# Quod Erat Demonstrandum

## 2009/09/29

### 今天雜記

Filed under: Life,mathematics,NSS,School Activities — johnmayhk @ 11:02 下午
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Given that $x$ and $y$ are purely imaginary numbers. If $(x + y) + (2x - y)i = 12 - i$, find (the values of) $x$ and $y$. (more…)

## 2009/09/27

### 奇異解

Filed under: Additional / Applied Mathematics,HKALE,HKCEE — johnmayhk @ 9:11 下午
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$\frac{dy}{dx} = f(x)$

$y = \int f(x)dx$ (more…)

## 2009/09/25

### Find dy/dx at a point not on the curve

Filed under: Additional / Applied Mathematics,HKCEE — johnmayhk @ 4:50 下午
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When distributing the marked test paper to students, one student, Carman, reminded me that there was a ‘question’ in the following question:

If $x^3 - 4x^2y + 3xy^2 - y^5 = 10$, find $\frac{dy}{dx}$ at the point ($-2,1$).

Carman said, ‘the point does NOT lie on the curve.’

Good observation! I had to say thank you to him. Although I’m not the setter, I should bear the responsibility of checking the paper.

But a natural follow-up question turns up: what is the meaning of the number $\frac{dy}{dx}|_{(-2,1)} = \frac{31}{33}$ we are obtaining? Is the number meaningless or standing for something?

Let’s consider a simple example. (more…)

## 2009/09/24

### Differentiation of parametric equations

It is extremely easy to set up questions about differentiation techniques (but good real life application questions are really rare, esp. at secondary school level), apart from tedious computation, when the differentiation involves parameter, students may have difficulties, like mistaking:

$\frac{d^2y}{dx^2} = 1/\frac{d^2x}{dy^2}$　（wrong！）
$\frac{d^2y}{dx^2} = \frac{d^2y}{dt^2}/\frac{d^2x}{dt^2}$　（wrong！）

Here is a question in recent quiz, which involves parametric equations: (more…)

## 2009/09/23

### .

Filed under: Uncategorized — johnmayhk @ 5:31 下午

## 2009/09/22

### Different formats of primitive functions

Filed under: HKALE,Pure Mathematics — johnmayhk @ 4:52 下午

Just take a rest from work, type something boring here…

$\int \frac{dx}{\sqrt{1 - x^2}} = \sin^{-1}x + C$

Students, you may regard the above as a formula or derive it by using trigonometric substitution $x = \sin\theta$ every time.

As you may know that the expression of a primitive is not unique, we may have other forms being a primitive of $\frac{1}{\sqrt{1 - x^2}}$, say (more…)

## 2009/09/14

### 類似地？

e.g. 1 循環小數

$0.3 \times 0.4 = 0.12$　正確，但不是「類似地」得到：

$0.\dot 3 \times 0.\dot 4 = 0.\dot 1\dot 2$（錯！） (more…)

## 2009/09/12

### [FW][Sad News] 香港數理教育學會主席吳重振先生辭世

Filed under: Report — johnmayhk @ 12:32 下午
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http://www.hkasme.org/CCNG.pdf

http://www.hkasme.org/Ng_CC2.pdf

## 2009/09/07

### 開學訓話

Filed under: Family,Life — johnmayhk @ 10:48 下午

《我對老師的訓話》

## 2009/09/06

### My first NSS math lesson

Filed under: Fun,Junior Form Mathematics,NSS — johnmayhk @ 12:01 上午

During the first NSS mathematics lesson in F.4D, I gave my students a piece of worksheet for practicing simple algebraic computation. (Well, I could collect money during the time when they were doing classwork)

Q.1 to 4 are factiorization problems: factorize the following

1. $1 - x^2(1 - 2x)^2$
2. $x^2 - 35x + 294$
3. $2x^2 + 6x - 15z - 2xy + 5yz - 5xz$
4. $x^4y^4 + x^2y^2 + 1$

Then I’d set the following questions in the worksheet and asked them to hand in as their homework:

5. What is your feeling about mathematics?
6. What is your expectation of mathematics teacher in this acedemic year?
7. What is your expectation on your own about the learning of mathematics? (more…)

## 2009/09/05

### 會議補充二則

Filed under: Additional / Applied Mathematics,HKALE,University Mathematics — johnmayhk @ 11:32 下午
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1.

“Introduction to Topological Manifolds" by John M. Lee (more…)

## 2009/09/01

### i 是開方負 1？

Filed under: mathematics,NSS — johnmayhk @ 11:58 下午
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$i = \sqrt{-1}$

$\sqrt{-1} = \sqrt{1} \times \sqrt{-1} = i$
but
$\sqrt{1} \ne \sqrt{-1} \times \sqrt{-1}$

$i$ 不是被定義為 $\sqrt{-1}$

$i$(more…)