QUESTION $1$ A cooling coil is integrated inside a stirred tank reactor to release heat generated during a reaction process. To maintain a constant reactor temperature, a proportional controller is utilized to control the flow rate of the coolant. The controller has been built in a manner that the regulated reactor displays common underdamped second$-$order temperature responses when it is disrupted by the flow rate of the feed or by the temperature changes of the coolant. $1\mathrm{.}1$ What is the gain of the transfer function $($under feedback control$)$ that relates the changes in the reactor temperature to that of the flow rate of the feed if the flow rate of the feed to the reactor changes from $0\mathrm{.}4$k$($g$)/($s$)$ to $0\mathrm{.}5$k$($g$)/($s$)$ and the reactor temperature changes from $100$deg C to $102$deg C$?1\mathrm{.}2$ What is the complete transfer function, if it is observed that the response is oscillating with a maximum of $102\mathrm{.}5$deg C and $102$deg C taking place at $1000$ seconds and $3060$ seconds after the change has started? $($NB$.$ Use the analytical approach to get your answers$).1\mathrm{.}3$ If the operator failed to notice the rise time, estimate t based on the results in question $1\mathrm{.}1$ and $1\mathrm{.}2.$ QUESTION $2$ Given the transfer function below: G$($s$)=($Y$($s$))/($U$($s$))=(10)/((5$s$+1)(3$s$+1))$ What is y$($t$->\backslash$infty $)$ for the subsequent inputs: $2\mathrm{.}1$ Step input of height M $2\mathrm{.}2$ Unit impulse input $$ta $($t$)2\mathrm{.}3$sin$($t$)2\mathrm{.}4$ Unit rectangular pulse, where h$=1)$ fi QUESTION $33$ Use the Taylor series to linearize the following $($show all your steps$)$: $3\mathrm{.}1$ f$($x$)=5$x$^(2)-2$x$,$ where x$=33\mathrm{.}2$ f$($x$)=-3$x$^(3)+30$x$,$ where x$=-3$