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\course \hfill Russell Brown \\
Review 1  \hfill \today

\begin{enumerate}
\item Let $f(x) = \sqrt {x-4} $ and $g(x) = x^2+1$.  Give the domain
of $f$.
Find $f\circ g$ and $g\circ f$.

\item Sketch the graph of $f(x)= 1/x$. Based on this graph, find
graphs of $f(x+1)$, $-2 f(x)$ and $f(-x+1)-1$. 



\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 $pi/4$.

\item Consider the parametric curve 
$$ x= \sin t , \qquad y = \cos t.$$
\begin {enumerate} \item Make a graph of the curve and give an arrow
to indicate the direction we move as $t$ increases.
\item Given an equation in $x$ and $y$ that every point on the curve
satisfies.
\item Find a value of $t$  where the $y$ coordinate is as large as
possible.
\end {enumerate}


\item A population doubles every 10 days and after 20 days, the
population is 400.
\begin{enumerate}
\item What is the population initially?
\item Give the size of the population after 23 days.
\item When does the population reach 20,000?
\end{enumerate}

\item \begin{enumerate}
\item Solve $2^{x^2}=7 $.
\item Find the exact value of $\log_3 1/9$.

      \end{enumerate}

\item Given the table of values fit a model of the form $N(t) = A
e^{kt}$.   First, make a plot of $t$ versus $\ln N$. What is the
relationship between the line these points lie near and the values of
$A$ and $k$.
Be careful, one of the values is wrong. Which value is wrong?

$$
\begin{array} 
{r|rrrrrr}
t& 0 & 1 & 2& 3& 4& 5\\
\hline
N & 3 & 6 & 12 & 15 & 48 & 56 
\end{array}
$$
  
\end{enumerate}
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