Q$3.$ Derive the deflection equations for the beam shown below and find the maximum deflection of the beam as well as the deflection at its center, $C.(35$ points$)$

Hint:

Shear and Moment diagrams are provided so you don't have to cut the beam. However, to derive the deflection equations, you may use the diagrams to help you in writing the shear and moment equations. For example, to write the equations for segment $AB$ when $-300lbV=-300M=-300x{O}^{\text{'}},$ you can see that shear $is$ a constant value $of-300lb.$ Therefore, the equation for shear will $beV=-300.$ For the same segment, the moment $is$ a line and one can write its equation $asM=-300x$

Following this approach, you should write the shear and moment equations for remaining segments and then integrate them $to$ get the deflection equations.

Obviously, you can ignore the given diagram and cut the beam $in$ the $3$ segments $to$ directly derive the shear and moment diagrams.