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\centerline{\textbf{\large Exam \#1 $\cdot$ Friday, February 2, 2007}}
\vspace*{.2in}
MATH 124 $\cdot$ Calculus I $\cdot$ Section 8 $\cdot$ Spring 2007  \hspace*{0.5in} Name \hrulefill
\vspace*{0.2in}

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\textbf{Problem 1.}  Let $f(x) = A(x-B)^2(x+C)$ with $A>0$, $B>0$, and $C>0$.
Using the following space, sketch a graph of $f(x)$.  Include and label the
$x$-intercept(s), if any, and the $y$-intercept.  (The $x$-intercepts and
$y$-intercept should be in terms of $A$, $B$, and $C$ --- please do not plug in
specific numerical values for $A$, $B$, and $C$.)

\vspace*{2.0in}

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\textbf{Problem 2.}  Sally's track coach tells her that to avoid injury,
she should increase her weekly mileage by 10\% per week.

\vspace*{0.2in}

\textbf{Part (a).} If she currently runs 2 miles per week, and if she follows
her coach's advice, by how many weeks from now will she be running 25 miles per
week?

\vspace*{2.0in}

\textbf{Part (b).} If she currently runs 2 miles per week, and if she instead
follows her friend's advice and increases her mileage by 1.5 miles per week, by
how many weeks from now will she be running 25 miles per week?

\vspace*{2.0in}

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\newpage
\textbf{Problem 3.}  Let
$$
	f(z) = \frac{(z-1)(z-2)}{(z-3)(z-4)}.
$$

\vspace*{0.2in}

\textbf{Part (a).}  Find the domain of $f(z)$.
\vspace*{0.7in}

\textbf{Part (b).}  Find the zeroes of $f(z)$.
\vspace*{0.7in}

\textbf{Part (c).}  Find the vertical asymptote(s), if any, of $f(z)$.
\vspace*{0.7in}

\textbf{Part (d).}  Find the horizontal asymptote(s), if any, of $f(z)$.
\vspace*{0.7in}

\textbf{Part (e).}  Find the hole(s), if any, of $f(z)$.
\vspace*{0.7in}

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\textbf{Problem 4.}  In your physics class, you attach an oscillosope to a
piece of electronic equipment and discover that the voltage you observe looks
like a sine wave.  In particular, you notice that
\begin{itemize}
\item The maximum voltage you observe is 1.8 volts.
\item The minimum voltage you observe is 0.0 volts.
\item This maximum voltage is found at the start of your data (i.e. $t=0.0$
	picoseconds); the first time it reoccurs is 2.0 picoseconds later (i.e.
	$t=2.0$ picoseconds).
\end{itemize}
Find an equation of the form $a+b\cos(ct)$ that describes the voltage $V$ as a
function of time $t$ in picoseconds.  (You don't need to know how many
picoseconds make a second.) (Hint:  You might want to sketch a graph.)

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\newpage
\textbf{Problem 5.}  In a lab experiment, you have microorganisms living in a
petri dish.  Assume that the microorganisms are dying at an exponential rate.
You have the following data:
\begin{itemize}
\item You forgot to count the initial population.
\item You count a population of 2500 organisms on day one.
\item You count a population of 1500 organisms on day two.
\end{itemize}
Find an exponential model of the form $y=Ce^{rt}$ for these data, where $y$ is
the number of microorganisms in the dish after $t$ days.  That is, find $C$ and
$r$ so this model matches the given data.

\vspace*{3.0in}

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\textbf{Problem 6.}  You leave on a car trip for San Diego.
The following events happen:
\begin{itemize}
\item[(i)] You drive at medium speed (in town) for the first 15 minutes.
\item[(ii)] Once you get out of town, you accelerate almost immediately
	to the speed limit and drive at that speed for 45 minutes.
\item[(iii)] Then, you hit a traffic jam.  You come to a stop almost
	immediately and sit at a stop for 30 minutes.
\item[(iv)] Once past the accident, you accelerate almost immediately back
	to the speed limit and drive at that speed until you reach the coast 4
	hours later.
\end{itemize}
Sketch a graph of distance $D$ from your house vs.
time $t$ in hours.  Label all the above events on the $t$ axis.

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\newpage
\textbf{Problem 7.}  The following graph shows the altitude $y$ of a rocket
(in miles) as a function of time $t$ since launch (in seconds):

\includegraphics{rocketfig.eps}

\textbf{Part (a).}  Estimate $f(18)$, and describe verbally what it means to
you.  Include units in your response.
\vspace*{1.0in}

\textbf{Part (b).}  Estimate $f'(18)$, and describe verbally what it means to
you.  Include units in your response.
\vspace*{1.0in}

\textbf{Part (c).}  Estimate $f^{-1}(40)$, and describe verbally what it means
to you.  Include units in your response.
\vspace*{1.0in}

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\textbf{Problem 8.}  Let $f(x) = 2x^2+1$.
Evaluate
$$
	\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}.
$$
using properties of limits.

%\vspace*{2.0in}

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\newpage
\textbf{Problem 9.}  The following are selected values of a function $f(x)$.
$$
	f(3) = \textrm{ is undefined} \qquad \qquad
	f(4) = -6 \qquad \qquad
	f(6) = 9
$$
$$
\begin{tabular}{|r|r|r|r|}
\hline \multicolumn{2}{|c|}{Table A} \\
\hline $x$ & $f(x)$ \\
\hline
\hline $     2.9    $ & $    -8.14286$ \\
\hline $     2.99   $ & $   -98.01493$ \\
\hline $     2.999  $ & $  -998.00150$ \\
\hline $     2.9999 $ & $ -9998.00015$ \\
\hline
\end{tabular}
\hspace*{0.2in}
\begin{tabular}{|r|r|r|r|}
\hline \multicolumn{2}{|c|}{Table B} \\
\hline $x$ & $f(x)$ \\
\hline
\hline $     3.1    $ & $   -12.15789$ \\
\hline $     3.01   $ & $  -102.01508$ \\
\hline $     3.001  $ & $ -1002.00150$ \\
\hline $     3.0001 $ & $-10002.00015$ \\
\hline
\end{tabular}
\hspace*{0.2in}
\begin{tabular}{|r|r|r|r|}
\hline \multicolumn{2}{|c|}{Table C} \\
\hline $x$ & $f(x)$ \\
\hline
\hline $     3.9    $ & $    -5.56566$ \\
\hline $     3.99   $ & $    -5.95070$ \\
\hline $     3.999  $ & $    -5.99501$ \\
\hline $     3.9999 $ & $    -5.99950$ \\
\hline
\end{tabular}
\hspace*{0.2in}
\begin{tabular}{|r|r|r|r|}
\hline \multicolumn{2}{|c|}{Table D} \\
\hline $x$ & $f(x)$ \\
\hline
\hline $     4.1    $ & $     6.57576$ \\
\hline $     4.01   $ & $     6.05071$ \\
\hline $     4.001  $ & $     6.00501$ \\
\hline $     4.0001 $ & $     6.00050$ \\
\hline
\end{tabular}
$$
$$
\begin{tabular}{|r|r|r|r|}
\hline \multicolumn{2}{|c|}{Table E} \\
\hline $x$ & $f(x)$ \\
\hline
\hline $     5.9    $ & $     6.32184$ \\
\hline $     5.99   $ & $     6.49616$ \\
\hline $     5.999  $ & $     6.49956$ \\
\hline $     5.9999 $ & $     6.49999$ \\
\hline
\end{tabular}
\hspace*{0.2in}
\begin{tabular}{|r|r|r|r|}
\hline \multicolumn{2}{|c|}{Table F} \\
\hline $x$ & $f(x)$ \\
\hline
\hline $     6.1    $ & $     6.63196$ \\
\hline $     6.01   $ & $     6.59234$ \\
\hline $     6.001  $ & $     6.50034$ \\
\hline $     6.0001 $ & $     6.50002$ \\
\hline
\end{tabular}
\hspace*{0.2in}
\begin{tabular}{|r|r|r|r|}
\hline \multicolumn{2}{|c|}{Table G} \\
\hline $x$ & $f(x)$ \\
\hline
\hline $   100$ & $     1.05285$ \\
\hline $  1000$ & $     1.00503$ \\
\hline $ 10000$ & $     1.00050$ \\
\hline $100000$ & $     1.00005$ \\
\hline
\end{tabular}
\hspace*{0.2in}
\begin{tabular}{|r|r|r|r|}
\hline \multicolumn{2}{|c|}{Table H} \\
\hline $x$ & $f(x)$ \\
\hline
\hline $   -100$ & $     2.95257$ \\
\hline $  -1000$ & $     2.99503$ \\
\hline $ -10000$ & $     2.99950$ \\
\hline $-100000$ & $     2.99995$ \\
\hline
\end{tabular}
$$
\vspace*{0.2in}

\textbf{Part (a).} Find the following limits, when they exist.  If a limit does
not exist, say why not.  For each, state which table(s) (A, B, C, D, E, F, G,
H) you use to support your answer.
$$
	\lim_{x \to 3}f(x) \qquad \qquad \qquad
	\lim_{x \to 4}f(x) \qquad \qquad \qquad
	\lim_{x \to 6^-}f(x) \qquad \qquad \qquad
	\lim_{x \to -\infty}f(x) \qquad \qquad
$$
\vspace*{2.5in}

\textbf{Part (b).} Determine if $f(x)$ is continuous or not continuous
at the following values.  If $f(x)$ is not continuous at the point,
describe what feature occurs there.
$$
	x = 3 \qquad \qquad \qquad \qquad
	x = 4 \qquad \qquad \qquad \qquad
	x = 6 \qquad \qquad
$$

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