The Exercise:
Calculate the Taylor polynomial of degree 3 of $f(x,y,z)=x^5y^4z^3$ at $(1,1,1)$ in an arbitrary direction $h$. Use Taylor's theorem to get a bound on the remainder when using this polynomial to estimate $f(2,1,1)$.
Note from the professor: "The bound required is for using the second order Taylor polynomial, so that the remainder is given by the third derivative."
Definitions:
My work:
$$T_{3,h}(1,1,1)(t)=f(1,1,1)+D_{1,h}f(1,1,1)(t)+D_{2,h}f(1,1,1)t^2/2+D_{3,h}f(1,1,1)t^3/6$$
I basically have nothing. I don't understand the definition of the monomial coefficient nor the definition of $D{r,h}f(x)$ (by the way, how would I format this correctly?). I would appreciate any hints, attempts to clarify the definitions, and/or links to/illustrations of similar example problems with solutions.