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 $[0,1]$?
Best Answer
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 $f(x)$.