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Re: MX2 2016 Integration Marathon $\noindent You don't need to bother writing $u$ and $v^\prime$ etc. You can just look at the product in the integrand, and pick one of them to be the one to differentiate, and the other to integrate. Then write the IBP as integrate one of them times leaving...

Re: HSC 2016 3U Marathon The symbol log is the natural log (well, one of the notations for it, the others being ln and loge).

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread $\noindent He used the fact that $S+D = \frac{1}{S-D}$, which is true because $(S+D)(S-D) = 1$ (this can be proved by using the definition of $S$ and $D$ in terms of $v$).$

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread It's basically just standard calculus Q's about tangents. Like for the second one, differentiate the function to find the derivative function. This tells us the slope of the tangent at a point x. To find the tangent to the curve...

$\noindent Let $\omega = \cos \frac{\pi}{3}+i\sin \frac{\pi}{3}$. Let $Q$ be the number $z_3$. Note that $\overrightarrow{RQ}$ is just $\overrightarrow{RP}$ rotated by $\frac{\pi}{3}$ counter-clockwise. Since $\overrightarrow{RP}$ represents the number $z_2 - z_1$ (since $P$ is $z_2$ and $R$ is...

In fact, the true spelling is neither "Avocado" nor "Avagadro". It is "Avogadro". :) His Wikipedia page is here: https://en.wikipedia.org/wiki/Amedeo_Avogadro .

$\noindent These aren't trinomials, they are simply cubes (they are also expandable via the binomial theorem). Expanding things like $\left(a_1 + a_2 + a_3\right) ^n$ is done via the trinomial theorem (a generalisation of the binomial theorem, and special case of the multinomial theorem). A...

$\noindent Use the factorisation rules $A^3 \pm B^3 = \left(A \pm B\right) \left(A^2 \mp AB + B^2\right).$

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread No

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread $\noindent Since $0\leq x <2\pi$, we have $\frac{5\pi}{3}\leq x +\frac{5\pi}{3} < 2\pi+\frac{5\pi}{3}$.$

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread $\noindent For part (ii), they said they want the answers in the range $0\leq x <2\pi$. So you need to convert your answer of $-\frac{\pi}{2}$ into the equivalent angle in the desired range, which is $\frac{3\pi}{2}$ (found by...

$\noindent Note that the graph of $-x^2 + 2x + 4$ is above that of $y=x^2$ for the region we need. First we find the points of intersection. They occur when and only when$ $$\begin{align*}-x^2 + 2x + 4 &= x^2 \\ \Longleftrightarrow -x^2 + 2x + 4-x^2 &= 0 \\ \Longleftrightarrow -2x^2 + 2x + 4...

Re: HSC 2016 4U Marathon $\noindent For the ellipse, the equation is $B^2 = A^2 \left( 1- e^2 \right)$, where $B$ is the semi-minor axis and $A$ is the semi-major axis (i.e. $A \geq B$, so $A$ is the bigger number).$

Re: HSC 2016 4U Marathon Correct!

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread $\noindent The answer is $y^\prime = -\frac{a}{x^2} -\frac{1}{a}$. Here we made use of the fact that $\frac{\mathrm{d}}{\mathrm{d}x} \left(\frac{1}{x}\right) =-\frac{1}{x^2}$. This is easy to show by using the power rule and...

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread $\noindent It should be $y^\prime = a^2 -2ax$. This is because $a^2$ is a constant, so the derivative w.r.t. $x$ of $a^2 x$ is $a^2$. Hence the answer is $a^2 -2a$.$

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread Haha. Quotient rule is something nice to avoid if possible as it is a bit tedious. :)

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread Those ones are easier done without the quotient rule. :)

Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread Split up each fraction and use index laws to make each term a power of x. Then differentiate using the power rule.

Re: MX2 2016 Integration Marathon $\noindent Let $f(x)=\frac{\alpha}{\lambda} \left(\frac{x}{\lambda}\right)^{\alpha-1}\mathrm{e}^{-\left(\frac{x}{\lambda}\right)^{\alpha}}$ for $x>0$, where $\alpha,\lambda$ are positive constants.$ $\noindent For $x>0$, find $F(x) \equiv \int _0 ^x...