Having learnt to compute derivatives, it is time to use them to understand functions: where they increase, where they decrease, where they have maxima or minima, where they change concavity. The derivative becomes a most powerful investigative tool, and the theorems of this chapter turn information about the derivative into information about the function. We start from Rolle’s theorem and Lagrange’s theorem (or the mean value theorem), move on to L’Hôpital’s rule for computing limits in indeterminate forms, and conclude with the two pillars of integral calculus: the integral mean value theorem and the fundamental theorem of integral calculus, which ties differentiation and integration together in a single logical chain.
Sections
- Rolle’s theorem
- Lagrange’s theorem (or the mean value theorem)
- L’Hôpital’s rule
- Integral mean value theorem
- Fundamental theorem of integral calculus