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\begin{center}
{\Large TEST 1

Math 2210 - 2}
\end{center}

Instructions:
\begin{itemize}
\item  You have 50 minutes to complete this exam.
\item  Show all your work.  If I can't follow your work or it is missing you WILL NOT get credit.
\item  Use the back of these sheets if you run out of room.  Make sure I can find your work.
\item  Make sure your work is neat.  You are responsible for me being able to read your work.
\item  No calculators or notes.
\item  Unless specified, you may use any method to solve the problems, just make sure the 
	mathematics and reasoning behind your answer is correct.
\item  The test is out of 60 points, there are two extra credit points you can also get.
\end{itemize}

\hrule

\begin{enumerate}

\item (8 points) Draw a plane curve that satisfies the conditions:
(so you will draw 4 curves, one for each set of conditions)
\begin{enumerate}
\item  Simple and closed
\item  Simple but not closed
\item  Not simple, but closed
\item  Not simple, not closed
\end{enumerate}


\item (10 points) Consider the parametric curve defined by the parametric equations:
\begin{eqnarray*}
x(t) & = & 2t-1 \\
y(t) & = & t-4
\end{eqnarray*}
\begin{enumerate}
\item  Eliminate the parameter $t$ to get an equation in $x$ and $y$.
\item  Graph the curve for $0 \leq t \leq 5$.  Indicate the initial and end points on your graph.
\end{enumerate}


\item (10+2 points) Consider the parametric curve given by the equations below:
\begin{eqnarray*}
x(t) & = & e^{2t}-5t \\
y(t) & = & \sin(t)+2
\end{eqnarray*}
\begin{enumerate}
\item  Find the slope of a line tangent to the curve when $t=0$.
\item  Are there any points on the curve where the tangent line is horizontal?  Justify your answer.
\item  Are there any points on the curve where the tangent line is vertical?  Justify your answer.
\item  (Extra Credit)  Is there an interval $a \leq t \leq b$ so that the curve is not a simple curve?
	Justify your answer!
\end{enumerate}

\item (10 points)  Consider the vectors below:
\begin{eqnarray*}
\mbf{v} & = & \mbf{i} + 3\mbf{j} \\
\mbf{u} & = & \sqrt{3}\mbf{i} - \frac{\sqrt{3}}{6}\mbf{j}
\end{eqnarray*}
Find the cosine of the angle between $\mbf{v}$ and $\mbf{u}$.


\item (10 points) Consider a particle moving in the plane with position vector given by
\[
\mbf{r}(t) = e^{-3t}\mbf{i} - (2t+1)^3 \mbf{j}
\]
\begin{enumerate}
\item  Find the position of the particle at time $t=0$.
\item  Find the velocity of the particle at time $t=0$.
\item  Find the speed of the particle at time $t=0$.
\item  Find the acceleration of the particle at time $t=0$.
\item  Is the particle moving at constant speed?  (Justify your answer!)
\end{enumerate}

\item  (12 points)  Consider the vector-valued functions below:
\begin{eqnarray*}
\mbf{F}(t) & = & -t^3 \mbf{i} + e^{3t}\mbf{j} \\
\mbf{G}(t) & = & t \mbf{i} + e^{-t} \mbf{j}
\end{eqnarray*}
Find $(\mbf{F} \bullet \mbf{G})'(0)$ in two different ways:
\begin{enumerate}
\item  By first computing $\mbf{F}(t) \bullet \mbf{G}(t)$ and then taking the derivative.
\item  By using the product rule for dot products.
\end{enumerate}


\end{enumerate}

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