Increment theorem

Increment theorem 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 {displaystyle Delta y=f'(x),Delta x+varepsilon ,Delta x} for some infinitesimal ε, where {displaystyle Delta y=f(x+Delta x)-f(x).} If {textstyle Delta xneq 0} then we may write {displaystyle {frac {Delta y}{Delta x}}=f'(x)+varepsilon ,} which implies that {textstyle {frac {Delta y}{Delta x}}approx f'(x)} , or in other words that {textstyle {frac {Delta y}{Delta x}}} is infinitely close to {textstyle f'(x)} , or {textstyle f'(x)} is the standard part of {textstyle {frac {Delta y}{Delta x}}} .

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 {displaystyle Delta y=f'(x),Delta x+varepsilon ,Delta x} holds with the same definition of Δy, but instead of ε being infinitesimal, we have {displaystyle lim _{Delta xto 0}varepsilon =0} (treating x and f as given so that ε is a function of Δx alone).

See also Nonstandard calculus Elementary Calculus: An Infinitesimal Approach Abraham Robinson References 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 Robinson, Abraham (1996). Non-standard analysis (Revised ed.). Princeton University Press. ISBN 0-691-04490-2. hide vte Infinitesimals History AdequalityLeibniz's notationIntegral symbolCriticism of nonstandard analysisThe AnalystThe Method of Mechanical TheoremsCavalieri's principle Related branches Nonstandard analysisNonstandard calculusInternal set theorySynthetic differential geometrySmooth infinitesimal analysisConstructive nonstandard analysisInfinitesimal strain theory (physics) Formalizations DifferentialsHyperreal numbersDual numbersSurreal numbers Individual concepts Standard part functionTransfer principleHyperintegerIncrement theoremMonadInternal setLevi-Civita fieldHyperfinite setLaw of continuityOverspillMicrocontinuityTranscendental law of homogeneity Mathematicians Gottfried Wilhelm LeibnizAbraham RobinsonPierre de FermatAugustin-Louis CauchyLeonhard Euler Textbooks Analyse des Infiniment PetitsElementary CalculusCours d'Analyse Categories: CalculusNonstandard analysis