道家, etc.

1. 道家

說這道家一教，乃是李老君騎青牛出關，關尹文始真人懇請留下道德真經五千言傳流至今。這家教門最上者，沖虛清靜，出有入無，超塵俗而上升，同天地而不老；其次者，修真煉性，吐故納新，築坎離以延年，煮鉛汞以濟物；最下者，行持符籙，役使鬼神，設章醮以通上界，建考召以達冥途。（拍案驚奇卷十七）

真簡潔.



2. Misconception: The path of a charge and the field lines.

Consider a positive charge moving in an electric field. Does the field line indicate the path it takes? No. The field line only shows the direction of the force acting on the charge, not the direction of the charge's motion. Think of a test charge rotating about a negative one.

How about a charge initially at rest? Will it move along the field line? The answer is no again, except when the charge happens to rest on a straight field line.



3. Log

Which of the following is the smallest?
a. 3^4000
b. 4^3000
c. 6^2000
d. 12^1000


解:

Consider their logarithm.
a => 4 ln 3
b => 6 ln 2
c => 2 ln 2 + 2 ln 3
d => 2 ln 2 + ln 3
Obvious, (d) is the smallest.



以下是另一解答, 很精采, 不是我做的:

Consider f(x) = x^{12/x} , where x is positive.
Noting x ln f(x) = 12 ln x , we get
f'(x) = (12/x - ln f(x)) f(x) / x ,
which equals 0 only when x ln f(x) = 12 ;
i.e. f(x) attains its maximum when x = e .
So, (d) is the smallest (x = 12 , most distant from e).

Further:
* 黎文傑老師: 卷二常考問題
* Solve a=5b and a^{log b}=2