A short note containing the definitions of a differential, linear approximation and some motivation for these concepts. This note also contains a worked example using differentials. The end of the note contains the statement of the MVT.

This note has three sections. The first section contains some advice on how to read and, in particular, understand some of the more formal/technical mathematical statements and assertions you are seeing. The second part of this section is in largely the same vein and is more akin to a large example to some of the material in the first section. The two sections are independent of each other, however. The second part of this section describes how to read off the conditions of the MVT from the theorem's statement but also gives some examples of what goes wrong if we don't assume the conditions of the MVT. Hopefully you will find these two sections useful in parsing some of the definitions and theorems you are receiving.

The second part describes using linear approximations to approximate values of functions and includes a worked example as well as how concavity determines whether a linear approximation over or under approximates. The third and last part contains several worked examples involving L'Hopital's rule as well as a collection of tricks to use to manipulate an expression into a form to which L'Hopital can be applied.

This note briefly describes an important property of power series and one manner of finding the radius of convergence of a power series. Contains a couple worked examples.