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Re: MX2 2015 Integration Marathon For any who liked the BoS trials question about the trapezoidal rule, the linked article explores those kind of approximations more. It's an interesting application of HSC integration theory and not enough teachers talk about these things imo...

The last question in this years BoS trials involved a closer look at the trapezoidal rule than we usually take. The below article of mine further explores this topic, and has a few exercises that are good practice for the interested HSC student. Nothing outside MX2 is used (except the...

Re: 2015 permutation X2 marathon Of course I don't mind :), everyone has their own ways of seeing things and it is a quite trivial matter to pass between the two in this case. There are several differing conventions for spherical coordinates even without the modification to preserve area...

Nope, the upper bound is not close to 4\pi. Its logic based (in that it is not lateral), but I wouldn't necessarily call it simple. I didn't use multiple variables, but there are probably lots of ways to think about it and phrase your working. It isn't hugely tedious but it is not trivial...

Yeah do it. Might be a while before full paper is uploaded.

Re: 2015 permutation X2 marathon No, I am mapping [0,1] to [0,pi], the full sphere (hence the normalisation by 4pi rather than 2pi). Your new formula is equivalent to mine, (as one might guess from their similar forms), you are just basing yours at the origin as opposed to mine which was...

Re: 2015 permutation X2 marathon Nope, for this particular question we don't need calculus, although calculus will certainly generalise better. The same method you proposed should work for the sphere should it not? $Let $\phi$ be the angle of latitude measured from the north pole at $\phi=0$...

It's clear that the sequence is increasing because u_{k+1}-u_k=u_k^2 \geq 0. This means that u_k\geq u_1=1/3 for all k. Hence \frac{T_{k+1}}{T_k}= \frac{1+u_k}{1+u_{k+1}}=\frac{u_{k+1}}{(1+u_{k+1})u_k}\leq \frac{1}{1+u_{k+1}}\leq \frac{3}{4}. So each term in our positive series S=\sum_k T_k...

Strictly speaking, one would have to justify the above with a proof that the sum is a convergent one, but I doubt this would be required in the HSC. This could be done (for example) by comparison to a geometric series.

S=\sum_{k=1}^\infty \frac{u_k}{u_{k+1}}=\sum_{k=1}^\infty\frac{u_{k+1}-u_k}{u_ku_{k+1}}=\sum_{k=1}^\infty \left(\frac{1}{u_k}-\frac{1}{u_{k+1}}\right)=u_1^{-1}=3.

Re: 2015 permutation X2 marathon (And quantities like dx are understood rigorously as things called differential forms.)

Re: 2015 permutation X2 marathon Judging by the way you have worded things you would like a way to randomly choose a point inside the unit disk given that you have a way of randomly choosing a points inside any given interval. (Ignoring the method of randomly choosing points inside the...

It has been ages since I solved this, and I did not write it myself so I cannot be 100% sure. But I believe: 1. It can only see the current position and direction of swimming, but it is also intelligent enough to plan ahead and anticipate the possible plans of the man. 2. Yes it can run as...

Re: HSC 2015 4U Marathon Haven't seen mechanics questions here for a while. $Suppose a particle moves in simple harmonic motion with amplitude $A$ and angular frequency $\omega$.\\ \\ Find a function $F(x)$ such that $\int_a^b F(x)\, dx$ is equal to the proportion of time that the particle...

I am fairly sure the lack of time is the dominant factor (and a relatively small number of olympians ever sitting it).

Re: MX2 2015 Integration Marathon That's because when you "learn" integrals in HS you do not do so rigorously. You are just shown a symbol for it, given a loose handwavy definition in terms of approximation by a finite number of rectangles (for what kinds of functions does this thing converge...

Re: MX2 2015 Integration Marathon Sure, but we are not integrating sin, we are integrating the log of sin. It is logical and indeed true that this does not change anything, but as we cannot explicitly compute these two integrals by antidifferentiation, a proof has to exploit properties of the...

Re: MX2 2015 Integration Marathon This is a VERY bad attitude to have in maths, there are so many counterintuitive facts and so many plausible statements that are decidedly false. You should always ask yourself the question "why?" about statements you make, and you should be able to answer...

Re: MX2 2015 Integration Marathon But why is the area under sin(x) between 0 and pi/2 the same as that between pi/2 and pi? This transformation is exactly how you would prove this rigorously (*). (*) At least as rigorously as possible in high school. A more subtle point here is that area is...

Re: MX2 2015 Integration Marathon No worries. Remember the pattern, because of log laws you can often exploit the x -> a-x substitution in ways like this.