%% Dwyer & Nollet
\magnification1200\footline{\ifnum\pageno=1 \hfil\else\hfil\number\pageno\hfil\fi}
\def\whichexam{Exam I}
\def\examdate{September 21, 1999}
\def\correctanswers{%
\ll 1 4
\ll 2 3
\ll 3 1
\ll 4 5
\ll 5 1
\ll 6 4
}
\font\small=cmr7
\def\newpage{\vfill\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}}}

\newcount\mc
\def\newchoice#1{\newpage\null\vskip10pt\centerline{{\bf#1}}\mc=0 %
\ifnum\pc<3\credit=5 \else\credit=10 \fi}
\newif\ifmultichoice
\def\problemspacing{%
\ifmultichoice%
\ifnum\mc>0 \ifodd\pc\relax\newpage\null\vskip10pt\else\vfill\fi\fi%
\else%
\ifnum\mc>0 \newpage\null\vskip10pt\fi\fi%
\mc=1}

\def\newp#1.{\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}%
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\setbox\Ad=\hbox to\Aspace{#4\hfil}\setbox\Ae=\hbox to\Aspace{#5\hfil}}

\def\ANp#1(a)#2(b)#3 (c)#4 (d)#5 (e)#6\Pra{%
\AN{#2}{#3}{#4}{#5}{#6}\Pra%
}

\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 \relax\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%
\immediate\write16{\number\pc - \number#1}%
}


\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}{Instructo:\ \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:\ \UL{2.3in}{#1}\hss}%
}}
\long\gdef\answersheet{
\null\vskip10pt
\centerline{\whichexam}
\centerline{\examdate}
\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!
\correctanswers
}\hss}}{10pt}{5in}
%%\vskip10pt

\settabs\+Total multiple choice: &\hskip 2in\cr
\hfill{{\vbox to0pt{\boxit{\vbox{\obeylines\hsize=3in\baselineskip26pt
DO NOT WRITE IN THIS BOX!
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\tabalign\hfill8.&\quad\ul{1in}\cr
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\tabalign\hfill12.&\quad\ul{1in}\cr
\tabalign\hfill13.&\quad\ul{1in}\cr

\tabalign\hfil {\bf Total:}&\quad\ul{1in}\cr
}}{10pt}{1in}\vss}}}}
\answersheet
\multichoicetrue
\newchoice{Multiple Choice}
\five
\newp 1.
Find the limit
$$\lim_{x \rightarrow 2} {{x^2-4}\over{x^2+x-6}}$$ 

\ANp
(a) ${{4}\over{5}}$
(b) ${1\over{2}}$
(c) $2 \over{3}$
(d) $-\infty$
(e) $\infty$ 
\Pra42315

\newp 2.
Find the {\bf average} rate of change of $f(x)=x^2+2$ on the interval 
$[1,3]$. 

\ANp
(a) $4$
(b) ${1\over{4}}$ 
(c) ${{3}\over{2}}$ 
(d) $7$ 
(e) ${{11}\over{2}}$ 
\Pra31254

\newp 3.
Find the limit 
$$\lim_{t \rightarrow 1^-} {{1}\over{t^2+t-2}}$$
\ANp
(a) $-\infty$
(b) $\infty$
(c) ${-1}\over{2}$
(d) ${1}\over{3}$
(e) $0$
\Pra15243

\two
\newp4.
Differentiate the function 
$$f(x)=\tan x + \sec x$$
\ANp
(a) $(\sec x)(\sec x + \tan x)$ 
(b) $\cot x + \csc x$ 
(c) $0$ 
(d) $\tan^2 x + \sec^2 x$ 
(e) $- \cot x$
\Pra54321

\five
\newp 5.
Evaluate the derivative 
$${d \over {dx}} (\sin x + 3 \cos x)^{100}$$ 
at $x = {\pi \over 2}$. 
\ANp
(a) $-300$  (b) $100$  (c) $1$  (d) $3^{99}$  (e) $0$
\Pra14352

\newp 6.
Let $$f(x)={x \over {x^2+3 x + 2}}$$
Calculate $f^\prime(x)$. 
\ANp
(a) ${-x^2 + 2 \over {(x^2+3x+2)^2}}$ 
(b) ${1 \over {2x+3}}$ 
(c) ${{3x^2+6x+2} \over {(x^2+3x+2)^2}}$ 
(d) ${1 \over {(x^2+3x+2)^2}}$ 
(e) ${{2x^2 + 5} \over {(x^2+3x+2)^2}}$ 
\Pra41325


\multichoicefalse
\newchoice{Partial Credit}

\newp
7. Compute the slope of the graph of the curve $y = \sqrt{x+1}$ at 
the point $(8,3)$ using the {\bf limit definition} of slope. 
Other methods will {\bf not} receive credit. 

\newp
8. The function $y=f(x)$ is defined by $$f(x) = \cases{\frac{\sin x}{x}& for $x > 0$\cr
 -4x& for $x \leq 0$\ .\cr}$$
For which values of $x$ is $f(x)$ continuous?
Justify your answer using limits.

\newp
9. Given the graph of the function below, sketch 
the graph of the derivative on the grid below. At which points is 
$g^\prime(x)$ not defined?  

\hbox{\vtop{\picture{6.81in} by {6.93in} ({function} scaled 300)}\hskip1.4in
\vtop{\picture{6.81in} by {6.93in} ({blankgraph} scaled 300)}}
\newp
10. Suppose that $f(x)$ and $g(x)$ are differentiable functions such that 
$f(2)=4$,  $f^\prime(2)=3$,  $g(2)=1$ and $g^\prime(2)=5$. If 
$h(x) = f(x) \cdot (g(x)+1)$, compute $h^\prime(2)$. 

\newp
11. A particle moves along the $y$-axis with its position being given 
by the function $y(t) = t^4 - 2 t^3 + 10 t$ for $-10 \leq t \leq 10$. 
Find the velocity of the particle at each time for which the 
acceleration is zero. 

\newp
12. Find an equation of the tangent line to the graph of the function 
$y = \sin (\pi x^4)$ at the point $(1,0)$. 

\newp
13. Given the function $f(x)=x^3 - 12x + 8$, 

(1) Calculate $f(2)$ and $f(-2)$ 

(2) Show that the equation $f(x)=0$ has a solution on the interval $[-2,2]$. 


\vfill\eject}

\exam{Dwyer}\end
%\exam{Cholak}
\exam{Taylor}
\exam{Wong}
\exam{Cao}
\exam{Jarre}
\exam{Nollet}
\exam{Bullwinkle}
\marked{}
\end
