$($a$)$ Consider the function defined by

$f(x)=\{\begin{array}{ccc}3x+2& if& x-1\\ x^2-2& if& -1=2\end{array}$

Determine all of the points where $f(x)$ is continuous. Justify your answer.

$($b$)$ What is the domain of the function $g(x)=ln(x+5)?$

Determine all of the points on the graph of this function where the tangent is parallel to the line $x-7y-8=0.$

$($c$)$ Evaluate ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{t^2-3\sqrt[\phantom{2}]{t}}{\sqrt[3]{t}}dt.$

$($d$)$ Sketch the graphs of $y=5x$ and $y=x^2-6$ on the same diagram, showing their points of intersection with each other and with the coordinate axes. Find the area enclosed between these two graphs.

$($e$)$ Evaluate ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{1}{\sqrt[\phantom{2}]{8-6x-9x^2}}dx.$

$($f$)$ Use integration by parts to evaluate ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}ln(3r+1)dr.$

$($g$)$ Solve the initial$-$value problem

$\frac{dx}{(d)t}=e^{-2x}(4t^3+3),x(0)=ln\sqrt[\phantom{2}]{5.}$

Express your answer in the form $x=f(t).$

$2.($a$)$ Find the horizontal asymptotes of the function $y=\frac{2x}{\sqrt[\phantom{2}]{x^2+3}}.$

$($b$)$ An object with mass $m$ is dragged along a horizontal plane by a force acting along a rope attached to the object. If the rope makes an angle $$ with the plane then the magnitude of the force is

$F=\frac{cmg}{csin+cos},$

where $c>0$ is the coefficient of friction, and $g>0$ is the acceleration due to gravity. Both $c$ and $g$ are constant.

For which value of $$ is the value of $F$ smallest? $[$Your answer should depend on $c.]$