How to study for an analysis test

[nietzsche]

Not sure if this is the right forum for this but...

I'm not sure how to study for my upcoming analysis test. It's a first-year course that is supposed to introduce you to more rigorous ways of doing mathematics. But I'm really not sure how to study for the test. I feel like the course really requires the ability to think creatively and to look for the answers.

I sort of feel like I'm not cut out for the course, which I don't like to think. I feel like I can learn anything, but I'm lacking a lot of the intuition or something, so it takes me a long time to learn.

But aside from that, this test is coming up and I want to do well. I compared the homework problems to my prof's previous midterms and they are completely different. This worries me, and I don't really know what to expect. Any suggestions at all on how to study would be great.

(We've studied properties of numbers, functions, graphs, limits, continuous functions, intermediate value theorem, extreme value theorem, and i suppose we're expected to know a bit more and have a good foundation...)

 

[naele]

Consider looking in your school's library for Problems in Mathematical Analysis series by WJ Kaczor and MT Nowak. It's a series of problems with solutions at the level of a first year course in analysis. Well, there are some quite challenging problems, no doubt about it. But I would just recommend going through that and seeing how they solve things.

 

[rasmhop]

I don't have that much experience with testing yet so take my advice with a grain of salt.

But aside from that, this test is coming up and I want to do well. I compared the homework problems to my prof's previous midterms and they are completely different. This worries me, and I don't really know what to expect. Any suggestions at all on how to study would be great.

In my experience it's common for professors to give you homework that requires a bit of creativity and may require you to do some proofs of your own. On tests however most professors seem to stick to routine questions that simply tests whether the student knows the material and has studied the examples, rather than whether he is creative. Therefore to score well on a test you need two things:
1) A comprehensive knowledge of the subject and the ability to apply this knowledge to fairly simple questions, often resembling problems done in class, in the book or in previous homework. If you have really kept up with the course, done the required reading, thought about the examples you didn't really get and know the general idea of the proofs, then I don't think this will be that bad.
2) Good test-taking skills. This includes working under time-pressure, the ability to check answers and management of time.
My opinion on how to study would be to first read your book till you're fairly confident that you know the theory. Then go through old mid-terms. If any question is hard, then ask yourself why. Did you forget about an important theorem? (then go read up on that) Did the question contain some small trick? (If so identify it and understand it, because it will likely reappear) If you really plan to ace a mid-term in my opinion you should be able to look at a question and almost immediately see what strategy almost certain works. You may for instance see a problem and say "if I make the substitution g(x) = f(x)-x then it should follow from the mean-value theorem", "I bet I can bound function f(x) by ae^x for some constant a, and then the result will follow".

Also if this is an introduction to rigorous mathematics, then it's likely that you're not used to real mathematical literature. Keep in mind that such books are read differently. You need to think a lot, write down notes, try to predict how a proof is done, read stuff over and over, etc. I have read books in which I could use 4 hours reading a single page, but this often just means that the material is extremely dense. Just keep in mind that the 3 pages you're struggling with may contain as much content as 150 pages of your calculus book (depending on the book of course and what 3 pages you're comparing with).

Now I have heard horror-stories from other students about professors setting extremely hard/long exams/mid-terms. I don't have any personal experience here so can't give any advice except: remember that the other students are likely having just a hard a time as you and 50% may be a fine score (if you get the impression that they're not speak to a TA or professor about it (whatever is appropriate) and see if you can figure it out).

 

[nietzsche]

thanks for your replies