Quod Erat Demonstrandum

2011/08/09

Past papers 1964, 1971

Filed under: Additional / Applied Mathematics,Fun,HKALE,Pure Mathematics — johnmayhk @ 11:38 上午


Matriculation Examination (1964)
Advanced Level

Pure Mathematics I,II
Applied Mathematics I,II


(more…)

2011/04/15

又談MVT

Filed under: Fun,HKALE,Pure Mathematics — johnmayhk @ 11:05 下午
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如果有 pure mathematics field trips,除了可以去 NYC,參觀:

也可以往 (more…)

2011/04/13

解微分方程

Filed under: Additional / Applied Mathematics,HKALE — johnmayhk @ 8:50 上午

[Hardsell 廣告腔] 解微分方程?用

http://www.wolframalpha.com/

啦!

y"+y=sin(x) (more…)

2011/03/24

use series instead of lhopital

Filed under: Additional / Applied Mathematics,HKALE,Pure Mathematics — johnmayhk @ 3:10 下午

利用洛必達法則計算

\displaystyle \lim_{x\rightarrow 0}(\frac{\sin^{-1}x}{x})^{\frac{1}{x^2}}

頗煩。

或以無窮級數,粗糙地 (more…)

2011/01/07

Exercise in counting balls and boxes

Filed under: Additional / Applied Mathematics,HKALE,NSS — johnmayhk @ 7:43 上午
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The following is just a basic counting exercise from a secondary mathematics textbook (used in Taiwan, I guess), enjoy.

Find the number of ways of (more…)

2010/11/20

唔打:唯一解與delta非零

Filed under: HKALE,NSS,Pure Mathematics — johnmayhk @ 5:56 下午

(more…)

2010/11/10

有解條件

Filed under: HKALE,Pure Mathematics — johnmayhk @ 3:10 下午

同事談到,原來以下問題,也可讓同學熱烈討論:

p,q 為常數,問在什麼條件下

px = q 有解? (more…)

2010/10/27

玩玩系列:二次方程

Filed under: HKALE,NSS,Pure Mathematics — johnmayhk @ 10:11 上午

問題

1. 設 a,b,c 是奇數(odd numbers),證明 ax^2 + bx + c = 0 無整數根(integral root)。
2. 已知 e\pi 是超越數,證明 e + \pie\pi 兩者中起碼一個是超越數。

提示 (more…)

2010/10/18

某插值法習題

Filed under: Additional / Applied Mathematics,HKALE — johnmayhk @ 2:56 下午
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一道普通的題:

f(x) = \sin(\frac{\pi}{2}x)

設多項式 p(x) 使 p(0) = f(0), p(1) = f(1), p(2) = f(2)

估計絕對誤差 |f(x) - p(x)| 在區間 (0.5,1) 上的上限。 (more…)

2010/09/27

series by wolfram alpha

Filed under: Additional / Applied Mathematics,HKALE — johnmayhk @ 5:40 下午

早前和中七同學談 Series,因書中習題:

Expand \sin^{-1}x about x = 0.

沒有答案,我在

http://www.wolframalpha.com/

輸入 (more…)

2010/07/26

MVT 某推廣

Filed under: HKALE,Pure Mathematics — johnmayhk @ 11:54 上午
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查 THE COLLEGE MATHEMATICS JOURNAL VOL. 35 NO.1 JANUARY 2004 有關 Cauchy’s Mean Value Theorem Involving n Functions 一文,當中以

證明

F(x) = -3x + \frac{\pi}{2}\cos(\frac{\pi}{2}x) + \frac{e^x}{e - 1} + \frac{1}{(x + 2)\ln 2} 在區間 (0,1) 存在零點 ………. (*)

時,

F(0) , F(1) 皆大於零,故不宜用介值定理(Intermediate Value Theorem)。

(more…)

2010/07/22

暑期無聊閱讀

Filed under: Additional / Applied Mathematics,HKALE,HKCEE — johnmayhk @ 9:02 上午
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學校圖書館檢來一本舊教科書,名為 Applied Numerical Analysis。

在解方程一章,以下例引起動機。

If you worked for a mining company the following might be a typical problem:

There are two intersecting mine shafts that meet at an angle of 123^o, as shown in the figure above. The straight shaft has a width of 7 feet, while the entrance shaft is 9 feet wide. What is the longest ladder that can be negotiate the turn? You can neglect the thickness of the ladder members, and assume it is not tipped as it is maneuvered around the corner. Your solution should provide for the general case in which the angle, A, is a variable, as well as the widths of the shafts. (more…)

2010/06/17

重複數算一例

Filed under: Additional / Applied Mathematics,HKALE — johnmayhk @ 11:05 上午
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擬了一道長題:

第一,我沒有說明該「骰子」的運作:所謂得出的「花」應是指在骰子底部沒有題示出來的那個花!sosad…

第二,明眼人知道我是迆出,因為,用「容斥原理」,兩步KO,變成最多值 4 分的短題,見

http://www.hkms-nss.net/discuz/home/space.php?uid=423&do=blog&id=38

但估計不會有太多同學會用「容斥原理」,所以我 (more…)

2010/06/16

單增區域

Filed under: HKALE,Pure Mathematics — johnmayhk @ 12:37 上午
Tags: ,

又是溫習純數的時間。參考下圖。

直觀地,函數在區間上是單增(strictly increasing)的。

那麼,若 f(x) 在區間上可導(differentiable),則對於區間內任何一點 a,恆有

f'(a) > 0

(見圖)

但逆向地 (more…)

2010/06/01

Just a revision on counting

Filed under: Additional / Applied Mathematics,HKALE,Teaching — johnmayhk @ 4:03 下午
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Here is just an ordinary question:

“An elevator starts carrying five persons at the ground floor and then goes up. It can stop at any floor of the building (from the first floor to the third floor). Events that people leaving the elevator are assumed to be independent. Let X be the number of ‘stop’ of the elevator during a “going-up” journey. Find the value of E(X)."

We may use “balls and boxes" (more…)

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