Given information:
Instruction to describe the n th Taylor polynomial of f(x) at x = a.
Explanation:
To approximate given function f(x) by a polynomial for values of x near some number a. Since the behavior of f(x) near x = a is determined by the values of f(x) and its derivative at
x = a.
To approximate f(x) by a polynomial p(x) for which the values of p(x) and its derivative at x = a are the same as those of f(x). It can be done if the polynomial is of the form
p(x) = a0 + a1(x − a) + a1(x − a)2 + . . . + an(x − a)n, which is the polynomial in x − a .
Where a0, a1, a2 . . . an are given numbers and an ≠ 0.
It is easy to compute, and so on, because setting x = a in p(x) or one of its derivative makes most of the terms equal to zero.
The following result yield,
The n th Taylor polynomial of f(x) at x=a is the polynomial pn(x) is defined by
pn(x)=f(a)+f'(a)1!(x − a)+f"(a)2!(x − a)2+........+fn(a)n!(x − a)n.