The document provides solutions for determining the third Taylor polynomial of two functions f(x) and f(x) = 1/x+3 at x = 0. For f(x), the polynomial is calculated to be 1 + (-1/2)x + (1/8)x^2+ (-1/48)x^3 based on finding the derivatives of f(x) at 0. For f(x) = 1/x+3, binomial expansion is used to derive the polynomial as 1/3 - x/9 + x^2/27 - x^3/81.

1. Determine the third Taylor polynomial of the given function at x = 0.
Solution
a)
f(0) = 1
f'(0) = -1/2
f''(0) = 1/4
f'''(0) = -1/8
so, third polinomial of f(x) = 1 + (-1/2)x + (1/8)x^2+ (-1/48)x^3
b)f(x) = 1/x+3
= 1/3(1+ x/3) = 1/3(1+x/3)^-1 = 1/3 - x/9 + x^2/27 - x^3/81