Math Help - Taylor expansion

1. Taylor expansion

Give the Taylor expansion order 1 for the function :

and dudect that graph of f accept asymtot of which they ask
his psition and graph of f .

2. Originally Posted by dhiab
Give the Taylor expansion order 1 for the function :

and dudect that graph of f accept asymtot of which they ask
his psition and graph of f .
$f(a)=a^2\arctan\left( \frac{1}{a+1}\right)$

$f'(x)=2x\arctan\left( \frac{1}{x+1}\right)+2x\left(\frac{1}{(\frac{1}{1+ x})^2+1} \right)\left( \frac{-1}{(x+1)^2}\right)=$

$f'(x)=2x\arctan\left( \frac{1}{x+1}\right)-\left(\frac{2x}{(1+x)^2+1} \right)$

$f'(a)=2a\arctan\left( \frac{1}{a+1}\right)-\left(\frac{2a}{(1+a)^2+1} \right)$

Since you didn't say where it was centered I centered it at "a"

$T_1(x)=f(a)+f'(a)(x-a)$

$T_1(x)=a^2\arctan\left( \frac{1}{a+1}\right)+\left[2a\arctan\left( \frac{1}{a+1}\right)-\left(\frac{2a}{(1+a)^2+1} \right) \right](x-a)$