About: Increment theorem

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In nonstandard analysis, a field of mathematics, the increment theorem states the following: Suppose a function y = f(x) is differentiable at x and that Δx is infinitesimal. Then for some infinitesimal ε, where If then we may write which implies that , or in other words that is infinitely close to , or is the standard part of . A similar theorem exists in standard Calculus. Again assume that y = f(x) is differentiable, but now let Δx be a nonzero standard real number. Then the same equation

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rdfs:label	Increment theorem (en) 増分定理 (ja) Teorema do incremento (pt)
rdfs:comment	数学の一分野、超準解析における増分定理（ぞうぶんていり、英: increment theorem; 増分の定理）は、無限小に対する可微分函数の増分が微分係数に無限に近いことを述べるものである。これを通常の微分積分学（標準解析）において述べたものは実質的に平均値の定理（有限増分の定理、あるいは一次の場合のテイラーの定理）である。 (ja) Em análise não padronizada, um campo da matemática, o teorema do incremento estabelece que: suponha que uma função y = f(x) é diferenciável em x e que Δx é infinitesimal. Então para um infinitesimal ε, sendo Se então podemos escrever implicando que , ou em outras palavras que é infinitamente perto de , ou é a de . (pt) In nonstandard analysis, a field of mathematics, the increment theorem states the following: Suppose a function y = f(x) is differentiable at x and that Δx is infinitesimal. Then for some infinitesimal ε, where If then we may write which implies that , or in other words that is infinitely close to , or is the standard part of . A similar theorem exists in standard Calculus. Again assume that y = f(x) is differentiable, but now let Δx be a nonzero standard real number. Then the same equation (en)
dcterms:subject	Nonstandard analysis Calculus
Wikipage page ID	12144610 (xsd:integer)
Wikipage revision ID	1059983151 (xsd:integer)
Link from a Wikipage to another Wikipage	Elementary Calculus: An Infinitesimal Approach Howard Jerome Keisler Function (mathematics) Nonstandard analysis Abraham Robinson Nonstandard analysis Calculus Differentiable function Infinitesimal Standard part function Nonstandard calculus
Link from a Wikipage to an external page	http://www.math.wisc.edu/~keisler/calc.html
sameAs	Increment theorem Increment theorem Increment theorem Increment theorem Increment theorem
dbp:wikiPageUsesTemplate	dbt:Cite_book dbt:Math dbt:Mvar dbt:Infinitesimals
has abstract	In nonstandard analysis, a field of mathematics, the increment theorem states the following: Suppose a function y = f(x) is differentiable at x and that Δx is infinitesimal. Then for some infinitesimal ε, where If then we may write which implies that , or in other words that is infinitely close to , or is the standard part of . A similar theorem exists in standard Calculus. Again assume that y = f(x) is differentiable, but now let Δx be a nonzero standard real number. Then the same equation holds with the same definition of Δy, but instead of ε being infinitesimal, we have(treating x and f as given so that ε is a function of Δx alone). (en) 数学の一分野、超準解析における増分定理（ぞうぶんていり、英: increment theorem; 増分の定理）は、無限小に対する可微分函数の増分が微分係数に無限に近いことを述べるものである。これを通常の微分積分学（標準解析）において述べたものは実質的に平均値の定理（有限増分の定理、あるいは一次の場合のテイラーの定理）である。 (ja) Em análise não padronizada, um campo da matemática, o teorema do incremento estabelece que: suponha que uma função y = f(x) é diferenciável em x e que Δx é infinitesimal. Então para um infinitesimal ε, sendo Se então podemos escrever implicando que , ou em outras palavras que é infinitamente perto de , ou é a de . (pt)
prov:wasDerivedFrom	wikipedia-en:Increment_theorem?oldid=1059983151&ns=0
page length (characters) of wiki page	2283 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Increment_theorem
is Link from a Wikipage to another Wikipage of	Elementary Calculus: An Infinitesimal Approach Increment Theorem List of theorems
is Wikipage redirect of	Increment Theorem
is foaf:primaryTopic of	wikipedia-en:Increment_theorem

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