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| In [[non-standard analysis]], a field of mathematics, the '''increment theorem''' states the following: Suppose a [[Function (mathematics)|function]] ''y'' = ''f''(''x'') is [[differentiable]] at ''x'' and that Δ''x'' is [[infinitesimal]]. Then
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| :<math>\Delta y = f'(x)\,\Delta x + \varepsilon\, \Delta x\,</math>
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| for some infinitesimal ε, where
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| : <math>\Delta y=f(x+\Delta x)-f(x).\,</math>
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| If <math>\scriptstyle\Delta x\not=0</math> then we may write
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| : <math>\frac{\Delta y}{\Delta x} = f'(x)+\varepsilon,</math>
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| which implies that <math>\scriptstyle\frac{\Delta y}{\Delta x}\approx f'(x)</math>, or in other words that <math>\scriptstyle \frac{\Delta y}{\Delta x}</math> is infinitely close to <math>\scriptstyle f'(x)\,</math>, or <math>\scriptstyle f'(x)\,</math> is the [[standard part function|standard part]] of <math>\scriptstyle \frac{\Delta y}{\Delta x}</math>.
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| == See also ==
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| *[[Non-standard calculus]]
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| *''[[Elementary Calculus: An Infinitesimal Approach]]''
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| *[[Abraham Robinson]]
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| == References ==
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| * [[Howard Jerome Keisler]]: ''[[Elementary Calculus: An Infinitesimal Approach]]''. First edition 1976; 2nd edition 1986. This book is now out of print. The publisher has reverted the copyright to the author, who has made available the 2nd edition in .pdf format available for downloading at http://www.math.wisc.edu/~keisler/calc.html
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| *{{cite book|last=Robinson|first=Abraham| title=Non-standard analysis|year=1996| edition=Revised edition | publisher=Princeton University Press| isbn = 0-691-04490-2}}
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| {{Infinitesimals}}
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| [[Category:Calculus]]
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| [[Category:Non-standard analysis]]
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Latest revision as of 00:03, 6 November 2014
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