A Mathematical Trivium (1991) [pdf]
physics.montana.eduVery enjoyable! I started working a few problems in my head, and realized I'd run out of scratch ram quickly.
I especially liked his points on examination, what we call the "orals" in Physics. There, the (in his words, defenseless) student is up in front of a board, while professors throw problems at them. I had a few good questions on mine, and one which was ... poorly specified. I remember thinking I simply had to crank on that problem, showing my thinking processes to try to answer the question. At the end, the prof nodded, pointed to something before my conclusion, and said "that was as far as I got."
I remember feeling relieved yet angry. Just smiled, nodded, thanked him for the question, and moved on.
Talk about an imbalanced power dynamic.
Seems pretty reasonable to me. If you look at these questions and learn how to do them, you'll have learned a fair bit. It's not like you can learn by rote all the transformations you'll need to answer these, you're better off just learning the theory.
I would have loved to have a simple 100 known-but-non-trivial questions like this. You avoid the lottery of having to remember some particular detail (say some integral that appears in some derivation that an adhoc question might contain), but you don't avoid having to actually know how the theory works, because it's a little bit too hard to memorize.
Admittedly you might still get stuck on a trivial step but at least you've had a chance to go over the questions, and it might be relatively fresh.
I think these questions are meant to really address mechanical skills. The second question took me 7 differentiations. It takes understanding to be able to do that but the rate of error is going to be very high even for those that understand differentiation perfectly.
Edit: someone posted a link showing that one can generalize this result for g, f and their inverses. Interestingly, after reading up on it, the point Arnold is trying to make is that modern students have a poor grasp of infinitesimals.
I like the (possibly, older) style of the integral sign that appears in this paper better than how it is usually typeset these days.
I've heard these called the German or Russian style integral sign. I don't know the actual history. I have seen this question [1] on the tex/latex stackexchange about making this integral sign.
I would expect a journal to not use a different font integral sign, however. Or rather, I would expect the "house style" to include font choices for integral signs as well. I'm not aware of any journal that uses this type of integral now, but I suppose I should also note that I mostly read papers from the arxiv now anyway.
[1]: https://tex.stackexchange.com/questions/170028/integral-sign...
The limit in question 2 is wickedly difficult to solve by traditional means. You have to apply l'Hôpital rule about seven times, and it becomes a monster formula. Or, you expand everything by Taylor up to order eight. In any case, the computation fills several pages. There must surely be a geometrical reasoning to compute that limit.
You can at least guess the answer must be 1 instantly based on the idea that tan(x) and sin(x) equal x up to first order, and so therefore their inverses do too.
(So the limit is essentially (x - x)/(x - x) )
To make this rigorous do as you say about expanding the series.
Edit: Up to beyond first order actually*
> the limit is essentially (x - x)/(x - x)
I agree with the intuition, but this is not so simple. A limit such as yours can still be anything, can't it? It depends on the higher-order terms. For example, the limit ((x+5x^2)-(x+2x^2))/((x+7x^2)-(x+6x^2)) is also "essentially" (x-x)/(x-x), but it turns out to be 3, not 1. In the given example, there are cancellations of even higher order terms, that you have to check that they cancel correctly. Following the links on the answer by admissionguy I found two nice arguments for the computation, an algebraic and a geometric one.
Curious how many of these can be tackled directly by a modern symbolic math package like Mathematica.
Question 2 in particular, I'd be curious if it can get there without guidance.
[EDIT]: Answer is yes: https://www.wolframalpha.com/input/?i=Limit%5B%28Sin%5BTan%5...
For anyone curious about the mean of the 100th power of the sin function: https://math.stackexchange.com/questions/24533/find-the-aver...
This one is a very beautiful question (it is called Wallis integral in other contexts). The computation exposed by Michael Lugo in that link is probably what Arnold intended, since it can almost be computed entirely in your head. The idea is that cos(x)^100 has almost exactly the same graph as a gaussian function e^(-50x^2), for which you know the integral.
What percentage of MIT grads could answer even half of these. Pretty hardcore questions it seems.
Is question 1 just asking for the student to sketch the derivative and integral curves of an arbitrary curve sketched by the teacher?
It seems a bit different to the other questions (but is question 1, so might be an "easy" starter.)
Trivium - Throes of Perdition