Stuff about the Math 131BH midterms
- The first midterm is on Friday, Apr 25, 12:00 pm, at MS 5148.
The second midterm is on Friday May 23, 12:00 pm, at MS
- There will be five questions, all worth 10 points; only your best
4 out of 5 will be graded, so read the questions beforehand to decide
which 4 to attempt (you may find that they are not all of equal
difficulty). They will all be short proof type questions (similar
to the homework questions, but somewhat shorter).
- The first midterm covers Weeks 1-3, up to uniform convergence but
not the Weierstrass approximation theorem. The second midterm
covers Weeks 4-7, covering differentiation but not any measure theory..
- The exam is open book, open notes; you may bring any course
material or your own written notes. Bringing the solutions to your
own homework may be a particularly good idea. Calculators are
permitted but are very unlikely to be useful. No computers or
other sophisticated electronic equipment is allowed.
- In addition to the open book and open notes, a page of
definitions will be supplied with the midterm. This is so that you
do not have to cycle through your notes just to find a particular
- I expect your proofs to be fairly detailed - it should be clear
to the grader which of the equations you write down are your
assumptions, which ones are the things you are trying to prove, which
ones are hypotheses, etc. Using English words such as "since",
"assume", "by hypothesis", "by a theorem in the notes", etc. is highly
recommended. On the other hand, you do not need to supply
explanations for each and every step; for instance, the manipulations of
high school algebra can be done without much comment. You do not
need to give precise references (i.e. page numbers or Proposition
numbers) for the results from the notes; it will be acceptable to say
things like "By a theorem in the notes, ..." without specifying exactly
which theorem. (This is, of course, assuming that the theorem you
are quoting IS indeed in the notes).
- Unless otherwise specified in the question, you are allowed to
use any result from the course notes or from the textbook when answering
a question. You may also use any material from other classes
(notably Math 131AH, but also Math 132, Math 121, etc.).
Some tips and tricks:
- There is no practice midterm; however I would recommend going
through the homework assignments carefully prior to taking the midterm.
In particular you should try to do as many of the homework
problems as possible by yourself, without external aids (except for
things like the class notes). You are encouraged to bring your own
homework solutions to the midterm. (Bringing someone else's
solutions (eg. the TA's) is much less likely to be helpful, unless you
understand how the solutions work, in which case you probably didn't
need to bring the solutions in the first place).
- Familiarity with the course notes is highly recommended; many of
the proofs in the midterm may be easier if you use some of the
propositions and theorems from the notes. Note that you should
take some care reading the hypotheses of each proposition before using
them; a proposition may apply to integers but not to rational numbers,
or to convergent sequences but not to divergent sequences, or to finite
sets but not to infinite sets, etc. This "fine print" is sometimes
very important, so don't just apply the conclusion of the proposition
blindly! You may find it helpful to create summaries of the
course notes for quicker reference when taking the midterm; also the
very process of creating summaries may help you organize the material
- Each midterm question is likely to bear some resemblance to a
homework question, or perhaps a lemma in the notes, which suggests that
the proof method is likely to be similar. For instance, if the
midterm question looks like a homework question, and you used induction
to do the homework question, it is likely that induction will be useful
for the midterm question as well. However, while the methods may be similar, the actual content of the proof may be somewhat
different, so merely copying a proof from the homework or notes word for
word into the midterm is unlikely to impress the graders.
- It may help to write down exactly what the hypotheses are, and
(in a separate location on the paper) write down what conclusions you
want to reach. This may help suggest what propositions or proof
methods may be helpful, and can help convince the grader that you know
what you are doing. Note that one should always take care to
separate the hypotheses from the conclusions; mixing the two up is
almost always a bad idea.
- It may be helpful to use the definitions of terms to break up a
complex statement into more elementary components. For instance,
if one of the hypotheses is that "a_1, a_2, .... is a Cauchy sequence",
it may help to use the definition of a Cauchy sequence to break this
statement up into more elementary statements (e.g. using
epsilon-steadiness). Of course, this can be taken too far... in
most cases you don't want to go all the way back to natural numbers and
++, as most statements become ridiculously lengthy when expanded that
- The most common method to try to prove something is the direct or forward approach: write down your
hypotheses, see what you can derive from them, and eventually work your
way toward the conclusion. Sometimes, however, it is better to try
a backward approach: look at
the conclusion and think about
the possible ways one could get to that conclusion. This allws
you to replace your conclusion with successively easier conclusions,
working your way back towards something which will follow easily from
the hypotheses. A typical use of the backwards approach is to
take the conclusion, and use some definitions to break up any complex
concepts in that conclusion to simpler concepts. Often this makes
the conclusion more elementary, and thus easier to reach.
- In many cases a statement may look obviously true, and yet you do
not know how to prove it. In this case you might ask yourself "is
there any possibility that the statement is in fact false?". Often
times you will find a good reason for why the statement cannot be false
- and this means that you should probably prove your statement via
- At certain times in a proof, you may want to use some result
which you are sure is true, but is not explicitly stated in the lecture
notes. In that case, you may wish to set that result up as a Lemma
(also called a Claim or an Auxiliary result), use that lemma to finish
the proof, and then go back and prove the lemma. Even if you don't
manage to prove the lemma, you can still get partial credit if you can
prove the original question assuming the lemma is true. One word of
caution, however - sometimes the lemma you wish to use is more
complicated than the original question, so this technique may not
necessarily save time. You should use your own judgement to see
if your lemma really is a more basic and fundamental fact than what you
are trying to prove.
- If you are completely stuck in proving a statement, you can at
least score partial credit by demonstrating to the grader that you at
least comprehend the statement.
Some ways in which you can do this is by drawing a picture, or
giving an example, or by explaining in informal English why you believe
the statement to be true, even if you can't explain it precisely enough
to qualify as a proof. Also, the very process of drawing a picture
or finding an example may lead you to understand better why the statement is true, and
perhaps give you an idea as to how to prove it.
- If you find that a proof is getting very complicated or very
frustrating, you should set it aside and work on a different problem
instead. The questions are intended to have relatively short proofs
(unlike the homework, which is intended to be much more
time-intensive), if you have the right approach and know what you are
doing; so if you find yourself getting nowhere then you are probably on
the wrong track. In which case it is often profitable to think about
another question for a while and come back to it later.
- In this course it is important to know the precise
definitions of terms; if you misquote a definition when using it in a
proof, you are likely to not be able to complete the proof correctly.
Many of the questions in the Java
Quiz will test your knowledge of various definitions; it might be
worth giving that quiz a try just to identify your strengths and