If $f$ and its derivatives, $f'$, $f''$, $f'''$, and so forth, exist in an interval containing a real number $a$, we may approximate $f(x)$ for $x$ near $a$ using a germ, short for polynomial germ of a smooth function; think “seed” not “pathogen”. The graph of a degree-one germ is a tangent line. As the degree increases, the approximation near $a$ only improves.
This program graphs germs up to degree eight at an arbitrary point, marked by a red dot. Type a function of $x$, and press TAB to graph. The left and right arrow keys move $a$. The up and down arrow keys increase or decrease the degree of approximation by one. The less-than and greater-than keys shrink or expand the domain.
Currently graphing the germ: