# Quod Erat Demonstrandum

## 2018/03/22

### a question about inequality with derivatives

Filed under: Fun,mathematics,NSS,Pure Mathematics — johnmayhk @ 3:46 下午
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Question

Let $p(x)$ be a polynomial with real coefficients. Prove that if $p(x)-p'(x)-p''(x)+p'''(x)\ge 0$ for any real $x$, then $p(x) \ge 0$ for any real $x$.

Solution (elementary) (more…)

## 2018/03/21

### 某求導題

Filed under: Additional / Applied Mathematics,NSS — johnmayhk @ 3:29 下午
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If $\displaystyle \sqrt{x^3+y^3}=6(xy+1)$, find $\displaystyle \frac{dy}{dx}$ at $(1,-1)$.

$\displaystyle x^3+y^3=36(xy+1)^2$

$\displaystyle \Rightarrow \frac{d}{dx}(x^3+y^3)=\frac{d}{dx}36(xy+1)^2$

$\displaystyle \frac{dy}{dx}=\frac{24xy^2+24y-x^2}{y^2-24x^2y-24x}$

$\displaystyle \frac{dy}{dx}|_{(1,-1)}=-1$

## 2018/03/14

### 黃金比某級數

Filed under: Fun,mathematics,NSS — johnmayhk @ 11:11 上午
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$\displaystyle \Phi=\frac{1}{\Phi}+\frac{1}{\Phi^2}+\frac{1}{\Phi^3}+\dots$

$\displaystyle \Phi=\frac{1+\sqrt{5}}{2}$

(more…)

## 2018/03/05

### 度數弧度微積分

Filed under: Additional / Applied Mathematics,NSS — johnmayhk @ 12:10 下午
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（免插聲明：本篇頗無聊，高手見諒）

$\displaystyle \frac{d}{dx}\sin x$　at　$x=0^o$

M2 學生應知

$\displaystyle \frac{d}{dx}\sin x=\cos x$

$\displaystyle \frac{d}{dx}\sin x=\cos 0^o=1$