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# Semi-monotonic functions

Whilst doing some work I stumbled upon the term "semi-monotonic", referring to approximations of mathematical functions. I couldn't find much information on this subject or even much explanation as to what it meant.

Eventually I found the definition in the Java Math class documentation.

Semi-monotonic functions are ones that increase or decrease in the same manner to their perfect mathematical equivalents, that is if the mathematical function $f(x)$ increases over a small change in $x$ then so does the approximation $f'(x)$, likewise $f'(x)$ decreases if $f(x)$ decreases over a small change in $x$.

## Example

Suppose that we are approximating $f(x)=\cos x$ as $f'(x)=1-\frac{x^2}{2}$. $f'(x)$ is semi-monotonic in the range $(-\pi,\pi)$ as shown; when $f(x)$ decreases, $f'(x)$ decreases.