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Arguably the fact that there is little as 40 minutes to write a full in depth essay in essence takes away whatever beauty literature studies had. I don't deny that for some other people memorising essays may be the best way to get marks, but I can't see it working for me, granted I haven't...

Yep that was silly of me For integrating its best to expand, the first term xsqrt(16-x^2) is easily dealt with reverse chain rule/easy sub, the second one is done better geometrically by taking the appropriate segment of the circle.

The areas on either side of the chord are the same, also the area underneath the x-axis is same as above x-axis, so we can split the volume into a quarter piece which is the volume generated by rotating the region from x=2 to x=4 and y=0 and y=sqrt(16-x^2) dv = 2\pi((x-2) \sqrt{16-x^2}) \ dx...

Re: HSC 2013 4U Marathon Well it is given that E_n approaches a finite limit, and H_n - ln(n) = E_n + 1/n, so the result is practically given, for my one it is not given.

Re: HSC 2013 4U Marathon $Define the following recurrence$ H_1= 1 \ \ H_n = H_{n-1} + \frac{1}{n} $i) Prove that$ \ \ H_n \rightarrow \infty \ \ $as$ \ n \ $increases without bound$ $ii) Prove through inequalities that the limit$ \ \ \gamma = \lim_{n\to \infty} \left(H_n - \ln n \right )...

True, my teacher doesn't have a problem with it fortunately. HSC markers on the other hand..... Good idea, thank you. I guess I was a bit too hasty with L'Hopitals, I should of probably never learnt it lol.

Say for a question like: \int_0^{\pi /2} \frac{\cos x}{1+ \cos x} \ dx It was Q1 Integation for a trial paper, I knew you just needed to do t=tan(x/2), but I decided to try and find another way: \int_0^{\pi /2} \frac{\cos x - \cos^2 x}{\sin^2 x} \ dx = \int_0^{\pi /2} \csc x \cot x -...

Re: HSC 2013 4U Marathon $A right circular hollow cone has surface area$ \ \ A \ \ $and volume$ \ V $The surface area of the cone remains constant while the radius and height vary$ $Show that the maximum volume is given by$ V_{max} = \sqrt{\frac{A^3}{72\pi}}

Re: MX2 Integration Marathon Dividing both top and bottom by x^2 then x, knowing that x is positive in the domain of integration. \frac{(x+1)^3 (x-1)}{(x^2+1)^2 \sqrt{x^4+x^2+1}} = \frac{\left(1- \frac{1}{x^2} \right)(x^2+2x+1)}{\frac{1}{x}(x^2+1)^2 \sqrt{x^2+ 1 + \frac{1}{x^2}}} =...

Re: HSC 2013 4U Marathon s=l\theta v=\frac{ds}{dt} = l \frac{d\theta}{dt} \frac{d \left(\frac{1}{2}v^2)}{d\theta} \cdot \frac{d\theta}{dv} = \frac{d\left(\frac{1}{2}v^2 \right )}{dv} = v \therefore l \frac{d \theta}{dt} = \frac{d}{d\theta} \left( \frac{1}{2}v^2 \right) \frac{d...

Re: HSC 2013 4U Marathon ah ok, well done. ===== $The sequence$ \ \{f_k \} \ \ $is defined by$ \ \ f_k=f_{k-1} + f_{k-2} \ \ k \geq 2 f_0 =0 \ \ f_1 = 1 $Assuming that the ratio$ \ \frac{f_{k+1}}{f_k} \ \ $approaches a finite limit as$ \ k \ $increases without bound$ $Prove that$ \ \...

Re: HSC 2013 4U Marathon The first inequality isn't always true for all positive integers n.

Re: MX2 Integration Marathon \int_0^{\pi /2 } \frac{\sin^2 x}{\sqrt{x^2 - \frac{\pi}{2}x + \frac{\pi^2}{16} + 1}} \ dx

Re: HSC 2013 4U Marathon Let the points be A(a,a^2), B(b,b^2), C(c,c^2) Find through differentiating that nromal to curve at A is: x+2ay=2a^3+a Intersection of normals at A and B are (-2ab(a+b), a^2+ab+b^2+1/2) Intersections of normals at A and C are (-2ac(a+c), a^2+ac+c^2+1/2) Equating...

Re: MX2 Integration Marathon \int_{\frac{1}{2}}^{2} \frac{\ln (x)}{1+x^2} \ dx

Re: HSC 2013 4U Marathon Let P_k \ $represent$ \ z_k \angle P_1 P_2 P_3 = \arg(z_1-z_2) - \arg(z_3-z_2) \angle P_3P_4P_1 =\arg(z_3-z_4) - \arg(z_1-z_4) Add them side by side, opposite angles of a cyclic quadrilateral add to pi so, and use the identity: \arg(ab) = \arg(a) + \arg(b)...

Re: MX2 Integration Marathon I didn't get it from there, its just the Beta function lol, The recurrence relation is quite simple for one that has multiple variables so I thought I'd just ask it straight away.

Re: HSC 2013 4U Marathon $Since$ \ \left{y_k \right} \ \ $is an arbitrary permutation of$ \ \left{x_k \right} x_1^2+x_2^2+ \dots + x_n^2 = y_1^2+y_2^2+\dots+y_n^2 x_1^2+y_1^2 \geq 2x_1y_1 x_2^2+y_2^2 \geq 2x_2 y_2 x_3^2+y_3^2 \geq 2x_3y_3 . . . x_n^2+y_n^2 \geq 2x_n y_n \therefore...

Re: MX2 Integration Marathon Yep. ================================== \int_2^{1+ \phi} \frac{(x+1)^3 (x-1)}{(x^2+1)^2 \sqrt{x^4+x^2+1}} \ dx $Where$ \ \phi = \frac{1+ \sqrt{5}}{2}

Re: HSC 2013 4U Marathon Do you mean x and y instead of a and b inside the sum?