%% Math 131, Fall, 2003
\documentclass[12pt]{article}
\usepackage{latexsym,amsmath,epic,eepic}

\pagestyle{empty}

\setlength{\textwidth}{6.5in}
\setlength{\oddsidemargin}{0in}
%% \setlength{\topmargin}{-.5in}
\setlength{\textheight}{8.25in}

\def\ds{\displaystyle}
\def\vp{$\vphantom{\big|}$}

\def\R{\textbf{R}}
\def\Item{\item{$\Box$}\quad}

\def\be{\begin{enumerate}}
\def\ee{\end{enumerate}}

\def\bd{\begin{description}}
\def\ed{\end{description}}

\def\Answer#1{$\phantom{.}$\hfill
        #1\quad\fbox{$\phantom{\Big|Answer}$}} 

\newcommand{\MyAnswer}[2]{$\phantom{.}$\hfill
        #1\quad\fbox{$\vphantom{\Big|}#2$}}

\newcommand{\soln}{\paragraph{Solution}}
\newcommand{\sep}{{\medskip\hrule\bigskip}}
\newcommand{\blank}{\underline{$\phantom{A blank}$}}

\newcommand{\argon}{{}^{39}\mathrm{Ar}}

\begin{document}

\begin{center}
\textbf{Holy Cross College, Fall Semester, 2003} \\
\textbf{Math 131, Review Sheet for Midterm 1}
Professor Hwang \\
\end{center}

\paragraph{Coverage}

Midterm~1 will be given in class on Friday, September~26. The test
will cover all the material through Section~2.2 (limits), with
emphasis on linear, exponential, and logarithmic functions. 

In addition to the odd-numbered questions from the homework and the
questions on the first group worksheet, the problems below will
provide good practice for the actual test. This is not to say that the
test questions will be as difficult as the problems below, but only
that it is better to be over-prepared than under-prepared. The
\emph{only} way to prepare is to work lots of problems until you're
fluent with the techniques. Trying to memorize formulas is not a good
study strategy. (Generally I will provide any formulas that can't be
deduced from a good understanding of the material.)

There will be a review session in the evening of Wednesday the~24th;
the location will be announced by early next week. The midterms for
the morning and noon sections will not be the same; nonetheless, if
you are in the morning section, it is not to your advantage to discuss
the test with students in the noon section.

\begin{enumerate}

\item Let $s(t)=100-16t^2$. (Recall that a stone dropped from a cliff
  of height 100~feet is $s(t)$~feet above the ground after
  $t$~seconds.) 

  \bd
  \item{(a)} Calculate the average velocity over the interval $1\leq
  t\leq 1+h$.

  \item{(b)} Calculate the average velocity over the interval $t_0\leq
  t\leq t_0+h$ for general $t_0<2.5$.

  \item{(c)} Use your answer to part~(b) to determine the
    instantaneous velocity of the stone as a function of time.
  \ed

\item Consider the graph $y=2^x$. Show that shifting the graph to the
  left by three units has the same effect as stretching the graph
  vertically by a factor of~8. Generalize.

\item Use a graphing calculator or Maple to graph the function
  $f(x)=(\sin x)(\cos x)$. Using the fact that the graph is
  sinusoidal, express~$f(x)$ in the form
  \[
  y = D \pm A\cos(Bx)\qquad\text{or}\quad y = D \pm A\sin(Bx)
  \]
  as appropriate.

\item A refrigerator costs~\$900. Each year, the value of the
  refrigerator depreciates by~25\%. 

  \bd
  \item{(a)} Find a formula for the value of the refrigerator after
    $t$~years have passed. (Hint: What kind of law are you seeking?)

  \item{(b)} Estimate the value of the refrigerator after one
    month. If the refrigerator continued to depreciate at this rate,
    how long would it be before the value is zero?

  \item{(c)} Find the average rate of depreciation during the first
    six months of ownership. Repeat for the second six months of
    ownership.
  \ed

\item An automobile travelling 70~mph requires 180~feet to stop on dry
  pavement. The same car travelling 50~mph requires 92~feet to
  stop. Assuming that speed and stopping distance are related by a
  power law, find the stopping distance if the car is travelling
  90~mph. 

\item Without graphing, find the horizontal and vertical asymptotes of
  the rational function
  \[
  y=\frac{2x^4-3x+1}{(x^2-1)(x^2-4)}.
  \]
  Then use a graphing calculator or Maple to check your answer.

\item Show that there is a real number~$c$ between 0~and 1 satisfying
  $c^5+c-1=0$.

\item (Challenge $\ddot\smile$) Find the inverse of the function
  $y=C(x)=\frac{1}{2}(e^x+e^{-x})$. (Recall that~$C$ is the ``even
  part'' of $y=e^x$.) Hint: Multiply both sides by~$2e^x$ and move
  everything to one side of the equation. Observe that the resulting
  equation is a quadratic in the variable~$e^x$; use the quadratic
  formula to solve for~$e^x$, then take logarithms.

\item Use properties of logarithms to show that
  \[
  \ln(x-\sqrt{x^2-1})=-\ln(x+\sqrt{x^2-1}),\qquad x>1.
  \]

\item The isotope Argon-39 ($\argon$) decays at a rate of about
  $0.25734\%$~per year.

  \bd
  \item{(a)} If there are $Q_0$~grams of Argon-39 in a sample, find a
    formula for the amount remaining after $t$~years.

  \item{(b)} Find the half-life of $\argon$.

  \item{(c)} Find the continuous rate of decay to five decimal places.

  \item{(d)} How long does it take for $90\%$~of a sample of~$\argon$
    to decay?

  \item{(e)} If one mole ($6.02\times10^{23}$~atoms) is present
    initially, how long does it take for the \emph{entire} sample to
    decay? Express your answer in both years and
    half-lives. (Mathematically the answer is ``never'', but the
    number of atoms is not really a continuous quantity! Instead,
    compute the time until there is 1/2~atom remaining. Beware of
    round-off error.)

  \item{(f)} In part~(e), does the ``total decay time'' in half-lives
  depend on the isotope in question? Explain.
  \ed

\end{enumerate}

\end{document}
