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Re: HSC 2016 4U Marathon A short question to test student understanding of differentiability: Suppose the function f is differentiable to all orders at the real number a. Explain why E(h):=\frac{f(a+h)-f(a)}{h}-f'(a)\rightarrow 0 and \tilde{E}(h):=\frac{f(a+h)-f(a-h)}{2h}- f'(a)\rightarrow 0...

Re: HSC 2016 4U Marathon Yep. What Integrand said. All you can deduce about the geometry of the set of four roots is that this set is closed under the map z->1/z. (In addition of course to being closed under conjugation, if our coefficients are real.)

Re: MX2 2016 Integration Marathon Cool, I will post my derivation of the expression for g(x) after my supervisor meeting tomorrow. Would like to take the time to explain it clearly. The main idea though: Check out the equation relating g(x) and x. It is quartic in g(x) but only quadratic in...

Re: MX2 2016 Integration Marathon Had a quick crack at the integral with the nested square roots in my break. So the main difficulty in the problem is handling the integrand, it is not at all clear that this formally written object will be a well defined function, and on which domain it will...

i)$\\ \\ $\phi(x):=\int_{\mathbb{R}^2}G(x,y)S(y)\, dy$\\ \\ solves this inhomogeneous problem, just from the definition of a Green's function.\\ \\ ii)\\ \\ We now need to construct a Green's function for the half-plane, which can be done via a reflection trick.\\ \\ We define \\ \\...

By symmetry about the y-axis, it suffices to show the circle just passes through S. Since AB is the diameter of our circle, it suffices to show ASB is a right angle, which you can do just by multiplying gradients of AS and BS. This does it v.quickly.

You really don't need one for biomath. For metric spaces, something like Searcoid's book, supplemented by Munkres' Topology for anything not in Seacoid / for a more abstract viewpoint. For Galois theory maybe Stewart's book is a good choice. If you want something broader/higher level you can...

Re: HSC 2016 4U Marathon - Advanced Level Which book/guy? And what exactly do you mean by saying that the concavity is towards the x-axis? That in a small neigbourhood of the root the second derivative is non-positive? That the second derivative is non-positive on the whole real line? (Note...

Re: HSC 2016 4U Marathon - Advanced Level What was the source of this question?

Re: HSC 2016 4U Marathon - Advanced Level Might as well post a solution to the conics question as it has been a while. Consider the rectangular hyperbola (ct,c/t) parametrised by t in the set of nonzero reals. Let P,Q,R be three points on this hyperbola with parameters p,q,r respectively...

The boredofstudies LaTeX formatting is different from traditional LaTeX, don't use the formatting on this site as guidelines. Instead, just look up LaTeX tutorials, there are many.

Re: HSC 2016 4U Marathon - Advanced Level The claim is true (and not particularly hard to prove), you must be doing something wrong in Geogebra.

Sure, they all do pretty much exactly the same thing. The power of LaTeX is in the code itself, frontends don't really matter.

Lol no, unsurprisingly the app in your screenshot is called TeXShop. But yes, yes you can. There are tons of possible editors. You could even use something notepad-esque if you really wanted to. (Not that that would be a good idea).

Open what file? mactex is just a backend. Download an editor like texmaker and work in that.

Which exact link did you download? For Windows it should really just be an executable. This executable will download the backend (all the stuff that interprets LaTeX and produces output), and often a frontend (the thing you use to write your documents, often with some tools to make it more...

A good chance to show what some integration by parts can achieve! $Suppose $f\in\mathcal{C}^{n+1}$, that is $f$ has continuous derivatives up to order $n+1$. Then:\\ \\ $f(x)=f(a)+\int_a^x f'(t)\,dt\\ \\ =f(a)+\int_a^x (-1)^n\frac{d^n}{dt^n}\left(\frac{(x-t)^n}{n!}\right)f'(t)\,dt\\ \\ =...

Re: HSC 2016 MX2 Combinatorics Marathon Of course, such solutions are often far nicer. The splitting into 4 subsets/teams was actually the first solution I spotted, I just wrote the other one as I figured it would take less writing to justify :).

Re: HSC 2016 MX2 Combinatorics Marathon It's not that tedious. 4^n=(1+3)^n=\sum_{r=0}^n \binom{n}{r}3^{n-r}\\ \\ = \sum_{r=0}^n \binom{n}{r} (1+2)^{n-r}\\ \\ =\sum_{r=0}^n\left(\binom{n}{r}\sum_{s=0}^{n-r}\binom{n-r}{s}2^{n-r-s}\right)

Re: HSC 2016 4U Marathon - Advanced Level Ah yes, this does it. :) What I must have been thinking of was the problem of computing the degree of of the interpolating polynomial for a given collection of x_i, y_i. This is equal to the rank of a certain matrix from memory, and I do not know if...