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Re: HSC 2013 4U Marathon For something like part i) of your most recent question, you would need to provide the students with a rigorous definition of integration from which to work ...as this is not assumed knowledge in 4U. You also need to state what kind of functions f you want this to be...

Re: HSC 2013 4U Marathon Proof that the statement is true for a=2, for all positive integers m. LHS-RHS=\frac{2^{m+1}-1}{m+1}-\frac{2^m-1}{m}=\frac{(m-1)2^m+1}{m(m+1)}>0.

Re: HSC 2013 4U Marathon It isn't.

Re: HSC 2013 4U Marathon I don't see how this is that surprising, PhD dropout rates are pretty significant...and many people who complete them don't always end up going into academia (the job market at the moment is a deterrent for one). "Cut out" in this context mostly refers to how strong...

Re: HSC 2013 4U Marathon Haha actually meant what was your approach to the problem? (Ignoring the calculation error). Cool though. The idea of teaching mathematics is very attractive to me if I decide at some stage down the road I am not cut out for academia.

Re: HSC 2013 4U Marathon Yeah I am not sure what Sy meant by that statement. Replacing -2/5 with a greater number removes the possibility of any real solutions existing, and replacing it with a smaller number leads to an infinitude of solutions as you said. This is (for example) a...

\infty.

Re: HSC 2013 4U Marathon Such an equation can have a finite number of solutions for the same reason that the equation x^2+y^2+z^2=0 has a finite number of solutions. The function on the LHS has a finite (in this case consisting of only one point) set of minima. This is the case here. Viewed...

Re: HSC 2013 4U Marathon But are you looking for real or complex solutions?

On grid paper, a lattice point is a point which is at the intersection of two perpendicular lines. Prove by induction or otherwise that a polygon drawn on grid paper with lattice point vertices has area given by: A = i + b/2 -1 where i is the number of lattice points interior to the polygon...

Re: HSC 2013 4U Marathon As no-one else seems to be answering it: P(z)=Q(z)(z^2+1)+(z^2-z+1)\\ \Rightarrow \sum_{n=1}^4 P(z_n)=\sum_{n=1}^4 z_n^2 -\sum_{n=1}^4 z_n +4\\=(\sum_{n=1}^4 z_n)^2-2\sum_{m<n}z_mz_n - \sum_{n=1}^4 z_n +4 = 6.

Re: HSC 2013 4U Marathon (and gives you the equality condition of all numbers being equal).

Re: HSC 2013 4U Marathon \displaystyle\prod_{k=2}^n \frac{k^2-1}{k^2}=\frac{\left(\displaystyle\prod_{k=2}^n (k-1)\right)\left(\displaystyle\prod_{k=2}^n (k+1)\right)}{(n!)^2}=\frac{n+1}{2n}>\frac{1}{2}.

Re: MX2 Integration Marathon Just inputting blah^{superscript}_{subscript} works fine.

BoS people should have some broodwar games.

+1. The number of stubborn and prideful high school maths teachers I have met is ridiculous. One major thing that makes science and in particular mathematics such an objective and pure intellectual pursuit is that there is no authority when it comes to establishing facts. One persons opinion...

Re: HSC 2013 4U Marathon Yep, its one of my favourite elementary applications of IBP, Taylor's theorem is very important in many fields of mathematics. The moral is that we can approximate smooth (or sufficiently differentiable) functions by polynomials, and the size of the error decays...

Re: HSC 2013 4U Marathon Here is a form of Taylor's theorem that can be easily proven by MX2 methods: $Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a function that is smooth (can be differentiated any number of times). Prove that:\\ \\ $f(x)=\sum_{k=0}^n \frac{x^kf^{(k)}(0)}{k!}+\int_0^x...

Re: HSC 2013 4U Marathon Yep, by proving that all partial sums taking more terms are greater/less than our partial sum, do this by pairing adjacent terms and noting that the sequence is decreasing in size. So you need series approximations with bounds on error. For e, we have...

Re: HSC 2013 4U Marathon Regarding the approximation of pi by an alternating series: Note that the series zig-zags by progressively smaller amounts. In fact we have every partial sum with an odd number of terms as an upper bound and every partial sum with an even number of terms as a lower...