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\begin{document}

\def\answerskip{\vskip0.25in}

\def\problemskip{\vskip1.6in}


\centerline{\bf Answers to Test 1}

\begin{enumerate}

\item Last month Mary rented a car twice from the Cars-R-Us Car Rental Company,
which charges a certain amount just for renting the car, plus a certain
amount per
mile.  The first time, she drove the car 200 miles and was charged \$120.  The
second time, she drove the car 50 miles and was charged \$45.  How much does
Cars-R-Us charge per mile?

\answerskip

(a) \$0.20 \hfill
(b) \$0.25 \hfill
(c) \$0.50 \hfill
(d) \$0.10 \hfill
(e) \$0.40

\problemskip

\item Find the $y$-intercept of the line $x + 2y = 3$.

\answerskip

(a) $\displaystyle {3 \over 2}$ \hfill
(b) $\displaystyle {1 \over 2}$ \hfill
(c) $\displaystyle 3$ \hfill
(d) $-3$ \hfill
(e) $\displaystyle -{3 \over 2}$

\problemskip

\item If $\displaystyle f(x) = {{(x-1)^2} \over {(x+1)^2}}$, find
$f(a+1)$.

\answerskip

(a) $\displaystyle {{(a-1)^2} \over {(a+1)^2}}$ \hfill
(b) $\displaystyle {{(a+1)^2} \over {(a+2)^2}}$ \hfill
(c) $\displaystyle {{(a-1)^2} \over {(a+1)^2}} +1$ \hfill
(d) $\displaystyle {{a} \over {a+2}}$ \hfill
(e) $\displaystyle {{a^2} \over {(a+2)^2}}$ \hfill

\problemskip

\item A rock is thrown from the top of a 48-foot tower straight up
into the air at time $t=0$ with an initial velocity of 32 feet per second.  Its
height $h$ at time $t$ is given by the formula
\[
h = -16 t^2 + 32 t + 48.
\]
At what time does the rock hit the ground?

\answerskip

(a) $t=2$ \hfill
(b) $t=3$ \hfill
(c) $t=4$ \hfill
(d) $t=1$ \hfill
(e) $t=5$

\vskip 2in

\item
It is determined that a certain company earns a profit of
$200x-2x^2$ (in dollars) per day for producing $x$ units of their product.
What is
the maximum possible profit that they can earn in one day?

\answerskip

(a) \$10,000 \hfill
(b) \$100 \hfill
(c) \$5,000 \hfill
(d) \$200 \hfill \\ \\
(e) No profit -- no matter what $x$ is, they lose money.

\pagebreak

\item  What is the natural domain of the function
\[
f(x) = \frac{5x}{(x^2+1)(\sqrt{x-2})} ?
\]

(a) $x \geq 2$, $x \neq -1$ \hfill
(b)  $x > 2$ \hfill
(c)  all real $x$ \hfill
(d)  $x \neq 0$ \hfill
(e)  $x \neq 2$

\vskip 2in

\item  Suppose the function $f$ is given by the multi-line definition

\[ f(x) = \left\{ \begin{array}{cl}
	x^2, & x \leq 1; \\
	x-1, & 1 < x < 3; \\
	-\frac{1}{3}x + 3, & x \geq 3.
	\end{array}
	\right. \]

Where does $f$ have discontinuities?

(a) At $x=1$ and $x=3$. \hfill
(b)  At $x=3$ only. \hfill
(c)  At $x=0$. \\ \\
(d) At $x=1$ only. \hfill
(e)  Nowhere; $f$ is continuous.

\pagebreak

\item  Which of the following could be the graph of $y = b^x$ for some $b > 1$?

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(a)
\begin{picture}(0,25)
\put(-70,-100){\usebox{\axes}}
\end{picture}
\hfill
(b)
\begin{picture}(0,25)
\put(-70,-100){\usebox{\axes}}
\end{picture}
\hfill
\ \vskip 1.5in
(c)
\begin{picture}(0,25)
\put(-70,-100){\usebox{\axes}}
\end{picture}
\hfill
(d)
\begin{picture}(0,25)
\put(-70,-100){\usebox{\axes}}
\end{picture}
\hfill
\ \vskip 1.5in
(e)
\begin{picture}(0,25)
\put(-70,-100){\usebox{\axes}}
\end{picture}
\hfill

\vskip 1.2in

\pagebreak

\item Jack invests \$5000 in an account that earns an annual rate of 7\%,
compounded monthly.  If he makes no further deposits or withdrawals, what
will be
the value of the account after one year (in dollars)?

(a) $\displaystyle 5000 $ \hfill
(b) $\displaystyle 5000 \left ( 1 + .07 \right )^{1} $  \hfill
(c) $\displaystyle 5000 \left ( 1 + {{.07} \over {12}} \right )^{1} $   \\
\begin{center}
 (d)  $\displaystyle 5000 \left ( 1 + .07  \right )^{12} $  \hskip
1.2in (e) $\displaystyle 5000 \left ( 1 + {{.07} \over {12}} \right )^{12} $
\end{center}

\vskip 1.4in

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\hbox to 6.5truein{\hrulefill}

\bigskip

\hskip -.3in {\bf Part II. Partial Credit}

\bigskip

Show all work, and put your final answer in the space provided.  You will
receive
no credit if your answer is not in the space provided, and no partial
credit for a
wrong answer if you do not show your work.

\vskip .4in

\item A widget company finds that in order to
make $x$ widgets in a given month, their cost function (in dollars) is
$C(x) = 1000
+ 3x$. Suppose they charge \$5.00 per widget.

\bigskip

\noindent (a) (3 points) Express their monthly profit function as a
function of $x$.

\vskip 1.2in

\hfill Answer:\hbox to 2.5truein{\hrulefill}

(continued on next page)

\pagebreak

\noindent (b) (2 points) How many widgets should they produce in order to break
even?

\vskip 1.2in

\hfill Answer:\hbox to 2.5truein{\hrulefill}

\item (a) (2 points) What is the slope of the line $2x + 3y = 4$?

\vskip 1.2in

\hfill Answer:\hbox to 2.5truein{\hrulefill}

\bigskip

\noindent (b) (3 points) Find the equation of the line
through the point $(-1,-2)$ and parallel to the line $2x + 3y = 4$.

\vskip 1.2in

\hfill Answer:\hbox to 2.5truein{\hrulefill}

\pagebreak

\item (5 points) The following is a table of values for some
function.

\bigskip

\begin{center}
\begin{tabular}{|c||c|c|c|c|c|c|} \hline
x & 0 & 2 & 4 & 6 & 8 & 10 \\
\hline
y & 18 & 15 & 12 & 9 & 6 & 3 \\
\hline
\end{tabular}
\end{center}

If the function is linear, write it explicitly in the form $y = mx + b$.  If it
is not linear, explain clearly why it is not.

\vskip 1.8in

\hfill Answer:\hbox to 2.5truein{\hrulefill}

\vskip .3in

\item (a) (4 points)
Complete the square of the quadratic equation $y = 2x^2 + 4x + 6$.

\vskip 1.5in

\hfill Answer:\hbox to 2.5truein{\hrulefill}

\bigskip

(b) (1 point) Does the graph of the equation in (a) open upward or downward?

\vskip 1.2in

\hfill Answer:\hbox to 2.5truein{\hrulefill}

\vskip .3in

\item (5 points) The following table of values does {\em not}
correspond to a linear function.

\begin{center}
\begin{tabular}{|c||c|c|c|c|c|c|} \hline
x & 0 & 2 & 4 & 6 & 8 & 10 \\
\hline
y & 18 & 16 & 16 & 14 & 13 & 10 \\
\hline
\end{tabular}
\end{center}

Using the method described in class, find {\bf just} the slope of the linear
function that approximates the data.

\vskip 1.2in

\hfill Answer: The slope is: \hbox to 1.5truein{\hrulefill}

\bigskip

\item  Consider the graph of the function \[ f(x) = \frac{x}{(x^2-9)}. \]

(a)  (3 points)  Find its {\em horizontal} asymptote, if there is one.

\vskip 1.2in

\hfill Answer:\hbox to 2.5truein{\hrulefill}

(b) (3 points)  Find its {\em vertical} asymptote(s), if there are any.

\vskip 1.2in

\hfill Answer:\hbox to 2.5truein{\hrulefill}

\item  (5 points)  The line $x = -2$ is a vertical asymptote for the function
\[
g(x) = \frac{x^2 + 6}{x+2}.
\]
Determine the behavior of the graph (i.e.\ positive or negative) as it
approaches the asymptote from either side.

\vskip 1.2in

\hfill Answer: From the right the graph is \hbox to 1.5truein{\hrulefill}

\hfill  From the left the graph is \hbox to 1.6truein{\hrulefill}

\vskip 1.2in

\item  Let $f(x) = x^{(-\frac{3}{2})}$.

(a)  (4 points)  What is the domain of $f$?

\vskip 1.2in

\hfill Answer:\hbox to 2.5truein{\hrulefill}

(b)  (2 points)  What is $f(9)$?  (Express your answer as a {\em fraction}, not
as a decimal.)

\vskip 1.2in

\hfill Answer:\hbox to 2.5truein{\hrulefill}

\item  The following tables give values from three functions.  One function is
linear, one is exponential, and one is neither.

\bigskip

\begin{itemize}
\item[I.]
\begin{tabular}{|c||c|c|c|c|c|c|}
\hline
$x$ & 0  & 1 & 2 & 3 & 4 & 5 \\ \hline
$f(x)$ & 6 & 12 & 24 & 48 & 96 & 192 \\ \hline
\end{tabular}

\item[II.]
\begin{tabular}{|c||c|c|c|c|c|c|}
\hline
$x$ & 0  & 1 & 2 & 3 & 4 & 5 \\ \hline
$g(x)$ & 8 & 20 & 50 & 100 & 160 & 220 \\ \hline
\end{tabular}

\item[III.]
\begin{tabular}{|c||c|c|c|c|c|c|}
\hline
$x$ & 0  & 1 & 2 & 3 & 4 & 5 \\ \hline
$h(x)$ & 5 & 7.3 & 9.6 & 11.9 & 14.2 & 16.5 \\ \hline
\end{tabular}
\end{itemize}

(a)  (5 points)  Determine which function is linear, which is exponential, and
which is neither.

\vskip 1.2in

\hfill Answer: Table I.\ is \hbox to 1.5truein{\hrulefill}

\hfill Table II.\ is \hbox to 1.5truein{\hrulefill}

\hfill Table III.\ is \hbox to 1.5truein{\hrulefill}

\bigskip

(b)  (3 points)  Find a formula $y = Ab^x$ for the exponential function.

\vskip 1.2in

\hfill Answer:\hbox to 2.5truein{\hrulefill}

\pagebreak

\item (5 points)  According to a recent {\em Headline News} broadcast, the
cost of
education at a state school may be as high as \$100,000 eighteen years from
now.

Suppose the parents of a newborn want to put some money away today for their
child's education.  If their savings account pays simple interest figured
annually
at a rate of 5\%, how much should they contribute now in order to have
\$100,000
in 18 years?

\vskip 1.2 in

\hfill Answer:\hbox to 2.5truein{\hrulefill}

\end{enumerate}

\end{document}
