[DSE 2019] 你地溫 數學 又溫成點呢? [3]

1001 回覆
24 Like 21 Dislike
2019-01-03 03:53:59
Solve x > i

A more normal question:

Is 94^93 > 93^94? Explain briefly.

Answer:
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
Which leads to contradiction that
x^x -x^(x+1) < 0 for all x>1

QED
2019-01-03 03:54:09
Solve x > i

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

Is 94^93 > 93^94? Explain briefly.
Answer:
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
Which leads to contradiction that
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
Solve x > i

又係你

想 invent 一下清 concept 的題目
2019-01-03 04:00:39
Solve x > i
A more normal question:

Is 94^93 > 93^94? Explain briefly.
Answer:
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
Which leads to contradiction that
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

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
Solve x > i
A more normal question:

Is 94^93 > 93^94? Explain briefly.
Answer:
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
Which leads to contradiction that
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

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)

用Binomial Theorem做得唔得?
2019-01-03 04:06:21
Solve x > i
A more normal question:

Is 94^93 > 93^94? Explain briefly.
Answer:
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
Which leads to contradiction that
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

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)

用Binomial Theorem做得唔得?

Core level
2019-01-03 04:08:27
Another intuitive question:

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

Answer

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
Another intuitive question:

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

Answer

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.

有錯請指正
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?
Answer

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.
有錯請指正

錯,仲有a^i = e^(i ln a)
2019-01-03 04:17:41
Another intuitive question:

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

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.
有錯請指正

錯,仲有a^i = e^(i ln a)

How to show that a^i is multi-valued by your method?
2019-01-03 04:25:16
Another intuitive question:

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

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.
有錯請指正

錯,仲有a^i = e^(i ln a)

How to show that a^i is multi-valued by your method?

a^i = e^(i*ln a) = e^(i*(ln a + 2n*pi))
this a^i is multi-valued
仲有龜蛇你對complex number 咁有興趣
你正正經經搵本complex variables and application 讀下啦
2019-01-03 04:33:41
Another intuitive question:

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

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.
有錯請指正

錯,仲有a^i = e^(i ln a)

How to show that a^i is multi-valued by your method?

a^i = e^(i*ln a) = e^(i*(ln a + 2n*pi))
this a^i is multi-valued
仲有龜蛇你對complex number 咁有興趣
你正正經經搵本complex variables and application 讀下啦

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
How to show that a^i is multi-valued by your method?

a^i = e^(i*ln a) = e^(i*(ln a + 2n*pi))
this a^i is multi-valued
仲有龜蛇你對complex number 咁有興趣
你正正經經搵本complex variables and application 讀下啦

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?

complex variables 已經係ug數,你點砌都砌唔入core level啦
除非數學科聽日改革啦
btw change of base 咪一早有囉
2019-01-03 04:38:22
Some tricky question:

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

My answers:

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
a^i = e^(i*ln a) = e^(i*(ln a + 2n*pi))
this a^i is multi-valued
仲有龜蛇你對complex number 咁有興趣
你正正經經搵本complex variables and application 讀下啦

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?

complex variables 已經係ug數,你點砌都砌唔入core level啦
除非數學科聽日改革啦
btw change of base 咪一早有囉

所以我係天貓訂了一堆UG數天書過完年慢慢玩
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).
My answers:

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
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?

complex variables 已經係ug數,你點砌都砌唔入core level啦
除非數學科聽日改革啦
btw change of base 咪一早有囉

所以我係天貓訂了一堆UG數天書過完年慢慢玩
DSE 來說得 past paper 有挑戰性

巴打覺得大陸高考數難過dse數學幾多?
2019-01-03 04:45:17
http://www.red-publish.com/big5/book/2619

本野好薄, 但好撚貴
2019-01-03 04:46:06
complex variables 已經係ug數,你點砌都砌唔入core level啦
除非數學科聽日改革啦
btw change of base 咪一早有囉

所以我係天貓訂了一堆UG數天書過完年慢慢玩
DSE 來說得 past paper 有挑戰性

巴打覺得大陸高考數難過dse數學幾多?

大陸文科人要識 Fourier Transform, 你話呢?

Note: BEng 可以當係 BA
2019-01-03 04:47:28
所以我係天貓訂了一堆UG數天書過完年慢慢玩
DSE 來說得 past paper 有挑戰性

巴打覺得大陸高考數難過dse數學幾多?

大陸文科人要識 Fourier Transform, 你話呢?

Note: BEng 可以當係 BA

高考文科數學卷都要Fourier Transform?

咪吹水好喎?
2019-01-03 04:49:06
巴打覺得大陸高考數難過dse數學幾多?

大陸文科人要識 Fourier Transform, 你話呢?

Note: BEng 可以當係 BA

高考文科數學卷都要Fourier Transform?

咪吹水好喎?

Bachelor of Engineering = BEng
Bachelor of Art = BA
2019-01-03 04:50:06
大陸文科人要識 Fourier Transform, 你話呢?

Note: BEng 可以當係 BA

高考文科數學卷都要Fourier Transform?

咪吹水好喎?

Bachelor of Engineering = BEng
Bachelor of Art = BA

讀MBA, MPAcc, MPA 就可以避過迪無窮級數課程
吹水台自選台熱 門最 新手機台時事台政事台體育台娛樂台動漫台Apps台遊戲台影視台講故台健康台感情台潮流台上班台財經台房屋台飲食台旅遊台學術台校園台汽車台音樂台創意台硬件台攝影台玩具台寵物台軟件台活動台電訊台直播台站務台成人台黑 洞