P$7\mathrm{.}1-$P$7\mathrm{.}7$

For the beams shown in Figures P$7\mathrm{.}1-$P$7\mathrm{.}7,$

$($a$)$ derive equations for the shear force $V$ and the bending moment $M$ for any location in the beam. $($Place the origin at point $A.)$

$($b$)$ use the derived functions to plot the shear$-$force and bending$-$moment diagrams for the beam.

$($c$)$ specify the values for key points on the diagrams.