Quod Erat Demonstrandum

2020/05/21

Basic question of differentiation

Filed under: Additional / Applied Mathematics,mathematics,NSS — johnmayhk @ 7:20 下午
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HKDSE 2020 M2 Q.9 (b), a 2-mark question:

Given \displaystyle f(x)=\frac{(x+4)^3}{(x-4)^2}, find f''(x).

How fast can you finish this part and obtain the correct answer, especially when you are under the pressure during the public examination?

3 minutes? (2/100 * total time allowed = 2/100 * 150 minutes)

(more…)

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:

(more…)

2020/02/28

正弦積


初中同學,請問下式何值?

\tan 1^o\tan 2^o\tan 3^o\dots \tan 88^o\tan 89^o

因為

\tan \theta \tan (90^o-\theta) \equiv 1

所以

\tan 1^o\tan 2^o\tan 3^o\dots \tan 88^o\tan 89^o
=(\tan 1^o\tan 89^o)(\tan 2^o\tan 88^o)\dots (\tan 44^o\tan 46^o)\tan 45^o
=1\times 1\times \dots \times 1
=1

冇難度。

\sin 1^o\sin 2^o\sin 3^o\dots \sin 88^o\sin 89^o

呢?
(more…)

2019/12/13

受保護的文章:F.4 Core Math Quiz Ch.2

Filed under: mathematics,NSS — johnmayhk @ 10:16 上午
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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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2019/05/10

線長乘積

Filed under: Pure Mathematics — johnmayhk @ 11:52 下午
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考慮單位圓內接正多邊形,比如正方形

由某點(比方說 A)出發,連起其他頂點,得出 3 條線段,其長度分別為 2, \sqrt{2}, \sqrt{2},故乘積(product)為 4。

對於五邊形

由某點出發連起其他頂點,得出 4 條線段,那麼線段長度的乘積如何? (more…)

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…)

2019/02/01

帕斯卡三角某結果

Filed under: NSS,Pure Mathematics — johnmayhk @ 5:35 下午
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早前貼了下圖

本文談如何推出上述結果。

對於函數 f(x)

f^{[1]}(x)=f(x)

f^{[2]}(x)=f(f(x)) (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)

對以上現象,小朋友給了不少有創意但錯誤的解釋,如「冷縮熱漲」。

所謂 (more…)

2018/11/02

受保護的文章:SFXC F.3A Mathematics Assignment 11 (Web Task)

Filed under: Teaching — johnmayhk @ 9:19 上午

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2018/09/16

長周素

Filed under: Fun,Junior Form Mathematics — johnmayhk @ 7:20 下午
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觀察一下:

\frac{1}{7}=0.\overline{142857} ,故循環周期(decimal period)為 6。

\frac{1}{17}=0.\overline{088235294117647} ,循環周期為 16。

\frac{1}{19}=0.\overline{052631578947368421},循環周期為 18。

對於所有質數 p\frac{1}{p} 的循環周期都是 p-1 嗎? (more…)

2018/09/15

用積分證 0=-1

Filed under: NSS — johnmayhk @ 6:01 下午
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用 integration by parts 處理 \int \tan xdx 如下

\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

於是

0=-1 (more…)

2018/07/18

畫 y=x^(1/n)

Filed under: Additional / Applied Mathematics,mathematics,NSS — johnmayhk @ 12:57 下午
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不用計算機,如何比較

\sqrt[3]{4}-\sqrt[3]{3}\sqrt[3]{3}-\sqrt[3]{2}

何值為大?

其中一個方法是考慮 y=\sqrt[3]{x} 的圖像,見下 (more…)

2018/07/13

pi的連分表達式

Filed under: mathematics,NSS — johnmayhk @ 2:52 下午
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\pi 寫成連分數(continued fraction),方式甚多,以下是其中一種:

可以變成 M2 習題如下:

Define \displaystyle I_n=\int_0^1\frac{x^{2n}}{1+x^2}dx. (more…)

2018/06/10

tan(89.99 度)

Filed under: Additional / Applied Mathematics,NSS — johnmayhk @ 7:19 下午
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那天觀課,同事教 trigonometric graphs。談到 tangent graph 在 90 度處斷開,著同學試 tan(89∘), tan(89.9∘), tan(89.99∘) 之類,可見結果愈來愈大,去到 90 就無限大云云。

N 日後,有學生問我,何解相鄰結果似乎有 10 倍變化?見下:

(more…)

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