\input amstex
\documentstyle{amsppt}
\magnification 1200
\NoRunningHeads
\pageno=1

% Page Setup

\raggedright
\raggedbottom
\everymath={\displaystyle}
\def\answerspace{\quad}
\def\answerskip{\vskip0.1in}
\def\problemskip{\vskip0.75truein}

% Macros

\catcode`\@=11\long\def\beginanswers#1\endanswers{{\count1=1\get@ns#1\end}}
\def\get@ns#1{\ifx#1\end\let\next=\relax\set@ns\else
\setbox\count1=\hbox{#1}\advance\count1by1\let\next=\get@ns\fi\next}
\def\set@ns{{\parindent=0pt\dimen2=\hsize\divide\dimen2by2
\dimen3=\hsize\divide\dimen3by3\dimen5=\hsize\divide\dimen5by5
\dimen0=\wd1\count0=1\loop\advance\count0by1\ifnum\count0<\count1\relax
\ifdim\dimen0<\wd\count0\dimen0=\wd\count0\fi\repeat
\ifdim\dimen0<\dimen5\dimen0=\dimen5\else\ifdim\dimen0<\dimen3\dimen0=\dimen3
\else\ifdim\dimen0<\dimen2\dimen0=\dimen2\else\dimen0=\hsize\fi\fi\fi
\spaceskip=0pt plus1pt minus1pt\xspaceskip=\spaceskip
\leavevmode\count0=0\loop\advance\count0 by1\ifnum\count0<\count1
\hbox to\dimen0{\vbox{\answerskip\unhbox\count0}\hss} \repeat}}
\catcode`\@=12\def\ansbox#1#2{\hbox{\vrule\vtop{\vbox{\hrule
\kern2pt\hbox to #1{\vphantom{/}\hfil#2\hfil}}\kern2pt\hrule}\vrule}}


\topmatter

\vfil\eject\output={}
\vbox{{\noindent
{\bf Name:}  \vrule height .4pt depth 0 pt width 1.4
in  \hskip .5 in {\bf Instructor:}  Viorel Nitica
\bigskip
\bigskip
\bigskip

\centerline{\bf {Math 108.02, Calculus II for Business}}
\centerline{\bf Fall Semester 1998}
\centerline{\bf  Exam 1 }
\centerline{\bf {Friday, September 18, 3:00-3:55 PM}}
\bigskip
\hrule
\bigskip
This Examination contains {\bf 13} problems, worth a total of  100  points,
on (10) sheets of paper including the front cover.
The first {\bf 9} problems are multiple choice with no partial credit, and
each is worth 5 points. Record your answers to these problems by placing
an $\times$ through one letter for each problem below:

% Blank answer sheet


\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
\answerskip\noindent\llap{2.}\enskip\ansbox{15pt}{a}\enskip\ansbox{15pt}
{b}\enskip\ansbox{15pt}{c}\enskip\ansbox{15pt}{d}\enskip\ansbox{15pt}{e}\par
\answerskip\noindent\llap{3.}\enskip\ansbox{15pt}{a}\enskip\ansbox{15pt}
{b}\enskip\ansbox{15pt}{c}\enskip\ansbox{15pt}{d}\enskip\ansbox{15pt}{e}\par
\answerskip\noindent\llap{4.}\enskip\ansbox{15pt}{a}\enskip\ansbox{15pt}
{b}\enskip\ansbox{15pt}{c}\enskip\ansbox{15pt}{d}\enskip\ansbox{15pt}{e}\par
\answerskip\noindent\llap{5.}\enskip\ansbox{15pt}{a}\enskip\ansbox{15pt}
{b}\enskip\ansbox{15pt}{c}\enskip\ansbox{15pt}{d}\enskip\ansbox{15pt}{e}\par
%\answerskip\noindent\llap{6.}\enskip\ansbox{15pt}{a}\enskip\ansbox{15pt}
%{b}\enskip\ansbox{15pt}{c}\enskip\ansbox{15pt}{d}\enskip\ansbox{15pt}{e}\par
}
\hfil\vtop{
\answerskip\noindent\llap{6.}\enskip\ansbox{15pt}{a}\enskip\ansbox{15pt}
{b}\enskip\ansbox{15pt}{c}\enskip\ansbox{15pt}{d}\enskip\ansbox{15pt}{e}\par
\answerskip\noindent\llap{7.}\enskip\ansbox{15pt}{a}\enskip\ansbox{15pt}{b}
\enskip\ansbox{15pt}{c}\enskip\ansbox{15pt}{d}\enskip\ansbox{15pt}{e}\par
\answerskip\noindent\llap{8.}\enskip\ansbox{15pt}{a}\enskip\ansbox{15pt}{b}
\enskip\ansbox{15pt}{c}\enskip\ansbox{15pt}{d}\enskip\ansbox{15pt}{e}\par
\answerskip\noindent\llap{9.}\enskip\ansbox{15pt}{a}\enskip\ansbox{15pt}{b}
\enskip\ansbox{15pt}{c}\enskip\ansbox{15pt}{d}\enskip\ansbox{15pt}{e}\par
%\answerskip\noindent\llap{11.}\enskip\ansbox{15pt}{a}\enskip\ansbox{15pt}{b}
%\enskip\ansbox{15pt}{c}\enskip\ansbox{15pt}{d}\enskip\ansbox{15pt}{e}\par
}}}

\bigskip
\bigskip
The last {\bf four} problems are partial credit problems worth
a total of 55 points. For these problems, {\bf show} your
computations and {\bf clearly} mark your answers on the page.
Books and notes are not allowed. You may use your calculator.
\bigskip
\hrule
\bigskip
\bigskip
\bigskip


{\bf Sign the pledge:}  \lq \lq {On my honor,
I have neither given nor received unauthorized aid on this Exam}":
\bigskip
\hskip .7 in\vrule height .10pt depth 0 pt width 2 in
\bigskip
\bigskip
\bigskip
\centerline{\bf GOOD LUCK}

\newpage

\noindent {\bf 1.} (5 pts.) The rate of change of an investment is 
$s(t)=500e^{0.4t}$ dollars per year. If $\$2,000$ was initially invested,
find the value of the investment after $t$ years.

\vskip .2in

a) \;\;$V(t)=1,250e^{0.4t}+750$;

\vskip .1in

b) \;\;$V(t)=2,000e^{0.4t}$;

\vskip .1in

c) \;\;$V(t)=1,250e^{0.4t}-750$;

\vskip .1in

d) \;\;$V(t)=400e^{0.4t}+2000$;

\vskip .1in

e) \;\;$V(t)=400e^{0.4t}+1,600$.

\vskip 2in

\noindent {\bf 2.} (5 pts.) Consider the initial value problem:
$$
	\frac{dy}{dx}=3x^{1/2}+2x,\;\;y(4)=33.
$$
Find $y(1)$.

\vskip .2in

a)\;\;-5;\;\;\;\;(b)\;\;4;\;\;\;\;(c)\;\;3;\;\;\;\;(d)\;\;2;\;\;\;\;(e)\;\;0.

\newpage

\noindent {\bf 3.} (5 pts.) The substitution $u=x+3$ changes the integral
$$
	\int x\sqrt{x+3}\;dx
$$
into which one of the following integrals?

\vskip .2in

a)\;\;$\int (u^{5/2}-3u^{1/2})\;du;$

\vskip .1in

b)\;\;$\int (u^{3/2}+3u)\;du;$

\vskip .1in

c)\;\;$\int \sqrt{u}\;du;$

\vskip .1in

d)\;\;$\int (u^{3/2}-3u^{1/2})\;du;$

\vskip .1in

e)\;\;$\int (u^2+3u)\;du.$

\vskip 2in

\noindent {\bf 4.} (5 pts.) Compute the definite integral: $\int^1_0 
\frac{2x+3}{x^2+3x+1}\; dx$.

\vskip .2in

a)\;\;$\ln 5-1$;\;\;b)\;\;$\ln 5$;\;\;c)\;\;$0$;\;\;d)\;\;$\frac{-24}{25}$;
\;\;e)\;\;$\frac{-51}{25}.$

\newpage

\noindent {\bf 5.} (5 pts.) Which one of the following methods would you use to
find the indefinite integral $\int x^2\ln(4x)\;dx$?

\vskip .2in

a)\;\;Substitution with $u=2x$;

\vskip .1in

b)\;\;Integration by parts with $u=x^2$, $dv=\ln(4x)dx$;

\vskip .1in

c)\;\;Integration by parts with $u=\ln(4x)$, $dv=x^2dx$;

\vskip .1in

d)\;\;Substitution with $u=x^2$;

\vskip .1in

e)\;\;Partial fractions.

\vskip 2in

\noindent {\bf 6.} (5 pts.) Using the method of integration by parts with 
$u=x^3$, $v'=e^x$, the integral
$$	
	\int x^3 e^x\; dx
$$
becomes which of the following expressions?

\vskip .2in

a) \;\;$x^3e^x-\int 3x^2e^x\; dx;$

\vskip .1in

b) \;\;$3x^2e^x- 3e^x\; dx;$

\vskip .1in

c) \;\;$3e^x+x^2e^x\; dx;$

\vskip .1in

d) \;\;$\int x^3 e^x\; dx-3x^2e^x\; dx;$

\vskip .1in

e) \;\;$x^3e^x-\int x^2e^x\; dx;$

\vskip .1in

\newpage

\noindent {\bf 7.} (5 pts.) Find the value of the definite integral from $x=-4$ to $x=4$ of the function whose graph is shown in the following picture.

\vskip .2in

a)\;\; -12;\;\;\;\;b)\;\; -8;\;\;\;\;c)\;\; 8;\;\;\;\;
d)\;\; -4;\;\;\;\;e)\;\; 4.

\vskip 2in

\noindent {\bf 8.} (5 pts.) Suppose the function $f(x)$ takes the 
following values

\vskip 1in

Estimate $\int^1_{-1}f(x)\;dx$ by computing the Riemann sum with 4 
subintervals and left end-points.

\vskip .2in
 
a)\;\;1;\;\;\;\;b)\;\;1/2;\;\;\;\;c)\;\;0;\;\;\;\;d)\;\;-1;\;\;\;\;
e)\;\;-1/2.

\newpage

\noindent {\bf 9.} (5 pts.) Which of the following expressions is a
Riemann sum of the integral
$$
	\int^1_0 e^{x^2}\;dx?
$$

\vskip .2in

a)\;\;$\frac{1}{3}(1+2e^{1/9}+2e^{4/9}+e);$

\vskip .1in

b)\;\;$\frac{1}{2}(e^{1/9}+e^{4/9}+e);$

\vskip .1in

c)\;\;$\frac{1}{3}(1+2e^{1/9}+e^{4/9}+2e);$

\vskip .1in

d)\;\;$\frac{1}{3}(1+e^{1/9}+e^{4/9});$

\vskip .1in

e)\;\;$\frac{1}{3}(1+e^{1/9}+e^{4/9}+e).$

\newpage


\noindent {\bf 10.} (15 pts.)\;a)Compute the definite integral
$$
	\int^1_0\frac{1}{1+x}\;dx.
$$

\vskip 1.8in

\;b) Compute the Riemann sum for the integral above using 4 subintervals
and right end-points.

\vskip 1.8in

\;c) Using a) and b), find an approximation for $\ln 2$. Explain your answer.

\newpage


\noindent {\bf 11.} (15 pts.) The marginal revenue for producing and selling 
$x$ pairs of boots is given by $MR=-3x+1500$. When 100 pairs of boots are 
sold, the revenue of the company is $\$40,000$.

\vskip .2in

a)\;\; Determine how many pairs of boots should be sold by the company
in order to maximize its revenue. Explain your answer.

\vskip 1.8in

b)\;\; Find the revenue function.

\vskip 1.8in

c)\;\; Compute the maximum possible revenue.


\newpage


\noindent {\bf 12.} (10 pts.)\;a) Find the partial fraction decomposition of
$$
	\frac{1}{x^2-5x+4}.
$$
That is find the numbers $A$ and $B$ such that
$$
	\frac{1}{x^2-5x+4}=\frac{A}{x-4}+\frac{B}{x-1}.
$$

\vskip 2in

\;b) Compute
$$
	\int^8_6 \frac{1/2}{x-3}.
$$

\newpage


\noindent {\bf 13.} (15 pts.)\;\; Consider the functions $f(x)=x^2-1$ and 
$g(x)=x+1$. 

\;\;a) Find the $x$-values of the points where the graphs of $f(x)$ and $g(x)$ intersect.

\vskip 1.8in

\;\;b) Draw the graphs of the functions $f(x)$ and $g(x)$ using the 
same coordinate system. 

\vskip 1.8in

\;\;c) Write the definite integral that gives the area of the region 
enclosed by the graphs of $f(x)$ and $g(x)$. Do not evaluate the integral.


\end

