$(70$pts$)$ A fixed$-$end beam AB in Fig. $4$ is loaded by a force $P$ acting at the midpoint $C.$

$($a$)(50$pts$)$ Determine the moments at the ends of the beam using the beam deflection table below.

$($b$)(20$pts$)$ Determine the deflection at point $C$ where the load $P$ is applied.

Beam Deflection Table

$v=-\frac{P{x}^2}{6EI}(3L-x),v^{\text{'}}=-\frac{Px}{2EI}(2L-x)$

${}_B=\frac{P{L}^3}{3EI},{}_B=\frac{P{L}^2}{2EI}$

$v=-\frac{P{x}^2}{6EI}(3a-x),v^{\text{'}}=-\frac{Px}{2EI}(2a-x),(0xa)$

$v=-\frac{P{a}^2}{6EI}(3x-a),v^{\text{'}}=-\frac{P{a}^2}{2EI},(axL)$

Atx$=a$:$,v=-\frac{P{a}^3}{3EI},v^{\text{'}}=-\frac{P{a}^2}{2EI}$

${}_B=\frac{P{a}^2}{6EI}(3L-a),{}_B=\frac{P{a}^2}{2EI}$

$v=-\frac{M_0{x}^2}{2EI},v^{\text{'}}=-\frac{M_{0}x}{EI}$

${}_B=\frac{M_0{L}^2}{2EI},{}_B=\frac{M_{0}L}{EI}$