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[M2] more M2 exercise

int |x| dx = int x d(|x|)

using integration by parts, the answer = B



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回覆 16# cwk7093942 的帖子

but isnt it d|x|/dx = x/|x| ?

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For Q3, it's perhaps clearer to separate into 2 cases x>=0 and x<0.Notation like d|x| will make your presentation quite messy.

Let I be the integral.

Case 1: x>=0
I=integrate x dx
=x^2/2 + C1, where C1 is a constant

Case 2: x<0
I=integrate -x dx
=-x^2/2 + C2, where C2 is a constant

If I is continuous, then C1=C2.
Combining the results, we get
I=x|x|/2 +C, where C is a constant

[ 本帖最後由 桃子 於 2014-4-4 01:50 PM 編輯 ]

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引用:
原帖由 lead113 於 2014-4-3 11:06 PM 發表
thnx, graph is like this, dse 未必會考, depends on the level of difficulties in 2014 dse, add oil
This question can be attempted without knowing the exact shape of the function.
However, you need to be familiar with the behaviour of elementary functions at the 2 ends.

The easiest asymptote to determine is normally the vertical asymptote.
There is no denominator or "hidden denominator" here. => no vertical asymptote

Examples of "hidden denominator":
tan x, which is actually sin x/cos x
sec x, which is actually 1/cos x

Now, check oblique asymptote (including horizontal) at the 2 ends.
On the +inf side, the function is dominated by e^x, which grows exponentially. => no oblique asymptote
On the -inf side, e^x -> 0 and arctan x -> -pi/2. There is an asymptote y=0-1+4(-pi/2)=2pi-1

Generally speaking, if f(x) -> ax+b and g(x) -> cx+d, then p f(x) + q g(x) -> p(ax+b)+q(cx+d).

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唉, 今日midterm 出左f(x)=x^(2/x), find x-y intercept, asymptote, limit x-->0+/- , dy/dx, d^2y/dx, plot graph真是gg啦   (P.S. 我連x, y intercept 都唔知)
仲有條limit

x-->0 計到

(x^2)(cot^2(x))/(In(x^2) 唔知點做



[ 本帖最後由 lead113 於 2014-4-4 07:52 PM 編輯 ]

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for me as an M2 student...
(x^2)(cot^2(x)) when x->0
(x^2)(cot^2(x)) = (xcosx/sinx)^2
                          = (x/sinx)^2 (cosx)^2
                          = (1)(1)
                          = 1

唔知岩唔岩呢...
我剩係識呢PART...
(x^2)(cot^2(x))/(In(x^2) = 1/(2 lnx)
      唔知係咪not exist le...

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回覆 21# MrBenwong 的帖子

我都是話not exist , sor actually u are correct, u get zero!!

[ 本帖最後由 lead113 於 2014-4-4 08:12 PM 編輯 ]

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回覆 21# MrBenwong 的帖子

indeed, 我教d野仲易過 dse, 他d question 就難到.....唉, 拿星都冇用   , 都唔知d人點讀落去, actually this course is the simplest foundation course

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會唔會因為when x-->0, ln x --> -inf
所以係0 ?_?
唔好咁啦...你入到HKU都已經好勁啦...
我呢家..唉...都唔知入唔入到大學

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回覆 24# MrBenwong 的帖子

sor, 睇錯,u are right, 果時我唸唔到 (被L'H rule occupy 左), 放心啦, 只要肯讀一定入到, 大把人hea 下 hea 下 都得啦, 其實hku 一d都唔勁, 你讀到就知, 只知自己好似廢左好多, 算啦, 溫書, again dse add oil, take it easy (唔是串, 其實入U is not as difficult as u think, 我之前都是你今唸)

[ 本帖最後由 lead113 於 2014-4-4 08:25 PM 編輯 ]

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引用:
原帖由 MrBenwong 於 2014-4-4 07:59 PM 發表
(x^2)(cot^2(x))/(In(x^2) = 1/(2 lnx)
Note that ln(x^2)=2ln|x|.
Otherwise, you will lose the left-hand limit.

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new question

update
附件: 您所在的用戶組無法下載或查看附件

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回覆 27# lead113 的帖子

1
solve : (1/x) dx = -((1 + y)/(1 + y^2)) dy

2.
using integration by parts

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回覆 28# cwk7093942 的帖子

btw 忽然之間覺得師兄變到好cute teeee...  

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回覆 27# lead113 的帖子

第一條似係solve ODE
【DSE Maths】中大教育文憑 × 數學系畢業◆助你奪*升Lv~
首次以課題分班,10月開班招生,快d黎睇下~

導師簡介:按此進入

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