Problem $3[27$ pts$]$

$(1)$ In circuit $A,$ the $s-$domain transfer function can be written as $G(s)=\frac{1+e^{-s}}{s^2+6s+8}.$ Please find the impulse response of the circuit. $[7$ pt$]$

$(2)$ In circuit B$,$ the voltage impulse response can be modeled by the triangular waveform shown below. The voltage input signal to this circuit $B$ is the step function $10u(t)[$V$].$ Please use the convolution integral to derive the expressions for the output voltage. $[17$ pt$]$ Then roughly sketch the output voltage over the interval $0$ to $15$ sec $.[3$ pt