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I just would like to check my work with someone else's:

The function f has derivatives of all orders for all real numbers, and the fourth derivative of f equals e^(sin(x)). If the third-degree Taylor polynomial for f about x=0 is used to approximate f on the interval [0,1], what is the Lagrange error bound for the maximum error on the interval [0,1]?

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We have that the remainder $R_4(x)$ is given by $$R_4(x)=\frac{1}{4!}f^{(4)}(\xi)(x-0)^4,$$ where $\xi$ is between $0$ and $x$.

Worst case for $x$ is $x=1$. An upper bound for the fourth derivative on the interval $[0,1]$ is $e^{\sin(1)}$. So an upper bound for the error is $\dfrac{e^{\sin(1)}}{4!}$.

Note that $f(x)$ is equal to the third degree approximation plus the term quoted at the beginning. So the third degree approximation underestimates $f(x)$.

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