Question

Find the Taylor polynomial T3(x) for the function f centered at the number a. f(x) = xcos(x), a = 0 ) O a. 1 X--X x3 O b. 1 2x -x3 O c. 2x– x3 2 O d. 1 x + 3x3 X

Answer 1

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.