% Version 2
% Correct Answers
%  1.a  2.d  3.b  4.b  5.c  6.b  7.e  8.c  9.e 10.b 11.a 12.c 13.d 14.e 15.e 16.c
% 17.a 18.b 19.d 20.a 21.a 22.c 23.b 24.b 25.c 26.e 27.c 28.e 29.b 30.a

% Page Setup

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% Macros

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% Preface

\input epsf

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% Marked answer sheet

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{\hfil\ansbox{70pt}{\ Answer Key 2\ }\hfil}\vss}
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{\noindent
{\bf Math 108: Calculus for Business}\hfill
Name:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\bf Exam 4}\quad{\sl December 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 4}\quad{\sl December 15, 2000}\hfill
Section:\hbox to 2.5truein{\hrulefill}

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% Problems

\vbox{
%1

\noindent\llap{1.\ }%
(5 pts.) Which of the following is the Riemann sum of the integral
$$
	\int^1_0 e^{x^3} dx
$$
if we use three subintervals and left-end points?

\beginanswers
{(a)~%
${1\over 3}(1+e^{1/27}+e^{8/27});$
\answerspace}
{(b)~%
${1\over 2}(e^{1/27}+e^{8/27}+e);$
\answerspace}
{(c)~%
${1\over 3}(1+2e^{1/27}+2e^{8/27}+e);$
\answerspace}
{(d)~%
${1\over 3}(1+e^{1/27}+e^{8/27}+e)$
\answerspace}
{(e)~%
${1\over 3}(1+2e^{1/27}+e^{8/27}+2e);$
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%2

\noindent\llap{2.\ }%
(5 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 $\$40,000$. Compute the maximum possible revenue.

\beginanswers
{(a)~%
617,500
\answerspace}
{(b)~%
250,000
\answerspace}
{(c)~%
300,000
\answerspace}
{(d)~%
280,000
\answerspace}
{(e)~%
270,000
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%3

\noindent\llap{3.\ }%
(5 pts.) Find the partial fraction decomposition of 
$$
	{1 \over {x^2-7x+6}}.
$$

That is find the numbers $A$ and $B$ such that:

$${1\over {x^2-7x+6}}={A\over {x-1}}+{B\over {x-6}}.$$

\beginanswers
{(a)~%
A=-1, B=1
\answerspace}
{(b)~%
A=-1/5, B=1/5
\answerspace}
{(c)~%
A=-1/3, B=1/3
\answerspace}
{(d)~%
A=1, B=-1
\answerspace}
{(e)~%
A=1/5, B=-1/5
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%4

\noindent\llap{4.\ }%
(5 pts.) Find the area bounded by the graphs of the functions $f(x)=x^2$ and
$g(x)=x+2$.

\beginanswers
{(a)~%
11/2
\answerspace}
{(b)~%
9/2
\answerspace}
{(c)~%
2
\answerspace}
{(d)~%
15/3
\answerspace}
{(e)~%
7/2
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%5

\noindent\llap{5.\ }%
(5 pts.) Using the method of integration by parts with $dv=x^3dx$, and $u=\ln (2x)$, the
integral
$$
	\int x^3 \ln (2x) dx
$$
becomes which one of the following expresions?

\beginanswers
{(a)~%
${x^4 \over 4}\ln (2x)-{1\over 2} \int x^3 dx$
\answerspace}
{(b)~%
none of the above
\answerspace}
{(c)~%
${x^4 \over 4}\ln (2x)-{1\over 4} \int x^3 dx$
\answerspace}
{(d)~%
${x^4 \over 2}\ln (2x)-{1\over 2} \int x^3 dx$
\answerspace}
{(e)~%
$x^4\ln (2x)- \int x^3 dx$
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%6

\noindent\llap{6.\ }%
(5 pts.) Suppose $f(x)$ is a function that takes the following values:

\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 & 0 & 1 & 2 & 3 & 4 \cr
\noalign{\hrule}
f(x) & 8 & 4 & 6 & 4 & -2  \cr
\noalign{\hrule}
}}


Evaluate $\int^4_0 f(x) dx$ using trapezoidal rule with 4 subintervals.

\beginanswers
{(a)~%
19
\answerspace}
{(b)~%
17
\answerspace}
{(c)~%
12
\answerspace}
{(d)~%
23
\answerspace}
{(e)~%
18
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%7

\noindent\llap{7.\ }%
(5 pts.) For a certain item, the demand curve is 
$D(q)=-2q+12$ and the supply curve is $S(q)=3q+2$. Find the producer surplus.

\beginanswers
{(a)~%
4
\answerspace}
{(b)~%
3
\answerspace}
{(c)~%
7
\answerspace}
{(d)~%
5
\answerspace}
{(e)~%
6
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%8

\noindent\llap{8.\ }%
(5 pts.) Find $y(2)$, if $y(t)$ is the solution of the following 
initial value problem:
$$
{{dy}\over {dt}}=2y^2t^3, \ \ y(0)=1.
$$

\beginanswers
{(a)~%
0
\answerspace}
{(b)~%
2
\answerspace}
{(c)~%
-1/7
\answerspace}
{(d)~%
-2
\answerspace}
{(e)~%
1/9
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%9

\noindent\llap{9.\ }%
(5 pts.) An individual opens an account with initial amout of 
$\$10,000$, and then makes continuous deposits at the rate of $\$2,000$ 
per year. We assume an interest rate of $8\%$ compounded continuously. Find the 
differential equation that models the amount of money $M(t)$ in the account.

\beginanswers
{(a)~%
${{dM}\over {dt}}=.08M-10000,\; M(0)=2,000$
\answerspace}
{(b)~%
${{dM}\over {dt}}=-.08M+2,000,\; M(0)=10,000$
\answerspace}
{(c)~%
${{dM}\over {dt}}=.08M-2000,\; M(0)=10,000$
\answerspace}
{(d)~%
${{dM}\over {dt}}=2000M-10,000,\; M(0)=10,000$
\answerspace}
{(e)~%
${{dM}\over {dt}}=.08M+2000,\; M(0)=10,000$
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%10

\noindent\llap{10.\ }%
(5 pts.) Find the present value of a perpetual income stream
of $\$10,000$ per year, if the interest rate is $8\%$.

\beginanswers
{(a)~%
100,000
\answerspace}
{(b)~%
125,000
\answerspace}
{(c)~%
150,000
\answerspace}
{(d)~%
120,000
\answerspace}
{(e)~%
80,000
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%11

\noindent\llap{11.\ }%
(5 pts.) Find the future value of a continuous income stream of
$\$20,000$ per year, that lasts 10 years, if the interest rate is $8\%$.

\beginanswers
{(a)~%
$250,000(e^{.8}-1)$
\answerspace}
{(b)~%
$550,125$
\answerspace}
{(c)~%
$200,000(1-e^{.8})$
\answerspace}
{(d)~%
$350,000$
\answerspace}
{(e)~%
$200,000(e^{.8}-1)$
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%12

\noindent\llap{12.\ }%
(5 pts.) For the function $f(x,y)=e^{3x+2y}$, find
${{\partial^2 f}\over {\partial y \partial x}}(1,1)$.

\beginanswers
{(a)~%
$2e^6$
\answerspace}
{(b)~%
$3e^5$
\answerspace}
{(c)~%
$6e^5$
\answerspace}
{(d)~%
$2e^3$
\answerspace}
{(e)~%
$3e^3$
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%13

\noindent\llap{13.\ }%
(5 pts.) Which of the following is a critical point of 
$f(x,y)=x^2y-2x-y+4$?

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

\vbox{
%14

\noindent\llap{14.\ }%
(5 pts.) Suppose that $f(x,y)$ is a function which satisfies
$$f(2,2)=3, {{\partial f}\over {\partial x}}(2,2)=4, {{\partial f}\over {\partial
y}}(2,2)=-1.$$ Use linear approximation to estimate $f(1.8, 2.1)$.

\beginanswers
{(a)~%
2.5
\answerspace}
{(b)~%
2.2
\answerspace}
{(c)~%
2.4
\answerspace}
{(d)~%
2.8
\answerspace}
{(e)~%
2.1
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%15

\noindent\llap{15.\ }%
(5 pts.) Find the equation of the plane passing through 
the points $$(0,1,2), (1,1,0), (-1,2,3).$$

\beginanswers
{(a)~%
$z=x-2y+1$
\answerspace}
{(b)~%
$z=-2x+y+2$
\answerspace}
{(c)~%
$z=x+y+3$
\answerspace}
{(d)~%
$z=-x+2y-1$
\answerspace}
{(e)~%
$z=-2x-y+3$
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%16

\noindent\llap{16.\ }%
(5 pts.) A function $f(x,y)$ satisfies
${\partial f \over \partial x}(1,2)=0={\partial f \over \partial y}(1,2)$.
Suppose also that $${{\partial^2 f} \over {\partial x^2}}(1,2)=-2,
{{\partial^2 f} \over {\partial y^2}}(1,2)=-3, {{\partial^2 f}\over {\partial x \partial y}}(1,2)=2.$$
Which of the following are true?

\beginanswers
{(a)~%
(1,2) is a saddle point
\answerspace}
{(b)~%
(1,2) is a relative minimum point
\answerspace}
{(c)~%
(1,2) is a relative maximum point
\answerspace}
{(d)~%
(1,2) is not a critical point
\answerspace}
{(e)~%
(1,2) is a critical point, but the second derivative test is inconclusive
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%17

\noindent\llap{17.\ }%
(5 pts.) According to the method of Lagrange multipliers,
at which of the following points $(x,y)$ could the function $f(x,y)=x^2+2y^2$
possibly obtain a maximum value subject to the constraint $x-2y=1$?

\beginanswers
{(a)~%
$x=1/3, y=-1/3$
\answerspace}
{(b)~%
$x=1/5, y=1/5$
\answerspace}
{(c)~%
$x=2/3, y=-1/3$
\answerspace}
{(d)~%
$x=1/3, y=2/3$
\answerspace}
{(e)~%
$x=-2/5, y=2/5$
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%18

\noindent\llap{18.\ }%
(5 pts.) The level curve(s) for the function $f(x,y)=(2x^2-2y^2)^2$
with height $z=0$ is which of the following?

\beginanswers
{(a)~%
a line
\answerspace}
{(b)~%
two lines
\answerspace}
{(c)~%
a circle
\answerspace}
{(d)~%
a hyperbola
\answerspace}
{(e)~%
a point
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%19

\noindent\llap{19.\ }%
(5 pts.) Let $Q(K,L)=20K^{0.8}L^{0.2}$ be the Cobb-Douglas
production function for a certain product, where $K$ and $L$ denotes the 
units of capital and labor respectively.
Each unit of capital cost $\$20$ and each unit of labor costs $\$40$. Use the 
Lagrange multiplier method to find the combination of capital and labor which gives 
the maximum output when the company's total budget for capital and labor is $\$500,000$.

\beginanswers
{(a)~%
L=25,000, K=3,000
\answerspace}
{(b)~%
L=10,000, K=2,500
\answerspace}
{(c)~%
L=2,000, K=25,000
\answerspace}
{(d)~%
L=2,500, K=20,000
\answerspace}
{(e)~%
L=5,000, K=10,000
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%20

\noindent\llap{20.\ }%
(5 pts.) Suppose that an experiment has sample space 
$S=\{s_1,s_2,s_3,s_4\}$, and \break $P(\{s_1,s_2\})=.7$. Which of the following 
{\bf could} be $P(\{s_4\})$?

\beginanswers
{(a)~%
.25
\answerspace}
{(b)~%
.7
\answerspace}
{(c)~%
-.1
\answerspace}
{(d)~%
.4
\answerspace}
{(e)~%
1.1
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%21

\noindent\llap{21.\ }%
(5 pts.) Santa has 10 reindeer. One has a red nose, 
four 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 black or brown?

\beginanswers
{(a)~%
9/10
\answerspace}
{(b)~%
1/2
\answerspace}
{(c)~%
4/5
\answerspace}
{(d)~%
7/10
\answerspace}
{(e)~%
1/10
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%22

\noindent\llap{22.\ }%
(5 pts.) Let $X$ be a random variable with values 0,1,2,5 and 6, with
probabilities given by the following table:


\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 & 0 & 1 & 2 & 5 & 6 \cr
\noalign{\hrule}
P(X=x) & .1 & .3 & .2 & .3 & .1  \cr
\noalign{\hrule}
}}

What is the expected value of $X$?

\beginanswers
{(a)~%
2.6
\answerspace}
{(b)~%
2.5
\answerspace}
{(c)~%
2.8
\answerspace}
{(d)~%
2.7
\answerspace}
{(e)~%
3
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%23

\noindent\llap{23.\ }%
(5 pts.) Consider the experiment of tossing two six-sided 
dice. Let $X$ be a random variable that associates to each outcome the 
sum of the top faces. Find $P(X=7)$.

\beginanswers
{(a)~%
1/12
\answerspace}
{(b)~%
1/6
\answerspace}
{(c)~%
1/18
\answerspace}
{(d)~%
1/9
\answerspace}
{(e)~%
5/36
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%24

\noindent\llap{24.\ }%
(5 pts.) Let $X$ be a continuous random variable with 
density function
$$f(x)={3\over 64}x^2, 0\le x \le 4.$$ Which one of the following expressions 
is $P(X \ge 2.5)$?

\beginanswers
{(a)~%
$\int^{\infty}_{2.5} {3\over 64}x^2dx$
\answerspace}
{(b)~%
$\int^{4}_{2.5} {3\over 64}x^2dx$
\answerspace}
{(c)~%
$\int^{2.5}_{-\infty} {3\over 64}x^2dx$
\answerspace}
{(d)~%
$\int^{2.5}_0 {3\over 64}x^4dx$
\answerspace}
{(e)~%
$\int^4_{2.5}{3\over 64}x^3dx$
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%25

\noindent\llap{25.\ }%
(5 pts.) Let $X$ be a continuous random variable with 
density function $$f(x)={1\over {2\sqrt{x}}}, 1\le x \le 4.$$ Which one of the following 
expressions is $E(X)$?

\beginanswers
{(a)~%
$\int_1^{\infty}{1\over 2}\sqrt{x}dx$
\answerspace}
{(b)~%
$\int^4_1 {1\over {2\sqrt{x}}}dx$
\answerspace}
{(c)~%
$\int_1^4{1\over 2}x^{{1\over 2}}dx$
\answerspace}
{(d)~%
$\int_0^{\infty}{1\over {2\sqrt{x}}}dx$
\answerspace}
{(e)~%
$\int_1^4 {1\over 2}x\sqrt{x}dx$
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%26

\noindent\llap{26.\ }%
(5 pts.) Assume that the density function of a continuous random
variable is given by $f(x)=ax+b, 0 \le x \le 3$. Find $a,b$ such that the expected value
of $X$ is 1.

\beginanswers
{(a)~%
$a=-1/2, b=2/3$
\answerspace}
{(b)~%
$a=12/5, b=1$
\answerspace}
{(c)~%
$a=-4/3, b=1$
\answerspace}
{(d)~%
$a=4/7, b=-2$
\answerspace}
{(e)~%
$a=-2/9, b=2/3$
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%27

\noindent\llap{27.\ }%
(5 pts.) The time in {\bf months} until the first breakdown of
a car is a random variable with exponential density with $\lambda=0.4$.
Determine the probability that no breakdown occurs for {\bf one year}.

\beginanswers
{(a)~%
$e^{.4}-1$
\answerspace}
{(b)~%
$1-e^{4.8}$
\answerspace}
{(c)~%
$e^{-4.8}$
\answerspace}
{(d)~%
$e^{-4.8}-1$
\answerspace}
{(e)~%
$1-e^{-4.8}$
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%28

\noindent\llap{28.\ }%
(5 pts.) A train arrives at a certain station every 60 minutes. If you
arrive randomly at this station, what is the probability that you will wait 
less than 15 minutes until the train comes?

\beginanswers
{(a)~%
.15
\answerspace}
{(b)~%
1/2
\answerspace}
{(c)~%
1/3
\answerspace}
{(d)~%
3/4
\answerspace}
{(e)~%
1/4
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%29

\noindent\llap{29.\ }%
(5 pts.) Let $Z$ be the standard normal random variable 
with $\mu=0, \sigma=1$. Find \break $P(0\le Z \le 1.9)$.

\beginanswers
{(a)~%
.9713
\answerspace}
{(b)~%
.4713
\answerspace}
{(c)~%
.1900
\answerspace}
{(d)~%
.8159
\answerspace}
{(e)~%
.4159
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

\vbox{
%30

\noindent\llap{30.\ }%
(5 pts.) Assume that the scores in a given exam are normally distributed
with expected value 75 and standard deviation 7.5. Also assume that the cutoff for
an A is 90. Find the probability that a student that will get an A in the exam.

\beginanswers
{(a)~%
.0228
\answerspace}
{(b)~%
.9772
\answerspace}
{(c)~%
.1587
\answerspace}
{(d)~%
.3413
\answerspace}
{(e)~%
.6828
\answerspace}
\endanswers\problemskip}\vfil\goodbreak

% Appendix

{}

\end
