Question

Question #07fe2

Answer 1

See explanation.

Explanation:

A Taylor polynomial of order m is simply a partial Taylor series, where the sum stops at the term of order m (i.e. when we have the 'm'th derivative of f(x) and where we have `(x-a)^m`

Recall that the formula of a Taylor series centered at `x=a` is:

`sum_(n=0)^(oo) (f^((n))(a))/(n!) (X-a)^n`

Thus, for a third order Taylor polynomial we would have:

`f(a)/1*(1) + (f'(a))/1(x-a) + (f''(a))/(2!)(x-a)^2 + (f'''(a))/(3!) (x-a)^3`

Of note, by taking the representation at a=, we very quickly get back our initial function