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(a) \\(\cos x + i\sin x)^5 = \cos^5 x -10\cos^3x \sin^2x+5\cos x \sin^4 x + i(5\cos^4 x \sin x - 10 \cos^2 x \sin^3 x + \sin^5 x) \\ \\ $Therefore$ \ \cos(5x) = \cos^5 x - 10\cos^3x \sin^2x + 5\cos x \sin^4x \ \ \ \ (1) \\ \sin(5x) = 5\cos^4x \sin x - 10\cos^2x \sin^3 x + \sin^5x \ \ \ \ \...

Re: HSC 2014 4U Marathon Just did the question, that is what I got

Re: HSC 2014 4U Marathon \\ $Define the notation$ \ f : \mathbb{R} \rightarrow \mathbb{R} \ $to mean that the function$ \ f \ $has inputs in the real numbers and outputs in the real numbers$ $Note$ \ \mathbb{C} = \ $complex numbers$ \\ $Also define a function$ \ f \ $ to have property$ \ S...

Oh ok cool Thanks!

Some fixed n > 2 (except for trivial cases like f(x) = 0 ) EDIT: I think you only need functions like f^(n) = f for prime n since if you have a function g with g^(3) = g and f^(2) = f, then h = f + g would be such that h^(3*2) = h

Are there any elementary functions with its 3rd, 4th etc derivative is itself? (i.e. f''''=f) If so how would we find these functions? (except for f(x) = 0)

Alternatively for the original limit as x goes to zero, we can see that \frac{x}{e^x-1} < 1 \\ $Since$ \ e^x > x+1 \ $proven by calculus or otherwise$ \\ \\ $And for small enough positive$ \ x , \ \ \ \ e^x < x+2 \ \ \ $proven with calculus or otherwise$ \\ \\ $Hence for small enough$ \ x ...

As for a rigorous proof, we note that \\ $For large enough$ \ x \ $note that$ \\ \\ 0 < \frac{x}{e^x-1} < \frac{1}{x} \\ \\ $The upper bound can be proven with calculus, as with the lower bound$ \\ \\ $Hence by taking$ \ x \ $to infinity$ \ 0 < \lim_{x \to \infty} \frac{x}{e^x-1} < \lim_{x \to...

Re: HSC 2014 4U Marathon I stole this question from a math tutorial on usyd, it had that restriction. I'm guessing because if b=0 then the function is a straight line and the only case for its own inverse is the trivial y=x

As x goes to infinity, e^x dominates the x on top, in heuristic terms e^x goes to infinity "faster" than x

Re: HSC 2014 4U Marathon \\ $For what values of$ \ f(x) = \frac{x-a}{bx-c} \ $constants$ \ a,b,c | b \neq 0 $Is the function its own inverse?$

$Consider the differentiation of$ \ f(x) = e^x \ $by first principles$ \\ \\ (e^x)' = \lim_{h\to 0} \frac{e^{x+h}-e^x}{h} = e^x \lim_{h \to 0} \frac{e^h-1}{h} \\ \\ (e^x)'=e^x \\ \\ $therefore$ \ e^x\lim_{h \to 0} \frac{e^{h}-1}{h} = e^x \ \rightarrow \ \lim_{h \to 0} \frac{e^h-1}{h} = 1 \\ \\...

Re: HSC 2014 4U Marathon \\ $Solve$ \ x^3-3x-1=0

Re: HSC 2014 4U Marathon \\ $Prove using complex numbers that for real$ \ x,y,u,v \\ \\ \sqrt{(x^2+u^2)(y^2+v^2)} \geq xu+yv

Re: MX2 Integration Marathon I think this integral has been posted at least 5 times lol

I think Conics should be in the 3U course being learnt in Y12 as an extension of Parametric Representation. It should replace the small topics in 3U that currently make up 1-2 marks each in the exams at the moment. (i.e. Simple trigonometric identities, Ratios, Angle between lines, Polynomial...

Re: MX2 Integration Marathon Well I mean't that it was cool, not sure if its difficult haven't tried it yet (probably is difficult considering the other questions you post lol)

Re: MX2 Integration Marathon \int e^{e^x+x} \ dx

Re: MX2 Integration Marathon \int \frac{1}{x^n - x} \ dx \ \ \ (n>1)

Re: MX2 Integration Marathon \int \sqrt{\csc x - \sin x} \ dx