% Version 1
% Correct Answers
%  1.e  2.e  3.e  4.b  5.c  6.a  7.d  8.c  9.a 10.a

% Page Setup

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% Preface

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% Marked answer sheet

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{\noindent
{\bf Math 108: Calculus for Business}\hfill
Name:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\bf Exam I}\quad{\sl September 15, 2000}\hfill
Section:\hbox to 2.5truein{\hrulefill}

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% Blank answer sheet

\vbox{\global\pageno=1
{\noindent
{\bf Math 108: Calculus for Business}\hfill
Name:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\bf Exam I}\quad{\sl September 15, 2000}\hfill
Section:\hbox to 2.5truein{\hrulefill}

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% Problems

\vbox{
%1

\noindent\llap{1.\ }%
(5 pts.) The rate of change of an investment is
$s(t)=1000e^{0.2t}$ dollars per year. If $\$10,000$ was initially invested,
find the value of the investment after $10$ years.

\beginanswers
{(a)~%
$5,000e^{2}$;
\answerspace}
{(b)~%
$1,000e^{2}-9,000$;
\answerspace}
{(c)~%
$1,000e^{2}+10,000$;
\answerspace}
{(d)~%
$1,000e^{5}+9,000$;
\answerspace}
{(e)~%
$5,000e^{2}+5,000$;
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%2

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

\beginanswers
{(a)~%
24
\answerspace}
{(b)~%
23
\answerspace}
{(c)~%
22
\answerspace}
{(d)~%
21
\answerspace}
{(e)~%
25
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%3

\noindent\llap{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

\beginanswers
{(a)~%
$\int (u^{3/2}+3u)\;du;$
\answerspace}
{(b)~%
$\int (u^{5/2}+3u^{1/2})\;du;$
\answerspace}
{(c)~%
$\int (u^2+3u)\;du;$
\answerspace}
{(d)~%
$\int \sqrt{u}\;du;$
\answerspace}
{(e)~%
$\int (u^{3/2}+3u^{1/2})\;du;$
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%4

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

\beginanswers
{(a)~%
$0$
\answerspace}
{(b)~%
$\ln 5$
\answerspace}
{(c)~%
${{-24}\over {25}}$
\answerspace}
{(d)~%
${{-51}\over {25}}$
\answerspace}
{(e)~%
$\ln 5-1$
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%5

\noindent\llap{5.\ }%
(5 pts.) Which one of the following methods would you use to
find the indefinite integral $\int x^3\ln(2x)\;dx$?

\beginanswers
{(a)~%
Substitution with $u=x^3$
\answerspace}
{(b)~%
Integration by parts with $u=x^3$, $dv=\ln(2x)dx$
\answerspace}
{(c)~%
Integration by parts with $u=\ln(2x)$, $dv=x^3dx$
\answerspace}
{(d)~%
Partial fractions
\answerspace}
{(e)~%
Substitution with $u=2x$
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%6

\noindent\llap{6.\ }%
(5 pts.) Using the method of integration by parts with
$u=x^2$, $dv=e^{2x}dx$, the integral
$$
        \int x^2 e^{2x}\; dx
$$
becomes which of the following expressions?

\beginanswers
{(a)~%
${{1}\over {2}}x^2e^{2x}-\int x e^{2x}\; dx;$
\answerspace}
{(b)~%
${{1}\over {2}}x^2e^{2x}- xe^{2x}\; dx;$
\answerspace}
{(c)~%
$\int x e^{2x}\; dx-{{1}\over {2}}x^2e^{2x}\; dx;$
\answerspace}
{(d)~%
$x^2e^{2x}-\int x e^{2x}\; dx;$
\answerspace}
{(e)~%
$x^2 e^{2x}-\int {{1}\over {2}}x e^{2x}\; dx;$
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%7

\noindent\llap{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 .5in

\epsfxsize=2.5truein\epsfbox{picture_gr1.eps}

\vskip .5in

\beginanswers
{(a)~%
8
\answerspace}
{(b)~%
5
\answerspace}
{(c)~%
-8
\answerspace}
{(d)~%
4
\answerspace}
{(e)~%
6
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%8

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

\vskip .5in

\moveright 1 in
\vbox{\offinterlineskip
\halign{\strut \vrule \quad $#$ \quad & \vrule \quad $#$ \quad 
& \vrule \quad $#$ \quad & \vrule \quad $#$ \quad & \vrule \quad $#$ \quad & 
\vrule \hfil \quad $#$ \quad \hfil \vrule \cr
\noalign{\hrule}
x & -1 & -.5 & 0 & .5 & 1 \cr
\noalign{\hrule}
f(x) & 8 & 4 & 6 & 4 & -2  \cr
\noalign{\hrule}
}}

\vskip 1in

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

\beginanswers
{(a)~%
-8
\answerspace}
{(b)~%
-1
\answerspace}
{(c)~%
-2
\answerspace}
{(d)~%
0
\answerspace}
{(e)~%
2
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%9

\noindent\llap{9.\ }%
(5 pts.) Use the trapezoidal rule with 3 subintervals to
approximate the integral
$$
        \int^1_0 e^{x^2}\;dx.
$$

\vskip .2in

\beginanswers
{(a)~%
${{1}\over {6}}(1+2e^{1/9}+2e^{4/9}+e);$
\answerspace}
{(b)~%
${{1}\over {2}}(e^{1/9}+e^{4/9}+e);$
\answerspace}
{(c)~%
${{1}\over {6}}(1+2e^{1/9}+e^{4/9}+2e);$
\answerspace}
{(d)~%
${{1}\over {3}}(1+e^{1/9}+e^{4/9}+e);$
\answerspace}
{(e)~%
${{1}\over {3}}(1+e^{1/9}+e^{4/9});$
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%10

\noindent\llap{10.\ }%
If $F(x)$ is an antiderivative of a function
$f(x)$ with $F(-1)=5$ and $F(2)=3$, then find $2\int^2_{-1}f(x)dx.$

\beginanswers
{(a)~%
-4
\answerspace}
{(b)~%
4
\answerspace}
{(c)~%
6
\answerspace}
{(d)~%
-6
\answerspace}
{(e)~%
16
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

% Appendix

{\vfil\eject\noindent
11. (10 pts.) a)Compute the definite integral
$$
        \int^2_0{{1}\over {1+x}}\;dx.
$$

\vskip 1.8in

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

\vskip 1.8in


\vfil\eject\noindent
12. (15 pts.) The marginal revenue for producing and selling
$x$ pairs of boots is given by $MR=-0.6x+300$. When 200 pairs of boots are
sold, the revenue of the company is $\$20,000$.

\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.

\vfil\eject\noindent
13. (10 pts.) a) Find the partial fraction decomposition of
$$
        {{1}\over {x^2-7x+12}}.
$$
That is find the numbers $A$ and $B$ such that
$$
        {{1}\over {x^2-7x+12}}={{A}\over {x-4}}+{{B}\over {x-3}}.
$$

\vskip 2in

b) Compute
$$
        \int^7_5 {{1}\over {x^2-7x+12}}.
$$


\vfil\eject\noindent
14. (15 pts.) Consider the functions $f(x)=x^2-4$ and
$g(x)=x-2$.

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 \break graphs of $f(x)$ and $g(x)$. Do not evaluate the integral.}

\end
