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\begin{document}

\noindent
{\bf Math 105} \hfil {\bf Answers to test 1}

\medskip
\noindent {\bf Part I.  Multiple Choice}

\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?
\quad
{\bf Ans.} \ 0.50.  Cost is a linear function of miles driven, and the slope is the charge per mile.

\item Find the $y$-intercept of the line $x + 2y = 3$. \quad {\bf Ans.}\  3/2.  Simply solve the equation for~$y$.

\item If $\displaystyle f(x) = {{(x-1)^2} \over {(x+1)^2}}$, find
$f(a+1)$.  \quad {\bf Ans.} \ $\displaystyle {{a^2} \over {(a+2)^2}}$. Replace $x$ by $a+1$ wherever it appears.


\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?

\noindent
{\bf Ans.} $t=3$.  Solve $h = 0$

\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?
\quad {\bf Ans.} \$5,000.  Completing the square expresses the profit as $-2(x - 50)^2 + 5000$, which is 
maximum when $x=50$ with a maximum value of 5000.

\item  What is the natural domain of the function
$\displaystyle f(x) = \frac{5x}{(x^2+1)(\sqrt{x-2})}$? \ 
{\bf Ans.} \   $x > 2$. To avoid having the square root of a negative quantity, we must have $x \ge 2$, and to avoid a zero in
the denominator we must have $x \ne 2$.  There are no other restrictions.

\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?
\quad {\bf Ans.} \ At $x=1$ only.  There are only two points we have to worry about, one with $x=1$ and the other with $x=3$.
At the former, the left-hand branch approaches height 1 and the right-hand brance approaches height 2, so there is a
discontinuity.  At the latter, both branches approach height 2, so there is no discontinuity.

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

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\begin{minipage}[t]{3.5in}
 \noindent {\bf Ans.}
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\begin{minipage}[t]{2.5in} It's the only graph that lies entirely above the $x$ axis, rises to the right, and falls asymptotically
to the $x$ axis to the left.
\end{minipage}

\vskip.5in
\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)?
\quad {\bf Ans.} \ $\displaystyle 5000 \left ( 1 + {{.07} \over {12}} \right )^{12}$.  That is the compound interest formula
with $r=0.07, \  n = 12, \ A=5000$, and $t=1$



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

\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.

\noindent (a)  Express their monthly profit function as a
function of $x$.
\quad  {\bf Ans.} \ $2x - 1000$, because $R(x) = 5x$ and $P(x) = R(x) - C(x)$.

\noindent (b)   How many widgets should they produce in order to break
even? {\bf Ans.} \ 500.  \  Solve $P(x) = 0$.

\item (a) What is the slope of the line $2x + 3y = 4$?
\quad {\bf Ans.} $-2/3$.  Solve the equation for $y$.

\noindent (b)  Find the equation of the line
through the point $(-1,-2)$ and parallel to the line $2x + 3y = 4$.
\quad {\bf Ans.} \ $y = -(2/3)x - (8/3)$ or $2x+3y = -8$.  Either write the equation of the line through $(-1,-2)$ with slope
$-2/3$ or write $2x + 3y = C$ and plug in $(-1,-2)$ to find $C$.

\item The following is a table of values for some
function.

\smallskip

\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. \quad {\bf Ans.} \  $y = -(3/2)x + 18$.  The change in $y$ divided by the change in
$x$ is constant, and that is the slope.

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

\noindent {\bf Ans.} \ $y = 2(x+1)^2 + 4$.

(b)  Does the graph of the equation in (a) open upward or downward?
\quad {\bf Ans.} Upward.  The coefficient of the second-degree term is positive.

\item 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.
\quad {\bf Ans.} \  $-4/5$.  The average $y$-change is $-8/5$, and the $x$-change is 2.

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

(a)  Find its {\em horizontal} asymptote, if there is one.
\quad {\bf Ans.} $y=0$.

(b)  Find its {\em vertical} asymptote(s), if there are any.
\quad {\bf Ans.} \ $x=3$ and $x=-3$.

\item  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.
\quad {\bf Ans.} \  From the right the graph is positive.  From the left the graph is negative.

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

(a) What is the domain of $f$? \quad {\bf Ans.} \  $x > 0$.

(b)   What is $f(9)$?  \quad {\bf Ans.} \ $1/27$ 

\vfil \eject

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

\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)   Determine which function is linear, which is exponential, and
which is neither.
\quad {\bf Ans.}  Table I.\ is exponential.
Table II.\ is neither.
Table III.\ is linear.

(b)  Find a formula $y = Ab^x$ for the exponential function.
\quad {\bf Ans.} \ $y = 6\cdot2^x$.  The ratio of each pair of successive $y$ values is 2, which equals $b$.

\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?

\noindent {\bf Ans.} \ \$41,552.07.  The compound interest formula gives $100,000 = A(1.05)^{18}$, 
so that $A = 100,000\big/(1.05)^{18} \approx 41,552.065$.



\end{enumerate}
\end{document}
