Q. 11 - Calculus

Question

What is Taylor’s Theorem?

Step-by-Step Solution

Answer

If f be a function that can be differentiated (n+1) times in some open interval I that contains the point $x_{0 }\mathrm{and} \mathrm{also} R_n\left(x\right)$ represents the nth remainder for the function at $x=x_0$.

Step 1. Given Information

The given term is Taylor's polynomial

Step 2. Explanation

Taylor's theorem state that if f be a function that can be differentiated (n+1) times in some open interval I that contains the point $x_0 \mathrm{and} \mathrm{also} R_n\left(x\right)$ represents the nth remainder for the function at $x=x_0$.

Then the nth remainder for function is given by,

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