$(13$ pts$)$ Consider the following function.

$f(x)=xln(2x),a=1,n=3$

$($a$)$ Approximate $f$ by a Taylor polynomial with degree $n$ at the number $a.$

$P_3(x)=$

$($b$)$ Use the Theorem we learned in class to approximate the error $(R_3(x))$ for $f(x)$~~$P_n(x)$ when $x=1\mathrm{.}4.($Round your answer to four decimal places.$)|R_3(x)|$