Quod Erat Demonstrandum

2011/03/31

To 6C students

Filed under: Pure Mathematics — johnmayhk @ 5:23 上午

Just a sketch of the solutions to 1-mark questions in the previous quiz.

Q.7

Denote [x] as the greatest integer not greater than x.

Does \displaystyle \lim_{x\rightarrow 0-}(\frac{x}{2011}[\frac{2012}{x}]-\frac{2011}{x}[\frac{x}{2012}]) exist?

Hint: (more…)

廣告

2011/03/30

M2 三點

Filed under: NSS,Pure Mathematics — johnmayhk @ 4:17 下午
Tags:

1.

今早,同學問了 M2 某習題:

Evaluate \int_{0}^{y} \ln(1+\tan x \tan y)dx.

隨便答: (more…)

[FW][TED]Marcus du Sautoy: Symmetry, reality’s riddle

Filed under: Fun,University Mathematics — johnmayhk @ 5:41 上午

2011/03/29

某古算題

Filed under: Junior Form Mathematics — johnmayhk @ 2:43 下午

今早同事快談一道(疑似)古算題,曰:

「今有共買璡(石之似玉者《說文》),人出半(即 \frac{1}{2}),盈四;人出少半(即 \frac{1}{3}),不足三。問璡價。」

確是中一數學習題的好材料。

Solution
Let n be the number of buyers.
\frac{n}{2}-4=\frac{n}{3}+3\Rightarrow n=42
Hence the price is \frac{42}{2}-4=17.

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…)

2011/03/20

[FW][song] Mandelbrot Set

Filed under: Fun,mathematics — johnmayhk @ 4:38 下午

2011/03/18

某兩題中四數

Filed under: NSS — johnmayhk @ 5:02 下午

1.

以下是中四上學期,教科書中某題:

A wire of 80 cm long is cut into two pieces and each wire is bent into a square.
The length of each side of the smaller square is x cm.

(a) Express the total area of the two squares in terms of x.
(b) Find the minimum area of the two squares in total.

易得答案 (more…)

2011/03/17

某有關開方根的題目

Filed under: Junior Form Mathematics — johnmayhk @ 3:44 下午
Tags:

由平凡的結果出發

(a+b)+(c+d)=(a+c)+(b+d)

如果 a,b,c,d 非負(non-negative),我們可以考慮

(\sqrt{a}+\sqrt{b})+(\sqrt{c}+\sqrt{d})=(\sqrt{a}+\sqrt{c})+(\sqrt{b}+\sqrt{d})

再考慮 (more…)

2011/03/15

混疊圖像

Filed under: Fun — johnmayhk @ 12:34 下午

《題西林壁》
(宋)蘇軾

橫看成嶺側成峰,
遠近高低各不同。
不識廬山真面目,
只緣身在此山中。

遠近,影響了人眼看事物 (more…)

2011/03/10

金比銀比銅比

Filed under: Junior Form Mathematics — johnmayhk @ 3:27 下午

香港人或比較熟悉 Golden Forum。

數學人當然熟悉 Golden ratio(黃金比),其實數學上還有 Silver ratio (more…)

2011/03/09

[FW][YouTube] Euler’s identity

Filed under: Fun — johnmayhk @ 8:51 下午

Easy but fun.

2011/03/08

[FW][YouTube] The Black-Scholes Formula

Filed under: Physics,University Mathematics — johnmayhk @ 5:27 下午

慚愧地,身為數學授課員,對物理學在財務金融上的應用,除了在普及書看過一丁點,所知近乎是零。這裡是有關介紹 Black-Scholes Formula 的電影,貼一貼吧。

(more…)

2011/03/04

[FW] An evolving landscape of a 3D fractal surface

Filed under: Fun — johnmayhk @ 3:45 下午

from
http://www.subblue.com/blog/2011/1/16/surface_detail

2011/03/03

Polynomial identity


Is the following an identity? Prove or disprove your claim:

(x+1)(x-3)+1=(x+2)(x-1)

This is a trivial (more…)

2011/03/02

當 x 接近零,(1+1/x)^x 如何?

Filed under: Pure Mathematics — johnmayhk @ 5:07 下午
Tags:

課堂上,我通常會談「非例子」。

比如教(講)了

\displaystyle \lim_{x\rightarrow \infty}(1+\frac{1}{x})^{x} = e

隨即問

\displaystyle \lim_{x\rightarrow \infty}(2+\frac{1}{x})^{x} = ?

這個易,再問

\displaystyle \lim_{x\rightarrow 0}(1+\frac{1}{x})^{x} = ?

學生 (more…)

在WordPress.com寫網誌.