Increment theorem
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In
nonstandard analysis The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using epsilon–delta ...
, a field of mathematics, the increment theorem states the following: Suppose a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
is
differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
at and that is
infinitesimal In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referr ...
. Then \Delta y = f'(x)\,\Delta x + \varepsilon\, \Delta x for some infinitesimal , where \Delta y=f(x+\Delta x)-f(x). If \Delta x \neq 0 then we may write \frac = f'(x) + \varepsilon, which implies that \frac\approx f'(x), or in other words that \frac is infinitely close to f'(x), or f'(x) is the
standard part In nonstandard analysis, the standard part function is a function from the limited (finite) hyperreal numbers to the real numbers. Briefly, the standard part function "rounds off" a finite hyperreal to the nearest real. It associates to every suc ...
of \frac. A similar theorem exists in standard Calculus. Again assume that is differentiable, but now let be a nonzero standard real number. Then the same equation \Delta y = f'(x)\,\Delta x + \varepsilon\, \Delta x holds with the same definition of , but instead of being infinitesimal, we have \lim_ \varepsilon = 0 (treating and as given so that is a function of alone).


See also

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Nonstandard calculus In mathematics, nonstandard calculus is the modern application of infinitesimals, in the sense of nonstandard analysis, to infinitesimal calculus. It provides a rigorous justification for some arguments in calculus that were previously considered m ...
*'' Elementary Calculus: An Infinitesimal Approach'' *
Abraham Robinson Abraham Robinson (born Robinsohn; October 6, 1918 – April 11, 1974) was a mathematician who is most widely known for development of nonstandard analysis, a mathematically rigorous system whereby infinitesimal and infinite numbers were reincorpo ...
*
Taylor's theorem In calculus, Taylor's theorem gives an approximation of a ''k''-times differentiable function around a given point by a polynomial of degree ''k'', called the ''k''th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the t ...


References

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Howard Jerome Keisler Howard Jerome Keisler (born 3 December 1936) is an American mathematician, currently professor emeritus at University of Wisconsin–Madison. His research has included model theory and non-standard analysis. His Ph.D. advisor was Alfred Tarski a ...
: '' 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 * {{Infinitesimals Theorems in calculus Nonstandard analysis