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A Mathematical Trivium (1991) [pdf]

physics.montana.edu

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Re: A Mathematical Trivium (1991) [pdf]

#11

Post by hervature »

lordnacho wrote: post #4

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 integra…

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.

Re: A Mathematical Trivium (1991) [pdf]

#14

Post by GPerson »

enriquto wrote: post #7

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*

Re: A Mathematical Trivium (1991) [pdf]

#15

Post by enriquto »

GPerson wrote: post #14
enriquto wrote: post #7

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.

Re: A Mathematical Trivium (1991) [pdf]

#16

Post by h_spacer »

I can't help but think his examples are symbolic manipulation and not actual mathematics.

These maybe of interest to physicists without access to a computer algebra system, but to actual mathematicians they are as meaningless today as being able to do your 12 times table in your head.

Not all of them mind you, the last 10 questions are still interesting, but again mainly to physicists.

Edit: which makes sense since he states the questions are a "mathematical minimum for a physics student", which is different to the minimum mathematics needed for a mathematics student.

Re: A Mathematical Trivium (1991) [pdf]

#17

Post by enriquto »

SquibblesRedux wrote: post #12

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.

Re: A Mathematical Trivium (1991) [pdf]

#18

Post by vmilner »

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.)

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