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By itself, this working does not show that your constructed P has the lowest degree possible.

A change of variables reduces it to evaluating the Gaussian integral (integral of e^{-x^2} over the real line). This is well known to be equal to sqrt(pi), and is most commonly computed using polar coordinates as in https://en.wikipedia.org/wiki/Gaussian_integral#By_polar_coordinates ). Your...

Re: Multivariable Calculus (Where, X,Y,Z are open subsets of arbitrary dimension Euclidean spaces, or more generally of curves/surfaces/manifolds.) Sidebar: Spivak has a less well known book "Calculus on Manifolds", that is amazing for learning multivariable calculus. From introducing partial...

Re: Multivariable Calculus Perhaps in some notations it will seem that way, but they can be unified remarkably well once one learns to think of the derivative of a function f at a point p as nothing more than a linear map between the vector space of tangents at p and the vector space of...

I think that's underselling blitz players to say that quick games are purely to test speed lol, there is still massive depth in tactics and strategy. But yes, you would most likely get bad habits and look at positions more superficially if you tried to learn the game only through playing blitz.

Casual game anyone? A bit high and just going to procrastinate for a bit before getting back to maths.

Yep. HSC students were the intended audience. I think it is not ideal that students are taught Simpsons rule, usually without any accompanying discussion of when it doesn't work well or any proofs of error bounds (which are basically just IBP exercises).

Re: First Year Uni Calculus Marathon Related: $Prove that $\lim_{n\rightarrow \infty }\left(1+\frac{x}{n}\right)^n$ and $\lim_{n\rightarrow \infty }\left(\sum_{k=0}^n \frac{x^k}{k!}\right)$ both exist and coincide for all real numbers $x$. $

Re: First Year Uni Calculus Marathon You kinda need to include the definition of e / exponentiation that you would like students to work from to make a first principles question like this well-defined.

Re: First Year Uni Calculus Marathon Here is a question related to a basic tool in the analysis I do. $For positive $\lambda$ let $I(\lambda):=\int_\mathbb{R} a(x)e^{i\lambda \phi(x)}\, dx$\\ \\where $a,\phi$ are smooth and $a$ is zero outside of a bounded interval $[-M,M]$.\\ \\ Expressions...

Don't rely too much on them, unless you have a handy supply of higher dimensional paper for when there are more than three sets. Way more important to just internalise how things like taking complements interacts with unions and intersections of sets (from a mathematical point of view outside...

Re: First Year Uni Calculus Marathon One can also consider the same problem dropping the condition of surjectivity. (For which positive integers k is it possible to find a continuous function f defined on R that takes every value in its range exactly k times?)

Re: First Year Uni Calculus Marathon Also unanswered. I will get people started: For k=2 it is not possible. Proof: Assume f is a cts function taking every value exactly twice. There must exist two zeros a < b of the function f. Between a and b f must have fixed sign (otherwise the...

So you have the integrand g(x)=x^{-1-s}e^{-x^{-1}} and we want to show it is integrable on the non-negative reals. The behaviour near zero is not a problem at all, since g(x)\rightarrow 0 as x\rightarrow 0 (essentially because exponentials dominate polynomials, there are many ways you could...

Re: First Year Uni Calculus Marathon Whilst it is fairly obvious that r^n/n! converges to zero if r is positive (in fact regardless of sign but we can wlog take it positive), I don't think it is a fact that should be glossed over by first year students. In any case, the methods used to bound...

Re: First Year Uni Calculus Marathon Unsolved.

Re: University Statistics Discussion Marathon Here is a exercise that some of you might enjoy from a probability course I am grading: $Let $X_j$ be independent uniformly distributed random variables on $[0,1]$. Prove that for every $a>1/2$, there exists a corresponding $b>0$ such that...

I=\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{k=1}^n f(\sqrt{x_{k-1}x_k})\\ \\ = \lim_{n\rightarrow\infty} \frac{1}{n}\sum_{k=1}^n \frac{n^2}{(n+k)(n+k-1)}\\ \\ = \lim_{n\rightarrow\infty} n\sum_{k=1}^n \left(\frac{1}{n+k-1}-\frac{1}{n+k}\right)\\ \\ = \lim_{n\rightarrow\infty}...

"No", if I had to say something but neither is really correct because the yes or no question is not a precise mathematical proposition. It's a silly question because it does not make sense to ask if a function f is differentiable/continuous at a point p outside its domain. A more sensible...

It means find \lim_{n\rightarrow\infty} T_n where T_n is the given expression in the two parts.