Quod Erat Demonstrandum

2008/09/29

Just a question of applied math. from a F.7 boy

Filed under: Additional / Applied Mathematics,HKALE — johnmayhk @ 6:13 下午
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Just discuss an easy AL Applied Mathematics (II) Question with students.

Let f(x), g(x), h(x) be twice differentiable functions such that f(x) = g^2(x) + x^3h(x).

(a) Let p(x) = \frac{f(x)}{g(x)}, where g(0) \ne 0. Show that p(0) = g(0), p'(0) = g'(0), p''(0) = g''(0).

(b) Using (a), or otherwise, find Taylor’s expansion of the function \frac{2x^4 - 3x^3 + x + 4}{\sqrt{x + 4}} about x = 0, up to the term in x^2. (more…)

廣告

2008/09/27

一本數學參考書

Filed under: Uncategorized — johnmayhk @ 10:40 上午

中六同學「大師」早前問我:甚麼是數學?我略略提"The Princeton Companion to Mathematics"一書,這可算是一本數學的百科全書,有興趣的同學可到以下連結,看看 sample pages。

http://press.princeton.edu/releases/m8350.html

作者之一 Timothy Gowers 在其網誌也談了一些,也順道看看

http://gowers.wordpress.com/2008/09/17/princeton-companion-errata/

2008/09/26

中二數學問題之有效數字

Filed under: Junior Form Mathematics — johnmayhk @ 9:37 上午

派完測驗,粱同學問:

如果 x = 0.25 (correct to 2 significant figures)

那麼 x 的 maximum value(最大值)是多少? (more…)

2008/09/24

比較係數

Filed under: Pure Mathematics — johnmayhk @ 10:05 下午
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以下是一道簡單的短題。

f(x) = ax^2 + bx + ca \ne 0),滿足 f(f(x)) \equiv f^2(x),求 a, b, c 的值。 (more…)

2008/09/22

論商餘 (一)

Filed under: HKCEE,Junior Form Mathematics,Pure Mathematics — johnmayhk @ 10:34 下午
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所謂「商」「餘」乃 quotient 及 remainder 是也。這個 post 只是一些有關除法的簡單討論。 (more…)

2008/09/20

利用插值法找餘式

Filed under: Additional / Applied Mathematics,HKALE,Pure Mathematics — johnmayhk @ 2:33 下午

近日,中六純數課談的是餘式定理,中七的應數課談的是插值法,恰巧,兩者有一點點關聯。 (more…)

2008/09/19

太空原子筆

Filed under: Fun — johnmayhk @ 6:01 下午

在下純粹因為知道 facebook 有寫 note 這個功能,所以寫了下篇,試用用該功能而已。 (more…)

2008/09/18

究竟「打從」個「打」字是什麼意思?

Filed under: Fun — johnmayhk @ 5:48 下午

印象中,「打從」這個說法在幾十年前的文章比較少見(或應是我孤陋寡聞),現在卻似乎成了慣用語。 (more…)

2008/09/16

什麼是無限

Filed under: Fun,Pure Mathematics — johnmayhk @ 5:14 下午

舊討論區有網友問:「什麼是無限?」我想答:「非有限也。」 (more…)

2008/09/13

太太入院

Filed under: Family — johnmayhk @ 10:28 下午

太太入院,正待 48 小時後抽血報告,看手術如何。不多說,為她禱告吧。這星期將是我的非人生活。

2008/09/10

一個多星期。推介網頁。IES

Filed under: Life,School Activities — johnmayhk @ 10:50 下午
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一個多星期

上課不過是個多星期,同事已稱工作生活好像渡日如年。 (more…)

2008/09/07

初中數學堂之:正負數加減

Filed under: Junior Form Mathematics,Teaching — johnmayhk @ 2:02 下午

每年,總有學生問:「如何證明一加一等於二?」下面是很久以前在網上找到的文章【為什麼 1 + 1 = 2】,我貼過又貼,未看過的同學,可讀讀黃毅青教授的介紹:

http://intranet.sfxc.edu.hk/it-school/homepage/nwc/temp/1_plus_1_equal_2.pdf

就是一些很基礎的東西,我們往往忘記它的因由,比如,為何會「負負得正」之類。讓我談一些初中數學的東西,希望 refresh 同學的一些記憶。 (more…)

2008/09/06

純數習題:代數數

Filed under: HKALE,Pure Mathematics — johnmayhk @ 1:04 上午

中六純數課,談了 proof by contrapositivity 和 proof by contradiction 的分別,再談證明「質數有無限多個」的簡單方法(同學,另外一些證明方法可見舊文:數學閒聊:素數有無限多個)以顯示反證法之威力。隨即,做習題,其中有一題是證明

\sqrt{2} + \sqrt{3} 是代數數(algebraic number)。 (more…)

2008/09/04

F.2 Mathematics: significant figures

Filed under: Junior Form Mathematics — johnmayhk @ 10:30 下午

F.2 students, please fill in the following blanks.

(a) -0.0012000 has __________ significant figures.

(b) 12000 has __________ significant figures.
(more…)

2008/09/03

Prerequisite of F.4 algebra

Filed under: Additional / Applied Mathematics,HKCEE — johnmayhk @ 5:17 下午

As usual, we start with basic algebraic computation drilling at the beginning of F.4 mathematics lessons. But, I just did some useless mathematical chatting and gave something about infinity (e.g. the discussion of 1 – 1 + 1 – 1 + … ) and they showed their excitment with hands clapping. Well, of course, I switched to serious matter very soon. As expected, most of the students did not know the fact that “zero is an even number" (even, they did not know that zero is an integer). I went through some ‘prerequisite’ questions with them, one is:

(a) \sqrt{9} = \pm3  [True or false?]
(b) If x^2 = 9, then x = 3. [True or false?]
(c) The square roots of 9 are 3 and -3. [True or false?]
(d) If x^2 = 9, then x = 3 and x = -3. [True or false?]

F.4 students, here are the answers. (more…)

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