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Re: MX2 2016 Integration Marathon Compute: I_n:=\int_{-\pi}^\pi \frac{\sin(nx)}{(2^x+1)\sin(x)}\, dx.

The UNSW contest could be regarded as one of the easier olympiad style contests. http://webee.technion.ac.il/people/aditya/www.kalva.demon.co.uk/ has a ton of past questions from various olympiads. (Aim for national rather than regional/international ones unless you are feeling pretty...

Re: University Statistics Discussion Marathon Yep good stuff, it remains to compute Var(s^2), but what you have done is enough to motivate the later parts of the question so don't worry about that if you don't want to.

(Personally I think it is easier to plot in polar form).

tan(t)=x/y, use this and trig manipulations to eliminate the t in one of the equations if you want to get a cartesian form.

Re: University Statistics Discussion Marathon Huh? Not all of statistics is just plugging numbers into memorised formulae, where do you think these formulae come from? As always, you have the option of ignoring any question not to your taste. In any case, this particular question is easier...

Re: University Statistics Discussion Marathon Here is something more theoretical. Suppose X_j\sim \mathcal{N}(\mu,\sigma^2) for j=1,...,n are i.i.d random variables for some unknown parameters \mu,\sigma^2. 1. Define \overline{X}:=\frac{1}{n}\sum_{j=1}^n X_j, s^2:=\frac{1}{n-1}\sum_{j=1}^n...

Re: University Statistics Discussion Marathon Sure, if you are using the t-test then use that formula instead, it will still decrease. You can compute the p-value in terms of n by using the t-distribution and show it decreases by using calculus or whatever else you like. Just think about it...

Re: University Statistics Discussion Marathon My stats knowledge is minimal, but wouldn't it decrease? I assume that whats going on is you have the hypothesis that the avg sleeping hours of a student are normally distributed with mean 8 and variance V. We are doing a one-tailed test with test...

Try to exploit the fact that you are given that that binomial expression is a pdf, ie it sums to 1 for any value of the parameter p. (*) Try to evaluate the sum that occurs in E(e^(ux)) by manipulating it to contain an expression that you can sum using (*) For the last bit, again try to...

Re: Multivariable Calculus 1. Approaching along the line x=0 gives you a limit of zero, and approaching along the line y=x gives you a limit of 1/2, so you cannot extend f continuously to the full plane. 2. Just literally partially differentiate by first principles, you get zero for both of...

No way, that's such a negative mindset. Even if you don't do a followup subject, pushing yourself beyond the minimum effort required in a maths/stats subject and doing the more difficult problems will definitely sharpen your general problem solving chops. The more difficult things you get...

Re: Multivariable Calculus This is exactly how you would prove the "easy" part of the inverse function theorem. Once you have established the differentiability of the inverse map, the differential of your inverse map is forced to be the inverse of the differential of your original map.

Re: Multivariable Calculus The logic behind it is that if f and g are inverse to each other, then the chain rule says that I=D(\textrm{Id})(p)=D(f\circ g)(p)=Df(g(p))\cdot Dg(p) so the differentials of functions inverse to each other are matrices inverse to each other.

Re: Multivariable Calculus Do you know how to differentiate x and y as functions of r and theta? Once you have the differential/Jacobian of the map (r,theta)->(x,y) (which will be a 2x2 matrix), just invert this matrix to get the differential of the inverse map.

Re: Multivariable Calculus Think about what partial f/partial u means, it just means the derivative of the function with respect to the first variable (and you can replace u with x or whatever your favorite greek letter is). You are differentiating f with respect to its first variable and then...

Re: Multivariable Calculus Explicitly: RHS=nf(x,y)=nt^{n-1}f(x,y)|_{t=1}=\frac{\partial}{\partial t}(t^nf(x,y))|_{t=1}=\frac{\partial}{\partial t}(f(tx,ty))|_{t=1}=(x\frac{\partial f}{\partial x}(tx,ty)+y\frac{\partial f}{\partial y}(tx,ty))|_{t=1}= LHS

Re: Multivariable Calculus Fix x and y, differentiate with respect to t (using the chain rule to deal with the LHS differentiation), and then set t=1 in the resulting identity.

The general principle of roots of polynomials having Galois conjugates does not depend on degree. Given an algebraic number s, it has a minimal degree monic rational polynomial that it is the root of. The other roots of this polynomial are said to be its conjugates. If we assume a certain...

x=f(x) is irrational? what is f(x) and what does it mean for an equation to be irrational? You just need to show that a quadratic with rational coefficients cannot have a root at this value. I have posted a proof for a similar (perhaps identical) question in the past on here, but will do so...