\magnification1200\footline{\ifnum\pageno=1 \hfil\else\hfil\number\pageno\hfil\fi}

\font\small=cmr7
\def\newpage{\eject}
\newdimen\pspacewidth
\newcount\credit\credit=6
\newcount\pspace\pspace=0
\newcount\pc\pc=1
\hfuzz=5pt
\let\xxsum=\sum
\edef\sum{\displaystyle\xxsum}

\newdimen\picH\newdimen\picV
\def\picture #1 by #2 (#3 scaled #4){
\picV=#2 \divide\picV by1000 \multiply\picV by#4 
\picH=#1 \divide\picH by1000 \multiply\picH by#4
\vtop to \picV{
      \hrule width \picH height 0pt depth 0pt
      \vfill
      \special{picture #3 scaled #4}}}

\def\newp{%%\problemspacing%
\noindent{\bf\number\pc.}(\number\credit\ pts.)
\global\advance\pc by1}

\newbox\Aa\newbox\Ab\newbox\Ac
\newbox\Ad\newbox\Ae
\newbox\aa\newbox\ab\newbox\ac
\newbox\ad\newbox\ae

\newdimen\Aspace
\newcount\Ax
\newcount\tmpPC
\def\rememberPC{\global\tmpPC=\pc\global\advance\tmpPC by-1 }

\hfuzz=20pt

\def\five{\global\Ax=5 \global\Aspace=.2\hsize\relax\global\advance\Aspace by-20pt}
\def\three{\global\Ax=3 \global\Aspace=.3\hsize\relax\global\advance\Aspace by-20pt}
\def\two{\global\Ax=2 \global\Aspace=.5\hsize\relax\global\advance\Aspace by-20pt}
\def\one{\global\Ax=1 \global\Aspace=\hsize\relax\global\advance\Aspace by-20pt}

\def\AN#1#2#3#4#5{%
\setbox\Aa=\hbox to\Aspace{#1\hfil}%
\setbox\Ab=\hbox to\Aspace{#2\hfil}\setbox\Ac=\hbox to\Aspace{#3\hfil}%
\setbox\Ad=\hbox to\Aspace{#4\hfil}\setbox\Ae=\hbox to\Aspace{#5\hfil}}

\newdimen\Ls\Ls=10pt
\def\pra{\ifcase\Ax \or\praone\or\pratwo\or\prathree\or \or\prafive\fi}

\def\prafive{\vskip\Ls\noindent(a)\quad\box\Aa(b)\quad\box\Ab%
(c)\quad\box\Ac(d)\quad\box\Ad%
(e)\quad\box\Ae}

\def\prathree{\vskip\Ls\noindent(a)\quad\box\Aa(b)\quad\box\Ab%
(c)\quad\box\Ac\vskip\Ls\noindent(d)\quad\box\Ad%
(e)\quad\box\Ae}

\def\pratwo{\vskip\Ls\noindent(a)\quad\box\Aa(b)\quad\box\Ab%
\vskip\Ls\noindent(c)\quad\box\Ac(d)\quad\box\Ad%
\vskip\Ls\noindent(e)\quad\box\Ae}

\def\praone{\vskip\Ls\noindent(a)\quad\box\Aa\vskip\Ls\noindent(b)\quad\box\Ab%
\vskip\Ls\noindent(c)\quad\box\Ac\vskip\Ls\noindent(d)\quad\quad\box\Ad%
\vskip\Ls\noindent(e)\quad\box\Ae}

\def\resetboxes{\setbox\Aa=\box\aa\setbox\Ab=\box\ab\setbox\Ac=\box\ac%
\setbox\Ad=\box\ad\setbox\Ae=\box\ae}
\def\switchbox#1#2{\ifcase#1 \or\setbox\aa=\box#2\or\setbox\ab=\box#2\or%
\setbox\ac=\box#2\or\setbox\ad=\box#2\or\setbox\ae=\box#2 \fi}

\def\Pra#1#2#3#4#5{\switchbox{#1}{\Aa}\switchbox{#2}{\Ab}%
\switchbox{#3}{\Ac}\switchbox{#4}{\Ad}\switchbox{#5}{\Ae}%
\resetboxes\ifcase\Ax \or\praone\or\pratwo\or\prathree\or \or\prafive\fi\vfill}


\def\dd#1#2{{{\textstyle d\> #1}\over{\textstyle d#2}}}
 \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}}}}

\long\def\boxit#1#2#3{\vbox{\hrule\hbox{\vrule\kern#2
\vbox{\hsize=#3\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
\newdimen\AA\AA=20pt
\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&
\hskip\AA\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#2#3#4#5#6{%
\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, Final}
\centerline{December 16, 1998}
\vskip10pt

\itemitem{$\bullet$}
The Honor Code is in effect for this examination.
All work is to be your own.
\itemitem{$\bullet$} Be sure that you have all 14 pages of the test. 
\itemitem{$\bullet$} No calculators are to be used.
\itemitem{$\bullet$} The exam lasts for two hours.
\itemitem{$\bullet$} You are to hand in just the front page.
\vskip5pt
\centerline{Good Luck!}
\vskip10pt
\def\leaderfill{\leaders\hbox to3pt{\hss.\hss}\hfill}
\def\pskip{\vskip -4pt\hbox to\pspacewidth{\leaderfill}\vskip-4pt}
\def\HH{\hskip.42in}
\settabs\+9.\qquad &\HH&\HH&\HH&\HH&\HH&\cr
\centerline{Please mark your answers with an {\bf X}!\qquad Do NOT circle them!}
\par\centerline{\small The dotted lines in the answer box indicate page breaks.}
\hskip-45pt\boxit{%
\noindent\hbox to0pt{%
\vtop{\obeylines\baselineskip20pt\hsize=2.5in\pspacewidth=2in
\parindent=0pt
\ll 1 3
\ll 2 5
\pskip
\ll 3 1
\ll 4 4
\pskip
\ll 5 3
\ll 6 3
\pskip
\ll 7 5
\ll 8 4
\pskip
\ll 9 2
\pskip
\ll 10 3
\ll 11 2
\pskip
\ll 12 4
\ll 13 2
\vskip10pt
}\AA=-50pt
\vtop{\obeylines\baselineskip20pt\hsize=2.5in\pspacewidth=2in
\parindent=0pt
\ll 14 2
\ll 15 5
\pskip
\ll 16 4
\ll 17 5
\pskip
\ll 18 3
\ll 19 4
\pskip
\ll 20 3
\ll 21 1
\pskip
\ll 22 2
\ll 23 1
\pskip
\ll 24 2
\ll 25 1
\vskip10pt
Score: \ul{1in}
}\hss}}{20pt}{5in}%
}

%% 4 a's 6 b's 6 c's 5 d's 4 e's

\answersheet
\vfill\eject\pc=1 \pageno=1 \pspace=0 %
\headline{\vbox{\rhs{3in}{Name:\ \ul{3in}\hss}\vskip10pt%
\rhs{3in}{Instructor-section:\ \UL{2.3in}{#2}\hss}%
}}
\answersheet
\vfill\eject\pc=1 \pageno=1 \pspace=0 %
\headline{\vbox{\rhs{3in}{Name:\ \ul{3in}\hss}\vskip10pt%
\rhs{3in}{Instructor-section:\ \UL{2.3in}{#3}\hss}%
}}
\answersheet
\vfill\eject\pc=1 \pageno=1 \pspace=0 %
\headline{\vbox{\rhs{3in}{Name:\ \ul{3in}\hss}\vskip10pt%
\rhs{3in}{Instructor-section:\ \UL{2.3in}{#4}\hss}%
}}
\answersheet
\vfill\eject\pc=1 \pageno=1 \pspace=0 %
\headline{\vbox{\rhs{3in}{Name:\ \ul{3in}\hss}\vskip10pt%
\rhs{3in}{Instructor-section:\ \UL{2.3in}{#5}\hss}%
}}
\answersheet
\vfill\eject\pc=1 \pageno=1 \pspace=0 %
\headline{\vbox{\rhs{3in}{Name:\ \ul{3in}\hss}\vskip10pt%
\rhs{3in}{Instructor-section:\ \UL{2.3in}{#6}\hss}%
}}
\answersheet

\vfill\eject\headline{\hfil}
\three
\newp
Find the unique number $a$ which makes the function
$$f(x)= \cases{%
\frac{\sin 4x \cos 2x }{ x}& $ -\pi/4< x\le \pi/4$, $x\neq 0$\cr\noalign{\vskip4pt}
\hfill a\hfill&$x=0$\cr}$$
continuous on the entire interval $[-\pi/4,\pi/4]$.
\AN{$4$}{$2$}{$\frac14$}{$\frac12$}{There is no such number}
\Pra 34215

\five
\newp
$$\Lim_{u\to\infty}\frac{u^3+u^{-3}}{u^5+u^{-5}}$$
\AN{$0$}{$\infty$}{$-\infty$}{$\frac35$}{$\frac53$}
\Pra 51234

\newpage%% page 3 top
\three
\newp
Partition the interval from $0$ to $2$ into $n$ equal parts.
In the $k$-th subinterval, choose the right-hand endpoint.
The resulting Riemann sum for the integral
$\Int_0^2 \sec x\ dx$ is
\AN{$\frac{2}{n}\sum_{k=1}^{n}\sec\left(\frac{2k}n\right)$}
{$\frac{2}{n}\sum_{k=1}^{n}\cos\left(\frac{2k}n\right)$}
{$\frac{3}{n}\sum_{k=1}^{n}\sec\left(\frac{k}n\right)$}
{$\frac{1}{2n}\sum_{k=1}^{n}\sec\left(\frac{k}n\right)$}
{$\frac{1}{n}\sum_{k=1}^{n}\cos\left(\frac{n}k\right)$}
\Pra 12345

\three
\newp
If
$$ x^3 + y^3 = 5xy-3\ ,$$
find $\frac{dy}{dx}$ at the point $(1,1)$.
\AN{$-1$}{$0$}{Does not exist}{$-3$}{$-4$}
\Pra 45123

\newpage%% page 4 top
\five
\newp %% tied to #4
\rememberPC
Suppose $f$ and $g$ are two functions which are differentiable at $5$.
If $f(5)=2$, $f^\prime(5)=1$, $g(5)=3$ and $g^\prime(5)=2$, 
$$\biggl({{\textstyle f}\over{\textstyle g}}\biggr)^\prime(5)=\ ?$$
\AN{$-\frac{1}9$}{$0$}{$\frac59$}{$\frac23$}{$\frac74$}
\Pra 34512

\newp
Suppose that the two functions in problem \number\tmpPC\ are differentiable everywhere.
What additional information do you need to compute the derivative
of $f(g(x))$ at $x=5$?
\AN{$f^\prime(3)$}{$f(3)$}{$f^\prime(2)$}{$g^\prime(1)$}{$g^\prime(2)$}
\Pra 34512

\newpage%% page 5 top
\two
\newp
$$\dd{(x^5+2x+1)^{45}}{x}=\ ?$$
\AN{$45(x^5+2x+1)^{44}(5x^4+2)$}{$45(x^5+2x+1)^{44}$}{$45(5x^4+2)^{44}$}{$45x^44(5x^4+2)$}
{$225x^{224}+90x^{44}$}
\Pra 51234

\five\newp
The absolute maximum of the function $x^3-3x^2+3x+5$ on the interval $[0,2]$ is
\AN{$7$}{$5$}{$3$}{$1$}{$0$}
\Pra 45123

\newpage%% page 6 top
\two
\newp
Suppose that the diagram below
is the graph of the function $f(x)$.

\hskip1in\picture 4.07in by 2.65in (function scaled 500)

\noindent Which graph below best represents the graph of the derivative of $f(x)$.
\AN{\picture 3.99in by 2.68in (d1 scaled 500)}
{\picture 4.08in by 2.68in (d2 scaled 500)}
{\picture 3.99in by 2.63in (d3 scaled 500)}
{\picture 4.11in by 2.74in (d4 scaled 500)}
{\picture 4.03in by 2.65in (d5 scaled 500)}
\Pra 23451


\newpage%% page 7 top
\two
\newp
What are the asymptotes (both vertical and horizontal) for the graph of
$$y=\frac{(2x-1)^2}{(x+1)(x-3)}\quad ?$$
\AN{$x=-1$, $x=3$ and $y=4$}{$x=\frac12$, $y=3$}{$x=\frac12$, $y=3$ and $y=-1$}
{$x=3$ and $y=\frac12$}{There are no asymptotes}
\Pra 34512

\three
\newp
Which of the following is the linearization of the function $f(x)=2x^3-2$ at $x=2$?
\AN{$y=24x-34$}{$y=12x-12$}{$y=6x^2(2x^3-2)$}{$y^2=24x-9$}{$y=6x^2-2$}
\Pra 23451

\newpage%% page 8 top
\five
\newp
How many solutions in real numbers does the equation $x^3-3x^2-1=0$ have?
\AN{$1$}{None}{$2$}{$3$}{$4$}
\Pra 45123

\newp
The graph $y=3x^5-5x^4$ has one point of inflection.
It occurs at $x=\>?$.
\AN{$1$}{$0$}{$-1$}{$2$}{$-2$}
\Pra 23451

\newpage%% page 9 top
\three
\newp
Find the derivative of the function
$\Int^{x^2}_2 \frac{t^3+1}{t^2+1}\>dt$
with respect to $x$.
\AN{$2x\frac{x^6+1}{x^4+1}$}{$0$}{$\frac{x^6+1}{x^2+1}$}
{$\frac{3x^4+1}{2x^2}-\frac{3x^2+1}{2x}$}{$\frac95$}
\Pra 23451

\five
\newp
The value of $\Int^1_0\frac{2x+3}{(x^2+3x+4)^2}\>dx$ is
\AN{$\frac18$}{$-\frac14$}{$8$}{$4$}{$\frac14$}
\Pra 51234

\newpage%% page 10 top
\newp
Suppose that the average value of a function $f(x)$ on the interval $[0,2]$
is $4$.
Suppose further that $f^\prime(x)>0$ on the same interval.
Which of the following statements {\sl must\/} be true?
\AN{$f(2)>4$}{$f(2)<4$}{$f(0)>4$}{$f(0)=4$}{$f(2)=4$}
\Pra 45123

\three
\newp
Find the area between the two curves
$y=x^2$ and $y=5x-4$.
\AN{$\Int^4_1 5x-4-x^2\>dx$}{$\Int^4_1 x^2-5x-4\>dx$}{$\Int^4_0 5x-4+x^2\>dx$}
{$\Int^1_0 5x-4-x^2\>dx$}{$\Int^1_0 x^2-5x+4\>dx$}
\Pra 51234

\newpage%% page 11 top
\five
\newp
$\Int_0^{\pi/4} \sec\theta\>\tan\theta\>d\theta=\>?$
\AN{$\sqrt2-1$}{$1$}{$\sqrt2$}{$\frac{\sqrt2}{2}-1$}{$\frac{\sqrt2}{2}$}
\Pra 34512


\three
\newp
Solve the initial value problem
$$\frac{dy}{dx}=\frac 1{x+2},\ \  y(1)=2\ .$$
\AN{$\Int^x_1 \frac {dt}{t+2}\> +2$}{$\Int^x_2 \frac {dt}{t+2}\> +1$}{$\Int^x_0 \frac {dt}{t+2}$}
{$\Int^x_1 \frac {dt}{t+2}$}{$\Int^x_2 \frac {dt}{t+2}\> +2$}
\Pra 45123

\newpage%% page 12 top
\two
\newp
\rememberPC
Consider the region between
$y=x+1$ and $y=-x^2+5x-2$.
Find the volume obtained by rotating this region about the $x$-axis.
\AN{$\pi\Int^3_1\bigl((-x^2+5x-2)^2-(x+1)^2\bigr)\>dx$}{$2\pi\Int^3_1\bigl((-x^2+5x-2)-(x+1)\bigr)\>dx$}
{$\pi\Int^3_1\bigl((x+1)^2-(-x^2+5x-2)^2\bigr)\>dx$}{$2\pi\Int^3_1\bigl((x+1)-(-x^2+5x-2)\bigr)\>dx$}
{$\Int^3_1\bigl((x+1)^2-(-x^2+5x-2)\bigr)\>dx$}
\Pra 34512

\newp
Find the volume obtained by rotating the region in problem \number\tmpPC\ about the $y$-axis.
\AN{$2\pi\Int^3_1x\bigl((-x^2+5x-2)-(x+1)\bigr)\>dx$}{$\pi\Int^3_1x\bigl((-x^2+5x-2)^2-(x+1)^2\bigr)\>dx$}
{$\pi\Int^3_1\bigl((x+1)^2-(-x^2+5x-2)^2\bigr)\>dx$}{$2\pi\Int^3_1\bigl((x+1)-(-x^2+5x-2)\bigr)\>dx$}
{$\Int^3_1\bigl((x+1)^2-(-x^2+5x-2)\bigr)\>dx$}
\Pra 12345

\newpage%% page 13 top
\three
\newp
Find the length of the curve $y=x^3+1$ between $x=0$ and $x=3$.
\AN{$\Int^3_0\sqrt{1+9x^4\>}\>dx$}{$\Int^3_0\sqrt{1+x^3\>}\>dx$}
{$\Int^3_0\sqrt{1+x^6\>}\>dx$}{$\Int^3_0\sqrt{1+3x^2\>}\>dx$}
{$\Int^3_0\sqrt{1+{{x^4}\over{4}}\>}\>dx$}
\Pra 23451

\two
\newp
Find the surface area of the surface of revolution obtained by revolving the curve
$y=\sin x$ from $x=0$ to $x=\pi$ around the $x$-axis.
\AN{$2\pi\Int_0^\pi \sin x\>\sqrt{1+\cos^2x\>}\>dx$}
{$\pi\Int_0^\pi \sqrt{1+\cos^2x\>}\>dx$}
{$2\pi\Int_0^\pi x\sqrt{1+\sin^2x\>}\>dx$}{$2\pi\Int_0^\pi \sin x\>\sqrt{1+\sin^2x\>}\>dx$}
{$2\pi\Int_0^\pi (x-\sin x)\>\sqrt{1+\cos^2x\>}\>dx$}
\Pra 12345

\newpage%% page 14 top
\three\newp
Find the moment about the origin of a rod which runs from $x=1$ to $x=4$
and for which the density is given by $\delta(x)=\frac1{x^2}$.
\AN{$\Int^4_1\frac1x\>dx$}{$\Int^4_1\frac1{x^2}\>dx$}
{$\frac{\Int^4_1\frac1{x^2}\>dx}{\Int^4_1\frac1x\>dx}$}
{$\Int^4_1\frac1{x^4}\>dx$}{${\Int^4_1\frac1{x^2}\>dx}{\Int^4_1\frac1x\>dx}$}
\Pra 23451

\five\newp
Find the center of mass  of the thin plate of constant density $1$ in the first quadrant below the line $x+y=2$.
\AN{$(\frac23,\frac23)$}{$(1,1)$}{$(-\frac32,-\frac32)$}
{$(\frac12,\frac12)$}{$(\frac13,\frac13)$}
\Pra 12345
\newpage

}

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