Final answer:
The degree 2 Taylor polynomial of f(x)=(1)/(x + 2) at a=0 is computed by evaluating the function and its first two derivatives at x=0. The resulting polynomial is P2(x) = 1/2 - (1/4)x + (1/8)x^2.
Step-by-step explanation:
Finding the Degree 2 Taylor Polynomial for f(x)=(1)/(x + 2) at a=0
To find the degree 2 Taylor polynomial of the function f(x)=(1)/(x + 2) at a=0, we need to compute the value of the function and its first two derivatives at x=0. Then, we'll use these values to construct the polynomial.
The function at x=0 is: f(0)=(1)/(0 + 2) = 1/2.
The first derivative of f(x) is: f'(x) = -1/(x + 2)^2, so f'(0) = -1/(0 + 2)^2 = -1/4.
The second derivative of f(x) is: f''(x) = 2/(x + 2)^3, thus f''(0) = 2/(0 + 2)^3 = 1/4.
Now, we use these values to construct the Taylor polynomial: P2(x) = f(0) + f'(0)x + (f''(0)x^2)/2.
Substituting the values we found, we get the Taylor polynomial: P2(x) = 1/2 - (1/4)x + (1/8)x^2.
This polynomial approximates f(x) near x=0.