Polynomial approximation for a function using Maclaurin's Theorem:

$f(x)=f(0)+x.f^{\prime }(0)+{\displaystyle \frac{x^2}{2!}}.f^{″}(0)+{\displaystyle \frac{x^3}{3!}}.f^{‴}(0)+\dots +{\displaystyle \frac{x^n}{n!}}.f^n(0)$

Example 01

Expand the function $f(x)=\sin x$ in ascending powers up to and including the term in $x^3$

$$\begin{array}{rl}& f^{\prime }(x)=\cos x,f^{″}(x)=-\sin x,f^{‴}(x)=-\cos x\\ \\ & \sin x=\sin 0+x.\cos 0+\frac{x^2}{2!}.-\sin 0+\frac{x^3}{3!}.-\cos 0\\ \\ & \sin x=0+x+0-\frac{1}{6}{x}^3+\dots \\ \\ & \sin x=x-\frac{1}{6}{x}^3+\dots \end{array}$$

Example 02

Expand the function $f(x)=e^x$ in ascending powers up to and including the term in $x^3$

$$\begin{array}{rl}& f^{\prime }(x)=e^x,f^{″}(x)=e^x,f^{‴}(x)=e^x\\ \\ & e^x=e^0+x.e^0+\frac{x^2}{2!}.e^0+\frac{x^3}{3!}.e^0+\dots \\ \\ & e^x=1+x+\frac{1}{2}{x}^2+\frac{1}{6}{x}^3+\dots \end{array}$$