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\bf Name:
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Teacher: \rm Karen S. Brown

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\head\bf Mathematics 108, Calculus II for Business\endhead

\head\bf Summer Session 1997\endhead

\head\bf Final Exam \endhead

\head\bf Friday, July 30, 1997\endhead

\rm\vskip2in

This Exam is worth a total of 150 points.  Point values are assigned next to the
problem numbers.  You should have a total of 11 sheets of paper - one cover page,
nine exam pages and one blank sheet for scratch work.  All the problems are 
partial credit problems.  Please show your work in the test booklet;  you do 
not have to turn in your scratch paper.  Calculators, books and notes are not
allowed.  The exam will begin at 8:35 and end at 10:35.  

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\bf Please sign the pledge: \rm \lq\lq On my honor, I have neither given nor received 
unauthorized aid on the test."

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\head\bf GOOD LUCK!\rm\endhead
\head\bf ENJOY THE REST OF THE SUMMER! \rm\endhead

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1.  Suppose that a company determines that its marginal profit function is
given by $MP(x)=6x^2-2x+3$, where x is the number of items produced.  If the 
company makes a profit of \$1000 by producing 10 items, determine the total
profit function.  (5 points)
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2.  Let \hskip .2in
$A=\left( \matrix -1 & 2 & 0\\3 & 1 & -2\\2 & -1 & 4 \endmatrix \right)$
\hskip .2in and \hskip .2in
$B= \left( \matrix 2 & -1\\0 & 5\\1 & -2 \endmatrix \right)$.
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Compute AB.  (5 points)

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Compute the determinant of A, det(A).  Does the matrix A have an inverse? (5 points)

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3.  A three-year-old business is analyzing its profit growth from the past three years.
You have been asked to help them with this assessment.  After 1 year the 
business made one profit unit;  after two years it made 3 profit units and
after 3 years it made 7 profit units.  Use the method of least squares to fit
a linear function to these data points.  (Hint:  Plot the data in a
graph so that you know you have the correct data points.)  (12 points)
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If the business asks you to predict what its profit would be at the end of the
fourth year, what would you tell them?  (3 points)

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4.  Joe has a lawn mowing job.  If he completes the work he earns \$40.  But
there is a 30\% chance it may rain, in which case he won't finish the job.  He
can pay Jane \$20 to help him and ensure that he finishes the job.  If X is 
the amount Joe will get if he does not get Jane to help, calculate E(X) and
thus decide whether Joe should hire Jane or not.  (If it rains, assume Joe 
will make no money and if Joe hires Jane assume they will be able to finish the 
job before it rains.)  (6 points)

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5.  Given the following table of outcomes and probabilities, compute the
expected value.  Then set up the equation for the variance, but \bf
do not evaluate it \rm. (6 points)

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 \bf Outcome:\rm \hskip.3in  1 \hskip .2in 2 \hskip.2in 3 \hskip.21in 4 
 
\bf Probability:\rm \hskip.13in $\frac15 \hskip .24in \frac25 \hskip.24in \frac15 \hskip.24in \frac15$

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\define\deriv#1{\dfrac{d#1}{dt}}
6.  Solve the following separable differential equation with the given initial
value. (7 points)

$$\deriv y = y(t^2+4t), \hskip .3in  y(0)=1$$

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7.  Solve the following system of equations using Gaussian elimination.  (10 points)

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$$\align
x-3y+2z &=6 \\
-3x+2y- \ \ z &=0 \\
-2x -\ \ y+3z &=8
\endalign$$

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8.  Evaluate the following integral.  (6 points)
$$\int x\cos5x\,dx$$

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9.  The riders of the New Town Elementary school bus consists of 5 five year
olds, 3 six year olds, 10 eight year olds, 1 nine year old, 4 eleven year
olds and a twelve year old.  A child is selected at random and her age is 
noted.  Let X be the outcome.  Construct a probability table of X. (6 points)

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What is the probability that the selected child is under 7 years of age?
(2 points)
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What is the probability that the selected child is over ten? (2 points)
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10.  Suppose that you have \$10,000 you would like to invest in a bank at a
rate of 10\% compounded continuously.  How much will this investment be worth
at the end of 5 years?  You may assume that $\sqrt{e}\approx1.65$.  (5 points)

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11. A business produces two products A and B.  Let x and y denote, respectively,
the quantity of A and B to be produced.  Limitations on the company resources
require that $500x^2+100y^2$ be at most 840,000.  Each unit of A yields a 
\$5000 profit and each unit of B yields a \$500 profit.  What should x and y
be to yield a maximum profit?  (Note:  You are maximizing the \bf profit \rm
in this problem under a certain constraint.) (20 points)

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12.  Suppose that the demand function for a certain item is given by 
$D(q)=\frac{20}{q+5}$ and the supply curve is given by $S(q)=q-3$. 
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a.  What is the equilibrium price and the equilibrium quantity?  (5 points)
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b.  Set up the integral for the consumer surplus for the given 
demand function, but \bf do not evaluate it.\rm

(3 points)
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13.  A random variable X is exponentially distributed with mean 2.

a).  What is the probability density function of this random variable?
(3 points)
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b).Determine $P(1\leq X \leq 3)$.  (3 points)

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14.  Your company produces two products, call them Item 1 and Item 2.  Suppose 
that you have determined that your costs are given by
 $C(x,y)=x^2+100x+y^2-300y+3000$, where x is the number of units of Item 1 
produced and y is the number of units of Item 2 produced.  If Item 1 sells
for \$500 and Item 2 sells for \$300, what production levels will maximize
the profit?  Hint:  Find the revenue function, then determine the profit
function from knowing the revenue and the cost.
  
\define\pd#1#2{\dfrac{\partial^2#1}{\partial#2^2}}
\define\tpd#1#2#3{\dfrac{\partial^2#1}{\partial#2 \partial#3}}
Recall that $D(x_0,y_0)=\left( \pd fx \right) \left( \pd fy \right) - 
\left( \tpd fxy \right)^2$.

(20 points)
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15.  Evaluate the following integral: (6 points) 
$$\int (3x^2+5)e^{x^3+5x}\,dx$$
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16.  Suppose $f(x)=k(x^2+2x)$ is a probability density function for a 
continuous random variable on the interval $0\leq x\leq 3$.  
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a.  Find the value for k.  (5 points)
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b.  Find the corresponding cumulative distribution function.  (5 points)

   
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