$8\mathrm{.}14.$ Consider the PID algorithm in equation $(8\mathrm{.}23$a$).$ For each of the individual modes$$proportional$,$ integral, and derivative$$describe with a sketch the result of its calculation when the error is each of the following idealized functions: $($a$)$ a constant, $($b$)$ an impulse, and $($c$)$ a sine $($consider one cycle$).$

$($This question provides a thought exercise to help understand the three PID modes; this type of analysis is not performed when monitoring a control system.$)$