Mean-Value Theorem, Taylor's formula, and error estimation

[mcastillo356]

TL;DR Summary

Can't conclude any error estimation from MVT, i.e., Taylor's formula for $n=0$

Hi, PF

Taylor's formula provides a formula for the error in a Taylor approximation $f(x)\approx{P_{n}(x)}$ similar to that provided for linear approximation.

Observe that the case $n=0$ of Taylor's formula, namely,

$f(x)=P_{0}(x)+E_{0}(x)=f(a)+\dfrac{f'(s)}{1!}(x-a)$,

is just the Mean-Value Theorem

$\dfrac{f(x)-f(a)}{x-a}=f'(s)$ for some $s$ between $a$

and $x$

The question is: to what extent, i.e. how, MVT provides a formula for the error in a Taylor approximation?

Attempt: let $f(x)=e^{x}$, $x=2$, $a=0$: for $n=0$

1- $\dfrac{e^{2}-1}{2}=e^{s}$
2- $\ln{\dfrac{e^{2}-1}{2}}=s\cdot{\ln{e}}\Rightarrow$
3- $\Rightarrow{s\approx{1.1614}}$
4- $\therefore{f(s)\approx{3.1944}}\Rightarrow$
5- All I can conclude is that the slope of the chord line joining the points $(0,1)$ and $(2,7.3890)$ is equal to the slope of the tangent line to the curve $y=e^x$ at the point $(1.1614,3.1944)$, so the two lines are parallel.
6- Which is here the error estimation, or how can I find out?

Greetings



PS: I post with no preview. Twice edited

 

[valenumr]

MVT says there is at least one point between x and a that has a tangent egual to the slope of the line through f(x) and f(a) (the secant). For a quadratic, that point will always be exactly be the mid point between x and a (on the x axis). For higher order functions, the error depends on the delta and the higher order terms in the function.

 

[mcastillo356]

MVT says there is at least one point between x and a that has a tangent egual to the slope of the line through f(x) and f(a) (the secant). For a quadratic, that point will always be exactly be the mid point between x and a (on the x axis).

Didn't know that fact for quadratics. Nice

For higher order functions, the error depends on the delta and the higher order terms in the function.

Sorry, could this be explained furthermore?

Love, thanks @valenumr

 

[fresh_42]

https://en.wikipedia.org/wiki/Taylor's_theorem#Explicit_formulas_for_the_remainder