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Here is Question 16 for 4u. Sorry for any blurriness, couldn't get proper lighting

You were so annoying in the exam, distracted me so much Plus the way I did the exam was different. I didnt do it in chronological order

If people really want to see question 16, ill take a pic and post it

Why weren't you there lol?

It wasnt about proving the trapezoidal rule, it was about justifying its geometrical significance

Actually it was different, it was about proving the inequality by using function derivatives, and substituting a suitable function that gives the AM-GM inequality

Actually question 16 consisted of proving the AM-GM inequality by using functions and the second part was to justify trapezoidal rule by using functions and integration techniques

Justifying trapezoidal rule for integration

Carrotsticks, destroying math dreams since 2012. In other words, exam was harder this year.

Re: MX2 2015 Integration Marathon $ Substitute: $ x= (\tan{\theta})^{\frac{2}{n}} \therefore \int_{0}^{1}{\frac{dx}{(1+x^n)^2 \sqrt[n]{1+x^n}}} dx = \frac{2}{n} \int_{0}^{\frac{\pi}{4}} \frac{(\tan{\theta})^{\frac{2}{n}-1}}{(\sec{\theta})^{\frac{2}{n} + 2}} d\theta = \frac{2}{n}...

I had a feeling that it was going to end up like this

Re: MX2 2015 Integration Marathon Ah yes, makes sense now. Thanks!

re: HSC Chemistry Marathon Archive Micelles can be formed without grease or dirt being in the centre of the molecule. This is because of the hydrophobic tail. When talking about micelle formation when cleaning grease or dirt, you would say: "The polar hydrophilic heads of the micelles decrease...

Re: MX2 2015 Integration Marathon \lim_{x \rightarrow 0} x\ln{x} = \lim_{x \rightarrow 0} \frac{\ln{x}}{\frac{1}{x}} $ Using L'Hopital's: $ \lim_{x \rightarrow 0} \frac{\ln{x}}{\frac{1}{x}} = \lim_{x \rightarrow 0} \frac{\frac{1}{x}}{\frac{-1}{x^2}} = \lim_{x \rightarrow 0} -x = 0

Re: MX2 2015 Integration Marathon You can prove it graphically or you can use L'hopital's rule. The reason why I just made it equal to 0 straight away was because I already knew that the limit went to 0, but under exam conditions I would of proved it via L'hopital's to make the working out...

re: HSC Chemistry Marathon Archive A micelle is a group of soap molecules that form into a spherical shape where the hydrophobic tails arrange themselves inside the sphere and the hydrophilic polar heads form the surface of the micelle sphere.

Re: MX2 2015 Integration Marathon Yes. Its a different form of IBP, where instead of establishing u, u' and v, v', you just find a function and shape it in a way such that when it is differentiated, one term will provide the integral desired.

Re: MX2 2015 Integration Marathon $ Let $ I = \int_{0}^{1} \frac{(-x)^n (\ln{x})^n}{n!} \left(\frac{-(-x)^{n+1} (\ln{x})^n}{n!(n+1)}\right)_{0}^{1} = I - \int_{0}^{1} \frac{(-x)^n (\ln{x})^{n-1}}{(n-1)!(n+1)} $ (Differentiate both sides if you want to confirm it) $ \therefore...

Re: 2015 HSC Economics Marathon Greater investor confidence. You wouldn't want to invest into a country that is politically corrupt and has a bad credit rating.

Well I guess we can all agree that BOP is a definite essay question, even the master essay predictor says that it is very very VERY likely