Provide an appropriate response.
-If $\sin x$ is replaced by $x - \frac { x ^ { 3 } } { 6 }$ and $| x |

Question

Provide an appropriate response.
-If $\sin x$ is replaced by $x - \frac { x ^ { 3 } } { 6 }$ and $| x | < 0.5$, what estimate can be made of the error?
A) $| \mathrm { E } | < 0.00052083$
B) $| \mathrm { E } | < 0.00026042$
C) $| \mathrm { E } | < 0.0026042$
D) $| E | < 0.0013021$

Quizplus · Accepted Answer

The error term can be found by taking the difference between the original function and the approximation: 
\begin{align*}
\sin x - \left(x - \frac{x^3}{6}ight) &= \sum_{n=1}^{\infty} (-1)^{n-1} \frac{x^{2n+1}}{(2n+1)!} - x + \frac{x^3}{6} \
&= \sum_{n=2}^{\infty} (-1)^{n-1} \frac{x^{2n+1}}{(2n+1)!}
\end{align*}
Since we are given that $|x|<0.5$, we can estimate the error by using the first term of the series:
\begin{align*}
\left|(-1)^{2-1} \frac{x^{2(2)+1}}{(2(2)+1)!}ight| &= \frac{|x^5|}{5!} \
&= \frac{|x|^5}{120} \
&< \frac{0.5^5}{120} \
&< 0.00026042
\end{align*}
Thus, the best choice for the estimate of the error is B.

Provide an Appropriate Response sin⁡x\sin xsinx Is Replaced By x−x36x - \frac { x ^ { 3 } } { 6 }x−6x3​

Provide an Appropriate Response $\sin x$ Is Replaced By $x - \frac { x ^ { 3 } } { 6 }$