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\course \hfill Russell Brown \\
Exam 1  \hfill 21 October 1997

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Answer all of the following questions.
Use the backs of the question papers for scratch paper.
Additional sheets are available if necessary.  No books
or notes may be used.  When answering these questions,
please be sure to 1) check answers when possible, 2)
clearly indicate your answer and the reasoning used to
arrive at that answer {\bf (unsupported answers may receive
NO credit)}, and 3) label all variables and equations.

If you use your calculator to solve an equation or produce a graph,
please indicate this on your test paper. Otherwise the answer will be
assumed to be ``unsupported''.
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Name \rule {2in}{0.01in}

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Section \rule{ 1in }{0.01in}

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\begin{center}
\begin{tabular}{|r|c|r|}
\hline
Question & Score & Total \\
\hline \hline
1 && 20 \\[0.1in]
\hline
2 && 15 \\[0.1in]
\hline
3 && 10 \\[0.1in]
\hline
4 && 10 \\[0.1in]
\hline
5 && 20 \\[0.1in]
\hline
6 && 10 \\[0.1in]
\hline
7 && 15 \\[0.1in]
\hline \hline
Total && 100 \\[0.1in]
\hline  
\end{tabular}
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\begin{enumerate}

\item (20 points) For each of the following limits, determine if it
exists. If the 
limit exists, give its value and use the limit laws or the squeeze
theorem to explain how  you obtained your result.
 If the limit  does not exist, determine if the limit is
$+\infty $ or $-\infty$ or if it does not exist and is not $+ \infty $
 or $-\infty$.

\begin{enumerate}
\item $\displaystyle{ \lim_{x\rightarrow 0} x^2 \sin( 1/x)}$

\item $\displaystyle {\lim _{x\rightarrow 2} \frac {x^2+4} { x^2-4}} $ 

\item $\displaystyle {\lim _{x\rightarrow 2} \frac {x^2 -4} { x-2}} $
\end{enumerate}

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\item (15 points)
 \begin{enumerate}\item State the intermediate value theorem. 
\item Prove that the equation $e^x= \sin x$ has a solution.
      \end{enumerate}



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\item (10 points) Find a rational function which satisfies
$$
\lim_{x \rightarrow \infty } f(x) =2 \qquad \mbox{and} \qquad \lim
_{x\rightarrow 3} f(x) = -\infty.
$$

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\item  (10 points) If  the price of tobacco is \$$p$ per pound, then a  farmer
produces $Q(p)$ pounds of tobacco. 
\begin{enumerate} \item What are the units for $Q'(p)$?

\item If $p=2$, do you expect that $Q'(p) > 0$, $Q'(p) =0$ or  $Q'(p) <
0$? Why?
\end{enumerate}

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\newpage

\item(20 points)
\begin{enumerate}
\item  
 If $f(x) = \sqrt{x+4}$, use the definition of the derivative to  find
$ f'(x) $. Give the  
domain of the $f'(x)$. 

\item Write out the equation of the  tangent line to the graph of
$f(x)$ at $x=0$.

\item Use the tangent line to approximate $\sqrt {4.1}$ and $\sqrt
{3.8}$. 

\end{enumerate}

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\item (10 points) Below is the graph of $f'$. On the same axes, sketch two
possibilities for the graph of $f$.

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\item (15 points) Below are the graphs of three functions, $g_1$, $g_2$ and
$g_3$.   These are the graphs of  a function $f$ and its first two
derivatives $f'$ and $f''$. Which function is $f$, $f'$ and $f''$?
Explain how you arrived at your answer. 

\end{enumerate}
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