Derive the nth order Taylor series expansion of f(x) = sin(x) expanded about x0 = 0. Call this polynomial Pn(x). Write a computer program to

evaluate Pn(x). Make a plot of Pn(x) versus f(x) for n = 10; 20; 40; 100;

0 x 2. Also derive a bound for the error Rn(x), call this function

Rbound

n (x), and plot jRbound

n (x)j versus the actual error E(x) = jf(x) ??

Pn(x)j.

Key constraints in your computer program are that I expect the code to evaluate Pn(x), for a given x, to be in its own subroutine and that Horner's algorithm (page 90, algorithm 2.7) is used to evaluate Pn(x).