Articles of calculus

Find the limit: $\lim\limits_{(x,y)\to (0,0)}\frac{x^3 – y^3}{x^2+y^2}$

We seek a solution to the limit question: $$\lim\limits_{(x,y)\to (0,0)}\frac{x^3 – y^3}{x^2+y^2}$$ I have an approach and think I have an answer but wanted to ask the community first. My approach is similar to the approach BabyDragon used at this link: What is $\lim_{(x,y)\to(0,0)} (x^3+y^3)/(x^2-y^2)$?

Holder continuity of power function

I need to compute the coefficient for the Holder continuity of $x^p$ with $x > 0$, that is $$ H(p) := \sup_{x\neq y}\frac{|x^p – y^p|}{|x – y|^p}. $$ I am actually going to apply this in numerical scheme, so I am interested in finding $H(p)$ itself, rather than upper bounds, or at least an upper […]

Horizontal tank with hemispherical ends depth to capacity calculation

I am trying to find an accurate way of calculating the capacity of an underground tank at a given depth. The tank manufacturer has provided a strapping table for the tank which tells me the capacity at various depths. Gauge Depth (cm) / Capacity (Liters) 2cm = 29.8 liters 12cm = 240.4 liters … 66cm […]

Darboux Theorem

Darboux’s Theorem states that” If $f$ is differentiable on $[a,b]$ and if $k$ is a number between $f^{\prime}(a)$ and $f^{\prime}(b)$, then there is at least one point $c\in (a,b)$ such that $f^{\prime}(c) = k$.” Most coomonly found proof goes as follows: Suppose that $f^{\prime}(a) < k < f^{\prime}(b)$. Let $F:[a,b]\rightarrow \mathbb{R}$ be defined by $F(x) […]

Infinite Product is converges

I am adding this problem since it is interesting and valuable to be verified here: Prove that the infinite product $\prod_{k=1}^{\infty}(1+u_k)$, wherein $u_k>0$, converges if $\sum_{k=1}^{\infty} u_k$ converges. What about the inverse problem? Thanks for any ideas.

$\lim_{y \rightarrow b} \lim_{x \rightarrow a} f \neq \lim_{(x,y)\rightarrow (a,b)} f \neq \lim_{x \rightarrow a} \lim_{y \rightarrow b} f$

Can someone give me an example to show that in general $\lim_{y \rightarrow b} \lim_{x \rightarrow a} f(x,y) \neq \lim_{(x,y)\rightarrow (a,b)} f(x,y) \neq \lim_{x \rightarrow a} \lim_{y \rightarrow b} f(x,y) $ I have been able to construct examples when the first and last limit exist but the middle one does not, but I can’t find […]

Can someone show me a proof of the general solution for 2nd order homogenous linear differential equations?

I thought I had it figured out but there’s a sort of ‘leap of faith’ at a pivotal point that annoys me. Can someone show me how to derive the general solution of an equation such as: $a\frac{d^2y}{dx^2}+b\frac{dy}{dx}+cy=0$ I want to avoid just saying, “let’s assume the solution is of the form $y=e^{mx}$”. I want […]

Monotonocity of the function

I am unable to find values of $x$ for which following function $$x^2 \cdot e^{-x}$$ When this function is increasing or decreasing? My normal approach is “differentiate” but I am stuck at some point…

Question About Orthoganality of Hermite Polynomials

It is known that is $m \ne n$: $$ \int_{-\infty}^{\infty} H_n(x) H_m(x) e^{-x^2}dx = 0 $$ Does this apply for any $f(x)$? $$ \int_{-\infty}^{\infty} H_n(f(x)) H_m(f(x)) e^{f(x)^2} dx = 0 $$

A naive example of discrete Fourier transformation

We know a discrete Fourier transformation with discrete $n$ and continuous $x_1,x_2$: $$ \sum_{n\in\mathbb{Z}} e^{-in(x_1-x_2)\frac{2\pi}{L}}=L\delta(x_1-x_2) $$ with Dirac delta function $\delta$. and a discrete Fourier transformation with discrete $n$ and discrete $x_1,x_2$: $$ \sum_{n\in\mathbb{Z}} e^{-in(x_1-x_2)\frac{2\pi}{L}}=L\delta_{x_1-x_2,0} $$ with Kronecker delta function $\delta$. Question: what is the discrete Fourier transformation with discrete $n$ and discrete $x_1,x_2$ below: […]