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

幫學生問的,urgent 的,先謝謝!

廣告

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

中四同學在學期初已接觸乘階(factorial) n! 的運算。比如

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

理論上,這個正整數 n,可以「要幾大,有幾大」;但實際上,我們日常接觸的運算工具,必有其限制;用 EXCEL 2003,只可以算出的最大乘階是

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

這個數字有多大? (more…)

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: ,

一些有關數學教師的對話

昨天,在上班的小巴上,兩名中壯年男士在我座位後高談:

「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

今天又是教學生使用計算機。

「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

剛開三角學,同學縱使寫對公式,卻因不小心運用計算機(指的是 CASIO fx-3650P 等相關型號),以致答案出錯,太「唔抵」了。

問題多數出於括弧的運用,實在太多同學忽略括弧了。

問題一:計算 3\sqrt{3} 的平方 (more…)

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》

講者及講題如下: (more…)

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…

qed

The drama may be interesting, no idea.

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

kwok-cc

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

晚上在校外的會議,得到別人的認同和稱許,實在是

超。開。心。 (more…)

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