Activity for Udi Fogiel
| Type | On... | Excerpt | Status | Date |
|---|---|---|---|---|
| Edit | Post #287077 | Initial revision | — | about 2 years ago |
| Answer | — |
A: Show that $\forall n \in \mathbb{Z}^{+}$, $25^n \equiv 25 \bmod{100}$. You can prove it using the binomial theorem. Assume that $1\leq n∈\mathbb{N}$, then: $$ \begin{align} 25^n & =(20+5)^n=\sum{k=0}^n\binom{n}{k}20^k\cdot 5^{n-k}=5+\sum{k=1}^{n-1}\binom{n}{k}20^k\cdot 5^{n-k}+20\\\\ & =25+5\cdot 20\cdot\sum{k=1}^{n-1}\binom{n}{k}20^{k-1}\cdot 5^{n-k-1}\equiv ... (more) |
— | about 2 years ago |
| Comment | Post #286450 |
While it is a nice example, i don't believe it is answering my question, as the set where $g'(x)> g^{1+\varepsilon}(X)$ might be of finite length. (more) |
— | over 2 years ago |
| Edit | Post #286428 | Initial revision | — | over 2 years ago |
| Question | — |
$g(x)\xrightarrow{x\to\infty}\infty$ Implies $g'(x)\leq g^{1+\varepsilon}(x)$ Recently in my ordinary differential equations class we were given the following problem: >Suppose $g:(0,\infty)\to\mathbb{R}$ is an increasing function of class $C^{1}$ such that $g(x)\xrightarrow{x\to\infty}\infty$. Show that for every $\varepsilon>0$ the inequality $g^{\prime}(x)\leq g^{1+\var... (more) |
— | over 2 years ago |
| Edit | Post #286154 | Initial revision | — | over 2 years ago |
| Answer | — |
A: equilateral triangle inscribed in an ellipse First of all, let's suppose that the points on the ellipse $(x1,y1),(x2,y2)$ are of the same distance from $(0,-2)$. We will get three equations from this assumption: $$ \begin{cases} x1^2+3y1^2=12 & (1)\\\\\\\\ x2^2+3y2^2=12 & (2)\\\\\\\\ x1^2+(y1+2)^2=x2^2+(y2+2)^2 & (3) \end{cases} $$ ... (more) |
— | over 2 years ago |
| Edit | Post #286141 |
Post edited: |
— | over 2 years ago |
| Edit | Post #286141 |
Post edited: |
— | over 2 years ago |
| Edit | Post #286141 |
Post edited: |
— | over 2 years ago |
| Edit | Post #286141 | Initial revision | — | over 2 years ago |
| Answer | — |
A: Finding distance to parabola's focus, given some points A parabola could be defined in the following way: >parabola is a set of points, such that for any point $P$ of the set, the distance $|PF|$ to a fixed point $F$, the focus, is equal to the distance $P\ell$ to a fixed line $\ell$, the directrix In part one of the question we were asked to find ... (more) |
— | over 2 years ago |
| Suggested Edit | Post #286130 |
Suggested edit: changed the formulas to be written with MathJax (more) |
declined | over 2 years ago |