Geometry has always been considered as an exact science, and indeed as the source of the exactness which is widespread among other parts of mathematics. . . But it seems that this feature of exactness does not reign anymore in geometry since the new system of infinitely small quantities has been mixed to it. I do not see that this system has produced anything for the truth and it would seem to me that it often conceals mistakes.
Assuming that all of the properties of limits we talked about in Section 14.1 can be proved, we have seen that all of the differentiation rules we developed intuitively using differentials in Chapter 4 can be made rigorous using limits.
The question we need to address now is this: Does the need for rigor, which prompted our definition of the derivative 13.2.3), get in the way of practical applications such as, say, the First Derivative Test? We will show that such a practical result can still be achieved while maintaining rigor. To do this, we will need a theorem that allows us to relate instantaneous changes to finite changes. The French name for this theorem is “le théorème des accroissements finis” (translated literally as “the Theorem of Finite Increments”). In English it is called the the Mean Value Theorem. We will see how this powerful theorem can be used to transition from theoretical to practical.
