function. So if you have $$f(x)$$ and its derivative $$f'(x)$$, the average value of $$f'(x)$$ is the mean value of $$f(x)$$. In essence, this means that $$f'(x)$$ between two points $$a) and \(b$$ must at some point equal its own average value between those two points.

7 years ago
OpenStudy (anonymous):

The average value formula, 1/(b-a) * Integral f(x)dx from a to b, gives the average y-value of function f between a and b. The mean value theorem (MVT) says take any smooth curve. Select two points on that curve and connect them with a straight line. MVT guarantees that somewhere between the two points, there will be a tangent line that is parallel to the line connecting the two points.

7 years ago
OpenStudy (anonymous):

thanks guys, i finally get it!

7 years ago
OpenStudy (anonymous):

Actually MVT of integration says that on (a, b) there exist a c such that f(c) = 1/(b-a) * Integral f(x)dx, and the proof is a lot easier than the derivative one

7 years ago