作者anoymouse (沒有暱稱)
看板Math
標題[微積] continuous derivative 問題
時間Thu Mar 21 16:37:52 2024
Thomas' calculus 關於積分弧長，有先做一個前提:
Suppose the curve whose length we want to find is the graph of the function
y = f(x) from x= a to x = b.
In order to derive an integral formula for the length of the curve, we assume
that ƒ has a continuous derivative at every point of [a, b]. Such a function
is called smooth,and its graph is a smooth curve because it does not have any breaks, corners,
or cusps.
最後說到:
(1 + [f'(x)]^2)^1/2 is continuous on [a,b]
想問f has a continuous derivative ... of [a,b]的意思是
(1) f'(x) is continuous on [a,b] 還是
(2) f在[a,b]每一點都有derivative
如果是(2) 舉一個example
https://imgur.com/Wv2wDLl
g(x)每一點都有derivative 但是 g'(0)在x = 0 並不連續。
這樣的話 "(1 + [f'(x)]^2)^1/2 is continuous " 就不一定會成立吧?
所以原文的意思是應該是(1)?
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1F：推 cuteSquirrel: 您po的原文敘述有強調是平滑曲線，排除斷點 折角 03/21 18:46
2F：→ cuteSquirrel: "not have any breaks, corners or cusps." 03/21 18:47
3F：→ anoymouse : 所以你認為是（2）？ 03/21 20:33
4F：→ anoymouse : 我的舉例g of x雖然靠近0會劇烈跳動 但應該每個點都 03/21 20:33
5F：→ anoymouse : 可以微分且g'在0不連續吧 這樣不符合圓滑曲線嗎 03/21 20:33
6F：推 cuylerLin : 這是英文問題不是數學問題，a continuous derivativ 03/21 23:06
7F：→ cuylerLin : e 的意思是 a derivative that is continuous 也就 03/21 23:06
8F：→ cuylerLin : 是導數是一個連續函數。 03/21 23:06
9F：推 arrenwu : 就 (1) 03/21 23:12
10F：→ cmrafsts : 我還以為是要問在端點微分是什麼意思XD 03/22 04:19
11F：→ anoymouse : 好的 謝謝各位 03/22 08:39
12F：推 alan23273850: 當然是 1，語意蠻明確的 03/22 11:05