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Consider the function $\sin(x)$. Give an upper bound on the error of this Taylor polynomial for $x \in[0, \pi / 2]$.
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$$\sin (x)=P_{T}(x)+E_{T}(x)$$
where
$$E_{T}(x)=-\frac{\sin (\xi)}{6}(x-\pi / 4)^{3}$$
Since $|\sin (x)| \leq 1$ and $\left|(x-\pi / 4)^{3}\right| \leq(\pi / 4)^{3}$ for $x \in[0, \pi / 2]$, it follows that
$$\left|E_{T}(x)\right|=\frac{\pi^{3}}{6 \times 64}=0.08075$$

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