Q. 8 - Calculus

Question

Given a function f and a Taylor polynomial for f at $x_0$, what is meant by the nth remainder $R_n\left(x\right)$? What does $R_n\left(x\right)$ measure? 

Step-by-Step Solution

Answer

For each point $x\in I$, there is atleast one c between $x_{0 }and x$ such that,

$R_n\left(x\right)=\frac{f^{n+1}\left(c\right)}{(n+1)!}{\left(x-x_0\right)}^{n+1}$

Step 1. Given Information

The given term is Taylor polynomial.

Step 2. Explanation

Consider a function f that can be differentiated (n+1) times in some open interval I that contains the point $x_0$ and $R_n\left(x\right)$ be the n remainder for f at $x=x_0$.

Hence, for each point $x\in I$, there is at least one point c between $x_{0 }and x$

such that $R_n\left(x\right)=\frac{f^{n+1}\left(c\right)}{(n+1)!}{\left(x-x_0\right)}^{n+1}$