\magnification1200\footline{\ifnum\pageno=1 \hfil\else\hfil\number\pageno\hfil\fi}
\input pics.tex
\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 %
\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 II}
\centerline{October 29, 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 1
\ll 3 3
\ll 4 4
\ll 5 5
\ll 6 2
\ll 7 3
\ll 8 1
}\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}


\pr(5pts) The Mean Value Theorem states that for a suitable function $y=f(x)$ 
on a closed interval $[a,b]$, there is a number $c$ in the interval where 
$f^\prime(c)$ equals the average rate of change of $f(x)$ on $[a,b]$.
For the function $f(x)=\frac1x$ on the interval $[3,12]$, what value
of $c$ satisfies the conclusion of the theorem?
\ans{5}{6}{7}{8}{9} %% (b)

\vskip2in
\pr(5pts)
Which of the following functions is an antiderivative for
$$x \cos x\quad ?$$
\ans{$x\sin x+ \cos x$}
{$\frac{x^2}{2}\cos x$}
%{$-\frac{x^2}{2}\sin x$}
{$x\sin x$}
{}{}%{$x\cos x+\frac{x^2}{2}$}
%% (a)

\vfill\eject\null\vskip10pt
\pr(5pts)
The equation $x^3+x-1=0$ has exactly one real solution.
Starting with the estimate $x_0=1$, use one step of Newton's method to find
a better approximation:
\ans{${{1}\over{2}}$}{$0$}{${{3}\over{4}}$}{${{3}\over{2}}$}{$2$}
%%(c)

\vskip2in
\pr(5pts)
Which of the following is the linearization of the function $f(x)=3x^2-2$ at $x=2$?
\ans{$y=3x+2$}{$y=-6x+2$}{$y=x+1$}{$y=12x-14$}{$y=6x-10$}
%% (d)

\vfill\eject\null\vskip10pt
\pr(5pts)
What are the asymptotes (both vertical and horizontal) for the graph of
$$y=\frac{(2x-1)^2}{(x-2)(x+3)}\quad ?$$
\ansB{$y=2$, $x=-2$, $x=3$}
{$y=2$, $y=-3$, $x=4$}
{$y=2x-1$, $y=2$, $y=3$}
{$x=-3$, and $x=5$}
{$y=4$, $x=2$, $x=-3$}
%%(e)

\vskip2in
\pr(5pts)
Let $f(x)=x^{20}-2x^{10}+5$.
Which of the following statements is true regarding $x=1$ (the point $(1,4)$)?
\ansA{$x=1$ is a point of inflection}
{$x=1$ is a local maximum}
{$x=1$ is a local minimum}
{$x=1$ is a critical point but neither a local maxima nor a local minima}
{$x=1$ is a not a critical point}
%%(b)

\vfill\eject\null\vskip10pt
\pr(5pts)
A continuous function $y=f(x)$ has the following properties:
\itemitem{(a)} $f^\prime(x)>0$ for $x>5$
\itemitem{(b)} $f^\prime(x)<0$ for $x<5$
\itemitem{(c)} $f(5)=10$.

How many solutions are there to the equation $f(x)=0$ ?
\ans{$2$}{$3$}{$0$}{$1$}{infinitely many}
%%(c)

\vskip2in
\pr(5pts)
The limit
$\displaystyle\lim_{x\to\infty}\frac{10x^3+6x+4}{2x^3+3x+2}$
is
\ans{$5$}
{$\displaystyle{{3}\over{4}}$}
{$2$}
{$\displaystyle{{10}\over{3}}$}
{$0$}%%(a)



\vfill\eject\null\vskip10pt

\centerline{\bf Partial Credit}
\null\vskip10pt
\pr(10pts)
Find the absolute minimum and the absolute maximum of the function $f(x)=x^2-2x$
on the interval $[0,3]$.
\vfill\eject\null\vskip10pt
\pr(10pts)
Consider the function $y=f(x)=-x^3-3x^2+2$.
\itemitem{(a)} Find the intervals where the function is increasing and where
the function is decreasing.
Locate any local extrema.
\itemitem{(b)} Find the intervals on which $f(x)$ is concave up and those on
which it is concave down.
Locate all inflection points.

\vfill\eject\null\vskip10pt
\pr(10pts)
A  function $y=f(x)$ on $(0,\infty)$ 
satisfies the following conditions:
\itemitem{}$f(1)=2$, $f(3)=1$, 
\itemitem{}$f^\prime(x)>0$ on $(0,1)$, $f^\prime(x)<0$ on $(1,\infty)$,
\itemitem{}$f^{\prime\prime}(x)<0$ on $(0,3)$, $f^{\prime\prime}(x)>0$ on $(3,\infty)$
\itemitem{}$\lim_{x\to\infty} f(x)=0$, $\lim_{x\to0^+} f(x)=-\infty$.

\noindent
Sketch the graph.

\hskip .2in\picture 4in by 5in (coor scaled 500)

\vfill\eject\null\vskip10pt
\pr(10pts)
A right triangle with base $10$ and adjacent angle $\theta$ 
has area $A=50\tan\theta$.
Find the differential $dA$ and use this to estimate the change in
$A$ as $\theta$ increases from ${{\pi}\over{4}}$ to  ${{\pi}\over{4}}+0.01$.
%\vskip.5in

\hskip 1in\picture 1in by 1in (triangle scaled 200)


\vfill\eject\null\vskip10pt
\pr(10pts)
Each point $P$ on the graph of $y=\frac{1}{4+x^2}$, $x>0$,
determines a rectangle in the first quadrant as shown in the diagram:
\vskip1in\hskip 1in
\picture 1in by 1in (graph scaled 300)

\noindent
Find the dimensions of the rectangle with the largest area.

\vfill\eject\null\vskip10pt
\pr(10pts)
Find the equation of the function $y=f(x)$ that satisfies the initial value problem
$y^{\prime}(x)=-\sin x+12x$, $y(0)=3$.
\vfill\eject
}
%\exam{Dwyer}
%\exam{Shaw}
%\exam{Taylor}
%\exam{Nollet}
%\exam{Budhiraja}
%\exam{Gekhtman \quad{\sl Section \#:}}
\exam{Bullwinkle}
\marked{}
\end

