Problem $3(15$ pts$.)$: Consider the RLC circuit given below, in which L and C have fixed values but R is variable. The switch in the circuit is closed at $t=0$ and the circuit is initially relaxed.

a$)(5$ pts$.)$ Use Laplace domain analysis to determine the expressions for $I(s)$ and $V_c(s).$

b$)(6$ pts$.)$ Select a numerical value for R such that the circuit to be overdamped. Explain your choice. Calculate $i(t)$ and $v_c(t)$ for $t>0$ and sketch the functions.

c$)(4$ pts$)$ Select a numerical value for R such that the circuit to be underdamped. Explain your choice. Calculate $i(t)$ for $t>0$ and sketch the function.