Increment theorem explained

In nonstandard analysis, a field of mathematics, the increment theorem states the following: Suppose a function is differentiable at and that is infinitesimal. Then$$\backslash Delta\; y\; =\; f\text{'}(x)\backslash ,\backslash Delta\; x\; +\; \backslash varepsilon\backslash ,\; \backslash Delta\; x$$for some infinitesimal, where$$\backslash Delta\; y=f(x+\backslash Delta\; x)-f(x).$$

If $\backslash Delta\; x\; \backslash neq\; 0$ then we may write$$\backslash frac\; =\; f\text{'}(x)\; +\; \backslash varepsilon,$$which implies that $\backslash frac\backslash approx\; f\text{'}(x)$, or in other words that $\backslash frac$ is infinitely close to $f\text{'}(x)$, or $f\text{'}(x)$ is the standard part of $\backslash frac$.

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$$\backslash Delta\; y\; =\; f\text{'}(x)\backslash ,\backslash Delta\; x\; +\; \backslash varepsilon\backslash ,\; \backslash Delta\; x$$holds with the same definition of, but instead of being infinitesimal, we have$$\backslash lim\_\; \backslash varepsilon\; =\; 0$$(treating and as given so that is a function of alone).

See also

• Nonstandard calculus
• Abraham Robinson
• Taylor's theorem

References

• Howard Jerome Keisler: . 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
• Book: Robinson, Abraham. Non-standard analysis. 1996. Revised . Princeton University Press . 0-691-04490-2.