Consider the function $f(x)=3x-1on$ the interval $[-,).$

$(a)$ Sketch the periodic extension $off(x)$ over the interval $[-3,3).$

$(b)On(-,),$ let the Fourier series $off(x)be$

$a_0+{\displaystyle \underset{m=1}{\overset{}{}}}[a_{m}cos(mx)+b_{m}sin(mx)]$

Determine the constants $a_0,a_m,$ and $b_m$ for $m1.$ You may find the following $an-$

tiderivatives helpful $(arbitrary$ additive constants have been excluded$)$:

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}xcos(mx)dx=\frac{cos(mx)+xmsin(mx)}{m^2},{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}xsin(mx)dx=\frac{sin(mx)-xmcos(mx)}{m^2}.$