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\course \hfill Russell Brown \\
Exam 1  \hfill 23 September 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{enumerate}

\item (14 points)
 \begin{enumerate}
\item If $f(x) = x^2$ and $g(x)= 2x+1$. Give $f\circ g$ and $g\circ f$.
\item Find an exact value for $\log_2 8$. Do not use your calculator.
\item  If $h(x) = x^3-x$, is $h$ even, odd or neither?
      \end{enumerate}


\newpage
\item (14 points)
 A  rectangular window has two dimensions $l$ and $w$. 


\begin{enumerate}  \item 
 Assume that the perimeter of the window is 30 feet.
Give the area as a function of the  $l$. Give the domain
of this function. 

\item Suppose the area of the window is 45 square feet. Give an
approximate value 
for the dimensions of the window.  Your answer should be correct to the first
decimal place. 

\end{enumerate}

\newpage

\item (14 points)
 Consider the graph below. 

\begin{enumerate}
 
\item Find $f(1)$, $f(2)$ and $f\circ f (4)$. 

\item Does $f^{-1} $ exist? If so, find $f^{-1}(1)$. If not, explain
why not.

\item On the axes provided, sketch $2f(x)$. Label the coordinates of
the point on the graph where $2f(x)$ is largest.

\item On the axes provided, sketch the graph of $f(-x+1)$.  Label the
coordinates of  the point on the graph where $f(-x+1)$ is largest.

\end{enumerate}

\newpage

\item  (14 points) Suppose that a particle moves along the path given by the
parametric equations:

\begin{eqnarray*}
x(t) &= & 2t+1\\
y(t) &=& t^2 -4t. 
\end{eqnarray*}


\begin{enumerate}
\item Eliminate the parameter to find an equation in $x$ and $y$ which
each point on the curve satisfies.  Simplify your answer.

\item Find the value of $t$ where the $y$ coordinate is smaller than
at any other point on the curve.

\item Find the $x$ and $y$ coordinates of the point where the $y$
coordinate is smaller than at any other point on the curve.
\end{enumerate}

\comment{
\newpage


\item Suppose that a wheel of radius 1 meter rolls along the $x$-axis without
slipping. 
If the bottom  of the  wheel is initially at the origin, find the
coordinates of the point $P$ after the wheel has rotated through an
angle of $\pi/4$. 
Hint: How far has the wheel travelled after it has rotated through an
angle of $pi/4$.
}

\newpage

\item (14 points)
Suppose 
$$
f(x) = 2+ \displaystyle { \frac {1} {x-1} } .
$$
\begin{enumerate}
\item Give a sketch of the graph of $f$. (You may use your
 calculator.)
 Why does $f^{-1}$ exist? 
\item  Give a formula for $f^{-1} (x)$.
\item Give the domain of $f$ and the range of $f$. 
\end{enumerate}

\newpage


\item  (15 points) 
Suppose a population grows exponentially. At $t=0$, the
population has 100 critters. After 2 days, the population has
doubled.  

\begin{enumerate}
\item Give formula for $N(t)$,  the size of the  population  after $t$
days.
\item Find the population after 5 days.
\item Find when the population reaches 1000 critters. 
\end{enumerate}

\newpage


\item (15 points) Consider the table of numbers.
$$
\begin{array} {r|r}
x & y \\ \hline
1 & 2 \\
2 & 2.83 \\
4 & 4.00 \\
8 & 5.66 \\
12 & 6.92 \\
\end{array}
$$
\begin{enumerate}
\item Make a plot of $\ln x $ against $\ln y$ on the axes. 
\item Estimate the slope and $y$-intercept of the line which passes near the points you
plotted in part a).
\item Using your answers in part b), find
 an  equation of the form  $y= a x^r$ which the numbers $x$ and $y$
 satisfy.
 You must explain how you found your answer. You
may not   use the power  regression function on your calculator. (I
suppose you could use this function to check your answer, but I expect
that there are easier ways to check your answer.)
\end{enumerate}

\end{enumerate}
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