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Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread $\noindent Since the derivative is 0 for all $x>0$, the function is some constant for $x >0$. Sub. in a convenient value of $x$ ($x=1$) to find this constant. Do $x=-1$ to find the constant value for negative $x$.$

Re: MX2 2016 Integration Marathon $\noindent The answer is $\ln ^{n+1} |x|+C$ $\Big{(}$ where by this notation we mean $\ln \Big{|}\ln \left |\ln \left| \cdots \ln |x| \right |\cdots \right | \Big{|}$, where the $\ln$ appears $n+1$ times $\Big{)}$. This can be confirmed by differentiation and...

Re: HSC 2016 4U Marathon $\noindent Values of $a$. E.g. If $a= 10$, the equation becomes $10^x = x$, which has no real solutions.$

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread Yeah, definitely. You can do it with your original substitution too, it's just more tedious when simplifying it. Maybe you made an algebraic mistake when doing it, so you might want to post your working.

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread $\noindent That formula is found on the standard integrals sheet of HSC papers $\Big{(}$written in the form $\int \frac{1}{\sqrt{a^2-x^2}}\text{ d}x =\sin ^{-1}\left(\frac{x}{a}\right)\Big{)}$. It should be in the Year 12 3U...

Re: MX2 2016 Integration Marathon $\noindent What do you mean? You mean for something like this?: $\mathrm{e}^{\ln x}$.$ $\noindent Edit: oh right, you mean raising logs to powers.$

Re: MX2 2016 Integration Marathon $\noindent Should there be an $x$ on the denominator? Otherwise, there's no elementary solution I believe. Also, it'd probably be better to put the nesting index as a superscript, since subscripts already have a meaning for logs (log base). Doesn't matter much...

Re: HSC 2016 4U Marathon $\noindent \textbf{NEW QUESTION}$ $\noindent (i) Let $a>1$. Find the value of $a$ for which the graph of $y=a^x$ is tangent to the graph of $y=x$.$ $\noindent (ii) Hence, deduce the set of all values of $a>1$ for which the equation $a^x = x$ has a real solution.$

You get to be on the front cover of the HSC Success One books. Also for topping HSC 4U Maths, there's a T.G. Room medal prize from UNSW (also has $500 prizemoney, as can be read here: https://www.maths.unsw.edu.au/news/2014-06/talented-students-day-unsw ).

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread $\noindent \textbf{Note.} We used the result that $\frac{\mathrm{d}}{\mathrm{d}u}\left(\sin ^{-1} \left(\frac{u}{a} \right) \right)=\frac{1}{\sqrt{a^2 - u^2}}$ (where $a>0$ is a constant). This result is a standard HSC 3U...

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread $\noindent Let $y=\sin^{-1} \left(\frac{2x-3}{4} \right)$. We want to find $\frac{\mathrm{d} y}{\mathrm{d} x}$. Let $u=2x-3$, so $\frac{\mathrm{d} u}{\mathrm{d} x}=2$. Also, $y=\sin^{-1} \left(\frac{u}{4}\right)\Rightarrow...

Yeah 2U maths is almost always 100 raw for 1st in state. No-one's gotten 100 raw in the HSC 4U paper since 1993 iirc (not sure about this year).

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread This'll work too, it's just a bit more algebraically tedious when simplifying it.

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread $\noindent To get my result, it's better to let $u=2x-3$ instead. :-)$

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread $We have$ $ $\begin{align*}\frac{\mathrm{d}}{\mathrm{d}x} \left( \sin^{-1} \left(\frac{2x-3}{4}\right) \right)&= \frac{1}{\sqrt{16-\left(2x-3\right)^2}}\times 2, \end{align*} $\noindent using the chain rule and the rule for...

Your raw internal mark is not what is counted (for any subject*); it's the moderated assessment mark that counts. * Except possibly really-low candidature subjects, e.g. those with fewer than 10 people in the state doing them. I'm not sure about those subjects.

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread $\noindent Your first derivative should be something like $\tan^{-1} x$, because the second derivative needs to be $\frac{1}{1+x^2}$, and this is the derivative of $\tan^{-1} {x}$.$

UOW is University of Wollongong.

Re: MX2 2016 Integration Marathon $\noindent What about limits like $\frac{\pi}{4}$ and $\frac{\pi}{4}+\frac{\pi}{2}$?$

Re: MX2 2016 Integration Marathon Actually, it is often the case for HSC integrals. Rarely though for the harder ones (which equally rarely appear in the HSC it seems!).