AP Calculus BC / Unit 10: Infinite Sequences and Series
10.12 Lagrange Error Bound
Notes
$$|f(x)-P_n(x)|\le\frac{M}{(n+1)!}|x-a|^{n+1}$$
where $M$ is the maximum of $|f^{(n+1)}|$ between $a$ and $x$.
Practice
Using $P_2$ for $e^x$ at $x=0.1$ (with $M\le e^{0.1}<2$), the error is at most about:
$n+1=3$.
In the Lagrange bound, $M$ bounds which derivative?
The first derivative not used in $P_n$.
For $\sin x$, a convenient $M$ for any derivative is:
Every derivative is $\pm\sin$ or $\pm\cos$.
Ready for more? Take the Unit 10 test.