Exam 3 study guide
- Calculators will not be allowed for this exam.
- For problems that have numerical answers, I expect the answer
in closed form (say, involving a few binomial coefficients ...).
- This exam will not be purely computational. You will have to do
some proofs as well.
- This exam will be mainly covering lectures 16-23 (March 16 to April 20)
and assignments 7-10.
- This corresponds to sections 7.1-7.6 and 8.1 in the book as well as some
lectures on Power series as indicated on the Tentative schedule.
- Section 7.5 in the book may not be helpful to you, and you can stick to
the lecture notes on this topic instead if you wish.
- In general, the lecture notes are more important than the book: if it's in the book, but not in the notes or assignments, then don't worry about it.
- You will not need to know section 8.2 for this exam and are not allowed to use
its techniques, or those found in any other later section, on this exam.
- Material from earlier lectures is still background material
even though I will not be "specifically" testing it. This means you can't
just forget what a multiset is, or what the the multiplication principle
does. This holds especially for identities we may have proved earlier.
- You should be familiar with all the terminology we have defined so far.
- You should know the correct statements of the Major results
we have covered so far. This includes basic results on recurrence relations
(such as the dependence of the general solution on the order of the recurrence,
and the relation between charactereristic polynomial and recurrence relation)
and generating functions/power series (arithmetic properties as well as
applications).
- Since we had many results you will be allowed to use a cheat sheet:
- Your sheet must be handwritten.
- Your sheet can use both sides of an 8.5 x 11 page.
- Write down all equation/theorem numbers, since you will
need those to justify your reasoning.
- Use the following notations:
(5.11) to denote equation (5.11) in the book,
Thm 5.6.1 to denote
Theorem 5.6.1 in the book (or you can use Newton's Binomial Theorem),
Cor 5.4.2 to denote Corollary 5.4.2 in the book,
and #7.1a) to denote Exercise
1 part a) from assignment 7.
- You may not quote problems from the book (since we didn't solve
all of those), only problems from the Homework.
- You should know how to apply these results to count the number
of various things: integral solutions to simple equations (with constraints),
combinations of n fruits in a bag, walks in a grid, ...
- You should understand the main idea of the proofs of
these results as well. I MAY ask you to reproduce one of
the shorter, straight forward proofs from lecture or homework.
- There will definitely be at least one proof of a result
you have not seen yet, so have your "tool box" ready.
- You should understand the solutions (either mine or yours)
to as many homework problems as possible.
- If you have the time for it, then doing extra problems
can also help. The more you work with the concepts,
the "readier" you are going to be.
- Some problems on the exam may throw you off at first.
Make sure that you get a good nights sleep, so that you are fresh,
rested and ready to go!