The inclusion of the Mean Value Theorem in a standard first-year calculus course, especially Advanced Placement Calculus AB, is a curious choice. Students are expected to know the necessary assumptions—the function is continuous on [a,b] and differentiable over (a,b)—and the conclusion—there is some point between a and b where the derivative is precisely equal to the average rate of change of the function over [a,b]. But why do we care? Perhaps the most important consequence is that two functions with the same derivative function must differ by a constant, but that hardly needs the full force of the mean value theorem. Another answer comes at the very end of Calculus BC: the Lagrange error bound. To explain the connection to Lagrange’s error bound, I will need some mathematical formulas that require me to switch to LaTeX.

Augustin Louis Cauchy (1789–857) and Joseph Louis Lagrange (1736–1813)