Increment theorem

In nonstandard analysis, a field of mathematics, the increment theorem states the following: Suppose a function Template:Math is differentiable at Template:Mvar and that Template:Math is infinitesimal. Then $$\mathrm{\Delta }y=f^{\prime }(x)\mathrm{\Delta }x+\epsilon \mathrm{\Delta }x$$ for some infinitesimal Template:Mvar, where $$\mathrm{\Delta }y=f(x+\mathrm{\Delta }x)-f(x).$$

If $\mathrm{\Delta }x\ne 0$ then we may write $$\frac{\mathrm{\Delta }y}{\mathrm{\Delta }x}=f^{\prime }(x)+\epsilon ,$$ which implies that $\frac{\mathrm{\Delta }y}{\mathrm{\Delta }x}\approx f^{\prime }(x)$, or in other words that $\frac{\mathrm{\Delta }y}{\mathrm{\Delta }x}$ is infinitely close to $f^{\prime }(x)$, or $f^{\prime }(x)$ is the standard part of $\frac{\mathrm{\Delta }y}{\mathrm{\Delta }x}$.

A similar theorem exists in standard Calculus. Again assume that Template:Math is differentiable, but now let Template:Math be a nonzero standard real number. Then the same equation $$\mathrm{\Delta }y=f^{\prime }(x)\mathrm{\Delta }x+\epsilon \mathrm{\Delta }x$$ holds with the same definition of Template:Math, but instead of Template:Mvar being infinitesimal, we have $$\underset{\mathrm{\Delta }x\to 0}{\lim }\epsilon =0$$ (treating Template:Mvar and Template:Math as given so that Template:Mvar is a function of Template:Math alone).

See also

• Nonstandard calculus
• Elementary Calculus: An Infinitesimal Approach
• Abraham Robinson
• Taylor's theorem

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
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