Tim Gowers (Fields Medalist) explains How to Answer Cambridge Tripos questions

Source:  Gowers blog, Apr 2012

Model answers have always seemed to me to be a bad idea in mathematics, because it is hard to learn how to think for yourself when you are given the answers to all the problems you tackle. So it might seem a bit odd that in this post I’m going to attempt to help people preparing for Part IA of the Cambridge Mathematical Tripos by providing some model answers.

However, my aim is not just to give the answers. Rather, I want to explain in as much detail as I can (without getting tedious) how I come up with the answers.

… what I’m hoping is that if I describe my thoughts as I tackle the questions, then I’ll transmit some general messages about exam technique and dealing with the rather characteristic kind of question that gets set in Cambridge. You will of course benefit far more from this if you do the questions first (or try hard to do them) before reading what I have to say about them.

3B. Define what it means for a function of a real variable to be differentable at x\in\mathbb{R}.

Absolutely no excuse for not being able to do this. The answer I would give is this.

A function f:\mathbb{R}\to\mathbb{R} is differentiable at x if \frac{f(x+h)-f(x)}h tends to a limit as h\to 0.

A couple of remarks about this. Note first that the question didn’t give a name to the function, so the first thing I did was remedy that by calling it f. That was obviously the right thing to do here (nobody is going to write phrases like “the number you get by dividing the difference between the value the function takes at x+h and the value it takes at x by h“) but in other questions it is easier to forget.

Secondly, I had various choices to make here. Should I write what I wrote, or should I have written something more like “there exists lsuch that \frac{f(x+h)-f(x)}h\to l as h\to 0“? And should I have explained what it means to converge to a limit?

In general, the answer to questions of the last type is that if the tripos question is about one part of the course, then you are free to use facts from earlier on in the course but not facts from later. Another principle to keep in mind is that if the fact or definition you assume leaves you with almost no work to do, then you may have assumed too much. (I say “may have” here because if you’re doing an early part of the question, it may be clear that it’s just a quick preliminary to be got out of the way.)

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