Quod Erat Demonstrandum

2008/02/29

Taylor’s polynomials@濟濟一堂

Filed under: Additional / Applied Mathematics,HKALE,Pure Mathematics — johnmayhk @ 3:14 下午

From『濟濟一堂學術討論區』2003-11-05 09:28:45

因 2005 年以前在我的舊論壇『濟濟一堂學術討論區』張貼的東西已不能直接連結,幸好我還有 backup,雖然不是什麼好東西,但相信對中學的同學也有一丁點的幫助吧。但不幸地,backup 不能顯示作者名稱,所以忘記問者,只記得答者是在下。

溫下AS A.Maths..發覺不對… Taylor’s polynomials。只是把一點造成好接近,卻不能把polynomials扮成好似個curve,咁個error應該是超大ga wo! d exmaples都是不是太大。如果error超大的話,我們是要找另一點再扮?定有其他方法?

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廣告

2008/02/28

Evaluate cube roots

Filed under: Additional / Applied Mathematics,HKCEE,Junior Form Mathematics — johnmayhk @ 4:02 下午

A F.5 student, Yan, asked me to do the following without using calculator

\sqrt[3]{10 + 6\sqrt{3}} + \sqrt[3]{10 - 6\sqrt{3}}

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2008/02/27

F.5 last day

Filed under: Life,School Activities — johnmayhk @ 5:45 下午

Today is the last normal school day of F.5 students. This is the song chosen

前程錦繡

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To senior form mathematics students

Filed under: Information — johnmayhk @ 4:15 下午

Just for your reference

Mathematics Curricula
http://www.edb.gov.hk/index.aspx?nodeid=2899&langno=1

Examination Syllabuses
http://www.hkeaa.edu.hk/en/hkale/Subject_and_Syllabuses/

Seminar and Workshop Handouts (For Mathematics Teachers)
http://www.edb.gov.hk/index.aspx?nodeid=4651&langno=1

In the link just above, it is quite interesting to find the NSS (New senior secondary 新高中) Enriching Knowledge for the Mathematics Curriculum (Advanced Topics in Mathematics) Rainbow in Mathematics, just click and read. If secondary students could come across and understand topics mentioned in the handout like

Homology
Pascal’s Theorem
Bézout’s Theorem
Cayley-Bacharach Theorem
Elliptic Curve
Mordell-Weil Theorem
Mazur’s Theorem
Birch and Swinnerton-Dyer conjecture

in NSS, then HK secondary mathematics education will take the lead over this lonely planet for sure! But there’s a long way to go, at least, I just heard about (not understand) some of the names mentioned in the list above. How poor I am.

Also read

http://www.hkedcity.net/ihouse_tools/forum/read.phtml?forum_id=27877&current_page=&i=955854&t=954351

2008/02/26

Book tomb

Filed under: Life — johnmayhk @ 10:48 下午

Recent weeks, whenever I checked the blog stats, I must find the following blog in the Top WordPress.com blog

http://lawchiwah.wordpress.com/

What an ironic tragedy that books became the tomb of a book lover. This event may be symbolic. Is that a sign for HK culture in future?

When talking about reading, just forword the following, saluting to Art Garfunkel!

http://www.wretch.cc/blog/giawgwan&article_id=7104149

Something to say about F.7 Pure Mathematics (II) Mock Exam (part 1)

Filed under: HKALE,Pure Mathematics — johnmayhk @ 6:08 下午

出中七 Mock 卷,又是因為時間不足,沒有細心地把問題淺化。

談有關 convexity (凸性) 那道短題目。

考慮某區間 I,當

f'' \ge 0 on I – – – – – – (*)

我們熟知 the graph of y = f(x) is concave upwards。但對於 convex function,更基本的描述,可表如下

f is concave upwards on I" means

pf(x) + (1 - p)f(y) \ge f(px + (1 - p)y) for all p \in [0 , 1] and x , y \in I" – – – – – – (**)

圖像意義是,在曲線 f 上任取兩點 A(x, f(x)), B(y, f(y)),則 chord AB 一定『高於』curve AB,見下:

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2008/02/24

Hilbert tells stories

Filed under: Family — johnmayhk @ 11:20 下午

Hilbert, a 2.5-year-old boy, was very exciting because my old friends from my old church visited us today. He had fun with sisters. We asked him to tell a story, he did. What a suprise that he told us a very short story created on his own. We all laughed and enjoyed the precious moment with the gift from God. I do want to record Hilbert’s every funny moment. However, he is really growing very fast and I’m too old to catch up with all the swift changes. Just picture some of his joyful times here.

p1080940.jpg

Hello to the world! Perhaps, someone had put that funny little green hat on your brain some days in your life… (more…)

2008/02/23

Quadratic formula program @ CASIO 3950p/3650p

Filed under: HKCEE,Junior Form Mathematics — johnmayhk @ 11:17 下午
Tags:

驚恐中。我班的中五同學不懂做

Factorize 4x^2 - 4xy + y^2.

(不變金句:教了,不等於學了)

在中二的課,談到 cross method,我教同學用計算機。Casio 3950p (or 3650p) 較以往好,是它顯示的根(roots)是分數,不是點數,這樣可方便同學寫出答案。中五的同學,若你的 Casio 3950p 還未有 quadratic formula,快快看:

http://intranet.sfxc.edu.hk/it-school/homepage/nwc/casio_%20fx_3950p_quadratic_formula_program.doc

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2008/02/20

Additional Mathematics 補底

Filed under: Additional / Applied Mathematics,HKCEE — johnmayhk @ 11:01 上午

更新日期:2008-02-21 (希望不斷更新。)
免插聲明:本文為修附加數學科的中五同學純粹補底之用,高手勿插,謝謝。

1. 口訣:一冇 trigo,即寫 general

有關 general solution,考生常犯毛病見下

\sin(2\theta) = \sin30^o
2\theta = 30^o
\theta = 15^o
\therefore \theta = 180^on + (-1)^n(15^o)

錯!最後一步才寫 general solution 的形式是錯的!

不要到最後一步,而是要一開始便寫,即是

\sin(2\theta) = \sin30^o
2\theta = 180^on + (-1)^n(30^o)

即口訣的:一冇 trigo function 符號(本例是 sin),立即 (係立即!!!) 要寫 general solution 的形式。於是,我們有

\sin(2\theta) = \sin30^o
2\theta = 180^on + (-1)^n(30^o)
\therefore \theta = 90^on + (-1)^n(15^o)

才是正確答案。

2. 又 degree 又 radian;個樣衰過 Edit 神

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2008/02/19

A beautiful question in binomial theorem

Filed under: Additional / Applied Mathematics,HKALE,HKCEE,Pure Mathematics — johnmayhk @ 5:02 下午

證明:對任何正整數 m,n 恒有

\frac{(m + n)!}{(m + n)^{m + n}} < \frac{m!}{m^m}\frac{n!}{n^n}

如何證?MI?

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2008/02/16

五稜鏡評陳冠希事件

Filed under: Life,Report,Teaching — johnmayhk @ 10:22 上午

2008/02/09

本誌消息

Filed under: Uncategorized — johnmayhk @ 3:19 上午

因我不在香港,網誌 (起碼) 要到年初十(2008-02-16)才會更新。祝願讀者(如有的話)鼠年安康!

Don’t you believe that I spent 3 or 4 days on marking the F.7C pre-mock paper! Just have a look at the results.

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[PM] Fixed point (不動點)

Filed under: Pure Mathematics,Teaching — johnmayhk @ 3:16 上午

年三十,湛同學回校問數:設 f 是『自己打自己』的連續函數,證明 f 存在不動點。

這是純數補習習作某題,詳表如下:

Let f : [a , b] \longrightarrow [a , b] such that f(x) is continuous on [a , b]. Prove that there exist c\in [a , b] such that f(c) = c

這是經典且漂亮的習題。

f 的值域(range of f)為 [a , b],故

對所有在 [a , b] 內的 x,恆有 a \le f(x) \le b

g(x) = f(x) - x,則

g(a) = f(a) - a \ge 0 – – – (1)
g(b) = f(b) - b \le 0 – – – (2)

若 (1) 的等號成立,取 c 為 a。
若 (2) 的等號成立,取 c 為 b。
若 (1) 及 (2) 的等號皆不成立,即 g(a)g(b) < 0
由於 g(x) 在 [a , b] 上連續,
故區間 (a , b) 內必存在 c,使 g(c) = 0
亦即 f(c) = c

有修 Applied Mathematics (II) 的同學應該對此法不會陌生,記得 Fixed point iteration 嗎?那個證明的開端就是這個。

還有很多關於『不動點』的東西想說,但,我實在太累了,還在生病中。

[Gag] 自貼

Filed under: Fun — johnmayhk @ 3:16 上午

疑似授課員在疑似數學堂上打疑似手槍

dsc00319.jpg

(設計對白)再嘈!打爆你個 \theta!!

2008/02/06

數學應用題:拍賣

Filed under: Junior Form Mathematics — johnmayhk @ 12:55 上午
Tags:

太太常以數學(普)書作我的(例如生日)禮物。其中一本是《數學謎工》(The Puzzler’s Elusion)作者是 Dennis Shasha。書面有宣傳句:『最靈活的數學題型,挑戰基本學力測驗』,或許將來的數學考題,比如 SBA(如有的話)可能也偏向這些形式,讓我抄一道題供大家參考。

【場景】

有兩家公司,A 和 B,生產類似的競爭產品。雖然 A 公司是比較小的公司,但市場佔有率正在擴大。A 公司的產品需要用到某種機器,其每月的月產量只有6 部,A 公司的陳生每個月必須採購 2 部。

這種機器透過一般的英式公開拍賣會銷售,價錢從某個設定的底價開始往上加。出價的規則是每次增加 1000 美元,而且出價不可超過自己的財務負擔能力。

直到不久前,陳生都是唯一的出價者,所以能夠以最低價錢每部 5000 美元投得。然而,B 公司也打算開始加入競標,據說是為了牽制陳生。因為陳生是個好客戶,他享有優先出價的特權,所以問題的重點圍繞在每次拍賣會上B 公司願意投注多少錢,與多少錢才足夠完全阻止陳生,假設 B 公司的高層只有這個目的。

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