Christian Ivicevic
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 Nov 21 comment Reconstructing a function from its critical points and inflection points Because it is just a sketch to show how derivatives behave in the neighbourhood of critical points I left out specific values of $x$ and $f(x)$ hence you can pick whatever values that make this easier. Nov 18 comment proof of limit explanation @Silas2033 For the proof we use the following definition: Let $(x_n)_n$ be a sequence then $a\in\mathbb{R}$ is the limit of $(x_n)_n$ if there is a $N\in\mathbb{N}$ for each $\varepsilon > 0$ s.t. $$|x_n-a|<\varepsilon\qquad \text{for each }n\in\mathbb{N}\text{ with }n\geq N.$$ In my example there is such an $N$ for $\varepsilon=0.5$ with $N=3$ because the inequality holds for each $n\geq N$. Nov 17 comment Prove that $\frac{b-a}{\cos^2 (a)}< \tan b - \tan a < \frac{b-a}{\cos^2 b}$ Did someone say Bill $\cos b$? Nov 7 comment Simplify $-2\sin(x)\cos(x)-2\cos(x)$ Factor out $-2\cos(x)$ and you get the desired result. Nov 5 comment Is this true (Natural Logarithm)? Keep in mind to not only include a negative, but a positive $\beta$ as well, i.e. $\beta=\pm\left(\ln(D)/\sqrt{\pi^2+\ln^2(D)}\right)$. Otherwise I see no error in your equivalence. Nov 5 comment Showing an inequality: $\sqrt{xy} \leq \frac{2xy}{x+y}$ @elfeck Assume you have $x\leq y$ then you multiply it by $1/x$ to obtain $1\leq y/x$ and then with $1/y$ which yields $1/y\leq 1/x$, hence $x\leq y\Leftrightarrow 1/x\geq 1/y$. Just keep the right direction of the inequality in mind. Nov 3 comment Is there a whole number $x\in\mathbb{Z}$ with $x\neq 0$ s.t. $\exp(x)$ is natural? @avid19 Please refer to my edit - it was a mistake to include 0. Nov 2 comment Prove that $x_{n+1}=\frac{x_n}{2}+\frac{1}{x_n}\geq\sqrt{2}$ @Dr.MV Yeah pretty similiar to my explained approach in the first part of my post. I was just wondering about the other approach and its correctness there. Oct 1 comment Prove $\sum_{k=0}^n\binom{2n+1}{2k}=4^n$ This proof definitely is as short as requested but easy to grasp nonetheless! Sep 23 comment Is there a subset of a non regular language that is regular @piperchester This comment is very old and it seems I deleted the file from my Dropbox - luckily I have an updated version. You will find the mentioned overview of languages on the last page of my cheatsheet. Apr 9 comment Infinite sets don't exist!? You could mention, that Von Neumann proposed that all natural numbers can be bootstrapped out of the empty set by the operation you described. Jan 16 comment When is $f(x) = \frac{ax}{e^{-b}-{(e^{c})}^x}$ monotonic? By the way, is this a homework related question? If so, then please consider adding the appropriate tag - furthermore show what you have done so far and what you don't understand so we see that you at least tried something. Jan 14 comment Parameterized curve describing trajectory of thrown object I don't know whether this is correct, but we never did such complicated computations in class. I tried to solve this problem now by not factorizing so that I have $\sqrt{v_0^2\cos^2\beta+(v_0\sin\beta-gt)^2}$ and simplify that with other (much easier) approaches. Do you have some other suggestions? Jan 14 comment Parameterized curve describing trajectory of thrown object Actually I never saw this method, but I think that this yields $\int \|k'(u)\|\,\mathrm du=\frac{1}{2}g\left(u-\frac{v_0\sin(\beta)}{g}\right)^2+v_0\cos(\beta)u+\text{c‌​onstant}$. Did I understand this substitution right? Jan 8 comment Proving that $\left(\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\cdots+\frac{1}{n!}\right)$ has a limit Careful, you say that $x_n$ is a finite sum and saying that it equals an infinite sum. Furthermore you ambiguously use $n$. These are just small mistakes with the notation. How about saying $\lim_{n\to \infty} x_n = \lim_{n\to \infty} \frac{1}{0!}+\ldots+\frac{1}{n!}=\sum_{k=0}^{\infty}\frac{1}{k!}=e$? Jan 7 comment If $\sum_{n=1}^\infty a_n$ is convergent, then $\sum_{n=1}^{\infty} a_n\sin(n)$ @GEdgar: I thought about that fact that $|\sin(n)|\leq 1$ like Andrew mentioned this could be an "easier" example to simplify $\sin$ if you understand. Jan 7 comment If $\sum_{n=1}^\infty a_n$ is convergent, then $\sum_{n=1}^{\infty} a_n\sin(n)$ Have you tried to think about $(-1)^n\cdot\frac{1}{n}\to 0$? This could be some motivation. Jan 7 comment Differentiation with respect to integral boundary @macydanim: Holy shit... this looks awesome. I will write this down as a possible alternative solution. Dec 22 comment Calculating a function limit Exact duplicate of: math.stackexchange.com/q/263306/12864 Dec 10 comment Antisymmetric and irreflexive relation which is not asymmetric I see it now. Let's forget what I was talking about.