We can use the mean value theorem to prove that linear approximations do, in fact, provide good approximations of a function on a small interval. The information the theorem gives us about the derivative of a function can also be used to find lower or upper bounds on the values of that function.
Lecture Video and Notes
Clip 1: The Mean Value Theorem and Linear Approximation
Clip 2: The Mean Value Theorem and Inequalities
The Mean Value Theorem and the Sine Function