@JohnDoe Otherwise, the result wouldn't necessarily hold, consider $$f_n(x) = \begin{cases} 1 &, n < x < 2n\\ 0 &, x \leqslant n \lor x \geqslant 2n.\end{cases}$$
@Studentmath: Only 9/27 got the first problem right (plus 2 more made small errors). Everyone else overcounted drastically (as did I one time thinking about it) :)
And #3a,b were exactly on my exam 1 (with answers posted), and people still missed it, @Studentmath. Only three or four understood why the answer to b) was irrelevant to c). :(
Stoichiometry is a good one, @Mike.
Game theory is full of it. Linear programming is used all over business.
At the moment, looking forward to only one semester more of giving/grading exams, @Chris'ssis. So disheartening to work so hard and have students do so badly :(
@TedShifrin In which year did you have the best students? Or maybe you have in mind a certain period of time expressed in years when you had the best students.
'Show that the quotient space of a finite collection of disjoint $2$-simplices obtained by identifying pairs of edges is always a surface locally homeomorphic to $\Bbb{R}^2$'
How does one prove this, other than 'it's obvious'?
I have no idea, @Chris'ssis. I've had some spectacular students throughout my career, including when I was a grad student. Still have some spectacular ones this year.
@JohnDoe If both fonts are used, usually one refers to the space of functions, and the other to the space of equivalence classes modulo "equal almost everywhere".
@robjohn $$\int_0^\infty\frac{ \log^3(x+1) }{ 2x^2+3x+1} \mathrm{d}x = 6\operatorname{Li}_4\left(\frac{1}{2}\right)$$ and this one can be generalized for $n$ power.
My Multivariable Math kiddies did worse than I would have liked, but for the most part the A students kept their A's. But the low end did appallingly ...
@JohnDoe: So both those had $\mathcal L$. I think it's only a typesetting issue. Some books use $L$, some use $\mathcal L$. I've honestly never noticed a distinction, but, as I said, I defer to @DanielF.
@Ted Yeah that's the kind of answer I like to hear (not meant in a sarcastic way) sometimes refreshing to hear "just ignore that it doesn't matter" in maths.
@JohnDoe In these cases, there's only one font used. Then it's probably the usual Lebesgue space of equivalence classes of functions, since the space where the a.e. $0$ functions aren't modded out is of less interest. If two distinct fonts are used in the same source, then they typically denote different spaces, but it's not uniform which font is used for which.
I'm still fairly confused, but I'll take it, @Behaviour. Was there any reason for the comment in particular? I don't feel like I've been doing anything special so I'm guessing it was a sarcastic comment.
@robjohn I think I'll send this version to some students I usually talk to $$\int_0^\infty\frac{ \log^{2015}(x+1) }{ 2x^2+3x+1} \mathrm{d}x$$ and the well-known $$\text{Happy New Year 2015!!!}$$
@TedShifrin: I don't drink a lot. For a certain time period, I drank a bottle of beer every evening, just because I felt like I needed it. But then I decided to stop. So now it's just a beer every once in a while when out with friends again. But there is an urge when correcting exams. :(
@Chris'ssis: Depends. Most of my high schoolers are not very motivated to learn maths which makes it frustrating sometimes. But there are also bright moments.
@TedShifrin Hey Ted... I was off helping someone get ready for a fridge delivery, then the incoming fridge was damaged. So we put everything back and try again.
@Chris'ssis: as I said, I had not worked on that integral, but I was simply saying that I would try integrating the function with a higher power around a keyhole integral. That often produces good results.
Are there any "mathematical" movies suited for high schoolers you could recommend? Something like Good Will Hunting, A Beautiful Mind, etc. ? @DanielFischer @MikeMiller
@MikeMiller: I have a rather cool (and well-known to those who know it) example of a phenomenon of quantum mechanics (or rather quantum information theory) we did in our third semester , but I'm not sure if it's still in the scope. Do you want me to write it up anyways?
Look at $\Sigma_3$ as three donuts squashed together; it's a covering map of order 2. The nontrivial deck transformation is rotating it around the center of the middle donut.
I'll abstract nonsense my group theoretic galois theory for arbitrary categories rather than that I guess and then apply it to category of topological spaces.
