碗粿 asked in 科學數學 · 1 decade ago

方程式所圍的面積與微分的關係..?

面積的微分就是原方程式

1.有證明嗎...

2.任何曲線下的面積都適用嗎?

Update:

我的課本是寫牛頓跟某某某發現的...

沒證明...不知道這本課本在想啥..

Update 2:

1.所以他是證明來的?

2.任何曲線下的面積都適用嗎?這你沒回答ㄟ!

Update 3:

喔喔~~!!

歹事ㄟ~!

讓你破戒

Update 4:

1.為什麼大部分課本不證明?

2.歹勢..因為書上是寫"fouction"音近方程式...而且我個人也沒有很注重函數跟方程式的差別...反正只要看到式子了解意思就好=.=

3 Answers

Rating
  • Anonymous
    1 decade ago
    Favorite Answer

    Fundamental theorem of calculus(Part I) says:

    圖片參考:http://img337.imageshack.us/img337/6302/calculus1r...

    If it is an Indefinite integral, and g'(x)=f(x), then

    圖片參考:http://img337.imageshack.us/img337/6544/calculus2m...

    If it is an definite integral, and a is the lower limit, b is the upper limit, then

    圖片參考:http://img412.imageshack.us/img412/7204/calculus3v...

    , g(b)-g(a) is a constant.

    About your 2th questions, if f is not a continuous function, the idea is wrong.

    If you find some mistakes, please tell me and I will add in the opinion column.

    2007-12-21 21:51:23 補充:

    Modify:

    About the area from a to b, and a, b be included in f, it is the type of definite integral which a and b is a constant. The result is a constant, so the differential is 0.

    2007-12-21 22:00:32 補充:

    What you say is an indefinite intrgral, so the differential is the ordinary function. If it is a definite integral, the type should be the type like the fundamental theorem of Calculus part 1.

    2007-12-21 22:00:53 補充:

    In fact, you say "面積的微分就是原方程式", it is strange, should modify"函數底下定積分面積的微分就是0", except the upper limit is x or u(x)

    else will add in the opinion column.

    2007-12-22 22:21:36 補充:

    任何曲線下的面積都適用嗎?→2007-12-21 22:00:32 補充

    After replying your question, I modify my answer.

    2007-12-21 22:00:53 補充→I say:

    "面積的微分就是原方程式" is wrong.

    First, the integral ∫f(x)dx, f(x) is a function, not an equation.

    Therefore, 面積的微分就是原方程式, 原"方程式" must be wrong.

    2007-12-22 22:45:41 補充:

    Second, the condition has two:

    1. It is a indefinite integral

    2. It is a definite integral, but the upper limit is x(if g'(u)=f(u), then is u)

    If the lower limit is x, then g'(x)= - f(x)

    Conclusion:

    面積的微分是原"函數",只在不定積分與符合微積分基本定理第一部份的前提下(這次是破戒,規定自己補充不能用中文)

    2007-12-23 20:34:52 補充:

    沒關係~反正這則第三次補充就破了

    或許我該說得更白一點

    證明因為你問了所以就寫了(貼圖內容),其實本來就可以證,一開始我也被你講的誤導了,想想不對~如果以上我們講的都是以積分來說,你的標題中方程式一定是錯的,方程式有等式,積分是對函數積不是對方程式,如果真是原函數就是符合基本定理1,∫f(x)dx可同∫f(t)dt應該不用說吧~不定積分看起來你一或應該比較少...因為就只是直接去掉,硬要證明也看到了,還蠻蠢的

    2007-12-25 12:17:20 補充:

    這我不清楚~不過關於微積分基本定理1的部分課本有說,他就是要證明這個性質的,g(x)只是方便指這個定積分罷了,由g'(x)=f(x)來說

    f(t)從a到x的定積分的微分是f(x)~對不定積分的部份,可能是講過反導函數的關係...我們課本先說反導函數再談積分,若求出反導函數後又微分回去就把動作倒回沒有積分的狀態了,差不多是這樣...證明可能是覺得不定積分的太簡單有概念就行了,要證明的確只有這樣而已~真要說哪本書有寫,我是翻我爸專科的用書才找到的...現在可能給你概念之後就自己處理了吧~除了基本定理還留著以外

    Source(s): Teacher's handout and me
  • 1 decade ago

    某某某....

    請記得萊布尼茲(Leibniz)

    http://www.mathland.idv.tw/history/leibniz.htm

  • 1 decade ago

    你問的問題應該是微積分中的微積分基本定理,你可以收尋一下相關的知識,相信你會找到你想要的解答。

    2007-12-21 10:41:03 補充:

    不知道你是用哪一本課本,是講關於哪方面的事情。台大數學系有個智識網站,你可以收尋一下微積分的發展歷史。應該可以找到你想要的答案。

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