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A Certain Consideration on Derivatives and Rolle's Theorem
https://kitami-it.repo.nii.ac.jp/records/6402
https://kitami-it.repo.nii.ac.jp/records/6402d40017f2-3415-4de2-bbad-128fb1477f4c
名前 / ファイル | ライセンス | アクション |
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7-2-13.pdf (1.1 MB)
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Item type | 紀要論文 / Departmental Bulletin Paper(1) | |||||
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公開日 | 2007-04-09 | |||||
タイトル | ||||||
タイトル | A Certain Consideration on Derivatives and Rolle's Theorem | |||||
言語 | ||||||
言語 | eng | |||||
資源タイプ | ||||||
資源 | http://purl.org/coar/resource_type/c_6501 | |||||
タイプ | departmental bulletin paper | |||||
著者 |
ISOBE, Kiro
× ISOBE, Kiro |
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抄録 | ||||||
内容記述タイプ | Abstract | |||||
内容記述 | Let f(x) be differentiable, α<x<β. We putr(x, y) = { (f(x)?f(y))/(x?y), x≠y, f´(x), x=y (1)We have the following :(i) r(x, y)=r(y, x),(ii) r(x, z)(x?z)=r(x, y)(x?y)+r(y, z)(y?z),(iii) <lim>___x→y r(x, y)=r(y, y),(iv) r is continuous at a point (x, y) of x≠y. Newly, having no relation to (1), let r(x, y) be a real function of 2-varia-bles, α<x, y<β. In [1], if r(x, y) satisfies the previous (i), (ii), (iii), (iv), then we obtain Rolle’s theorem and mean value theorem, namely, (v) if r(a, b)=0,a<b, then there exists c(a<c<b) such that r(c, c)=0,(vi) for a,b(a<b), there exists c(a<c<b) such that r(a, b)=r(c, c).The purpose of this note is to point out that if r(x, y) satisfies (i), (ii), (iii), then we have (iv), (v), (vi) and Cauchy's theorem, namely,(vii) if a function s(x, y) satisfies (i), (ii), (iii) and s(x, y)≠0, (a≦x, y≦b) for a<b, then there exists c(a<c<b) such that (r(a, b))/(s(a, b)) = (r(c, c))/(s(c, c)). | |||||
書誌情報 |
北見工業大学研究報告 巻 7, 号 2, p. 441-443, 発行日 1976-06 |
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内容記述タイプ | Other | |||||
内容記述 | application/pdf | |||||
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値 | publisher | |||||
出版者 | ||||||
出版者 | 北見工業大学 |