\magnification1200\footline{\ifnum\pageno=1 \hfil\else\hfil\number\pageno\hfil\fi}
\input pics.tex
\newcount\credit\credit=5
\newcount\pspace\pspace=0
\newcount\pc\pc=1
\hfuzz=5pt
\let\xxsum=\sum
\edef\sum{\displaystyle\xxsum}

\def\problemspacing{%
\ifcase\pspace\relax%
\null\vskip10pt\global\pspace=1 \relax\or%
\vfill\global\pspace=2 \or%
\vfill\eject\null\vskip10pt\global\pspace=1 \or%
\vfill\eject\null\vskip10pt\or
\global\pspace=3 %
\fi%%
\ifnum\pc=8 \global\pspace=4 \fi%
}

%%\def\problemspacing{\vskip10pt}
\def\newp{\problemspacing%
\noindent{\bf\number\pc.}(\number\credit\ pts.)
\global\advance\pc by1}
\def\onea#1{\noindent#1}
 \def\twoa#1#2{\noindent\hbox to\hsize{%
\hbox to.5\hsize{#1\hfil}\hbox to.5\hsize{#2\hfil}}}
 \def\stwoa#1#2{\noindent\hbox to\hsize{%
\hbox to.38\hsize{#1\hfil}\hbox to.62\hsize{#2\hfil}\hfil}}

\def\threea#1#2#3{\noindent\hbox to\hsize{%
 \hbox to.33\hsize{#1\hfil}%
\hbox to.33\hsize{#2\hfil}%
\hbox to.33\hsize{#3\hfil}%
\hfil}}
\def\twothreea#1#2{\noindent\hbox to\hsize{%
 \hbox to.33\hsize{#1\hfil}%
\hbox to.33\hsize{#2\hfil}%
\hfil}}

\def\sfivea#1#2#3#4#5{\noindent\hbox to\hsize{%
 \hbox to.17\hsize{#1\hfil}%
 \hbox to.17\hsize{#2\hfil}%
 \hbox to.17\hsize{#3\hfil}%
 \hbox to.17\hsize{#4\hfil}%
 \hbox to.32\hsize{#5\hfil}%
\hfil}}
\def\fivea#1#2#3#4#5{\noindent\hbox to\hsize{%
 \hbox to.2\hsize{#1\hfil}%
 \hbox to.2\hsize{#2\hfil}%
 \hbox to.2\hsize{#3\hfil}%
 \hbox to.2\hsize{#4\hfil}%
 \hbox to.2\hsize{#5\hfil}%
\hfil}}



 \def\frac#1#2{{{\textstyle#1}\over{\textstyle#2}}}
\def\Lim{\displaystyle\lim}
\def\Int{\displaystyle\int}


\def\frac#1#2{{\displaystyle{{\textstyle#1}\over{\textstyle#2}}}}
\def\ans#1#2#3#4#5{\par\vskip2pt\noindent\hbox to\hsize{%
(a)\ #1\hfil (b)\ #2\hfil (c)\ #3\hfil (d)\ #4\hfil (e)\ #5}\par}
\def\ansA#1#2#3#4#5{\par\vskip2pt\noindent\hbox to\hsize{%
(a)\ #1\hfil }\par\hbox to\hsize{(b)\ #2\hfil }%
\par\hbox to\hsize{(c)\ #3\hfil }
\par\hbox to\hsize{(d)\ #4\hfil }\par\hbox to\hsize{(e)\ #5\hfil}\par}

\def\ansB#1#2#3#4#5{\par\vskip2pt\noindent\hbox to\hsize{%
\hbox to .4\hsize{(a)\ #1\hfil}\hbox to .45\hsize{(b)\ #2\hfil}}%
\par\hbox to\hsize{\hbox to .4\hsize{(c)\ #3\hfil}\hbox to .4\hsize{(d)\ #4\hfil}}\par
\hbox to\hsize{(e)\ #5\hfil}\par}

\long\def\boxit#1#2{\vbox{\hrule\hbox{\vrule\kern#2
\vbox{\kern#2#1\kern#2}\kern#2\vrule}\hrule}}
\newcount\pc
\def\pr{\vskip10pt\noindent\number\pc.\global\advance\pc by 1}
\def\dd#1#2{{{\textstyle d\> #1}\over{\textstyle d#2}}}

\def\rhs#1#2{\par\noindent\hbox to\hsize{\hfill\hbox to#1{#2\hfill}}}
\def\ul#1{\vrule width#1 height -3pt depth3.1pt}
\def\UL#1#2{\hbox to0pt{\qquad #2\hss}\ul{#1}}
\newif\ifnotmarkanswers\notmarkanswerstrue
\newcount\ansmark
\def\ll#1#2{\tabalign {#1.}&
\global\ansmark=#2 \relax%
\ifnotmarkanswers \global\ansmark=0 \fi%
\ifnum\ansmark=1 \relax($\bullet$)\else(a)\fi&
\ifnum\ansmark=2 \relax($\bullet$)\else(b)\fi&
\ifnum\ansmark=3 \relax($\bullet$)\else(c)\fi&
\ifnum\ansmark=4 \relax($\bullet$)\else(d)\fi&
\ifnum\ansmark=5 \relax($\bullet$)\else(e)\fi&
\qquad\cr}

\def\marked{\pageno=1 %
\global\notmarkanswersfalse
\headline{\vbox{\rhs{3in}{Name:\ \ul{3in}\hss}\vskip10pt%
\rhs{3in}{Instructor-section:\ \UL{2.3in}{Bullwinkle}\hss}%
}}
\answersheet}


\def\+{\tabalign}
\long\def\exam#1{%
\pc=1 \pageno=1 \pspace=0 %
\headline{\vbox{\rhs{3in}{Name:\ \ul{3in}\hss}\vskip10pt%
\rhs{3in}{Instructor-section:\ \UL{2.3in}{#1}\hss}%
}}
\long\gdef\answersheet{
\null\vskip10pt
\centerline{\bf Math 125, Test III}
\centerline{December 1, 1998}
\vskip10pt

\itemitem{$\bullet$}
The Honor Code is in effect for this examination.
All work is to be your own.
\itemitem{$\bullet$} No calculators.
\itemitem{$\bullet$} The exam lasts for one hour.
\itemitem{$\bullet$} Be sure that your name is on every page in case
pages become detached. 
\itemitem{$\bullet$} Be sure that you have all 11 pages of the test.
\vskip5pt
\centerline{Good Luck!}
\vskip4pt
\def\HH{\hskip.75in}
\settabs\+9.\qquad &\HH&\HH&\HH&\HH&\HH&\cr
\hskip.5 in\boxit{\hbox to 3.8in{\vbox{\obeylines\baselineskip20pt
\noindent
PLEASE MARK YOUR ANSWERS WITH AN X, not a circle!
\ll 1 2
\ll 2 5
\ll 3 3
\ll 4 1
\ll 5 3
\ll 6 4
\ll 7 5
\ll 8 3
}\hss}}{10pt}
%%\vskip10pt

\settabs\+Total multiple choice: &\hskip 2in\cr
\hfill{{\vbox to0pt{\boxit{\vbox{\obeylines\hsize=3in\baselineskip26pt
DO NOT WRITE IN THIS BOX!
\tabalign Total multiple choice: &\quad\ul{1in}\cr
\tabalign\hfill9.&\quad\ul{1in}\cr
\tabalign\hfill10.&\quad\ul{1in}\cr
\tabalign\hfill11.&\quad\ul{1in}\cr
\tabalign\hfill12.&\quad\ul{1in}\cr
\tabalign\hfill13.&\quad\ul{1in}\cr
\tabalign\hfill14.&\quad\ul{1in}\cr

\tabalign\hfil {\bf Total:}&\quad\ul{1in}\cr
}}{10pt}\vss}}}}
\answersheet
\vfill\eject\null\vskip10pt
\centerline{\bf Multiple Choice}

\newp Suppose that $\Int_0^{-3} f(x) dx =4$
and $\Int_{-3}^0 g(x) dx =2$.
Find $\Int_{-3}^0 \left(3f(x) + 4g(x)\right) dx$.

\sfivea{(a) 4}{(b) -4}{(c) 18}{(d) -17}{(e) cannot be determined}
%%% b

\newp Partition the interval from $0$ to $1$ into $n$ equal parts.
In the $k$-th subinterval, choose the right-hand endpoint.
The resulting Riemann sum for the integral
$\Int_0^1 3x^2 dx$ is

\twoa{(a) $\frac{1}{n}\sum_{k=1}^{n}\left(3k^2\right)$}
{(b) $\frac{1}{n}\sum_{k=1}^{n}\left(3k^2n^2\right)$}

\twoa{(c) $\sum_{k=1}^{n}\left(\frac{k}{n}\left(3k^2n^2\right)\right)$}
{(d) $\frac{1}{n}\sum_{k=1}^{n}\left(\frac{3}{n^2}\right)$}

\onea{(e) $\frac{1}{n}\sum_{k=1}^{n}\left(\frac{3k^2}{n^2}\right)$}
%%% e

\newp Suppose that $f$ is a continuous function on $[a,b]$ such
that
$\Int_a^b f(x) dx = 0$.
Which of the following must be true?

\noindent (a) There exists exactly one $c$ in $[a,b]$ such that $f(c)
=0$.

\stwoa{(b) $f(c) = 0$ for all $c$ in $[a,b]$}
{(c) There exists at least one $c$ in $[a,b]$ such that $f(c)
=0$}

\stwoa{(d) $f(x) \le 0$ for all $x$ in $[a,b]$}
{(e) $f'(c) =0$ for some $c$ in $[a,b]$}
%%% c

\newp Find $\Int \frac{\sin x}{(1+\cos x)^3}dx$.

\threea{(a)$\frac{1}{2(1+\cos x)^2} + C$}
{(b)$\frac{1+C}{2(1+\cos x)^2}$}
{(c)$\frac{1}{2(1+\sin x)^2} + C$}
\twothreea{(d)$\frac{1+C}{2(1+\sin x)^2}$}
{(e)$\frac{1}{(\sin x+\cos x)^2} + C$}
%%% a

\newp  Find the derivative 
$\frac {dy}{dx}$ of the function $y=\Int_0^{
x^2 }\sqrt {1+t^2}dt$.

\fivea{(a) $\sqrt{1+x^2}$}
{(b) $\sqrt{1+x^4}$}
{(c) $2x\sqrt{1+x^4}$}
{(d) $x^2\>\sqrt{1+x^2}$}
{(e) $x^2\>\sqrt{1+x^4}$}
%%% c

\newp Solve the initial value problem
$$\frac{dy}{dx}=\frac 1x,\ \  y(1)=2.$$

\threea
{(a) $-\frac 1{x^2}+3$}
{(b) $ \frac 1{x^2}  +1$}
{(c) $-\Int_1^x\frac 1tdt+2$}

\twothreea{(d) $\Int_1^x\frac 1tdt +2$}
{(e) $\Int_1^x\frac 1tdt$}
%%% d

\newp Find the area of the region between the curves  
$y=1+\cos x$ and the $x$-axis, $0\le x\le \pi$.

\fivea{(a) $1$}
{(b) $2$}
{(c) $0$}
{(d) $\frac 12$}
{(e) $\pi$}
%%% e

\newp If $\Int_0^xf(t)dt=x \sin x$, find $f(x)$.

\threea{(a) $x\sin x$}
{(b) $x\cos x$}
{(c) $\sin x + x\cos x$}
\twothreea{(d) $\cos x -x\sin x$}
{(e) $(\sin x)(\cos x)$}
%% a

\vfill\eject\null\vskip10pt
\credit=10
\centerline{\bf Partial Credit}

\newp  A solid lying above the $xy$-plane 
is being sliced by planes perpendicular to the $y$-axis.
The base of the solid is the circle of radius $3$ in the $xy$-plane
given by $x^2+y^2=9$.
Each slice is a rectangle with one side a chord of this circle and
the other side a constant height $5$.
Set up a definite integral which will yield the volume.
Do NOT evaluate this integral.


\newp  Let $f(x) = 3\sqrt{x}$.  Find the average value of $f$
on $[1,4]$.

\newp Find the area of the region enclosed by the parabola
$y=x^2+x-1$ and the line $y=x$.

\newp Integrate 
$$ \Int_0^1 \frac {x^2}{\sqrt {x^3+1}}dx.$$

\newp Solve the initial value problem
$$\frac{dy}{dx}=6x^2(x^3-1)^3,\ \  y(1)=3.$$

\newp  Estimate the  integral 
$$\Int_0^2\frac 1{1+x^2}dx $$

\itemitem{(a)} using  the trapezoidal rule with $n=4$
\itemitem{(b)}  using  the Simpson's rule with $n=4$

\noindent We are looking for the formulas for these two rules.
In particular 
{{\bf DO NOT}} attempt to add the fractions and no error
estimates are required.
\vfill\eject}

%\exam{Dwyer}
%\exam{Shaw}
%\exam{Taylor}
%\exam{Nollet}
%\exam{Budhiraja}
%\exam{Gekhtman \quad{\sl Section \#:}}
\exam{Bullwinkle}
\marked{}
\end
