How to graph Taylor polynomials by hand?

Question

What techniques exist for drawing Taylor polynomials by hand?

I've been pointed to have a look at a theorem about the $n$th derivative test and its proof (e.g. p.3. here), but I am unsure about how to use it.

Answer 1

There is no specific method to draw Taylor polynomials, which can be handled like ordinary polynomials.

Just one thing: at the point of evaluation, the Taylor polynomial (at least quadratic) has the same value, the same slope and the same curvature. Higher order constraints have no simple geometric interpretations better than "staying close to the true curve".

Below an illustration with the exponential and constant, linear and quadratic expansions.

The cubic approximation starts being undistinguishable.