Determine the values of $x$ for which the given linear approximation is accurate to within $0.07$ at $a = 0$.
$\tan x \approx x$
A) $- 0.19

Question

Determine the values of $x$ for which the given linear approximation is accurate to within $0.07$ at $a = 0$.
$\tan x \approx x$
A) $- 0.19 < x < 0.28$
B) $- 0.57 < x < 0.57$
C) $0.06 < x < 0.68$
D) $- 1.04 < x < 1.55$
E) $- 0.71 < x < 0.48$

Quizplus · Accepted Answer

The linear approximation $\tan x \approx x$ is accurate near $x = 0$. The accuracy of this approximation depends on the difference between $\tan x$ and $x$ being less than $0.07$. Since $\tan x$ grows faster than $x$ as $x$ moves away from $0$, the range of $x$ for which the approximation is accurate is symmetric around $0$. The given options suggest testing or understanding the behavior of $\tan x - x$ around these intervals. Option B, $-0.57 < x < 0.57$, is a reasonable range where the difference between $\tan x$ and $x$ is likely to stay within the given tolerance of $0.07$ for values of $x$ close to $0$, considering the gradual increase of $\tan x$ from its value at $0$.

Determine the Values Of xxx For Which the Given Linear Approximation Is Accurate to Within

Determine the Values Of $x$ For Which the Given Linear Approximation Is Accurate to Within