09 May 2014
Edit

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 $x$.
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.