# [DSE 2019] 你地溫 數學 又溫成點呢？ 

## dse杏林

1001 回覆
24 Like 21 Dislike
2019-01-03 03:53:59
Put x=93, we have:
(x+1)^x > x^(x+1)
(x+1)^x - x^(x+1) >0
divide both side by (x+1)^x/x^x, we have:
x^x - x^(2x+1)/(x+1)^x > x^x - x^(2x+1)/x^x = x^x -x^(x+1) >0
x^x -x^(x+1) < 0 for all x>1

QED
2019-01-03 03:54:09

2019-01-03 03:57:30
Solve x > i
A more normal question:

Is 94^93 > 93^94? Explain briefly.
Put x=93, we have:
(x+1)^x > x^(x+1)
(x+1)^x - x^(x+1) >0
divide both side by (x+1)^x/x^x, we have:
x^x - x^(2x+1)/(x+1)^x > x^x - x^(2x+1)/x^x = x^x -x^(x+1) >0
x^x -x^(x+1) < 0 for all x>1

QED

Manipulation mistake I mean the joint inequality:
0< x^x - x^(2x+1)/x^x< x^x - x^(2x+1)/(x+1)^x
2019-01-03 03:57:49

2019-01-03 04:00:39
Key point:

If

x^x - x^(2x+1)/x^x <0

Then since (x+1)^x/x^x must be positive, we have:

(x+1)^x < x^(x+1)
2019-01-03 04:05:07
Another intuitive question:

Let f(x) = a^x where a >2
Then we have f(i) = a^i.
Is it correct?
2019-01-03 04:05:25

2019-01-03 04:06:21
Core level 2019-01-03 04:08:27

i = i^(4n+1)
a^i = a^i^(4n+1) = a^(4n+1)^i
for any real integer n>0

Then a^i is multi-valued, which defies the definition of function
So f(i) is not defined.
2019-01-03 04:09:28

2019-01-03 04:14:44
Some tricky question:

Without out-C method, show that the shortest distance between y = 10^x and y = log_10(x) is sqrt(2).
2019-01-03 04:16:35
Another intuitive question:

Let f(x) = a^x where a >2
Then we have f(i) = a^i.
Is it correct?

i = i^(4n+1)
a^i = a^i^(4n+1) = a^(4n+1)^i
for any real integer n>0

Then a^i is multi-valued, which defies the definition of function
So f(i) is not defined.

2019-01-03 04:17:41
How to show that a^i is multi-valued by your method?
2019-01-03 04:25:16
a^i = e^(i*ln a) = e^(i*(ln a + 2n*pi))
this a^i is multi-valued

2019-01-03 04:33:41
I just exploring how my question can be incorporated into core level. You know, core syllabus has no Euler's number and natural logarithm, so shall we introduce change of base?
2019-01-03 04:36:56
complex variables 已經係ug數，你點砌都砌唔入core level啦

btw change of base 咪一早有囉
2019-01-03 04:38:22

By inspection:

10^x converges when 10^x <1 as x is decreasing
For y = log_10(x),
we can see x = 10^y converges when y<1 as y is increasing.

By method of locus:

The perpendicular distance is given by the locus y=x.

So the points (1,0) and (0,1) yields the smallest distance
2019-01-03 04:40:33

DSE 來說得 past paper 有挑戰性
2019-01-03 04:43:51
Some tricky question:

Without out-C method, show that the shortest distance between y = 10^x and y = log_10(x) is sqrt(2).

By inspection:

10^x converges when 10^x <1 as x is decreasing
For y = log_10(x),
we can see x = 10^y converges when y<1 as y is decreasing.

By method of locus:

The perpendicular distance is given by the locus y=x.

So the points (1,0) and (0,1) yields the smallest distance

typo 2019-01-03 04:45:00

2019-01-03 04:45:17
http://www.red-publish.com/big5/book/2619

2019-01-03 04:46:06

Note: BEng 可以當係 BA
2019-01-03 04:47:28

2019-01-03 04:49:06
Bachelor of Engineering = BEng
Bachelor of Art = BA
2019-01-03 04:50:06