Determine the degree of the Maclaurin polynomial required for the error in the approximation of the function $\sin ( 0.6 )$ to be less than $0.001$.
A) 4
B) 2
C) 5
D) 1
E) 3

Question

Quizplus · Accepted Answer

The Maclaurin series for $\sin(x)$ is $x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots$. To ensure the error is less than 0.001, we use the remainder term $R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!}x^{n+1}$. For $\sin(x)$, the maximum value of the derivative is 1. Testing $n=4$, the error term is $\frac{0.6^5}{5!} \approx 0.00096$, which is less than 0.001. Thus, a degree of 4 is required.

Determine the Degree of the Maclaurin Polynomial Required for the Error