The following lecture notes are ones that I have developed over the past years as I have been teaching Calculus. You will see that there are three types of documents enclosed. The first, the long lecture notes, are meant to help a first time instructor. These provide motivation for the material and how it’s being taught. Furthermore, they include suggestion for how to present topics and what your focus should be on.
The short lecture notes are what I normally bring to class. By the time I lecture on material, I have a good understanding of what I want to say and accomplish. The short notes remind me of the examples I picked to go through with class and highlights that I need to hit. They are really meant to remind me in the case I forget where I am.
Lastly, I have included a power point presentation for the material. While I don’t always use power point, I have actually found this extremely useful during this spring 2020 semester when I have had to teach remotely. By writing the examples and definitions down for the students on power point, it has saved time and board space over trying to write everything out by hand. I have only used power point in the classroom when I have had enough board space left over after the using the projector that I can still write on the board. Even though the examples are on the power point, I will work through the examples by hand on the board. I also make these available to my students ahead of time so that, if they wish, they can use the a print out for note taking purposes allowing them to focus more on what I am saying.
Goals of This Chapter
This chapter will cover the first major topic of Calculus, that of limits. As I work this this section, I want to make sure that the students truly understanding what is meant by a limit. That is, I want to make sure that we are looking at what happens to a function value as we get close to but not equal to an \(x\) value. Students commonly want to associate limits with function values at the point, so make sure that they see the difference here.
By properly describing limits, you can set the tone for rigor and accuracy that you expect through the rest of the class. Doing this now will make it much easier for you to explain derivatives and integrals based off the limit definition. This really helps students from falling into the trap of just thinking that they will just need to memorize a lot of formulas.
For this reason, you will see that I also include proofs of many of the results I show my students. While I don’t expect them to give these proofs on exams, I want to take this opportunity to open their eyes to what mathematics is really like if you continue yours studies beyond Calculus. Furthermore, these really emphasize that we not only want to find answers, but we want to be able to explain where these answers come from.