# How do you use a Taylor series to expand: f(x) = x^2 + 2x + 5 about x = 3?

Jun 2, 2015

If $f \left(x\right) = {x}^{2} + 2 x + 5$
then
$\textcolor{w h i t e}{\text{XXXXX}}$$f ' \left(x\right) = 2 x + 2$
$\textcolor{w h i t e}{\text{XXXXX}}$$f ' ' \left(x\right) = 2$
$\textcolor{w h i t e}{\text{XXXXX}}$$f ' ' ' \left(x\right) \text{ and beyond} = 0$

The Taylor Series about $x = 3$ is
$\textcolor{w h i t e}{\text{XXXXX}}$f(3)+(f'(3))/(1!)(x-3)+(f''(3))/(2!)(x-3)^2

$\textcolor{w h i t e}{\text{XXXXX}}$$= 20 + 8 \left(x - 3\right) + {\left(x - 3\right)}^{2}$