\documentclass[12pt]{article}
\input{rmb_local}
\input{rmb_macros}
\pagestyle{empty}
\oddsidemargin -0.25in

\textwidth 6.5in
\textheight 9in
\topmargin -.5in
\begin{document}

\begin{flushleft}



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

The first hour exam is from 7:30 to 9:30 pm in CB 114 on Tuesday, 22
September.  You may use a calculator during the exam. You may not use
a laptop computer
a calculator which can perform symbolic manipulation, such as the
TI-92.  If you have any questions as to whether your calculator is
allowed, please see Brown before the test.

The exam will cover sections 1.1 to 1.6 of Stewart and the principle
of mathematical induction. There will be no questions taken from
appendix C. (Though the trigonometric functions do arise in problems
from Chapter 1.)

To review for the exam, you should review all assigned homework, quizzes, the
problem sheets from recitation and  material from the book and
lecture. The review sheet below provides additional problems for you
to work.


\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 Sketch the graph of this function $f(x) = \sin x$ with domain $[\pi/2,
3\pi/2]$.  Give the range of this   function.  Graph $f^{-1}$ and give
its domain and range.  Find $f^{-1}(0)$.  Why is your answer different
than your calculator's value for $\sin^{-1}(0)$?

\item State the principle of mathematical induction.

\item Use induction to prove that $5^n-1$ is a multiple of 4 for
$n=1,2,\dots$.

\item Use induction to prove that $\sum_{k=1}^n 4k =  2n^2+2n $ for
$n=1,2,\dots$. 

\item If $f$ is odd, is $f\circ f$ odd?  Is $2f$ odd? Is $f^2$ odd?
\item Which of the following functions are even?  
$$
f(x)= x^4+x^2, \qquad g(x) = x^5- x^2, \qquad h(x) = \cos x \sin x.
$$

\end{enumerate}
\end{flushleft}

\end{document}