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http://lichess.org/kTXP5mMD :). I'm not as good as I used to be, you might do well.

Some chill games today anyone? Need some procrastination from maths.

That WA input is being interpreted as the principal cube root (take cube root of modulus, and divide principal argument by 3). This doesn't map R to R though. Because cubing is a bijection on R, if we are only talking about reals, the natural definition of the cube root is just the inverse. Ie...

x^{1/3} is perfectly well defined for x<0, even if you have never heard of complex numbers. x^3 is a bijection from R to itself.

A remark that is useful in some situations: You can work your way around the "closed" part of the assumption in the extreme value theorem to some extent in some situations. Say you had a continuous function f(x) on the interval (a,b) that tends to limiting values at a and b. Then you can...

Anyway, worry more about the ideas than the notation. As long as you can clearly express your arguments, that is what is more important. Have seen plenty of papers/proofs in which the roles of epsilon and delta are switched.

You probably wouldn't have encountered it as much but its common enough in first year calculus/analysis books. Like I said, its not realllly convention but its unsurprising that Integrand and I chose the same letter.

M is often used to represent something large and delta is very often used to represent something small, so they are pretty natural to use. Wouldn't say the M one is really a convention though.

It's kind of simpler than that. All you need to show that the limit exists and is 1, is that for all e > 0 we can find M(e) > 0 such that |f(x)-1| < e for all x > M(e). Just take M(e)=sqrt(1/e-1).

Re: HSC 2016 4U Marathon A classic: 1. Use the trapezoidal approximation to show that there exists a positive constant A such that An^{n+1/2}e^{-n}\leq n! for all positive integers n. 2. Using another inequality coming from integration, show there exists a positive constant B such that n...

Just remember that sinh and cosh don't change sign when differentiated, and deduce all sign things from that. I don't memorise this stuff despite using it pretty often.

Re: HSC 2016 4U Marathon Yes simple harmonic motion is just the case where g is a linear polynomial in time, ie the case treated by Integrand.

Re: HSC 2016 4U Marathon For simple things like power functions, you would still be able to get arbitrarily large EV, but the calculations are messier. Try to show this yourself. To find a g where sin(g(t)) (with g(t) increasing) did not have this property, you could cook up g so that the...

Yes, best to do this for now. A sketch of a possible proof though (about the lowest-tech one I can think of off the top of my head, although also the least general): Given a sequence of real numbers, we define the limit superior of the sequence as \lim_{m\rightarrow\infty}(\sup_{n\geq m}...

Re: HSC 2016 4U Marathon Good stuff Integrand, that is exactly correct. Of course all this question really amounts to is the statement that the PDF of the particles position at a random large time tends to infinity as you approach +-1.

This post makes no sense. How can f(x) tend to infinity at b? By continuity it has to tend to f(b) there. And how have you deduced that a minimum exists? That is almost precisely what you need to prove. Also b and a are the endpoints of a fixed interval, why are you talking about b -> a ?

Call it the extreme value theorem, the term "max-min" and "min-max" occur a ton in other contexts. How you should prove it will depend on exactly what you have learned, typically you assume this theorem without proof in most first year courses. (For good reason, discussion of things like...

Re: Extracurricular Integration Marathon Swap the order of integration to arrive at \int_0^1 (y-y^3)\sqrt{1-y^4}\, dy. This is killed by the substitution y^2=\sin(\theta), and the result is \frac{\pi}{8}-\frac{1}{6}.

When lambda=-2, the bottom two rows will give you contradictory expressions for z. Not sure if you have learned about determinants yet, but they give you a v. quick way of determining when a matrix is invertible (ie when Ax=y has a unique solution). This reduces such problems to checking the...

Eh. I don't think it's that big a deal if students don't know what things are called in high school. I'd much rather them develop good problemsolving skills than learn the names of theorems. FTC in particular refers to so many different theorems as we generalise notions of...