Quod Erat Demonstrandum

2019/12/08

受保護的文章：F5 M2 RT 20191206

Filed under: Additional / Applied Mathematics,NSS — johnmayhk @ 11:49 上午
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2019/05/05

What’s wrong?

Filed under: Additional / Applied Mathematics,NSS — johnmayhk @ 7:05 下午
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Here is a basic level M2 question:

Given that $\sqrt{xy}=7+2y$, find $\frac{dy}{dx}$ at ($-\frac{1}{3}$,$-3$).

Student 1 gave

$\frac{1}{2\sqrt{xy}}(x\frac{dy}{dx}+y)=2\frac{dy}{dx}$

$\frac{1}{2}\sqrt{\frac{x}{y}}\frac{dy}{dx}+\frac{1}{2}\sqrt{\frac{y}{x}}=2\frac{dy}{dx}$

$\frac{dy}{dx}=\sqrt{\frac{y}{x}}\cdot\frac{1}{4-\sqrt{\frac{x}{y}}}$

Thus, at ($-\frac{1}{3}$,$-3$),

$\frac{dy}{dx}=\sqrt{\frac{-3}{-1/3}}\cdot\frac{1}{4-\sqrt{\frac{-1/3}{-3}}}=\frac{9}{11}$

Student 2 gave (more…)

2018/11/20

費氏講

Filed under: Additional / Applied Mathematics,Fun,Junior Form Mathematics — johnmayhk @ 6:36 下午
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N 年前往中一班代堂，必談「64 = 65」謎題：

(圖片來源：https://i.stack.imgur.com/fWdMd.jpg)

2018/07/18

畫 y=x^(1/n)

Filed under: Additional / Applied Mathematics,mathematics,NSS — johnmayhk @ 12:57 下午
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$\sqrt[3]{4}-\sqrt[3]{3}$$\sqrt[3]{3}-\sqrt[3]{2}$

2018/06/10

tan(89.99 度)

Filed under: Additional / Applied Mathematics,NSS — johnmayhk @ 7:19 下午
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N 日後，有學生問我，何解相鄰結果似乎有 10 倍變化？見下：

(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 $(1,-1)$.

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

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

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

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

2018/03/05

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

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

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

2017/11/14

as gs

Filed under: Additional / Applied Mathematics,mathematics,NSS — johnmayhk @ 12:29 上午
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Derive the formula for the sum of first $n$ terms of the following sequence in terms of $a,b,d,r,n$, where $r \ne 1$.

$ab,(a+d)br,(a+2d)br^2,(a+3d)br^3,\dots$

2017/11/08

作正五邊形

Filed under: Additional / Applied Mathematics,Fun,Junior Form Mathematics — johnmayhk @ 10:49 上午
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2017/08/13

最小值

Filed under: Additional / Applied Mathematics,NSS,Pure Mathematics — johnmayhk @ 6:19 下午

$x\ge 3$

$x$ 的最小值是 3，

(more…)

2017/06/10

正多邊形方程

Filed under: Additional / Applied Mathematics,Fun,mathematics,NSS — johnmayhk @ 12:24 下午
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https://www.desmos.com/calculator/vv7stc4nl0

2017/03/19

盛水水深

Filed under: Additional / Applied Mathematics,HKCEE,NSS — johnmayhk @ 12:43 下午
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$(H-\sqrt[3]{H^3-h^3})$ 單位。

2016/03/21

用Ｄ證trigo

Filed under: Additional / Applied Mathematics,NSS — johnmayhk @ 11:13 上午
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Prove the following identity

$\cos^2x+\cos^2(x+y)-2\cos y\cos x\cos(x+y)=\sin^2y$.

2015/08/28

sum of 1/k^2 from 1 to infinity

Filed under: Additional / Applied Mathematics,Fun — johnmayhk @ 4:30 下午
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…………………………………………….

$ax^n+bx^{n-1}+\dots +cx+d=0$ ………. (*)

$-\frac{b}{a}$(more…)

2015/08/16

某舊習題

Filed under: Additional / Applied Mathematics,Pure Mathematics — johnmayhk @ 4:25 下午
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$a+b+c=0$

$\frac{a^5+b^5+c^5}{5}=\frac{a^3+b^3+c^3}{3}\cdot \frac{a^2+b^2+c^2}{2}$