Find the third Taylor polynomial about x = 1 for f(x) =1+2(X-1)-(X-1)^2+(X-1)^3

The Taylor polynomial of a function about a point can be found using the formula:

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

Given the function f(x) = 1+2(x-1)-(x-1)^2+(x-1)^3, we want to find the third Taylor polynomial about x = 1.

First, we need to find the derivatives of the function at x = 1.

f'(x) = 2 - 2(x-1) + 3(x-1)^2 f''(x) = -2 + 6(x-1) f'''(x) = 6

At x = 1, these become:

f'(1) = 2 - 2(0) + 3(0)^2 = 2 f''(1) = -2 + 6(0) = -2 f'''(1) = 6

Then, we substitute these values into the Taylor polynomial formula:

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

P(x) = 1 + 2(x-1) - (x-1)^2 + 2(x-1)^3

This is the third Taylor polynomial for the given function about x = 1.