$\text { Determine the values of } x \text { for which the function } f ( x ) = \sin x \text { can be replaced by the }$ Taylor polynomial $f ( x ) = \sin x \approx x - \frac { x ^ { 3 } } { 3 ! }$ if the error cannot exceed $0.008$. Round your answer to four decimal places.
A) $- 0.9919

Question

$\text { Determine the values of } x \text { for which the function } f ( x ) = \sin x \text { can be replaced by the }$ Taylor polynomial $f ( x ) = \sin x \approx x - \frac { x ^ { 3 } } { 3 ! }$ if the error cannot exceed $0.008$. Round your answer to four decimal places.
A) $- 0.9919 < x < 0.9919$
B) $- 1.9838 < x < 1.9838$
C) $- 0.2480 < x < 0.2480$
D) $- 1.9919 < x < 1.9919$
E) $- 0.4960 < x < 0.4960$

Quizplus · Accepted Answer

The error in the Taylor series approximation of a function can be estimated using the remainder term of the series. For the sine function approximated by $f(x) = x - \frac{x^3}{3!}$, the next term in the series (which gives an estimate of the error) is $\frac{x^5}{5!}$. To ensure the error does not exceed 0.008, we set $\frac{x^5}{5!} \leq 0.008$ and solve for $x$. Solving this inequality gives the range for $x$ that satisfies the condition, which corresponds to the range in option A.

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