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Jan
27
comment Multivariable taylor series expansion of $\exp(-(x^2+y^2))$
Am I right in assuming that $-\exp(-5)(4x+2y-11)$ would be the right polynomial in my case?
Jan
21
comment Eigenvectors of $\begin{pmatrix}6&2\\-10&-1\end{pmatrix}$ (linear equations with complex numbers)
@Prospect I do know the algorithm but I am struggling with the particular steps in this specific case.
Dec
9
comment Continuous function taking each of its values twice
It seems I am misunderstanding something in regards to the claim from the paper as all examples that come to mind have a finite amount of discontinuities.
Dec
9
comment Continuous function taking each of its values twice
Every function that satisfies this property has an infinite number of discontinuities. Refer to ams.org/journals/proc/1986-098-02/S0002-9939-1986-0854049-8/…
Dec
9
comment Using L Hopital's Rule Using $e$
$e=\exp(1) \rightarrow (\exp(1))'=(1)'\cdot\exp(1)=0$ as AndreasT mentioned.
Dec
8
comment Definite integral $\int _{0}^{1}\sqrt [3] {2x^{3}-3x^{2}-x+1}~\mathrm dx$
What have you tried so far?
Dec
7
comment Show that $\int_1^\infty \frac{\ln x}{\left(1+x^2\right)^\lambda}\mathrm dx$ is convergent only for $\lambda > \frac{1}{2}$
Could you please elaborate why the term $[-\ln(x)/(\mu x^\mu)|_1^M$ remains finite as $M\to\infty$?
Dec
2
comment Trying to understand some terminology
Short version: $\mathcal{O}(n)$ is the class of functions which behave asymptotically similiar to $f(n)=n$. You would say that a function is in $\mathcal{O}(n)$ if it does not grow much faster than $n$. Examples would be $2n, 14n,\ldots$. Refer to this article for more detailed information about Big O notation.
Dec
1
comment How does one come up with the anti-derivative?
In short: Let $F$ be the anti-derivative of $f$ then $F'=f$.
Nov
28
comment Give me hints for evaluating this limit
If I am not overlooking anything this is true indeed. I guess you just need to explain your reasoning more detailed. Moreover you have to check whether you are allowed to apply my hint - if it is for a homework, then the question is whether you have already proven the hint in a lecture etc.
Nov
28
comment Give me hints for evaluating this limit
Hint. $e^x=\lim_{n\to\infty}(1+x/n)^n$
Nov
21
comment Reconstructing a function from its critical points and inflection points
@kccu Well you can just add a constant to $f$ s.t. $f(x_2)=0$ and it should be $7/7680$ if you work with rajb245's points $x_1,x_2,x_3$ - I am currently experimenting with other numbers for a better scaling but the general idea of this approach is definitely good.
Nov
21
comment Reconstructing a function from its critical points and inflection points
@whatever Well saddle points are just inflection points with the added property that $f'(x)=0$. I guess there are multiple names that are being used for both types hence the confusion. :)
Nov
21
comment Reconstructing a function from its critical points and inflection points
@whatever Every saddle point is an inflection point but not the other way round if I am not mistaken. The slope at a saddle is always zero while it does not have to be zero for inflection points.
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.