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You are in the Mathematics (Extension 1) forum. Mathematics (Extension 2): http://community.boredofstudies.org/14/mathematics-extension-2/

Sorry just confused: Shouldn't we multiply your answer for question 4 by 2 (taking into account area bounded by the y-axis in quadrant 1 and quadrant 2)?

\frac{d}{dx}\, e^{f(x)} = f'(x) \times e^{\,f(x)}

\int \frac{1}{\sqrt{y}} = \int (y)^{-\frac{1}{2}} = \frac{y^{\frac{1}{2}}}{\frac{1}{2}\times1} + C = 2\sqrt{y} + C

Re: HSC 2015 3U Marathon How do you work with the domain and know it's undefined? I solved to only get \frac{2}{1+x^{2}}

Re: HSC 2015 3U Marathon The given answer is: \frac{2}{1+x^{2}} \,\,\,\, $if$\,\,\,\, x>0, \frac{-2}{1+x^{2}}\,\,\,\, $if$\,\,\,\, x<0,\,\,\, $undefined at$ \,\,\, x=0

Re: HSC 2015 3U Marathon New Question $Differentiate$ \,\,\,\, y=cos^{-1}(\frac{1-x^{2}}{1+x^{2}})

Re: HSC 2015 3U Marathon Here is my attempt: \\ y=2x-x^{2} \\ y=-(x^{2}-2x) \\ y=-[(x-1^{2}-1)] \,\,\,\,\,($by completing the square$) \\ y=-(x-1)^{2}+1 \\ -y+1=(x-1)^{2}\\ 1\pm\sqrt{-y+1}=x \rightarrow\,\,\,\, $We will need both branches.$ \\\\\\ $Region A: (Using branch 1)$ \\...

Re: HSC 2015 4U Marathon Just looking at both of your last few posts, i think these would be helpful: \pm = \ pm (no space) $ And all the other values which i cant be bothered coding $ = $ And all the other values which i cant be bothered coding $ If you want space between your...

Re: HSC 2015 3U Marathon Is the answer found by working out $ V=\pi \int_{0}^{2} \, [(x)(x-2)]^{2} \, \, dx $

Do you mean 'k' is negative? I know it has the same effect but just making sure.

Re: HSC 2015 3U Marathon So we must prove by induction that 2^{11n} -1 is divisible by 2047 first? P(1)\rightarrow = \, \, 2^{11\times1} -1 =2047 \,\,\,\,\,\,\,\,\,\,\, $Therefore P(1) is true.$ \\ $Assuming P(k) is true \rightarrow 2^{11k} -1 = 2047M \\ \begin{align*}...

Re: HSC 2015 3U Marathon So, knowing 2^{11}-1 = 2047 , should my solution be: \begin{align*} &= 2^{2047}-2 \\ &= 2\left ( 2^{2046} -1\right ) & \\ &= 2\left ( 2^{11 \times 186} -1 \right ) \, \,\,\textit{which is in the form}\,\,\,\, (2^{11n}-1) \\ &= 2 \times 2047n \\ &= 4094n \\ &=...

Re: HSC 2015 3U Marathon \begin{align*} &= 2^{2047}-2 \\ &= 2\left ( 2^{2046} -1\right ) & \\ &= 2\left ( 2^{11 \times 186} -1 \right ) \end{align*} \\ =2\left ( 2^{11n}-1 \right )\\=2\times 23n \\ =46n i've been here for a while

Re: HSC 2015 2U Marathon \lim_{h\rightarrow 0} \frac{(x+h)^{3}-x^{3}}{h} \\ \\ = \lim_{h\rightarrow 0}\frac{(x+h)(x^{2}+2hx+h^{2})-x^{3}}{h} \\ \\ = \lim_{h\rightarrow 0}\frac{x^{3}+2x^{2}h+h^{2}x+hx^{2}+2h^{2}x+h^{3}-x^{3}}{h} \\ \\ = \lim_{h\rightarrow 0}\frac{3x^{2}h+3h^{2}x+h^{3}}{h} \\...

Re: HSC 2015 2U Marathon ah i see \tan^{2}(x) -3\tan(x) +2 =0 \\ (\tan(x)-2)(\tan(x)-1) =0 \\ \tan(x)=2 \, \, \, \, \, \tan(x) = 1 \\ \\ \therefore x=\frac{\pi}{4} \, ,\tan^{-1}2,\, \frac{5\pi}{4} \, , \tan^{-1}2 +\pi

Re: HSC 2015 2U Marathon Oops. I was thinking it couldn't equal twice at once. sinx=2cosx Tanx=2 x=63.43,243.43 x=1.107 radians, 4.249 radians Thank you.

Re: HSC 2015 2U Marathon $Solve for x:$ \, \, \, \sin^{2}(x) -3\sin(x)\cos(x) +2\cos^{2}x =0 \,\,\,\,\,\, $where 0 \leq x\leq 2\pi \\ \\ $Rewriting:$ \, \, \, \\ sin^{2}x -2\sin(x)\cos(x) +\cos^{2}(x) +\cos(x)\cos(x) -\sin(x)\cos(x) =0 \\ (\sin(x)-\cos(x))^{2} + \cos(x)(\cos(x) - \sin(x))...

Re: HSC 2015 2U Marathon \begin{align*}V &= 2\pi\int_{0}^{1} (x)^{2} -(x^{3})^{2} \,dx \\&= 2\pi\int_{0}^{1}x^{2}-x^{6} \,dx\\ &= 2\pi\left[\frac{x^{3}}{3} -\frac{x^{7}}{7} \right]_{0}^{1}\\&= 2\pi(\frac{1}{3} -\frac{1}{7}) \\&= \frac{8\pi}{21} units^{3} \end{align*} \\

Re: HSC 2015 2U Marathon $Make a quick sketch of y=x and y=x^3. $ \\ $Now, first find x-coordinates of intercepts$ \\ x^{3} = x \\ x^{3} - x =0 \\ x(x^{2}-1)=0 \\ \therefore x=0, \pm 1 \\ \\ $Volume Generated is found by: \pi\p\int_{a}^{b}f(x)^{2} -g(x)^{2} \, dx \\