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Re: HSC 2016 4U Marathon Well my definition of algorithm time etc was very simplistic. There might be a provably optimal solution to the problem I posed, but I am certainly not asking anyone to obtain or prove an optimal algorithm. Just hoping the competitive aspect of "beating someone elses...

Re: HSC 2016 4U Marathon Perhaps more suitable for the advanced thread, but there are enough unanswered questions there for now: A friend of yours (who knows very little about maths) is building a calculator and has programmed in the following features: -It can display 20 digits on it's screen...

Re: MX2 2016 Integration Marathon The integral actually does converge because it decays as x^(-2) and does not blow up anywhere finite. Notice that sometimes we work with integrals that aren't convergent in the traditional sense though, like the sin(x)/x integral. when we talk about...

Wait till I knew the course well enough to be very sure of when I was right/wrong, then I did a bunch of past papers. Before then I mostly just sought out standalone hard / interesting / extracurricular problems and did them in class / at home instead of my classwork/homework. I thought a lot...

Re: MX2 2016 Integration Marathon So I guess a natural followup question then if it hasn't been done recently is to prove that: \int_0^\infty \frac{\sin{x}}{x}\, dx = \frac{\pi}{2} which I ask people to attempt separately (with MX2 appropriate methods).

Re: MX2 2016 Integration Marathon Key ingredients (in my solution at least): 1.\int_0^\infty \frac{\sin{x}}{x}\, dx=\frac{\pi}{2} You can assume this if you don't know how to prove it, but it has been done before in this and similar threads, and is within MX2 scope although slightly tricky...

1. Read the theory from the textbook, but read it actively. When you learn the definition of a new class of objects (eg complex numbers), or a new technique (integration by parts), play around with it a bit! See what you can do with these things that you couldn't before. Similarly, when you read...

Re: MX2 2016 Integration Marathon Bump.

A beautiful problem if one knows some real analysis: F is a smooth function defined on the interval [0,1] such that at every x in [0,1], a derivative of some order of F vanishes. (Using one-sided differentiation at the boundary.) Prove that F is a polynomial.

Re: MX2 2016 Integration Marathon Probably. I can't remember / didn't see it. Anyway, will leave it up as I am sure some students haven't done it.

Re: MX2 2016 Integration Marathon On the easy side, but: \lim_{R\rightarrow\infty}\int_{-R}^R \frac{\sin(ax)\sin(bx)}{x^2}\, dx \qquad (a,b\in\mathbb{R}).

Re: Extracurricular Integration Marathon \int_0^\infty x^3 e^{-x^2}\log(1-e^{-x^2})\, dx

Re: MX2 2016 Integration Marathon Rings a bell with me too lol.

Re: MX2 2016 Integration Marathon :p. Thanks for texing it though! I did get the same.

Re: MX2 2016 Integration Marathon Are you sure there is a "nice" closed form? (It is easy to express it as a sum of logs, with nice coefficients.)

Re: Extracurricular Integration Marathon Excellent solution to the integral :). One minor remark is that another way of proving the summation result used is taking the real part of the geometric series summation (common ratio z=a*cis(x)). I.e. find the real part of 1/(1-a*cis(x)), which is...

Re: Extracurricular Integration Marathon Nice :), it's a bit lengthier than you can do it using complex analysis but I like the fact that a high school student could understand it.

Re: Extracurricular Integration Marathon A=\int_0^\infty -\log(1-e^{-x})\, dx \\ \\ = \int_0^\infty \left(\sum_{j\geq 0} \frac{e^{-x(j+1)}}{j+1}\right)\, dx \\ \\ = \sum_{j\geq 0} \left(\frac{1}{j+1}\int_0^\infty e^{-x(j+1)}\, dx\right) \\ \\ = \sum_{j\geq 0}\frac{1}{(j+1)^2}\\ \\=\zeta(2)...

Re: Extracurricular Integration Marathon For 3, the swapping of order of integration is fine can be justified by your choice of the many variants of Fubini's/Tonelli's theorem. You don't need an especially powerful one because the function is smooth and absolutely integrable. For 4, I think...

Re: Extracurricular Integration Marathon One more before I wait for the ones already mentioned to be answered: 5. i) Find an expression for the volume of an n-dimensional ball of radius r. (Hint: This is like computing volumes of 3-d solids in MX2 by slices, but the cross sections of n-balls...