Our second term exam will look a lot like the first, but notice the
The material for the second exam is all of chapters 7 through 12 of
Spivak's book ("Three Hard Theorems" through "Inverse Functions"). Unlike
what I said before to some students, the material in chapters 1-6 is not
officially included, though, of course, what chance have you got answering
questions about the continuity of inverse functions (say), if you aren't
yet absolutely fluent with the notion of continuity?
Some of the questions may have a part in which you will be required
to reproduce an example or a definition or a proof given in full in the
class or in the text. The class material is important; I put proofs on
the blackboard because I really want you to understand them. Doing lots
of exercises is great, but the most important exercises are the ones
that are called "theorems" and are shown in class; that's precisely why
they are shown in class. You may want to prepare a list of all topics
touched in class (you may reach 50 or even 100), and you may want to go
over this list several times until you are sure you understand
everything in full.
Remember, you really understand a mathematical definition
/ theorem / claim / lemma / anything only when you have fully
internalized it and made it your own. Check if you can say to yourself
one of the following:
"Gosh this is so right. I would have done it in just the same
way" (sometimes add: "if I was a little smarter when the issue first
"Hey, I can do it better! Here's how...".
It's worthwhile! Your grades will be higher, you will have gained more
from this (and other) classes, and there is a lot of satisfaction and
joy when you succeed. I internalized this sometime in my second year as
an undergrad and it was the most important thing I learned that year.