Hi! To find the Taylor polynomial T3(x) for the function f(x) = xcos(x) centered at a = 0, we will need to find the first few derivatives of f(x) and evaluate them at a = 0.
1. f(x) = xcos(x)
f(0) = 0 * cos(0) = 0
2. f'(x) = cos(x) - xsin(x)
f'(0) = cos(0) - 0 * sin(0) = 1
3. f''(x) = -2sin(x) - xcos(x)
f''(0) = -2 * sin(0) - 0 * cos(0) = 0
4. f'''(x) = -3cos(x) + xsin(x)
f'''(0) = -3 * cos(0) + 0 * sin(0) = -3
Now we can write T3(x) using the first 3 terms of the Taylor polynomial:
T3(x) = f(0) + f'(0)(x - 0) + f''(0)(x - 0)^2 / 2! + f'''(0)(x - 0)^3 / 3!
T3(x) = 0 + 1 * x + 0 * x^2 / 2 - 3 * x^3 / 6
T3(x) = x - x^3/2
So, the Taylor polynomial T3(x) for the function f(x) = xcos(x) centered at a = 0 is x - x^3/2, which corresponds to option c. 2x– x3 2.