# Quod Erat Demonstrandum

## 2009/03/31

### 請問有什麼免費軟件可以繪畫 vector field？

Filed under: Uncategorized — johnmayhk @ 10:00 上午

## 2009/03/30

### Simple questions about mean value theorem

Filed under: HKALE,Pure Mathematics — johnmayhk @ 5:20 下午
Tags: , ,

For your revision, students.

Question 1

Suppose f(1) = f(2) = 0, f(3) = 1 and f is twice differentiable on [0,3].

Show that $f''(c) > \frac{1}{2}$

for some $c \in (0,3)$.

Question 2

Suppose f(0) = 0, f(1) = 1, f is differentiable on [0,1].

Show that $\frac{1}{f'(a)} + \frac{1}{f'(b)} = 2$

for some $a, b \in (0,1)$. (more…)

## 2009/03/29

### 大數值的乘階

$1! = 1$
$2! = 1 \times 2 = 2$
$3! = 1 \times 2 \times 3 = 6$
$4! = 1 \times 2 \times 3 \times 4 = 24$

$170! \approx 7.2574 \times 10^{306}$

## 2009/03/28

### 有關數學教師的話-堂上gag-Kai聞

Filed under: Fun,Life,Teaching — johnmayhk @ 12:01 上午
Tags: ,

「數學老師最緊要識 (more…)

## 2009/03/27

### 老師考學生，家長考老師

Filed under: School Activities — johnmayhk @ 12:02 上午

2009-03-21 (SAT) 是濟記的中一入學面試日。家長對我說，頗滿意面試安排，因為： (more…)

## 2009/03/26

### Just answer an easy additional mathematics question

Filed under: Additional / Applied Mathematics,HKCEE — johnmayhk @ 12:09 上午

A F.5 student asked me the following question some days ago, reply now.

A(-3,0) and B(-1,0) are two points and P(x,y) is a variable point such that $PA = \sqrt{3}PB$. Let C be the locus of P.

(a) Show that the equation of C is $x^2 + y^2 = 3$.

(b) T(a,b) is a point on C. Find the equation of the tangent to C at T.

(c) The tangent from A to C touches C at a point S in the second quadrant. Find the coordinates of S.

(d) L is a straight line which passes through point A and makes an angle $\theta$ with the positive $x$-axis, where $-\frac{\pi}{2} \le \theta \le \frac{\pi}{2}$. Q(x,y) is a point on L such that $AQ = r$. (See the figure below)

(i) Write down the coordinates of Q in terms of r and $\theta$.

(ii) L cuts C at two distinct points H and K. Let $AH = r_1$, $AK = r_2$.

(1) Show that $r_1, r_2$ are roots of the quadratic equation $r^2 - 6r\cos\theta + 6 = 0$.

(2) Find the range of possible values of $\theta$, giving your answers correct to three significant figures.

(HKCEE 1999) (more…)

## 2009/03/25

### 教學現場之中二數學之又玩計算機

Filed under: Junior Form Mathematics — johnmayhk @ 12:14 下午

「大家取出計算機，幫我計一計 $\theta$ 的數值是多少。」

$\cos\theta = \frac{\cos35^o}{\sqrt{5}}$ (more…)

## 2009/03/23

### 中二數學：小心使用計算機

Filed under: Junior Form Mathematics — johnmayhk @ 12:23 下午

## 2009/03/20

### Just a question about limit of elementary function

Filed under: HKALE,Pure Mathematics — johnmayhk @ 10:31 上午

The following theorem appears in secondary school pure mathematics textbook.

Let $F(x)$ be an elementary function. If $F(a)$ is well-defined, then

$\lim_{x \rightarrow a}F(x) = F(\lim_{x \rightarrow a}x) = F(a)$

Fine. Then a student, chan, asked,

How about $F(x) = \sqrt{x}$? Is it an elementary function? (more…)

## 2009/03/19

### Reply to a F.7 student

Filed under: Additional / Applied Mathematics,HKALE — johnmayhk @ 11:42 上午

To estimate $\int_0^1e^{-x^2}dx$, please refer to the normal table, you may see (more…)

## 2009/03/18

### Continuity of composite functions

Filed under: HKALE,Pure Mathematics — johnmayhk @ 7:50 上午

It is well-known that the following is NOT true in general,

$\lim_{x \rightarrow a}f(g(x)) = f(\lim_{x \rightarrow a}g(x))$

Just give an example, let (more…)

## 2009/03/17

### 簡報：數學教育研討會 2009

Filed under: Report — johnmayhk @ 5:37 上午
Tags: ,

2009-03-14 (SAT) 出席香港教育學院文理學院主辦《數學教育研討會 2009 數學學與教 – 東方的理論框架》

## 2009/03/16

### 是日劇照：證明終了

Filed under: Fun — johnmayhk @ 8:16 下午
Tags:

Too busy (and meaningless) to decorate the q.e.d. blog here, just come across the following occasionally…

The drama may be interesting, no idea.

When tidying up old files, I found the following old news:

May be some vistors here know him, I guess.

## 2009/03/13

### Kai の事記：微觀工作記錄

Filed under: Life,School Activities — johnmayhk @ 11:21 下午

2009-03-12（THU）教育局學位分配組把「學生成績次第表」（rank order list）發送給中學。 (more…)

## 2009/03/11

### 超開心

Filed under: Family — johnmayhk @ 10:42 下午