$(20$ points$)$ Consider the first$-$order analogue lowpass filter in the $s-$domain:

$H(s)=\frac{64}{s+8}$

$($a$)$ What is the maximum value of the magnitude of $H(j)?$

$($b$)$ Find the analog cut$-$off frequency ${}_c$ analytically? Show your work step by step.

$($c$)$ Assuming $T_s=0\mathrm{.}1$ second, use the impulse invariance method to find the approximate $H(z).$

$($d$)$ Assuming $T_s=0\mathrm{.}05$ second, use the step invariance method to find the approximate $H(z).$ please don't use ai