Question

Let T₂(x) be the Taylor polynomial of degree 2 for the function : f(x) = cos(x) at a = 0. Suppose you approximate f(x) by T₂(x). If |x| ≤ 1, what is the bound for your error of your estimate?

Answer 1

The Taylor polynomial T₂(x) of degree 2 for the function f(x) = cos(x) at a = 0 is 1 - x². The bound for the error of the estimate |R₂(x)| is f'''(c)/6, where c is some value between 0 and x.

Step-by-step explanation:

The Taylor polynomial T₂(x) of degree 2 for the function f(x) = cos(x) at a = 0 is given by:

T₂(x) = f(0) + f'(0)(x - 0) + f''(0)(x - 0)²

For cos(x), we have f(0) = cos(0) = 1, f'(0) = -sin(0) = 0, and f''(0) = -cos(0) = -1.

Therefore, the Taylor polynomial is:

T₂(x) = 1 + 0(x - 0) + (-1)(x - 0)² = 1 - x²

The error of the estimate can be determined using the remainder term in the Taylor series expansion, which is given by:

R₂(x) = f'''(c)(x - 0)³/3!

where c is some value between 0 and x. Since |x| ≤ 1, the maximum value of |x - 0| is 1. Therefore, the bound for the error of the estimate, |R₂(x)|, is:

|R₂(x)| ≤ f'''(c)(1)³/3! = f'''(c)/6