Taylor series and their application

3.5 Taylor series and their application

We have a ”normal” function $f(x)$

$f(x)\approx f(x_0)+(x-x_0)f'(x)\;\rightarrow\;$ linear approximation!!

Approximation of this function by a power series: \begin{eqnarray*}f(x)&=&f(x_0)+\frac{1}{1!}(x-x_0)f'(x_0)+\frac{1}{2!}(x-x_0)^2f''(x_0)+\ldots+\frac{1}{n!}(x-x_0)^nf^{n}(x_0)+\ldots\\ &=&\sum_{n=0}^\infty\frac{1}{n!}(x-x_0)^nf^{n}(x_0)\;\;\mbox{Taylor Series around the point $x_0$}\\ \Rightarrow f(x)&\approx&f(x_0)+(x-x_0)f'(x_0)+\mbox{O}((x-x_0)^2)\;\;\mbox{linear approximation}\\ f(x)&\approx&f(x_0)+(x-x_0)f'(x_0)+\frac{1}{2}(x-x_0)^2f''(x_0)\;\;\mbox{quadratic approximation}\\ &\Rightarrow&\mbox{two most important approximations in physics because:} \end{eqnarray*}

Potential:

3.5.1 Example: Taylor expansion of exponential function
3.5.2 Example: Taylor expansion of sin function
3.5.3 Example: Taylor expansion of cos function
3.5.4 Example: Taylor expansion of the logarithm function