Problem #$3.(40$ points$).$ For the simply supported beam shown in Figure #$2$:

Figure #$2.$ Simply Supported Beam for Problem #$3.$

Find:

$(10$ points$).$ The reactions at the supports A and $B.$

$(10$ points$).$ The slope or rotation at the point $A,{}_A=\frac{dy}{dx},$ in radians.

$(10$ points$).$ Draw the beam's deflection along the length of the beam, from $x=0$ to $x=L.$

$(10$ points$).$ The beam's maximum deflection of the beam, $y_{\text{m}\text{a}\text{x}}.$

The governing equation of the elastic curve $($or deflection curve$)$ of a beam is given by:

Where:

$\frac{d^{2}y(x)}{d{x}^2}=\frac{M(x)}{EI}$

$y(x)$:$=$ is the deflection of the beam.

$M(x)$:$=$ is the internal moment as a function of $x.$

$EI$:$=$ is the beam's flexural rigidity $($assumed known and constant$).$