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\documentclass[12pt]{article}
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\def\easy{4}
\def\medium{15}
\def\hard{8}

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\pagestyle{myheadings}
\def\la{\langle}  \def\ra{\rangle}
\def\header{{\hfill Final \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
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  \immediate\write\ans{\string\bigskip \the\probno . }
  \copytoblankline}
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  \ifx\next\empty\let\next=\endgroup%
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\def\printanswers{\newpage\begin{center} Answers \end{center}
    \immediate\closeout\ans\input answers}

\begin{document}

\bigskip

\begin{center}
{%\Large
No calculators. \\
No notes.\\
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}

\prob{\medium} % c6.4 17 exam 1
Find the derivative of $f(t)=\log_2(t^4-t^2+1)$.
\answer $${ {4t^3-2t} \over {(t^4-t^2+1)\ln(2)}}$$

\prob{\medium} % exam 1
The half-life of Carbon-14 is 5730 years.   A prehistoric human bone
is analyzed and found to contain only $25$ percent of the Carbon-14
which it had when the creature walked the earth.  How many years
ago did this person live?
\answer $2\times 5730=11460$ years ago.

\prob{\medium} % exam 1
Suppose $g$ is the inverse function of a differentiable
function $f$ and let $h(x)=(g(x))^2$.  Suppose $f(3)=2$,
$f(2)=1$, $f^\prime(2)=3$, and
$f^\prime(3)={1\over 9}$. \par  Find $h^\prime(2)$.
\answer $54$

\prob{\medium} % exam 2
In certain engineering groups it is customary to let
90 degrees be refered to as 100 engineering degrees.  So,
for example,  25 engineering degrees is the same
as 25\% of 90 degrees, or 22.5 degrees.
The function engsin$(x)$ is the ordinary sine function using
engineering degrees instead of radians.  Thus, for example,
engsin$(50)=\sin({\pi\over 4})={\sqrt{2}\over 2}$.
Find the derivative of engsin$(x)$ with respect to $x$.
\answer ${\pi\over 200}\mbox{engcos}(x)$

\prob{\medium} % exam 2
Find the following limit:
$$\lim_{n\to\infty}\sum_{k=1}^n ({k\over n})^6({1\over n})$$
\answer $\int_0^1 x^6\;dx=1/7$

\prob{\medium} % exam 2 (also done in lecture on volumes)
Find the definite integral:
$$\int_0^1 x\cos(x^2)\;dx $$
\answer $1/2\sin(1)$.  Substitute $u=x^2$.

\prob{\medium} % exam 2
Sketch the region bounded by the curves $x=1-y^2$ and $x=y^2-1$
and find the area.
\answer $4\int^1_0(1-y^2)dy={8\over 3}$


\prob{\medium} % volume
Graph the region bounded by the curve
$$x^2+(y-3)^2=4$$
This region is rotated about the $x$-axis.
Find the volume of the resulting solid.
\answer The region is a disk of
radius $r=2$ centered at the point $(0,3)$.
The solid is a torus.  The volume of a torus is
$(2\pi R)(\pi r^2)$ where a disk of radius
$r$ is rotate about a point $R$ units away.
In this case $R=3$ and $r=2$ so the volume is $24\pi^2$.

\prob{\medium}  % volume
Define the function $f$ by
$$f(x)=\left\{\begin{array}{ll}
  2x   & \mbox{ if } 0\leq x\leq 1\\
x^2+1  & \mbox{ if } 1 < x\leq 2  \\
\end{array}\right.$$
Sketch the graph of $f$.
Find the volume of the solid obtained by rotating
the region under the graph of $f$ about the $x$-axis.
\answer $\int_0^1 \pi(2x)^2\;dx +\int_1^2 \pi (x^2+1)^2\;dx=
{198\over 15}\pi$

\prob{\medium} % volume
Sketch the region in the plane bounded by
$$y=x^2$$
$$y=0$$
$$x=1$$
$$x=2$$
Use the method of cylindrical shells to find the volume
of the solid obtained by rotating this region
about the line $x=1$.
\answer $\int^2_1 2\pi(x-1)x^2\;dx={17\over 6}\pi$

\finishtbl
\printanswers

\end{document}