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Re: Extracurricular Integration Marathon There has to be a less potentially dodgy way of doing it, I also thought of expanding out a series in e^(ix) to start with, but I couldn't see any way of justifying that so I left it. Gotta be careful with these things, because functions/sequences...

Re: Extracurricular Integration Marathon The imaginary part of the integrand is the original integrand, which is \frac{e^{\cos(x)}\sin(\sin(x))}{x}. This is not absolutely convergent. We have something continuous and periodic divided by x. Upon taking absolute values, integrating this is like...

Re: Extracurricular Integration Marathon I don't think that integral is absolutely convergent...there might be a nice way of justifying that interchange, but its not a trivial matter.

Re: HSC 2016 3U Marathon By order, I mean the "matching order" referred to in the question. Congruence of polygons means you can rotate, translate, and flip one till it is "on top" of the other, and in doing so you get a pairing between vertices of the first polygon and the vertices of the...

Re: HSC 2016 3U Marathon Also you have cut off the text so idk, but the question itself might even be asking you to state the vertices in the order of correspondence.

Re: HSC 2016 3U Marathon I don't know how strict the HSC is on such things, but it is generally a bad idea to state congruences and similarities without listing the vertices in order of their correspondence (which is not the case with your statement, the angles are not in the same order)...

Re: Multivariable Calculus Another exercise without much required knowledge. Suppose we have a smooth function X:\mathbb{R}^n\rightarrow \mathbb{R}^n. Suppose further, that given any starting point x\in\mathbb{R}^n, we can find a smooth curve c(t) with c(0)=x and c'(t)=X(c(t)). (Intuitively...

I agree!

1. g8 (N) b5 2. Ne7 Kxb4 3. Nc6 double check and mate. (all moves in this sequence are forced too) I don't think you can do it in three with a queen promotion, so forcing the king into a double check in this manner seems to be the only solution.

Re: Multivariable Calculus In the integrand we are applying g (a function of a single variable) to the expression v+2t. Partially differentiating g(v+2t) with respect to v and partially differentiating with respect to t just differ by a factor of 2, as computed using the chain rule. (Write...

I would be very wary of quoting results without proof (apart from essential and very basic stuff like the quadratic formula or exact trig ratios etc). Use your common sense, but I would urge you to err on the side of caution.

Emphasise correctness over speed until you are near perfect. Be super critical of your working and of your proofs, think of every possible place where things could go wrong and be painstakingly thorough in ensuring that they don't. Obviously this style of solving problems will be very slow to...

Re: HSC 2016 MX2 Combinatorics Marathon Note: Some of the possible sums that can arise in the above problem might be very unwieldy, making it difficult to compute exact probabilities for a given finite n. You can leave such expressions in this messy form if you like, and move straight to...

Re: HSC 2016 MX2 Combinatorics Marathon Here is a fun one to think about. Please refrain from answering if you have seen this question before in some guise. Shadowdude has n first dates with n of his crushes over the course of a day. He would like to ask one of them out to a second date, and...

Lol, it always catches me off guard when people edit the things they quote.

If you need to estimate the integrand anyway, I would just go ahead and use the Taylor bounds you mentioned. F(x)=\frac{1}{x}\int_x^{4x} \cos(1/u)\, du\Rightarrow \\ \\ 3=\frac{1}{x}\int_x^{4x}\, du\geq F(x)\geq \frac{1}{x}\int_x^{4x}(1-\frac{1}{2u^2})\, du = 3-\frac{3}{8x^2}\\ \\ \Rightarrow...

Re: HSC 2016 4U Marathon Suppose you have never heard of exponentials or logarithms before. Define for positive reals x, the function L(x):=\int_1^x \frac{dt}{t}. Prove: a) L(1)=0 L(xy)=L(x)+L(y) L(x^r)=rL(x) for rational r L is differentiable in its domain and L'(1)=1. b) Prove...

pq=(p+q)/2 \geq \sqrt{pq} \Rightarrow pq \geq 1$ with equality iff $p=q. $So $4/pq \leq 4$ with equality iff $p=q.

Re: First Year Uni Calculus Marathon $A series $S_n:=\sum_{k=1}^n x_k$ converges as $n\rightarrow \infty$ iff for every $\epsilon >0$ there exists a $N>0$ such that for every $m\geq n\geq N$ we have $|S_m-S_n|<\epsilon.$ (You can prove this if you like, but you don't have to. It just comes from...

Re: HSC 2016 4U Marathon (In this question you need to know how tennis scoring works.) In a basic model of a tennis match, every point is won by the server with probability p and the receiver with probability 1-p. a) What is the probability of winning a service game in terms of p? b) Suppose...