Quod Erat Demonstrandum

2020/05/21

Similar-looking formula

Filed under: Junior Form Mathematics,mathematics,Physics — johnmayhk @ 4:01 下午
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The equivalent resistance $R$ of a parallel circuit

can be determined by

$\displaystyle \frac{1}{R}=\frac{1}{R_1}+\frac{1}{R_2}$.

A similar-looking formula found in a basic mathematics question involving parallel lines as shown below:

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…)

2018/09/15

用積分證 0=-1

Filed under: NSS — johnmayhk @ 6:01 下午
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$\int \tan xdx=\int \frac{\sin xdx}{\cos x}=-\int \frac{d\cos x}{\cos x}=-\frac{\cos x}{\cos x}+\int \cos xd(\sec x)=-1+\int \tan xdx$

$0=-1$ (more…)

2018/04/17

互斥與獨立

Filed under: NSS — johnmayhk @ 12:37 下午
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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

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

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

2017/04/21

三垂線定理

Filed under: mathematics,NSS,Pure Mathematics,University Mathematics — johnmayhk @ 12:56 下午
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（一）前言

2016/07/10

論商餘（三）

Filed under: mathematics,NSS,Teaching — johnmayhk @ 9:45 下午
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(a) 多項式與數字

2015/09/23

兩數相等

Filed under: Fun — johnmayhk @ 5:43 下午
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$8\times 9=9\times 8$

$\Rightarrow 8\times (17-8)=9\times (17-9)$

$\Rightarrow 8\times 17-8^2=9\times 17-9^2$

$\Rightarrow 8^2-8\times 17=9^2-9\times 17$

$\Rightarrow 8^2-8\times 17+(\frac{17}{2})^2=9^2-9\times 17+(\frac{17}{2})^2$

（配方法了） (more…)

2015/09/09

平行四邊形面積

Filed under: NSS — johnmayhk @ 7:54 下午
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2015/08/03

小學三角題

Filed under: Junior Form Mathematics — johnmayhk @ 4:34 下午
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Whatsapp 傳來同事阿仔的暑期功課：在釘板上圍出一個等邊三角形

2015/07/03

帶分數惹的禍

Filed under: Junior Form Mathematics — johnmayhk @ 9:53 上午
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$2\frac{1}{3}-2=\frac{1}{3}$

$2\sqrt{3}-2$

2015/03/05

三角形內切圓

Filed under: NSS — johnmayhk @ 10:48 上午
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2015/02/21

表面面積

Filed under: Fun,Junior Form Mathematics — johnmayhk @ 7:47 下午
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(more…)

2015/02/20

鈄截柱體體積

Filed under: Junior Form Mathematics,NSS,Physics — johnmayhk @ 8:25 下午
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1.緣起

$V=A\frac{h_1+h_2+h_3}{3}$

$V=A\frac{0+0+h}{3}=\frac{1}{3}Ah$ (more…)