How do you find the largest interval #(c-r,c+r)# on which the Taylor Polynomial #p_n(x,c)# approximates a function #y=f(x)# to within a given error?

Question

How do you find  the largest interval #(c-r,c+r)# on which the Taylor Polynomial #p_n(x,c)# approximates a function #y=f(x)# to within a given error?

William Abdalla · Accepted Answer

Let us assume that there is #M>0# such that

#|f^{(n+1)}(x)| le M# for all #x#.

If we want the error to be less than #epsilon>0#, then

#|R_n(x;c)|=|{f^{(n+1)}(z)}/{(n+1)!}(x-c)^{n+1}|#,

where #z# is between #x# and #c#.

by replacing #|f^{(n+1)}(z)|# by #M#,

#le M/{(n+1)!}|x-c|^{n+1} < epsilon#

By solving for #|x-c|#,

#Rightarrow |x-c| < root(n+1){{epsilon (n+1)!}/M}#

Hence,

#r=root(n+1){{epsilon (n+1)!}/M}#

I hope you found this useful.

Paisley Johnson · Answer

undefined To find the largest interval $(c - r, c + r)$ on which the Taylor Polynomial $p_n(x, c)$ approximates a function $y = f(x)$ to within a given error, you can use the Taylor Remainder Theorem or Taylor's Inequality.

1. **Using Taylor Remainder Theorem**:
   Taylor Remainder Theorem states that if $f$ has $n+1$ continuous derivatives on an interval containing $c$ and $f^{(n+1)}(x)$ exists on $(c - r, c + r)$, then for each $x$ in $(c - r, c + r)$, there exists a number $z$ between $x$ and $c$ such that:
   $$ R_n(x) = \frac{f^{(n+1)}(z)}{(n+1)!}(x - c)^{n+1} $$
   Here, $ R_n(x) $ is the remainder term of the $n$th degree Taylor polynomial. You can choose $r$ such that the error $|R_n(x)|$ is less than or equal to the desired error.

2. **Using Taylor's Inequality**:
   Taylor's Inequality provides an upper bound on the error of the Taylor polynomial approximation. It states that if $f$ has $n+1$ continuous derivatives on an interval containing $c$ and $f^{(n+1)}(x)$ exists on $(c - r, c + r)$, then for each $x$ in $(c - r, c + r)$:
   $$ |f(x) - p_n(x, c)| \leq \frac{M}{(n+1)!}|x - c|^{n+1} $$
   where $ M $ is an upper bound for $ |f^{(n+1)}(x)| $ on $(c - r, c + r)$. You can choose $ r $ such that $ \frac{M}{(n+1)!}|x - c|^{n+1} $ is less than or equal to the desired error.

In both cases, you may need to analyze the behavior of the function and its derivatives to find suitable values for $r$.