1. The final will be about the same length as the midterms, maybe slightly longer.

2. There will be a true-false page like the midterms, so it's useful to read all the statements of the theorems, so as to have some feel as to what is true and what is not.

3. I'll try to make about half the exam on the newer material. There are not so many theorems as on the earlier exams, but they are important ones, especially MVT/Rolle's and FTC I and FTC II. The definitions of derivative and integral are clearly important and theorem 5.2.4. There was no homework for section 7.5, so you might look at 7.5.2, 7.5.4, 7.5.7, 7.5.9 for practice.

4. Please look over your homework and earlier exams, as I sometimes repeat questions. Some of the true-false questions on the midterms were taken directly from the homework. You could expect a few problems on the final to be mild variations of problems from the first two exams.

Chapters 1-4: what was important?

Chapter 1. There was background on sets and functions (which we have used plenty) and information about countability (mainly to emphasize that the reals are much larger than the rational numbers), but the completeness axiom is the most important thing, as everything else follows from it. You should know the statement/language of the completeness axiom and the immediate consequences: the Archimedean property, the Nested Interval Property, and density of the rationals (and irrationals).

Chapter 2. The e-N definition of convergence of sequences is essential here. MCT, BW, and CC are all important, as well as the ALT and OLT, which we have used many times since. It is unlikely that I will ask a series question on the final, unless it comes up in a question focused on sequences.

Chapter 3. Most of this chapter gives language for describing subsets of the reals: open, closed, bounded, compact, connected, isolated points and limit points - you should be able to recognize these when you see them, along with the closure of a set. These notions are important later when we see how continuous functions interact with these properties. The most important theorem in chapter 3 is 3.3.8 (or theorem 3.3.4), which characterizes the compact subsets as those that are closed and bounded (and also those for which every open cover has a finite subcover).

Chapter 4. Finally the functions arrive. Be sure to know the e-d definitions of limits of functions and continuity of a function at a point (these definitions are almost the same) and the equivalent versions in terms of sequences (thm. 4.2.3 and the class version of thm. 4.3.2). The most important theorems are 4.4.2 (the continuous image of a compact set is compact, hence continuous functions achieve maximums and minimums on compact sets) and 4.5.1 (intermediate value theorem, equivalent to theorem 4.5.2 which says the continuous image of a connected set is connected).

Chapters 5 and 7, the newer material

Chapter 5. Know the definition of derivative, as I'll almost certainly ask for it, as well as some evidence that you can use the definition (for example to compute a derivative or to prove part of theorem 5.2.4). Theorem 5.2.6 is an important part of Calc I, giving a method to find max/mins of a differentiable function. I could imagine asking for a proof or to apply it. The MVT is easy to remember from the picture that accompanies it. I'm not sure if I'd ask for a proof of it, but the proof of Rolle's theorem is short, as are the corollaries that relate to Calc I (5.3.3, 5.3.4). L'Hospital's rule is another application.

Chapter 7. As I already indicated, I'll almost certainly ask you for the definition of the integral. I may ask you to use the definition to compute one (recall exercises 7.2.2, 7.2.3, 7.2.4 bc, 7.3.1). I will not ask for any proofs of the properties of integrals (section 7.4, I was sketchy about them in class) unless I give a good hint. It will be useful to know what they are, especially if I ask for a proof of any part of FTC (not sure how likely this is). I might also ask about content zero sets (Exercise 7.3.6) and the fact that a function is integrable on an interval if its discontinuities have content zero.