Mean Value Theorem

Recall that the MEAN VALUE THEOREM states: If f is a function that is both CONTINUOUS over the closed interval [a,b] and DIFFERENTIABLE over the open interval (a, b), then THERE EXISTS a value "c" in the open interval (a, b) for which the instantaneous rate of change of function f at x = c EQUALS the average rate of change of function f over the interval (a,b). This applet was designed to help you see a clear illustration of this theorem. Simply type in any continuous and differentiable function in the green input box in the top left hand corner. Adjust your "a" and "b" values using either the sliders or input-boxes next to these sliders. The X-COORDINATE of the PINK POINT C you see is the value of "c" described in the Mean Value Theorem above. Can you explain why?