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\begin{document}
\thispagestyle{empty}

{\bf Name:}  \vrule height .4pt depth 0 pt width 2.0in

\bigskip

\hfil {\bf Math 108 - Calculus II for Business} \hfil

%\vskip{0.05in}
\hfil {\bf Final Exam - Fall Semester 1998} \hfil

%\vskip{0.05in}
\hfil {\bf Wednesday, December 16, 1:45-3:45 p.m.} \hfil

\medskip
\hrule
\medskip
This Examination contains {\bf 34} problems, worth a total of  200  points.
Books and notes are not
allowed. You may use your calculator.

The first {\bf three} problems are partial credit problems worth
a total of 45 points. For these problems, {\bf show} your
computations and {\bf clearly} mark your answers on the page.
The remaining 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:

\bigskip
% Blank answer sheet

\begin{large}

{\bf ~~4.}\hss\ba\hss\bb\hss\bc\hss\bd\hss\be \hs {\bf
15.}\hss\ba\hss\bb\hss\bc\hss\bd\hss\be \hs
{\bf 25.}\hss\ba\hss\bb\hss\bc\hss\bd\hss\be

{\bf ~~5.}\hss\ba\hss\bb\hss\bc\hss\bd\hss\be \hs {\bf
16.}\hss\ba\hss\bb\hss\bc\hss\bd\hss\be \hs
{\bf 26.}\hss\ba\hss\bb\hss\bc\hss\bd\hss\be

{\bf ~~6.}\hss\ba\hss\bb\hss\bc\hss\bd\hss\be \hs {\bf
17.}\hss\ba\hss\bb\hss\bc\hss\bd\hss\be \hs
{\bf 27.}\hss\ba\hss\bb\hss\bc\hss\bd\hss\be

{\bf ~~7.}\hss\ba\hss\bb\hss\bc\hss\bd\hss\be \hs {\bf
18.}\hss\ba\hss\bb\hss\bc\hss\bd\hss\be \hs
{\bf 28.}\hss\ba\hss\bb\hss\bc\hss\bd\hss\be

{\bf ~~8.}\hss\ba\hss\bb\hss\bc\hss\bd\hss\be \hs {\bf
19.}\hss\ba\hss\bb\hss\bc\hss\bd\hss\be \hs
{\bf 29.}\hss\ba\hss\bb\hss\bc\hss\bd\hss\be

{\bf ~~9.}\hss\ba\hss\bb\hss\bc\hss\bd\hss\be \hs {\bf
20.}\hss\ba\hss\bb\hss\bc\hss\bd\hss\be \hs
{\bf 30.}\hss\ba\hss\bb\hss\bc\hss\bd\hss\be

{\bf 10.}\hss\ba\hss\bb\hss\bc\hss\bd\hss\be \hs {\bf
21.}\hss\ba\hss\bb\hss\bc\hss\bd\hss\be \hs
{\bf 31.}\hss\ba\hss\bb\hss\bc\hss\bd\hss\be

{\bf 11.}\hss\ba\hss\bb\hss\bc\hss\bd\hss\be \hs {\bf
22.}\hss\ba\hss\bb\hss\bc\hss\bd\hss\be \hs
{\bf 32.}\hss\ba\hss\bb\hss\bc\hss\bd\hss\be

{\bf 12.}\hss\ba\hss\bb\hss\bc\hss\bd\hss\be \hs {\bf
23.}\hss\ba\hss\bb\hss\bc\hss\bd\hss\be \hs
{\bf 33.}\hss\ba\hss\bb\hss\bc\hss\bd\hss\be

{\bf 13.}\hss\ba\hss\bb\hss\bc\hss\bd\hss\be \hs {\bf
24.}\hss\ba\hss\bb\hss\bc\hss\bd\hss\be \hs
{\bf 34.}\hss\ba\hss\bb\hss\bc\hss\bd\hss\be

{\bf 14.}\end{large}\hss\ba\hss\bb\hss\bc\hss\bd\hss\be

%\end{large}

\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 3.0 in

\newpage
\hfil {\bf PARTIAL CREDIT PART} \hfil

In problems 1-3, show all work on the paper. You {\bf MUST} show your work
to receive
full credit for these problems.

\bigskip
{\bf 1.} (15 points) Consider the functions $f(x)=2x$ and  $g(x)=x^2-x$.

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

\vskip2.5in
(b) Sketch the graphs of $f(x)$ and $g(x)$ using the same coordinate
system. Shade
the region enclosed by the graphs of $f(x)$ and $g(x)$.

\vskip2.5in
(c) Write in simplified form the definite integral that gives the area of
the region
enclosed by the graphs of $f(x)$ and $g(x)$.  You do {\em not} need to
compute the integral.

\newpage
{\bf 2.} (15 points)  For a certain item, the demand curve is
$D(q)=-2q+12$, and the supply
curve is $S(q)=3q+2$.

(a) Find the equilibrium price and the equilibrium quantity.

\vskip2.5in
(b) Find the consumer surplus.

\vskip2.5in
(c) Find the producer surplus.

\newpage
{\bf 3.} (15 points) In the weeks before Christmas, the number of toys made
each day at
Santa's workshop is normally distributed with a mean of 500 and standard
deviation of 50.
Compute the probability that on a random day between 475 and 550 toys are
made.

\newpage
\centerline{\bf MULTIPLE CHOICE SECTION - EACH PROBLEM IS WORTH  5 POINTS }

In problems 4-34, mark the correct answer on the front cover.

\bm
{\bf 4.} An investment is growing at the {\em rate} of $R(t)=20t+5$ dollars per
year. If $\$1000$ was initially invested, find the value of the investment
after 5 years.

\mm
(a) \$1325 \hfil (b) \$1300 \hfil (c) \$1025 \hfil (d) \$1275 \hfil (e) \$1225

\vskip2.0in
{\bf 5.} Solve the initial value problem
$$
		\frac{dy}{dx}=3x^2+1,\ \ y(1)=2.
$$

\mm
(a) $y=x^3+x+2$ \hfil (b) $y=3x^3+x+2$ \hfil (c) $y=x^3+x$

(d) $y=3x^3+x$ \hs\hs\hs\hh (e) $y=x^3+x-2$.

\vskip2.0in
{\bf 6.} Which of the following methods would you use to compute
$$
	\int xe^{x^2} dx.
$$

\mm
(a) Integration by parts with $u=e^{x^2}$ and $v'=x$.\\
(b) Integration by parts with $u=x$ and $v'=e^{x^2}$.\\
(c) Partial fractions.\\
(d) Substitution with $u=x^2$.\\
(e) Direct integration using basic formulas.

\newpage
{\bf 7.} Write the equation of the plane that contains the
points $(0,0,3)$, $(2,0,1)$, $(0,2,1)$.

\mm
(a) $z=x+y+3$ \hfil (b) $z=-x-y+3$ \hfil (c) $z=x+y-3$

(d) $z=-3x-3y-3$ \hs\hh~ (e) None of the preceding

\vskip2.0in
{\bf 8.} For which value(s) of $k$ will the following system
of equations have no solution:
$$
\aligned
	       x+2y+z&=3\\
         2x+5y+2z&=5\\
         2x+4y+kz&=5
\endaligned
$$

\mm
(a) $k = 0, -1$ \hfil (b) $k = 1$ \hfil (c) $k = -1$ \hfil (d) $k = 2$ \hfil
(e) None of the preceding

\vskip2.0in
{\bf 9.} Find the equation of the tangent plane to the graph
of the function $f(x,y)=x^2-y$ at (1,0).

\mm
(a) $z=2x-y$ \hfil (b) $z=2x-y+1$ \hfil (c) $z=2x-y-1$

(d) $z=x-2y+1$ \hs\hs (e) $z=x-2y$.

\newpage
{\bf 10.} Compute $\int x\ln x dx$ using integration by parts.

\mm
(a) $\ds x^2\ln x - \frac{x^2}{2}+C$ \hfil (b) $\ds \frac{x^2}{2}\ln
x+\frac{x^2}{4}+C$ \hfil
(c) $\ds\frac{x^2}{2}\ln x-\frac{x^2}{4}+C$

(d) $\ds\frac{x^2}{2}-\frac{x^2}{4}\ln x +C$ \hs\hs
(e) None of the preceding

\vskip2.0in
{\bf 11.} Suppose $f(x)$ is a function which takes the following values:
$$
\begin{large}
\begin{tabular}{|c||c|c|c|c|c|} \hline
x & 0 & 1 & 2 & 3 & 4 \\ \hline
f(x) & .1 & .4 & .5 & 1 & -.1\\ \hline
\end{tabular}
\end{large}
$$
Estimate $\ds\int^4_{0}f(x) dx$ by computing the Riemann sum with 4
subintervals and left endpoints.

\mm
(a) 1 \hfil (b) 1.9 \hfil (c) 1.45 \hfil (d) 1.1 \hfil (e) 2

\vskip1.8in
{\bf 12.} Find the present value of a perpetual income stream flowing
continuously at a
rate of $\$20,000$ per year, which is earning continuously compounded
interest at $8\%$.

\mm
(a) \$140,000 \hfil (b) \$160,000 \hfil (c) \$250,000 \hfil (d) \$200,000 \hfil
(e) \$14,000

\newpage
{\bf 13.} Find the partial fraction decomposition of
$$
	\frac{1}{x^2-6x+5}.
$$
That is find the numbers $A$ and $B$ such that:
$$
	\frac{1}{x^2-6x+5}=\frac{A}{x-5}+\frac{B}{x-1}.
$$

\mm
(a) $A=1,B=-1$ \hfil (b) $A=-\frac{1}{4},B=\frac{1}{4}$ \hfil
(c) $A=\frac{1}{4},B=-\frac{1}{4}$

(d) $A=-\frac{1}{3},B=\frac{1}{3}$ \hs\hs\hh (e) $A=\frac{1}{3},B=-\frac{1}{3}$

\vskip2.0in
{\bf 14.} Find the solution of the following
initial value problem:
$$
	\frac{dy}{dt}=4y^2t^3, \ \ y(0)=-1.
$$

\mm
(a) $\ds y=-\frac{1}{4t^4+1}$ \hfil (b) $\ds y=-\frac{1}{t^4+1}$ \hfil
(c) $\ds y=\frac{1}{4t^4+1}$

(d) $\ds y=\frac{1}{t^4+1}$ \hs\hs\hs\hh~ (e) None of the preceding

\newpage
{\bf 15.} Compute the improper integral:
$$
	\int^{\infty}_{1}\frac{2x}{(x^2+1)^2}dx.
$$

\mm
(a) 1/2 \hfil (b) 1/4 \hfil (c) 1 \hfil (d) 0 \hfil (e) None of the preceding

\vskip2.0in
{\bf 16.} Solve the following equation for $p$ in terms of $t$.
$$
	\frac{1}{2}\ln \left(\frac{p-1}{p}\right )=2t+1.
$$

\mm
(a) $\ds p=\frac{1}{1-e^{4t + 2}}$ \hfil (b) $\ds p=\frac{1}{1+e^{4t}}$ \hfil
(c) $\ds p=\frac{1}{1-e^{4t}}$ \hfil (d) $\ds p=\frac{1}{1+e^{4t+2}}$ \hfil
(e) $\ds p=e^{-2-4t}$

\vskip2.0in
{\bf 17.} The level curve for the function $f(x,y)=(2x^2+2y^2)^2$
with height $z=1$ is which of the following?

\mm
(a) a line \hfil (b) a circle \hfil (c) a point \hfil (d)  a parabola \hfil
(e) does not exist

\newpage
{\bf 18.} An individual opens an account with initial amout of
$\$100,000$, and then makes continuous withdrawals at the rate of $\$2,000$
per year.
We assume an interest rate of $5\%$ compounded continuously. Find the
differential
equation and initial condition that models the amount of money $M(t)$ in
the account.

\mm
(a) $\frac{dM}{dt}=.05M-2000,\; M(0)=100,000$ \hfil
(b) $\frac{dM}{dt}=-.05M-2000,\; M(0)=100,000$

(c) $\frac{dM}{dt}=.05M+2000,\; M(0)=98,000$ \hfil
(d) $\frac{dM}{dt}=-.05M+2000,\; M(0)=98,000$

(e) $\frac{dM}{dt}=2000M-50000,\; M(0)=100,000$

\vskip1.0in
{\bf 19.} According to the method of Lagrange multipliers, at which of the
following points $(x,y)$ could the function $f(x,y)=2x^2-y^2$ possibly
obtain a
maximum value subject to the constraint $x+y=2$.

\mm
(a) $(-2, 4)$ \hfil (b) $(2, -4)$ \hfil (c)  $(0,-2)$ \hfil (d)  $(1,-2)$
\hfil
(e) $(-1,2)$

\vskip2.0in
{\bf 20.} A function $f(x,y)$ satisfies $\ds\frac{\partial f}{\partial x}=0=
\frac{\partial f}{\partial y}$ at the point $(0,1)$. Suppose also that
$\ds\frac{\partial^2 f}{\partial x^2}=1$, $\ds\frac{\partial^2 f}{\partial
y^2}=4$,
$\ds\frac{\partial^2 f}{\partial x \partial y}=3$ at the point $(0,1)$.
Which of the
following is true?

\mm
(a) (0,1) is a relative maximum point\\
(b) (0,1) is a saddle point\\
(c) (0,1) is a relative minimum point\\
(d) (0,1) is not a critical point\\
(e) (0,1) is a critical point, but the second derivative test is inconclusive

\newpage
{\bf 21.} Suppose that $f(x,y)$ is a function which satisfies $f(1,2)=4$,
$\ds\frac{\partial f}{\partial x}(1,2)=2$, and $\ds\frac{\partial
f}{\partial y}(1,2)=3$.
Use linear approximation to estimate $f(.8,2.1)$.

\mm
(a) 4.1 \hfil (b) 3.9 \hfil (c) 4.2 \hfil (d) 3.8 \hfil (e) 4.3

\vskip1.0in
{\bf 22.} What bowl game are the Irish going to?  {\em Hint:} It begins
with a {\em G}.

\mm
(a) Orange \hfil (b) Alamo \hfil (c) GATOR \hfil (d) Sugar \hfil (e) Rose

\bm
{\bf 23.} Suppose that an experiment has sample space
$S =
\{s_1,s_2,s_3\}$ and
$P(s_1) = .7$. Which of the following {\em could} be $P(s_2)$?

\mm
(a) $-.5$ \hfil (b) .1 \hfil (c) .4 \hfil (d) .8 \hfil (e) 1.2

\vskip1.0in
{\bf 24.} Santa has 9 reindeer.  One has a red nose, three have brown
noses, and five have
black noses.  If Santa picks one reindeer at random, what is the
probability that the
reindeer's nose is brown?

\mm
(a) 1/9 \hfil (b) 1/5 \hfil (c) 3/5 \hfil (d) 1/3 \hfil (e) 5/9

\vskip1.0in
{\bf 25.} Suppose that the sample space for an experiment is $S =
\{s_1,s_2,s_3,s_4\}$ and
that\\ $P(s_1) = .2$, $P(s_2) = .1$, and $P(s_2,s_3) = .5$.  Find $P(s_4)$.

\mm
(a) .2 \hfil (b) 0 \hfil (c) .7 \hfil (d) .4 \hfil (e) .3

\newpage
{\bf 26.} Let $X$ be a random variable with values 1, 2, 5, and 7, with
probabilities
given by the following table:
$$
\begin{large}
\begin{tabular}{|c||c|c|c|c|} \hline
X & 1 & 2 & 5 & 7 \\ \hline
P & .3 & .2 & .4 &.1\\ \hline
\end{tabular}
\end{large}
$$
What is the expected value of $X$?

\mm
(a) 3.75 \hfil (b) 3.5 \hfil (c) 3.4 \hfil (d) 2.8 \hfil (e) 3

\vskip1.5in
{\bf 27.} Consider the experiment of choosing an integer $i$ at random from
$\{1,2,3,4\}$.
Let $X$ be the random variable given by $X(i) = (i-3)^2$ for each integer
$i$.  Find
$P(X = 1)$.

\mm
(a) 3 \hfil (b) 1/4 \hfil (c) 0 \hfil (d) 3/4 \hfil (e) 1/2

\vskip2.0in
{\bf 28.} Let $X$ be a continuous random variable with density function
$\ds f(x) =
\frac{1}{2\sqrt{x}}$ on $1 \leq x \leq 4$.  Which of the following
expressions is $P(X \leq 3)$?

\mm
(a) $\ds\int_1^3\frac{1}{2\sqrt{x}}dx$ \hfil
(b) $\ds\int_3^{\infty}\frac{1}{2\sqrt{x}}dx$ \hfil
(c) $\ds\int_1^3\sqrt{x}dx$

(d) $\ds\int_3^4\sqrt{x}dx$\hs\hs\hs\hs
(e) $\ds\int_3^4\frac{1}{2\sqrt{x}}dx$

\newpage
{\bf 29.} Let $X$ be a continous random variable with density function $\ds
f(x) =
\frac{c}{4x^2}$ on the interval $1 \leq x \leq 5$.  Find the constant $c$.

(a) $c = -4$ \hfil (b) $c = 1$ \hfil (c) $c = -5$ \hfil (d) $c = 5$ \hfil
(e) $c = 4$

\vskip2.0in
{\bf 30.} Let $X$ be a continuous random variable with density function
$\ds f(x) = \frac{1}{2\sqrt{x}}$ on $1 \leq x \leq 4$, which of the
following expressions is
$E(X)$?

(a) $\ds\int_1^4\frac12\sqrt{x}dx$ \hfil
(b) $\ds\int_1^4\frac{1}{2\sqrt{x}}dx$ \hfil
(c) $\ds\int_0^{\infty}\frac12\sqrt{x}dx$

(d) $\ds\int_0^{\infty}\frac{1}{2\sqrt{x}}dx$\hs\hs\hs\hh
(e) $\ds\int_1^4\frac12x^{\frac32}dx$

\vskip1.0in
{\bf 31.} Let $X$ be a {\em continuous} random variable on $1 \leq x <
\infty$, what is $P(X
= 3)$?

\mm
(a) $-1$ \hfil (b) 1 \hfil (c) 0 \hfil (d) 1/3 \hfil (e) 1/2

\newpage
{\bf 32.} Suppose that $X$ is a continuous random variable on $1 \leq x
\leq 5$ and that
the {\em cumulative distribution} function is $F(x)$.  Suppose further that
$F(3) = 1/3$.  Find
$P(3 \leq X \leq 5)$.

\mm
(a) 1/3 \hfil (b) 3/5 \hfil (c) 2/5 \hfil (d) 2/3 \hfil (e) 1

\vskip1.5in
{\bf 33.} Let $Z$ be the standard normal random variable with $\mu = 0$ and
$\sigma = 1$.\\
Find $P(0 \leq Z \leq 2.25)$.

\mm
(a) .4878 \hfil (b) .4861 \hfil (c) .9878 \hfil (d) .9938 \hfil (e) .0122

\vskip2.0in
{\bf 34.} Let $X$ be a normal random variable with a mean of 6 and standard
deviation of
4.  In terms of probabilities of the standard normal random variable $Z$,
which of the
following is $P(-2 \leq X \leq 10)$?

\mm
(a) $P(-2 \leq Z \leq 10)$ \hfil (b) $P(-1 \leq Z \leq 1)$ \hfil (c) $P(4
\leq Z \leq 6)$

(d) $P(-6 \leq Z \leq 6)$ \hs\hs\hh (e) $P(-2 \leq Z \leq 1)$
\end{document}

