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

Question

shadowfiend · Answer

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.

anonymous · Answer

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.

anonymous · Answer

thanks guys, i finally get it!

anonymous · Answer

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