Question

use the error bound for taylor polynomials to give a good bound for the error for the nth degree taylor polynomial about x = 0 approximating sin(x) on the interval [0, 1.4]

Answer 1

To find a good bound for the error of the nth degree Taylor polynomial approximating sin(x) on the interval [0, 1.4], we can use the Lagrange Error Bound formula.

Step-by-step explanation:

To find a good bound for the error of the nth degree Taylor polynomial approximating sin(x) on the interval [0, 1.4], we can use the Lagrange Error Bound formula. The Lagrange Error Bound for the nth degree Taylor polynomial is given by: E ≤ M(x-a)^(n+1)/(n+1)!, where M is the maximum value of the (n+1)st derivative of sin(x) on the interval [0, 1.4] and a = 0. Since sin(x) is always between -1 and 1, the value of M will depend on the (n+1)st derivative of sin(x). The (n+1)st derivative of sin(x) is equal to sin(x) if (n+1) is even, and cos(x) if (n+1) is odd.

Let's consider the case where (n+1) is even. In this case, the maximum value of the (n+1)st derivative of sin(x) is 1, since sin(x) is always between -1 and 1. Therefore, we have: E ≤ (1)(1.4)^(n+1)/(n+1)!.

Let's consider the case where (n+1) is odd. In this case, the maximum value of the (n+1)st derivative of sin(x) is also 1, since the absolute value of cos(x) is always between 0 and 1. Therefore, we have: E ≤ (1)(1.4)^(n+1)/(n+1)!.