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\begin{document}

% Blank answer sheet
\vfil\eject\output={}
\vbox{{\noindent
{\bf Math 108 }\hfill
Name:\hbox to 2.5truein{\hrulefill}\medskip\noindent

{\bf Exam 1 }\hfill
Instructor:\hbox to 2.5truein{\hrulefill}\medskip\noindent

{\sl September 26, 2002}\hfill
Section:\hbox to 2.5truein{\hrulefill}

\bigskip \noindent
Calculators are  allowed.  Do not remove this answer page -- you will return
the whole exam.  You will be allowed 1 hour and 15 minutes to do the test. You may leave earlier if you are finished.


\smallskip \noindent
Part I consists of 13 multiple choice questions worth 6 points each. 
Record your answers  by  placing an~$\times$ through one
letter for each problem on this answer sheet. 

\smallskip \noindent
Part II consists of 2 partial credit problems worth a total of 11 points. 
Write your answer and show {\bf all} your work on the page on which the
question appears.


\vskip .15in 
\centerline {\bf
You are taking this exam under the honor code.}
\vskip .1in 
\centerline {\bf GOOD LUCK}

\vskip0.3in

\divide\hsize by3
\leavevmode\hfil\vtop{

\answerskip\noindent\llap{1.}\enskip\ansbox{15pt}{a}\enskip\ansbox{15pt}{b}\enskip\ansbox{15pt}{c}\enskip\ansbox{15pt}{d}\enskip\ansbox{15pt}{e}\par

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\par
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%\answerskip\noindent\llap{14.}\enskip\ansbox{15pt}{a}\enskip\ansbox{15pt}{b}\enskip\ansbox{15pt}{c}\enskip\ansbox{15pt}{d}\enskip\ansbox{15pt}{e}\par
}}}


\vskip0.8in

\begin{center}
For grading use:

\renewcommand{\arraystretch}{1.5}
\begin{tabular} 
	{| c | p{.5in} |} \hline
	1-13 & \\ \hline
	14 & \\ \hline
	15 & \\ \hline
	Total & \\ \hline	      
\end{tabular}
\end{center}







\newpage



\centerline{\bf \large Part I: Multiple choice questions (6 points each)}

\begin{enumerate}

\item %1
Find the solution to the initial value problem $\ds \frac{dy}{dt}=
\frac{4}{t^3}$, $y(1)=1$.

\vspace{2mm}

\noindent
(a) $y=\frac{2}{t^2}+1$ \hfill
(b) $y=\frac{1}{t^4}$ \hfill
(c) $y=\frac{-1}{t^4}+2$ \hfill
(d) $y=\frac{16}{t^4}-15$ \hfill
(e) $y=\frac{-2}{t^2}+3$ 

\vskip3.2in



\item %2
Compute the average value of
  $\ds f(x)= \sqrt{2}x^{3/2} +2 e^{x/2}$
over the interval $[0, 2]$.
\vskip 0.2in

\noindent
(a)  $-\frac{11}{10}-\frac{1}{2}e$\hfill
(b) $-\frac{2\sqrt{2}}{5}+4e-4$ \hfill
(c) $-\frac{2}{5}+2e$ \hfill
(d) $\frac{4}{5}+4e$ \hfill
(e) $\frac{8}{5}+2e$

\vskip1.6in

\newpage

\item %3
  Estimate the integral
$\int^1_0\sqrt{x+2} \ dx$
by dividing the interval [0,1] into five equal segments and
using the {\bf Riemann sum} corresponding to {\bf left endpoints}.

\bigskip
\noindent
(a) $\frac{1}{5}( \sqrt{2.2} + \sqrt{2.4} + \sqrt{2.6} + \sqrt{2.8}
+ \sqrt{3.0})$ \hfill 
(b) $\frac{1}{5}(\sqrt{2.0} + \sqrt{2.2} + \sqrt{2.4} + \sqrt{2.6} +
\sqrt{2.8})$

(c) $\frac{1}{10}( \sqrt{2.0} + \sqrt{2.25} + \sqrt{2.5} + \sqrt{2.75}
+ \sqrt{3.0}) $ \hfill
(d) $\frac{1}{5}(\sqrt{2.1} + \sqrt{2.3} + \sqrt{2.5} + \sqrt{2.7} +
\sqrt{2.9})$

(e) none of the above
\vskip 1in



\medno

\item %4
Find the {\bf total area} (not the definite integral) between the function
$f(x) = x^2 - 2$ and the x-axis over the interval $[0,2]$.
\vskip 0.25in
(a) $\frac{8}{3} \sqrt{2} - \frac{4}{3}$ \hfill
(b) $\frac{4}{3}$ \hfill
(c) $\frac{-4}{3}\sqrt{2} + \frac{22}{3}$ \hfill
(d) $2$ \hfill
(e) none of these

\vskip 2.2in

\item %5
Compute the following indefinite integral
$\ds \int x^2 (x^3-1)^{5/3} \, dx$.




\vskip 0.10in
(a) $\frac{3}{8}(x^3-1)^{8/3}+C$ \hfill
(b) $\frac{1}{8}(x^3-1)^{8/3}+C$ \hfill
(c) $\frac{1}{6}(x^3-1)^2+C $ 

(d) $\frac{1}{8}x^3(x^3-1)^{8/3}+C$
\hskip 1in
(e) $\frac{2}{3}(x^3-1)^2+C$

\newpage

%\item 
%Find the area between the curves $y=x$ and $y =x^2$
%over the interval $[0, 2]$.

%\vskip 0.25in
%(a) ${2 \over 3}-2$ \hfill
%(b) $ $ \hfill
%(c) $ $ \hfill
%(d) $ $ \hfill
%(e) $ $
%\vskip2.2in

\item %6

Compute the following definite integral
$\ds \int_4^6  \frac{2}{3x^2+3x-6} \, dx$.
\vspace{2mm}

\noindent
(a) $\frac{2}{3}\ln {\frac{5}{4}}$ \hfill
(b) $\frac{2}{3}\ln {\frac{5}{8}} $ \hfill
(c) $\frac{2}{9}\ln {\frac{15}{8}}$ \hfill
(d) $2\ln {\frac{20}{9}}$ \hfill
(e) $\frac{2}{9}\ln {\frac{5}{4}}$ 

\vskip2.2in


\item %7
Compute the following indefinite integral: 
$\ds \int 2t \ \ln{t} \ dt =$

\vskip 2mm
\noindent
(a) $\frac{t^2}{2} \ln{t} - \frac{t^2}{4} + C$ \hfill
(b) $2 \ln{t} - t^2 + C$ \hfill
(c) $2\ln{t} + C$

(d) $t^2 \ln{t} - \frac{t^2}{2} + C$ \hskip 1.3in
(e) $2-2 \ln{t} + C$

\vskip2.1in

\begin{minipage}[b]{2.5in}
\item %8 
 The demand and supply curves for a product are given in the diagram.
Use geometry to compute the {\bf consumer surplus}.
\end{minipage} \ \hskip.15in \
\begin{minipage}[b]{2.5in}
  $\epsfxsize=1.5in\epsfbox{ex1pict1temp.eps}$
\end{minipage}

\vspace{2mm}
\noindent
(a) $120$ \hfill
(b) $240 $ \hfill
(c) $60$ \hfill
(d) $360 $ \hfill
(e) $180 $ 

\newpage

\item %9
The marginal cost at a given steel mill is $MC=45t^{-1}$ in dollars per
ton.  If production is increased from $80$ tons to $100$ tons, what is the
total change in cost?
\vspace{2mm}

\noindent
(a) $45 \ln{\frac{5}{4}}$ \hfill
(b) $162000$ \hfill
(c) $45\ln{20}$ \hfill
(d) $900$ \hfill
(e) $4580$ 

\vskip 2.2in

%\item %9
%Mark is saving to send his first-born child to Notre Dame in 25 years. 
%He decides to save at a rate of \$$1000$ a year. Assuming he can get an
%interest rate of $6$\% compounded continuously, how much will he have after
%the 25 years?

%\vspace{2mm}

%\noindent
%(a) $\frac{50000}{3} (e^{1.5}-1)$ \hfill
%(b) $\frac{50000}{3} (1-e^{1.5})$ \hfill
%(c) $60000(e^{1.5}-1)$

%(d) $1000e^{1.5}$ \hskip1.8in
%(e) none of these 

%\vskip3in






\item %10
The demand function for designer shoe laces is $D(q)=-q^2 - 50 q + 800$ and
the supply function  is $S(q)=q^2 + 10 q$ (in thousands of units).
Find the {\bf equilibrium quantity} for designer shoelaces.
\vspace{2mm}

\noindent
(a) $q_e = 2000$ \hfill
(b) $q_e = 10$ \hfill
(c) $q_e = 40$ \hfill
(d) $q_e = 200$ \hfill
(e) $q_e = 20$ 

\vskip2.2in
\medno

\item %11
The demand function for designer shoe laces is $D(q)=-q^2 - 50 q + 800$ and
the supply function  is $S(q)=q^2 + 10 q$ (in thousands of units).
Compute the {\bf producer surplus} for designer shoelaces.
\vspace{2mm}

\noindent
(a) $\frac{2000}{3}$ \hfill
(b) $\frac{15500}{3}$ \hfill
(c) $\frac{9500}{3}$ \hfill
(d) $\frac{3500}{3}$ \hfill
(e) $\frac{500}{3}$ 

\newpage

\item %12
If Kate knows she can get $15$\% interest compounded continuously, how
much money must Kate deposit now so that she can have \$1,000,000 in $
40 $ years?

\vspace{2mm}

\noindent
(a) $1000000e^{6}$ \hfill
(b) $\frac{1000000}{.15}e^{6}$ \hfill
(c) $1000000e^{-6}$ \hfill
(d) $\frac{1000000}{.15}(e^{6}-1)$ \hfill
(e) $\frac{1000000}{40}e^{-.15}$ 

\vskip 3in

\item %13
The Jones family wants to buy a house to keep up with the neighbors.  They
decide to get a $ 30 $ year loan. If they think that they can
find a $10$\% interest rate (compounded continuously) and make payments at
a rate of \$$5000 $ a year, how much can they afford to borrow now?

\vspace{2mm}

\noindent
(a) \$$50000(1-e^{-3})$ \hfill
(b) \$$5000(e^{3}-1)$ \hfill
(c) \$$5000(1-e^{-3})$

(d) \$$50000(e^{3}-1)$ \hskip 1.3in
(e) \$$5000e^{3}$ 

\vskip2.2in


\newpage
\centerline{\bf \large Part II: Partial credit questions (8 points)}

\medno
\item %14
Now integrate $\int_1^2 2x \ e^{-2x} dx$.  SHOW YOUR WORK\

\newpage
\medno
\item %15
Consider the following integral: $\ds \int_1^2 e^{2r} \, dr$.

\medno
(a) Estimate the integral using a {\bf Riemann sum} with {\bf
midpoints} and $n=4$.

\vskip 3in
\noindent
(b) Now integrate to calculate the exact value of the integral.


\vskip 3in
\noindent
(c) Use the answers from (a) and (b) to calculate how far off the
approximation in (a) actually was from the correct answer.





\end{enumerate}
\end{document}