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It's essentially the technique of separation of variables for solving an ordinary differential equation: https://en.wikipedia.org/wiki/Separation_of_variables#Ordinary_differential_equations_.28ODE.29 .

∫ dt = ∫ 1 dt = t + C.

$\noindent Rearranging and integrating, $\int \left(x+4\right) \text{ d}x = \int \text{ d}t \Rightarrow \frac{1}{2}x^2 + 4x = t + C$. When $t=0$, $x=0$, so $C=0$. So $\frac{1}{2}x^2 + 4x = t$. So when $x=2$, $t=\frac{1}{2}\times 2^2 + 4\times 2 = 2+8 = 10$.$ Edit: Done above. (Btw, it's SHM...

Re: HSC 2016 3U Marathon Oh. I asked what was the function being optimised rather than the derivative you obtained (I didn't bother differentiating it). But assuming you differentiated correctly, the derivative is 0 iff -t^2 + t +1 = 0, i.e. t^2 - t - 1 = 0. This is the famous defining...

Re: HSC 2016 3U Marathon If you just used xa – xb, in general, maximising this would give us the maximum displacement of A relative to B. So it'd give us the maximum displacement A is ever right of B. So say A at most 1 metre right of B, but at some other time, A was 100 m left of B. Using...

Re: HSC Chemistry Marathon 2016 The symbols on the right tell us the net charge, as mentioned here: https://en.wikipedia.org/wiki/Ion#Notation.

Re: HSC 2016 3U Marathon This is not the difference between the two functions you gave. The two position functions were: xA(t) = 4te^(-t) and xB(t) = -4(t^2)e^(-t). The squared-difference is: s(t) := (xA(t) – xB(t))2. Try maximising that. (The distance isn't simply xA – xB, it is |xA –...

Re: HSC 2016 3U Marathon What was the function you tried to maximise?

Re: HSC 2016 3U Marathon Troll power rule. (It'd be the answer if the integral was de instead of dx, where e is a variable rather than the number e, and x is a parameter independent of this variable e, and assuming x is not equal to -1.)

Re: HSC 2016 3U Marathon Maximise the squared difference between the two particles' distance functions.

Re: MATH1131 help thread $\noindent I'll sketch the idea (I assume you managed to do part (a)). Assume the result is true for $n-1$, and consider the $n\times n$ matrix, which'll essentially be of this form: $A=\begin{bmatrix} a_1 &a_2 &\cdots &a_n \\ a_1 & a_2 &\cdots &a_n \\ \vdots &...

Re: HSC 2016 2U Marathon = e^x + C.

Re: HSC 2016 3U Marathon Most of us can do this; will we all get a rep from you? :p

In the previous years' 4U ones, no-one broke the 70 mark I think.

^ Take a look at one (I think they can be found in the Resources section). Basically each question is hard and there's probably not enough time to finish the exam in exam conditions (realistically speaking).

Re: HSC 2016 2U Marathon Haha – you actually ended up getting the right line. And why would Paradoxica put you up to this?

Re: HSC 2016 2U Marathon Which x are you referring to?

Re: HSC 2016 2U Marathon The emboldened part is correct, it's just he forgot to include the (x.ln(x)+1) factor in the next line. Edit: oh wait, saw he had an unnecessary x. Should just have been (ln(x)+1) rather than (x.ln(x)+1)

I suppose logarithmic differentiation was explicitly taught before?

Don't need to think of it mathematically, can just think of it intuitively: a (non-hollow) cylinder of foam will clearly weigh less than a (non-hollow) cylinder of wood of the same or even slightly smaller size (this is because foam is less dense than wood).