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c) Yep. This approximate equality comes from just the definition of the derivative, letting h be small (so we are close to our limit) and multiplying out by h. So this equation tells you approximately how Q changes given an small change in x. (*) If we are making a sizable change in x, this...

Re: HSC 2014 4U Marathon - Advanced Level Correct. You should have a crack at the second :).

Re: HSC 2014 4U Marathon - Advanced Level 1. Do there exist functions f and g defined on the real numbers such that: f(g(x))=x^2\textrm{ and } g(f(x))=x^3\textrm{ for all }x. 2. Do there exist functions f and g defined on the real numbers such that: f(g(x))=x^2\textrm{ and }...

Another way of thinking about such questions: Another way of seeing it is that the chain rule tells you that the derivative of an inverse is the inverse of the derivative. So, assuming Q is locally invertible, we can write t as a function of Q such that dt/dQ=1/(dQ/dt)=1/(kQ). This is...

Well obviously a computer can do it, or you could ask someone else as well. I meant that any method involving "graphing" requires work like Sy's to justify rigorously in exam conditions.

Well I knew what the graph looks like, my solution was similar to Sy's, and you need to do something like that to come up with the graph.

Care to elaborate on your own proof? What are you saying is equal to cos(x+75)? Perhaps I am missing something basic, but I do not see how this can be done by a simple rearrangement and graph.

ur boi bangs.

Yep, all good. Your solution is the same as mine, just worded differently.

-y^2 can be 3, eg y=2.

Still nfi what you are trying to do. x^2 + y^2 can be any multiple of 7, it don't see why it suffices to consider M=1 or M=7.

huh?

It's not clear to me what you are saying here...

Go to open days (ideal) / go for a walk around campus, check out facilities, talk to academics in your area and hopefully one of them "feels" right to you. Usyd was a much better fit for me in undergrad. (A BSc (Adv Maths) degree consisting mostly of maths, science and philosophy.) I had some...

Quadratic residues (modulo n) are just what squares can be modulo n. Eg. In this case (mod 7) we have 0^2=0 1^2=1 2^2=4 3^2=2 4^2=(-3)^2=2 5^2=(-2)^2=4 6^2=(-1)^2=1 (note the mirrored pattern due to the negatives, this will save you time in future questions.) So the quadratic...

Do you know what quadratic residues are?

Re: HSC 2014 4U Marathon - Advanced Level :). Right idea and pretty close to being done, but we can have a pair of the k's being -1 instead of all of them having to be 1.

Re: HSC 2014 4U Marathon - Advanced Level $Let $P(x)$ be a polynomial with integer coefficients.\\ \\ Is it possible to find three distinct integers $a,b,c$ with $P(a)=b$, $P(b)=c$ and $P(c)=a$ ?\\ \\ Justify your answer.$

Re: HSC 2014 4U Marathon - Advanced Level Haha that's a cute question. (Although I am not sure if you are are talking about a) the length of the block before the first copy of the repeating block, b) the length of the repeating block or c) the decimal at which the second copy of the...

Re: HSC 2014 4U Marathon - Advanced Level 1. I would have to cook up a different counterexample for a constraint like x^2+y^2=1 as we would then necessarily get a maximum. The claim is still not true though. But as I said before, I am not going to bother to provide any more counterexamples...