Provide an appropriate response.
-For approximately what values of $x$ can $\sin x$ be replaced by $x - \frac { x ^ { 3 } } { 6 } + \frac { x ^ { 5 } } { 120 }$ with an error of magnitude no greater than $5 \times 10 ^ { - 6 } ?$
A) $| x |

Question

Provide an appropriate response.
-For approximately what values of $x$ can $\sin x$ be replaced by $x - \frac { x ^ { 3 } } { 6 } + \frac { x ^ { 5 } } { 120 }$ with an error of magnitude no greater than $5 \times 10 ^ { - 6 } ?$
A) $| x | < 0.59106$
B) $| x | < 0.45942$
C) $| x | < 0.54146$
D) $| x | < 0.40357$

Quizplus · Accepted Answer

The expression $x - \frac { x ^ { 3 } } { 6 } + \frac { x ^ { 5 } } { 120 }$ is the Taylor series expansion of $\sin x$ around $x = 0$ up to the $x^5$ term. The error in approximating a function by its Taylor series is given by the remainder term in Taylor's theorem. For the sine function, the next term in the series (which gives an indication of the error) is of the form $\frac{x^7}{7!}$. The condition for the error to be less than $5 \times 10^{-6}$ can be derived by setting $\frac{x^7}{7!} < 5 \times 10^{-6}$ and solving for $x$. The exact boundaries depend on solving this inequality, and among the given options, $|x| < 0.59106$ is the correct choice, indicating that within this range, the approximation will have an error less than the specified amount.

Provide an Appropriate Response xxx Can sin⁡x\sin xsinx Be Replaced By x−x36+x5120x - \frac { x ^ { 3 } } { 6 } + \frac { x ^ { 5 } } { 120 }x−6x3​+120x5​

Provide an Appropriate Response $x$ Can $\sin x$ Be Replaced By $x - \frac { x ^ { 3 } } { 6 } + \frac { x ^ { 5 } } { 120 }$