differentiation

👨🏻‍🏫 學術台數學

[匿名]

2012-05-20 12:57 am

(A)
1.lim h->0 (h/h) = lim h->0 1 = 1
2.lim h->0 (h/h) = [lim h->0 (h)] / [lim h->0 (h)] = 0/0

Two different answers are obtained,seems that both are correct.
What's happened?

(B)
Find the slope of tangent to the curve x^2 + y^2 = -xy+4y+11 at (3,-1)
solution:
d/dx(x^2 +y^2) = d/dx(-xy+4y+11)
2x + 2y(dy/dx) = -x(dy/dx) -y +4(dy/dx)
dy/dx = (-2x-y)/(x+2y-4)
Next step:substitude x=3,y=1 into dy/dx.After that,the slope of tangent at (3,-1) will be found

I want to ask why dy/dx is the slope of tangent of the equation.
x^2 + y^2 = -xy+4y+11 is an implicit function,which cannot be expressed in terms of x,so why differentiating y with respect to x will give the slope of tangent?

更新1:

typo,substitude x=3,y=-1

回答 (4)

50418129

2012-05-21 9:23 pm

✔ 最佳答案

其實 (A) 2 第一個等號都可以是對的。問題是 0 / 0 等於甚麼呢？關於這個問題數學家做了多方面的解說，我只解說它可以等於 0，可以等於無限大，可以等於1，可以等於任何數，視乎它出現在甚麼情況。我們都統稱它為不定型。所以 (A) 1 與 (A) 2 是無衝突的。

參考: knowledge

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hung

2012-05-20 5:26 am

你所說的THM 前提是分母不可以為 0

2012-05-19 21:31:23 補充：
個THM 係 if lim x->a f(x) & lim x->a g(x) exist , and lim x->a g(x) does not equal zero.
then ,
lim x->a [f(x)/g(x)] = [lim x->a f(x)]/[lim x-> g(x)]

2012-05-19 21:35:58 補充：
Implicit Function Theorem 應該係大專程度 ( REAL ANALYSIS ) 會提及
運用係依題既話可以知道係 (3,-1) 附近 , y 可以 x 表示 , 即 y = f(x)

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Crystal

2012-05-20 4:38 am

恩,是我問的,謝謝你的意見
為甚麼A2的第1個等號是錯的?
by limit property
if lim x->a f(x) = L and lim x->a g(x) = M
then
lim x->a [f(x)/g(x)] = [lim x->a f(x)]/[lim x-> g(x)] = L/M
套在這裡,就是 lim h->0 (h) = 0,lim h->0 (h/h) = [lim h->0 (h)] / [lim h-> 0 (h)]
=0/0?

2012-05-19 20:48:44 補充：
(B)..不明白= =不好意思,只有高中程度

2012-05-19 22:13:34 補充：
(A)i have made a very careless mistake..thx hung

2012-05-19 22:18:03 補充：
(B)雖然不知道原因,
不過,謝謝回答

2012-05-21 17:38:51 補充：
根據definition
lim h->0 (h/h) = [lim h->0 (h)] / [lim h->0 (h)] 是錯的因為分母=0,
應該說根本就不能運用這個property
計算limit的目的是要找x趨向某一點時,y相應的數值
依你所說,0/0並沒有一個實質的數值,
所以,A2是不成立的.

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Copestone

2012-05-20 3:36 am

A2 第一個等號錯的。

2012-05-19 19:40:34 補充：
(B) 好問題。微分是 local properties，你可以證明 Implicit Function Theorem，詳細陳述請看高微課本。

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收錄日期: 2021-04-20 12:56:29
原文連結 [永久失效]:
https://hk.answers.yahoo.com/question/index?qid=20120519000051KK00434

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