QuestionFind $T_{5}(x)$, the degree 5 Taylor polynomial of the function $f(x)=\cos (x)$ at $a=0$. $$T_{5}(x)=$$ $\square$ Find all values of $x$ for which this approximation is within 0.004794 of the right answer. Assume for simplicity that we limit ourselves to $|x| \leq 1$. $$|x| \leq \square$$ ote: You can earn partial credit on this problem.

Studdy Solution
To find where the approximation is within 0.004794 of the right answer, we need to solve:
|x|⁶/720 ≤ 0.004794
720 * 0.004794 = |x|⁶ 3.45168 = |x|⁶ |x| ≤ ∛∛3.45168 |x| ≤ 0.9999...
Since we're already limited to |x| ≤ 1, our final answer is:
|x| ≤ 1
This means the approximation is within 0.004794 of the true value for all x in the given interval [-1, 1].
The Taylor polynomial T₅(x) for cos(x) at a = 0 is:
T₅(x) = 1 - (1/2)x² + (1/24)x⁴
And the range of x for which this approximation is within 0.004794 of the right answer, given |x| ≤ 1, is:
|x| ≤ 1