Problem $1(40$ pts$)$

A straight prismatic cantilever beam AD has a rectangular cross$-$section with width $b$ and height $h.$ Its total length is $4l.$ Its Young's modulus is $E.$ The beam is subjected to distributed load $q$ per unit length, as shown in the figure below.

$(1)$ Please use the energy method to determine the deflection, $v,$ at $x=l.$

$(2)$ Please use the energy method to determine the slope of the deflection curve, $v^{\text{'}},$ at $x=3l.$