The mean value theorem is one of many useful tools in mathematics. The following may be interesting for you.

1. Show that the mean value theorem can be written as f(x+h)-f(x) = hf '(x + th), where 0 < t < 1.
2. Determine t as a function of x and h for the functions f(x) = x², e^x, and ln(x).
3. Keep x fixed and determine the limit of t as h approaches zero for each of these functions.