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Re: HSC 2016 2U Marathon $\noindent Actually, it still works if we blindly apply our familiar log and algebra laws.$

Re: HSC 2016 4U Marathon Is this an unsolved problem (i.e. there may be faster ways than the currently fastest known way?)?

$\noindent So the takeaway message from this theorem is that if we are faced with a limit of a quotient like in Shuuya's original question, and we know some functions that are asymptotically equivalent to the numerator and to the denominator, when taking the limit, we can simply replace the...

$\noindent Basically, we have the following general result, which we can prove using the limit laws.$ $\noindent \textbf{Theorem.} Suppose $f(x) \sim F (x)$ and $g(x) \sim G (x)$ as $x\to a $ ($a$ can be a real number or $\pm \infty$, or even a one-sided limit) (see the note on notation at the...

You should show working out haha. But with the quick method above, you can at least immediately see what the answer has to be, so you essentially know what kinds of manipulations to use.

$\noindent Here's how they want us to formally show it in the HSC:$ $$\begin{align*}\lim_{x\to 0}\frac{\sin 2x}{\tan 3x} &= \lim_{x\to 0}\frac{\sin 2x}{2x} \cdot \frac{3x}{\tan 3x}\cdot \frac{2x}{3x} \quad (\text{manipulating the limitand}) \\ &=\lim_{x\to 0}\frac{\sin 2x}{2x} \cdot...

It's just 2/3. The quick way is to see that sin(2x) is "like" 2x and tan(3x) is "like" 3x when x is close to 0.

Re: HSC 2016 2U Marathon $\noindent Yeah, it's basically this rule: $\frac{\mathrm{d}}{\mathrm{d}x}\left(\ln y \right) = \frac{y^\prime}{y}=\frac{f^\prime (x)}{f(x)}$, where $y=f(x)$. This comes from the chain rule.$

Re: HSC 2016 2U Marathon $\noindent Not sure what you mean exactly. We don't say $\ln y = \ln x$ (since that'd be true if and only if $y=x$). Rather, we do $\ln y = \ln \left( f(x)\right)$, where $y=f(x)$.$

Re: HSC 2016 4U Marathon $\noindent Let $R$ be a real number and $n$ be a positive integer.$ $\noindent (i) Suppose $R<0$, so $-R>0$. For any integer $k$, let $\omega_k := \sqrt [n]{-R}\,\mathrm{c\textit{i}s}\left(\frac{(2k+1)\pi}{n} \right)$, so that $\omega_k$ is an $n^{\text{th}}$ root of...

Re: HSC 2016 2U Marathon $\noindent Correct!$

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread $\noindent The shaded region below the $x$-axis becomes a cone when rotated, and the volume of this is $V_\text{cone} =\frac{1}{3}\pi r^2 h$, where $r = \frac{\sqrt{3}}{2} -\frac{\pi}{6}$ (the $x$-intercept of the tangent line)...

Re: MX2 2016 Integration Marathon $\noindent In reality, it does converge. Maybe try actual values for $a$ and $b$ if you use WolframAlpha, because maybe it can't do it for generic $a,b$.$

Re: HSC 2016 2U Marathon $\noindent For any curious students, this technique is known as \textit{logarithmic differentiation}. It's especially useful if the base is a function of $x$. Here's an example of a question where logarithmic differentiation can be helpful.$ $\noindent \textbf{ANOTHER...

Re: HSC 2016 2U Marathon And his original question was to find the second derivative. :)

Re: HSC 2016 2U Marathon Guessing he (Drsoccerball) meant it to be a product rule and typo'ed, since there's not much point just randomly putting a factor next to it for no reason, as it'd just be "carried along for the ride", so to speak.

Re: HSC 2016 2U Marathon $\noindent To differentiate something like $\pi ^{\sin ^{n} {x}}$ with respect to $x$, we would use the chain rule. Try letting $u =\sin ^n x$ (and remember that this is just $\left(\sin x \right)^n$ when differentiating $u$).$

Re: HSC 2016 2U Marathon Here are the ones about averages, mostly unanswered: . Here's Drsoccerball's one about finding a derivative: .

Re: MX2 2016 Integration Marathon Yeah that second integral can't be found using elementary functions (requires a Fresnel Integral). The C in the WolframAlpha output refers to a Fresnel Integral (a non-elementary function). More about Fresnel Integrals...

Re: MX2 2016 Integration Marathon $\noindent I don't think omegadot was telling you off $\ddot{\smile}$.$