Q. 10 - Calculus

Question

If f(x) is an nth-degree polynomial and $P_n\left(x\right)$ is the nth Taylor polynomial for f at $x_0$, what is the nth remainder $R_n\left(x\right)$? What is $R_{n+1}\left(x\right)$? 

Step-by-Step Solution

Answer

Thus, the required remainders are $R_n\left(x\right)=\frac{f^{n+1}\left(c\right)}{(n+1)!}{(x-x_0)}^{n+1} and R_{n+1}\left(x\right)=\frac{f^{n+2}\left(c\right)}{(n+2)!}{(x-x_0)}^{n+2}$

Step 1. Given Information

The given data is If f(x) is an nth-degree polynomial and $P_n\left(x\right)$ is the nth Taylor polynomial for f at $x_0$

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 c between $x_{0 }and x$ such that,

$$\begin{gathered} R_n\left(x\right)=\frac{f^{n+1}\left(c\right)}{(n+1)!}{(x-x_0)}^{n+1} \\ \mathrm{Now}, (n+1)th \mathrm{remainder} \mathrm{is}, \\ {R}_{n+1}\left(x\right)=\frac{f^{n+1+1}\left(c\right)}{(n+1+1)!}{(x-x_0)}^{n+1+1} \\ {R}_{n+1}\left(x\right)=\frac{f^{n+2}\left(c\right)}{(n+2)!}{(x-x_0)}^{n+2} \end{gathered}$$