%=========================================================================%
% initialize TeX
%=========================================================================%

\magnification=\magstep1
\baselineskip=14pt
\hsize = 6.5truein
\hoffset = 0.0truein
\vsize = 9.1truein
\voffset = -0.25truein
\emergencystretch = 0.05\hsize

\newif\ifblackboardbold

% comment out the following line if AMS msbm fonts aren't available
\blackboardboldtrue

%=========================================================================%
% select fonts
%=========================================================================%

\font\titlefont=cmbx12 scaled\magstephalf
\font\sectionfont=cmbx12
\font\netfont=cmtt9

% Establish AMS blackboard bold fonts without using amssym.def, amssym.tex

\newfam\bboldfam
\ifblackboardbold
\font\tenbbold=msbm10
\font\sevenbbold=msbm7
\font\fivebbold=msbm5
\textfont\bboldfam=\tenbbold
\scriptfont\bboldfam=\sevenbbold
\scriptscriptfont\bboldfam=\fivebbold
\def\bbold{\fam\bboldfam\tenbbold}
\else
\def\bbold{\bf}
\fi

%=========================================================================%
% exam macros (use \practice or \exam to initialize)
%=========================================================================%

\newcount\probcount
\probcount=0

\def\practice{
\def\prob##1\par
{\medskip\global\advance\probcount by 1\noindent
{\bf[\number\probcount]}~##1}}

\def\exam{
\def\prob##1\par
{\global\advance\probcount by 1\noindent {\bf[\number\probcount]}~##1
\vfill\eject
\noindent{\bf Problem: \hbox to 30pt{\hrulefill}}
\vfill\eject
}
\footline={\tenbf Page \folio\hfil Continued on page: \hbox to
30pt{\hrulefill}}
% set up name, id, school fields for exam
\vskip 0.1in 
{\bf \noindent Name: \hrulefill \hbox{}}
\vskip 0.1in 
{\bf \noindent ID: \hrulefill ~ School: \hrulefill \hbox{}}
\vskip 0.2in
}

% set up those boxes to put the scores in

\def\boxAsub#1{\vbox{\hbox{#1}}}
\def\boxA#1#2{[{\bf #1}] (#2 pts)}
\def\boxT{\bf TOTAL}
\def\boxB{\vbox to 25pt{\hbox{\kern 56pt}}}

\def\fiveboxes#1{
\centerline{
\vbox{\offinterlineskip\halign{
 \strut\vrule\hfil##\hfil\vrule
 &&\hfil##\hfil\vrule\cr
 \noalign{\hrule}
 \boxA 1{#1} &
 \boxA 2{#1} &
 \boxA 3{#1} &
 \boxA 4{#1} &
 \boxA 5{#1} &
 \boxT \cr
 \noalign{\hrule}
 \boxB & \boxB & \boxB & \boxB & \boxB & \boxB  \cr
 \noalign{\hrule}
}}}}

\def\eightboxes#1{
\centerline{
\vbox{\offinterlineskip\halign{
 \strut\vrule\hfil##\hfil\vrule
 &&\hfil##\hfil\vrule\cr
 \multispan4{\hrulefill}\cr
 \boxA 1{#1} &
 \boxA 2{#1} &
 \boxA 3{#1} &
 \boxA 4{#1} \cr
 \multispan4{\hrulefill}\cr
 \boxB & \boxB & \boxB & \boxB \cr
 \noalign{\hrule}
 \boxA 5{#1} &
 \boxA 6{#1} &
 \boxA 7{#1} &
 \boxA 8{#1} &
 \boxT \cr
 \noalign{\hrule}
 \boxB & \boxB & \boxB & \boxB & \boxB\cr
 \noalign{\hrule}
}}}}

\def\tenboxes#1{
\centerline{
\vbox{\offinterlineskip\halign{
 \strut\vrule\hfil##\hfil\vrule
 &&\hfil##\hfil\vrule\cr
 \multispan5{\hrulefill}\cr
 \boxA 1{#1} &
 \boxA 2{#1} &
 \boxA 3{#1} &
 \boxA 4{#1} &
 \boxA 5{#1} \cr
 \multispan5{\hrulefill}\cr
 \boxB & \boxB & \boxB & \boxB & \boxB \cr
 \noalign{\hrule}
 \boxA 6{#1} &
 \boxA 7{#1} &
 \boxA 8{#1} &
 \boxA 9{#1} &
 \boxA{10}{#1} &
 \boxT \cr
 \noalign{\hrule}
 \boxB & \boxB & \boxB & \boxB & \boxB & \boxB\cr
 \noalign{\hrule}
}}}}

%==========================================================================
% macros
%==========================================================================

\def\bmatrix#1{{\left[\vcenter{\halign
  {&\kern 4pt\hfil$##\mathstrut$\kern 4pt\cr#1}}\right]}}

% surround with $ $ if not already in math mode
\def\enma#1{{\ifmmode#1\else$#1$\fi}}

\def\mathbb#1{{\bbold #1}}
\def\mathbf#1{{\bf #1}}

% blackboard bold symbols
\def\FF{\enma{\mathbb{F}}}
\def\NN{\enma{\mathbb{N}}}
\def\RR{\enma{\mathbb{R}}}
\def\CC{\enma{\mathbb{C}}}
\def\QQ{\enma{\mathbb{Q}}}
\def\ZZ{\enma{\mathbb{Z}}}
\def\bbPP{\enma{\mathbb{P}}}

% caligraphic symbols
\def\cAA{\enma{\cal A}}
\def\cBB{\enma{\cal B}}

% bold symbols
\def\aa{\enma{\mathbf{a}}}
\def\bb{\enma{\mathbf{b}}}
\def\cc{\enma{\mathbf{c}}}
\def\ee{\enma{\mathbf{e}}}
\def\mm{\enma{\mathbf{m}}}
\def\pp{\enma{\mathbf{p}}}
\def\uu{\enma{\mathbf{v}}}
\def\vv{\enma{\mathbf{v}}}
\def\ww{\enma{\mathbf{w}}}
\def\xx{\enma{\mathbf{x}}}
\def\yy{\enma{\mathbf{y}}}
\def\XX{\enma{\mathbf{X}}}
\def\KK{\enma{\mathbf{K}}}
\def\PP{\enma{\mathbf{P}}}
\def\LL{\enma{\mathbf{L}}}
\def\II{\enma{\mathbf{I}}}
\def\zero{\enma{\mathbf{0}}}
\def\blambda{\enma{\mathbf{\lambda}}}

% misc

\def\abs#1{\enma{\left| #1 \right|}}
\def\set#1{\enma{\{#1\}}}
\def\setdef#1#2{\enma{\{\;#1\;\,|\allowbreak
  \;\,#2\;\}}}
\def\idealdef#1#2{\enma{\langle\;#1\;\,
  |\;\,#2\;\rangle}}

%=========================================================================%
% exam
%=========================================================================%

\centerline{\titlefont Exam 1}
\smallskip
\centerline{Calculus IIIA, Dave Bayer, February 15, 2001}
\medskip

%\practice
\exam\fiveboxes{6}

Please work only one problem per page, starting with the pages provided, and
number all continuations clearly. Only work which can be found in this way will
be graded.

Please do not use calculators or decimal notation.

\bigskip

\prob
Find the area under the parametrized curve
$$x=e^t, \quad y=e^{2t}, \quad 0 \le t \le \ln(2).$$

\prob
Find the area of one petal of the polar curve $$r = \cos(2\theta).$$

\prob
Find the surface area generating by rotating around the $x$-axis the parametrized curve
$$x = \cos(t), \quad y = \sin(t), \quad 0 \le t \le \pi.$$

\prob
Find the arc length of the parametrized curve
$$x=t \,\cos(t), \quad y=t \,\sin(t), \quad 0 \le t \le 2\pi.$$
Simplify the integral as far as possible, but do not solve it.  Instead, guess its value.

\prob
Find the arc length of the polar curve
$$r=\theta, \quad 0 \le \theta \le 2\pi.$$
Simplify the integral as far as possible, but do not solve it. Instead, guess its value.

\bye