Consider the circuit shown below. Express the transfer function tilde$(A{)}_v(s)$ in the following form.

$]=[tilde(V{)}_v\frac{s}{t}$ilde$(V{)}_i(s)]=[A_{v(LF)}[tilde(N)\frac{s}{t}$ilde$(D)(s)]]=[a_{N_z}{s}^{N_z}+$cdots$+a_3{s}^3+a_2{s}^2+a_{1}s+1]=[b_{N_P}{s}^{N_P}+$cdots$+b_3{s}^3+b_2{s}^2+b_{1}s+1$

$($a$)$ Derive an expression for the low$-$frequency gain $A_{v(LF)}$ and an expression for the high$-$frequency gain $A_{v(HF)}.(100$ pts$)$

$($b$)$ How many poles does the transfer function tilde$(A{)}_v(s)$ have? $(40$ pts$)$

$($c$)$ How many finite zeroes does the transfer function tilde$(A{)}_v(s)$ have? Which capacitors contribute to $a_1,a_2,a_3,$ etc.? Justify your answers. $(60$ pts$)$

$($d$)$ Using the $n$ Extra Element Theorem $($ nEEET $),$ derive an expression for the transfer function tilde$(A{)}_v(s)($i$.$e$.$ derive an expression for $b_1,a_1,$ etc.$).(200$ pts $)$