Q. 7 - Calculus

Question

What is a difference between the Maclaurin polynomial of order n and the Taylor polynomial of order n for a function f ?

Step-by-Step Solution

Answer

$$\begin{gathered} \mathrm{From} \mathrm{Maclaurin} \mathrm{series} \mathrm{and} \mathrm{Taylor} \mathrm{series}, \mathrm{a} \mathrm{Maclaurin} \mathrm{polynomial} \mathrm{is} \mathrm{approximation} \\ \mathrm{to} \mathrm{the} \mathrm{function} \mathrm{f} \mathrm{at} \mathrm{x}=0 \mathrm{while} \mathrm{a} \mathrm{Taylor} \mathrm{polynomial} \mathrm{at} {\mathrm{x}}_0 \mathrm{is} \mathrm{approximation} \mathrm{to} \mathrm{the} \\ \mathrm{function} \mathrm{f} \mathrm{at} \mathrm{x} = {\mathrm{x}}_0. \end{gathered}$$

Step 1. Given information is:

A function f with the Maclaurin polynomial of order n and the Taylor polynomial of order n 

Step 2. Maclaurin polynomial

$$\begin{gathered} \mathrm{For} \mathrm{any} \mathrm{function} \mathrm{f} \mathrm{with} \mathrm{derivatives} \mathrm{of} \mathrm{all} \mathrm{orders} \mathrm{at} \mathrm{the} \mathrm{point} {\mathrm{x}}_0 = 0, \\ \mathrm{then} \mathrm{the} \mathrm{Maclurin} \mathrm{polynomial} \mathrm{of} \mathrm{degree} \mathrm{n} \mathrm{is}, \\ \mathrm{f}\left(\mathrm{x}\right)=\mathrm{f}\left(0\right)+\mathrm{f}\text{'}\left(0\right) \mathrm{x}+\frac{\mathrm{f}\text{'}\text{'}\left(0\right)}{2!} {\mathrm{x}}^2+\frac{\mathrm{f}\text{'}\text{'}\text{'}\left(0\right)}{3!} {\mathrm{x}}^3+....+\frac{{\mathrm{f}}^{\mathrm{n}} \left(0\right)}{\mathrm{n}!} {\mathrm{x}}^{\mathrm{n}} \\ \mathrm{The} \mathrm{general} \mathrm{form} \mathrm{Maclurin} \mathrm{polynomial} \mathrm{of} \mathrm{degree} \mathrm{n} \mathrm{of} \mathrm{the} \mathrm{function} \mathrm{f} \mathrm{is}: \\ \mathrm{f}\left(\mathrm{x}\right)=\sum _{\mathrm{k}=0}^{\mathrm{n}}\frac{{\mathrm{f}}^{\mathrm{k}} \left(0\right)}{\mathrm{k}!} {\mathrm{x}}^{\mathrm{k}} \end{gathered}$$

Step 3. Taylor polynomial

$$\begin{gathered} \mathrm{For} \mathrm{any} \mathrm{function} \mathrm{f} \mathrm{with} \mathrm{derivative} \mathrm{of} \mathrm{order} \mathrm{n}, \mathrm{then} \mathrm{the} \mathrm{Taylor} \\ \mathrm{polynomial} \mathrm{of} \mathrm{degree} \mathrm{n} \mathrm{at} \mathrm{x}={\mathrm{x}}_0 \mathrm{is}, \\ {\mathrm{P}}_{\mathrm{n}}(\mathrm{x}={\mathrm{x}}_0)=\mathrm{f}\left({\mathrm{x}}_0\right) +\mathrm{f}\text{'}\left({\mathrm{x}}_0\right)(\mathrm{x}-{\mathrm{x}}_0)+\frac{\mathrm{f}\text{'}\text{'}\left({\mathrm{x}}_0\right)}{2!}{(\mathrm{x}-{\mathrm{x}}_0)}^2+....+\frac{{\mathrm{f}}^{\mathrm{n}}\left({\mathrm{x}}_0\right)}{\mathrm{n}!}{(\mathrm{x}-{\mathrm{x}}_0)}^{\mathrm{n}} \\ \mathrm{The} \mathrm{general} \mathrm{form} \mathrm{Taylor} \mathrm{polynomial} \mathrm{of} \mathrm{degree} \mathrm{n} \mathrm{of} \mathrm{the} \mathrm{function} \mathrm{f} \mathrm{is}: \\ {\mathrm{P}}_{\mathrm{n}}\left(\mathrm{x}\right)=\sum _{\mathrm{k}=0}^{\mathrm{n}}\frac{{\mathrm{f}}^{\mathrm{k}}\left({\mathrm{x}}_0\right)}{\mathrm{k}!}{(\mathrm{x}-{\mathrm{x}}_0)}^{\mathrm{k}} \end{gathered}$$

Step 4. Result

$$\begin{gathered} \mathrm{From} \mathrm{Maclaurin} \mathrm{series} \mathrm{and} \mathrm{Taylor} \mathrm{series}, \mathrm{a} \mathrm{Maclaurin} \mathrm{polynomial} \mathrm{is} \mathrm{approximation} \\ \mathrm{to} \mathrm{the} \mathrm{function} \mathrm{f} \mathrm{at} \mathrm{x}=0 \mathrm{while} \mathrm{a} \mathrm{Taylor} \mathrm{polynomial} \mathrm{at} {\mathrm{x}}_0 \mathrm{is} \mathrm{approximation} \mathrm{to} \mathrm{the} \\ \mathrm{function} \mathrm{f} \mathrm{at} \mathrm{x} = {\mathrm{x}}_0. \end{gathered}$$