\documentstyle[12pt]{article}

\def\easy{4}
\def\medium{6}
\def\hard{8}

\oddsidemargin 5pt \evensidemargin 5pt \marginparwidth 68pt
\marginparsep 10pt \topmargin 10pt \headheight 10pt
\headsep 15pt \footskip 35pt
\textheight = 625pt  \textwidth 440pt \columnsep 10pt \columnseprule 5pt

\pagestyle{myheadings}
\def\la{\langle}  \def\ra{\rangle}
\def\header{{\hfill Exam 1 \hfill A. Miller \hfill Spring  99
                                       \hfill Math 272\hfill}}
\markboth\header\header
\newcount\probno
\newcount\total
\newwrite\examTBLFile

\probno=0
\def\prob#1{\advance\probno by 1  \vfill\newpage  % \par
    \noindent\the\probno . (#1 pts)\advance\total by #1
    \immediate\write\examTBLFile{\the\probno
       \string& #1 \string&
       \string\qquad \string\\ \string\hline}}

\def\probx#1{\advance\probno by 1  \vfill  % \par
    \noindent\the\probno . (#1 pts)\advance\total by #1
    \immediate\write\examTBLFile{\the\probno
       \string& #1 \string&
       \string\qquad \string\\ \string\hline}}

\def\finishtbl{
    \immediate\write\examTBLFile{Total \string& \the\total
    \string& \qquad \string\\ \string\hline}
     }

\newwrite\ans
\immediate\openout\ans=answers
\outer\def\answer{\par\medbreak
  \immediate\write\ans{}
  \immediate\write\ans{\string\bigskip \the\probno . }
  \copytoblankline}
\outer\def\putanswer{\par\medbreak
  \immediate\write\ans{}
  \copytoblankline}
\def\copytoblankline{\begingroup\setupcopy\copyans}
\def\setupcopy{\def\do##1{\catcode`##1=12}\dospecials
     \catcode`\|=12\obeylines}
{\obeylines \gdef\copyans#1
  {\def\next{#1}%
  \ifx\next\empty\let\next=\endgroup%
  \else\immediate\write\ans{\next}\let\next=\copyans\fi\next}}
\def\printanswers{\newpage\begin{center} Answers \end{center}
    \immediate\closeout\ans\input answers}

\begin{document}
\noindent  \header

\bigskip

\begin{center}
{%\Large
No calculators. \\
Show all work.\\
Circle your answer.\\
Hand in to your TA.\\
}
\end{center}

\par\vskip .25in
\begin{center}
Name\makebox[3in]{\hrulefill} \\
\end{center}

\begin{center}
Circle the time of your TA section: \\
\end{center}
\begin{center}
   8:50-Laghi
   9:55-Laghi
   11:00-Basamakov
   12:05-Basamakov
   1:20-Jenkins
   2:25-Jenkins
\end{center}

% end file with \finishtbl
%\bigskip

\begin{center} \Large
 \begin{tabular}{||c|c|c||} \hline\hline
  Problem & Points & Score \\  \hline\hline
  \input{exam.tbl}
  \hline \hline
 \end{tabular}
\end{center}

\immediate\openout\examTBLFile=exam.tbl

% \setcounter{page}{0}

%p9.1
\prob{\easy}
Find the equation for the parabola that has its vertex at the
origin and its focus at $(0,-5)$.
\answer $-20y=x^2$

\probx{\easy} Sketch the graph of the equation
$$4(x-2)^2+y^2=16$$
\answer This an ellipse thru the points $(0,0),(4,0),(2,4),(2,-4)$.

%p9.4
\prob{\easy}
Classify the following according to whether they are a parabola,
ellipse, or hyperbola, by drawing a connecting line.

\def\doit#1{\par\vspace{.5in}\fbox{{#1}}}
\def\doitx#1{\par\vspace{1.25in}\fbox{{#1}}}

\begin{minipage}[t]{3.5in}
  \doit{$12x^2 +(y-3)^2=144$}
  \doit{$xy=1$}
  \doit{$x+2y^2-y+3=0$}
  \doit{$x^2-y^2=1$}
  \doit{$12x^2 -(y+3)^2=144$}
  \doit{$x^2+y^2=1$}
  \doit{$y=x^2$}
  \doit{$4x-2x^2-y^2+17=0$}
\end{minipage}
\begin{minipage}[t]{1.5in}
  \doitx{parabola}
  \doitx{ellipse}
  \doitx{hyperbola}
\end{minipage}
\answer ehphhepe

%p4.1
\prob{\easy}
Suppose $f(x)=Ca^x$ is an exponential function such that
the graph of $f$ meets the $y$-axis at the point $(0,3)$ and
contains the point $(2,12)$.  What is $f(x)$?
\answer $f(x)=3\cdot 2^x$


\probx{\easy}
Find the inverse function of $f(x)={{x-2}\over{x+3}}$.
\answer $f^{-1}(y)={{3y+2}\over{1-y}}$


%p4.2
\prob{\easy}
Suppose you invest 1234 dollars at 6 percent yearly interest compounded
continuously.  After 40 days how much will you have? (You need
not simplify your answer.)
\answer $1234e^{{40\over 365}.06}$

% p4.4 laws of logs
\probx{\easy}
Use the Laws of Logarithms to expand the expression below as
much as possible.
$$\ln\left(\sqrt{{{{x^e}\over{(1-x^2)}}}}\right)$$
\answer $({1\over 2})e\ln(x)-({1\over 2})\ln(1-x)-({1\over 2})\ln(1+x)]$

\prob{\easy} Rewrite the expression below as a single logarithm.
$$3\log(xy)-2\log(x-y)+5\log(x+y)$$
\answer $\log\left((xy)^3(x-y)^{-2}(x+y)^5\right)$

%c6.4
\probx{\easy}
Suppose
$$f(x)={1\over{(1+\ln(x))^2}}$$
What is $f^\prime(e)$?
\answer $-{1\over 4e}$

%p4.5
\prob{\medium} Find all solutions of the equation
$$\log(x^2+1)=1+2\log(x)$$
\answer $x={1\over 3}$.  Note that $x=-{1\over 3}$ is not a solution.

%application
\probx{\medium}
The half-life of Carbon-14 is 5730 years.   A prehistoric human bone
is analyzed and found to contain only $12.5$ percent of the Carbon-14
which it had when the creature walked the earth.  How many years
ago did this person live?
\answer $3\times 5730$ years ago.  After 5730 years there was 50\% left,
after another 5730 years there was 25\% left, and after the third
5730 there was 12.5\% left.


\prob{\medium}
Find $\lim_{x\to 0}({1\over{\ln(1+x)}}-{1\over x})$.
\answer ${1\over 2}$

\probx{\medium}
Find $\lim_{x\to 1}x^{4\over {x-1}}$.
\answer $e^4$

% p4.3
\prob{\hard}
What is the domain of the function
$$\log_{\log(x)}(x)$$
\answer All $x>1$.  $\log_a(x)$ is only defined for $a>0$ and
$a=\log(x)>0$ only when $x>1$.

%c6.1
\probx{\hard}
Suppose $g$ is the inverse function of a differentiable
function $f$ and let $h(x)={1\over g(x)}$.  Suppose $f(3)=2$,
$f(2)=1$, $f^\prime(2)=3$, and
$f^\prime(3)={1\over 9}$. \par  Find $h^\prime(2)$.
\answer It's $-1$.
$f(g(x))=x$ so $f^\prime(g(x))g^\prime(x)=1$ and so
$$g^\prime(x)={1\over f^\prime(g(x))}.$$
Since $f(3)=2$ we know that $g(2)=3$, hence
$$g^\prime(2)={1\over f^\prime(g(2))}={1\over f^\prime(3)}=9$$
and so by the quotient rule:
$$h^\prime(x)=-{1\over (g(x))^2}g^\prime(x)$$
thus
$$h^\prime(2)=-{1\over (g(2))^2}g^\prime(2)=-{1\over9}9=-1$$

%application
\prob{\hard}
On a hot day a thermometer is taken outside from an airconditioned room.
The room temperature is 70 degrees.  After one minute the thermometer
reads 80 degrees and after two minutes it reads 85 degrees.  What
is the temperature outside?
\answer 90 degrees.  Let $T(t)$ be the temperature of the thermometer
after $t$ minutes.  By Newton's Law of cooling
$$T(t)=A+Ce^{kt}$$
where $A$ is the temperature outside and $C$ and $k$ are unknown constants.
We are given
$$70=T(0)=A+C$$
$$80=T(1)=A+Ce^k=A+Cx$$
$$85=T(2)=A+Ce^{k2}=A+Cx^2$$
where we have let the unknown $e^k$ be written $x$.
Subtracting the first equation from the second and the third for the second
gives
$$10=Cx-C=C(x-1)$$
$$5=Cx^2-Cx=x[C(x-1)]$$
Substituting $10$ for $C(x-1)$ into the second equation gives $5=x10$ and
so $x=1/2$, $C=-20$, and $A=90$.



\prob{\hard}
The function $y=e^x$ satisfies the differential equation
$${dy\over dx}=y$$
So does any constant multiple of it, $y=Ce^x$ for some
constant $C$.
\par Explain why, if $y$ is any solution of this differential equation,
then there exists a constant $C$ such that $y=Ce^x$.
(Hint: I did this twice in lecture.)
\answer Suppose ${dy\over dx}=y$.  Let $u=ye^{-x}$.
By the product rule
$${du\over dx}={dy\over dx}e^{-x}+ye^{-x}(-1)=ye^{-x}-ye^{-x}=0$$
This means the derivative of $u$ is always zero, hence $u$ is a
constant function.  So there exists a constant $C$ such that
$u\equiv C$ and so $ye^{-x}=u=C$ and hence $y=Ce^x$.

\prob{\hard}
State L'Hospitol's rule for computing
$$\lim_{x\to a}{f(x)\over g(x)}$$
and explain why it is true in the special
case that $g^\prime(a)\not=0$ and the indeterminate form
is $0 \over 0$.
\answer See pages 420-421.  Note 3 on page 421 was done twice
in lecture.

\finishtbl
\printanswers

\end{document}