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\course \hfill Russell Brown \\
Exam 1  \hfill 22 September 1998

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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)}.

If you use your calculator to solve an equation or produce a graph,
please indicate this on your test paper. Otherwise the answer may
receive no credit.
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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 && 5 \\[0.1in]
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2 && 5 \\[0.1in]
\hline
3 && 5 \\[0.1in]
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4 && 5 \\[0.1in]
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5 && 15 \\[0.1in]
\hline
6 && 15 \\[0.1in]
\hline
7 && 20 \\[0.1in]
\hline
8 && 15 \\[0.1in]
\hline
9 && 15 \\[0.1in]
\hline \hline
Total && 100 \\[0.1in]
\hline  
\end{tabular}
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\begin{enumerate}


\item  
If $f(x) = x+1$ and $g(x) = \sqrt x$, find a formula for $g\circ
f$. Give the domain of $g\circ f$.  

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\item Solve the equation 
$$
2^{x^2 + 1} = 8.
$$

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\newpage

\item Give a parametric curve $x(t), y(t)$ which traces out the circle
centered at $(0,0)$ and with radius 2.

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\item Give the definition of odd function. 
For each of the following two functions, determine if the function is odd.
$$
f(x) = x^3+ \cos x, \qquad g(x) = x^3 + \sin x.
$$

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\newpage
\item
Consider the parametric curve:
\begin{eqnarray*}
x(t) & = & 2t\\
y(t) & = & t^2 - 4t, \qquad -2 \leq t \leq 6.
\end{eqnarray*}

\begin{enumerate}
\item What are the values of $t$ when the curve is on the $x$-axis?
\item What is the value of $t$ when the $y$ coordinate attains its
smallest value?
\item Eliminate $t$ to find an equation in $x$ and $y$ which is
satisfied by  all
points on the curve.
\end{enumerate}

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\item A population of critters doubles every 4 hours. At $t=0$, there
are 200 critters.
\begin{enumerate}
\item How many critters are there after 12 hours?
\item Write a function $N(t)$ which gives the population after $t$
hours.
\item What is the population after 13 hours?
\item When does the population contain 500 critters?
You must explain how to answer this question algebraically. Do not use
the \tt Solve \rm feature on your calculator.
\end{enumerate}

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\item Below is the graph of a function $f(x)$.  Use the graph to
answer the following questions.
\begin{enumerate}
\item What is $f(0)$? What is the domain of $f$?
\item On the same axes, sketch the graph of $f(x+2)$. Label the
coordinates of the corner on the graph of $f(x+2)$.
\item   On the same axes, sketch the graph of $f(2x+2)$.  Label the
coordinates of the corner on the graph of $f(2x+2)$.
\item On the second set of  axes, sketch the graph of $f^{-1}(x)$.
\end{enumerate}



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\item Let  $$f(x) = 2+ \frac 1 x .$$ 
\begin{enumerate}
\item
Find a formula for $f^{-1}$.
\item What is the domain of $f$ and what is the domain of $f^{-1}$?
\end{enumerate}

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\item \begin{enumerate}
\item State the principle of mathematical induction.
\item Prove that for $n=1,2, \dots$,
$$
\sum_{k=1} ^n (2k-1) = n^2.
$$
      \end{enumerate}




\end{enumerate}
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