Final answer:
A Taylor polynomial of degree 3 for the function f(x) = x^(-1/2) centered at a = 4 involves calculating the first, second, and third derivatives of the function at x = 4 and plugging them into the formula for the Taylor polynomial.
Step-by-step explanation:
You have asked to approximate the function f(x) = x-1/2 by a Taylor polynomial of degree n = 3 centered at a = 4. The general formula for the Taylor polynomial of a function at a point a is given by:
P(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)2/2! + f'''(a)(x-a)3/3! + ... + f(n)(a)(x-a)n/n!
First, we need to compute the derivatives of f(x) at x = 4. Here's the first derivative:
f'(x) = (-1/2)*x-3/2
Plugging x = 4 gives f'(4) = (-1/2)*4-3/2 which simplifies to -1/(4√4).
Similarly, you can find the second and third derivatives and evaluate them at x = 4. Finally, you substitute these values into the Taylor polynomial formula to get your approximation.