Accuracy of Fundamental thereom compared to MVT

[nobahar]

Hello!
The Mean Value thereom gives F(b)-F(a) = f(c).(b-a) Where f is F' and c is the value of x at which it's derivate is equal to the average rate of change over the interval a to b for F.
The Fundamental thereom of calculus also gives F(b)-F(a) = $$(\frac{1}{b-a} \left \left \int _{a}^{b}f(x)dx)(b-a)$$
Why is the second more accurate than the first, surely both give the average rate of change multiplied by the same interval, (b-a)? Is it to do with practical applications, and determining the f(c) value?
Any pointers/help much appreciated.

 

[lippyka]

The second theorem is more accurate because it takes into account the actual shape of the function, while the first theorem only considers the average rate of change. This means that the second theorem can give more accurate results when dealing with functions that have nonlinear shapes. For example, if we take a function that has a parabolic shape, the first theorem will only consider the average rate of change, while the second theorem can take into account the actual shape of the function and give a more accurate result.