Find the Taylor polynomials ....

Question

Find the Taylor polynomials of degree 2, 3 generated by f(x)= (x+2)^-1/2 about x = -1

Answer 1

The Taylor Series up to degree 3 is as follows for f(x) about a:

f(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)²(1/2!) + f'''(a)(x-a)³(1/3!)

Here a is -1.

You need to find f, f', f'', and f''' evaluated at a (= -1)

f = (x+2)^-1/2

f' = -1/2 (x+2)^-3/2

f'' = 3/4 (x+2)^-5/2

f''' = -15/8 (x+2)^-7/2

Note that the derivatives at x = -1 will just be the coefficients. As a check: cubic term is (-15/48)(x+1)³

Good luck!