If we compose two functions f and g to get the function h, how do Taylor polynomials of h compare to the Taylor polynomials of f and g?

To explore this, let f(x) = sin(x), g(x) =x²+ 3 and h(x)=f(g(x))= sin(x²+ 3).

(a) What is the 4th degree Taylor polynomial of f around x= 4?

(b) What is the 4th degree Taylor polynomial of g around x= 1?

(c) What is the 4th degree Taylor polynomial of h around the point x= 1? Graph this and h.

(d) What is the composition of the above Taylor polynomials for f and g? Graph the composite polynomial and h.

(e) Which approximation of h seems better?

(f) Why is f approximated around x= 4? Graph the composition of f's Taylor polynomial about x= 1 with g(x), and comment on its usefulness.