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Yes. The function with domain {0} and codomain {e}. For something less silly, I will assume you are asking whether or not there exists a bijection from the transcendentals to the reals. This is also true. In fact we can replace the transcendentals with any co-countable set S. Picture the...

Less often than you might think. A pretty large proportion of my ex-students had not heard that name, and just knew that "differentiation and integration were inverse to each other."

I assume so, am not familiar with the exact process. Yes, the question about the irrationality of pi was provided by a lecturer from UNSW. (p brown) (I think the original proof was just a short article written by someone else, but the lecturer parsed it into a form suitable for an MX2...

Well that's the same with the intermediate value theorem and the fundamental theorem of calculus. Theres no point formalising things and naming them properly if students aren't doing rigorous analysis. (Which they cannot even begin to do without proper construction of the reals and discussion of...

I reckon it's fairly quickly justifiable in some applications, MX2 students know how to squeeze things and use the fact that the integral of something non-negative is non-negative. From here it suffices to find upper and lower bounds for difference quotients which definitely wouldn't be a...

Yeah, I am sure it has come up in trials, maybe not recent HSC then. In any case, it is definitely a trick a good mx2 student should have in his/her toolbox.

Re: HSC 2016 4U Marathon - Advanced Level Easier than my unanswered summation problem: Consider the solid of revolution S formed by rotating the disk (x-R)^2+y^2\leq r^2 \quad (r < R) about the y-axis. a) Find it's volume by standard MX2 methods. b) Show that this volume is also equal to Ad...

Will have a look when I have time if you really want, don't have any high school books with me and don't have time to look through past papers right now.

Idk, some series/integrals have to be evaluated in super counterintuitive ways, but I wouldn't consider this to be one of them. It has a pretty clear visual similarity to the geometric series, and differentiation of the geometric series is used plenty of times in MX2 questions. Eg evaluating...

I just thought I'd ask because you said you can't evaluate the second one by "simple things alone". Surely the geometric series and knowing how to differentiate rational functions are pretty simple things!

Just differentiate the geometric series a few times. Or was that what you were referring to by "famous proof"?

\frac{-1}{1}=\frac{1}{-1}\\ \\ \Rightarrow \sqrt{\frac{-1}{1}}=\sqrt{\frac{1}{-1}}\\ \\ \Rightarrow \frac{\sqrt{-1}}{\sqrt{1}}=\frac{\sqrt{1}}{\sqrt{-1}}\\ \\ \Rightarrow \frac{i}{1}=\frac{1}{i}\\ \\ \Rightarrow i^2=1\\ \\ \Rightarrow -1=1.

It's just a name. Lots of old terminology like that is ambiguous and different mathematicians mean different things by it. I don't hear that particular name much in modern mathematics so I doubt you would get universal agreement on what objects to use that name for.

Re: HSC 2016 4U Marathon - Advanced Level A little sum I whipped up. For a fixed positive integer n, evaluate...

For anyone that doesn't remember, the integral referred to in my above post was \int_{-\infty}^\infty \frac{\sin(ax)\sin(bx)}{x^2}\, dx where a,b\in\mathbb{R}.

So I did have another think about it and couldn't come up with a much nicer way. I wrote out my proof though, so perhaps that will be of some help to people who want to understand it more. Remark: You might expect with an answer as simple as b(n-a)(-1)^{a+b}, there must be a much quicker way...

Where did you find this sum in the first place? That should provide a good clue as to the cleanest known way to compute it. Anyway, if you find a satisfactory explanation for why it works out the way it does, please post it here. I will probably have another think about it this evening to see...

Re: HSC 2016 4U Marathon A game of chance uses a simple harmonic oscillator. Its amplitude of oscillation is 1 but its frequency is unknown to you. To play the game, you have to pay $1, and you get to choose a finite number of closed subintervals of [-1,1]. You win if after a random large...

A pretty ghetto way of doing it: For a point (x,y,z) on the line with parameter t, we have: 3x-3y-z=6t-3t-3t=0. But the set of ALL (x,y,z) such that 3x-3y-z=c is a plane parallel to the given plane for any constant c (*). (Indeed that's why we evaluated this particular linear combination.)...

Re: Mistakes from Text Book Thread You realise how much of a pain this will be for people to actually read through if it get big and a student is usure if his book is correct? I highly recommend someone just forms a central google doc for some sort of collaborative BoS errata for HS textbooks...