taylor series

$f(a)+f^{\prime }(a)(x-a)+\frac{f^{\prime \prime }(a)}{2!}(x-a{)}^2+\cdots +\frac{f^n(a)}{n!}(x-a{)}^n$

problems

first, second taylors for $\sqrt{x}$ at $x=4$

$f(x)=\sqrt{a}+\frac{1}{2}{a}^{-\frac{1}{2}}(x-a)-\frac{\frac{1}{4}{a}^{-\frac{3}{2}}}{2}(x-a{)}^2$
$f(x)=2+0.25(x-4)-0.015625(x-4{)}^2$
$f(6)=2.437$, $\sqrt{6}=2.449$

$err\le \frac{max(|f^{(n+1)}(x)|)}{(n+1)!}|x-a{|}^{(n+1)}$
$|\frac{3}{8}{x}^{-\frac{5}{2}}|$ @ $x=4$ (for max deriv) $\to 0.0117$
$\frac{0.0117}{6}\cdot 2^3=0.0156$

fourth maclaurin polynomial for $cos(x)$

$f(x)=cos(0)+sin(0)x-\frac{cos(0)}{2!}{x}^2-\frac{sin(0)}{3!}{x}^3+\frac{cos(0)}{4!}{x}^4$
$f(\frac{\pi }{12})=0.965$, $cos(\frac{\pi }{12})=0.965$