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Business Mathematics II
Sample Final Examination
This test is only presented as a sample of the types of questions which might appear on an
examination. The instructor of each section will set his or her own examinations, which may differ from the
sample in both length and content.
Show all of the steps and all of the work that you use to solve each problem.
A list of formulas that you may find useful is provided on the last page of this test.
1. Different types of trendlines have been fitted to the data from six test markets.
Linear Trendline
Quadratic Trendline
$60
$40
$40
$40
$20
$0
D(q)
$60
$60
D(q)
D(q)
Polynomial Trendline
$20
$0
0
100
200
q
300
400
$20
$0
0
100
200
q
300
400
0
100
200
q
300
400
(i) Which trendline gives the best approximation for the actual demand function, D(q)?
(ii) Use the trendline that you selected in part (i) to estimate the number, q, of units that would be sold at
a price of $30.
2. The fixed costs for a good are $12,000 and the marginal cost is $25 per unit. Give a formula for the
total cost function, C(q).
3. Graphs of the demand and profit functions for a good are shown below.
Profit Function
P(q)
D(q)
Demand Function
$50
$40
$30
$20
$10
$0
0
100
200
300
q
400
500
$3,000
$2,000
$1,000
$0
-$1,000 0
-$2,000
-$3,000
100
200
300
400
500
q
(i) Use the graphs to estimate the number of units that should be produced and sold in order to maximize
the profit.
- Business Mathematics II, Sample Final Examination: page 2 -
(ii) Use the graphs to estimate the price per unit for the number of units that you found in part (i).
4. The graph of a demand function is shown at the right.
$ 60
(i) Shade in the area that corresponds to the Consumer
Surplus, when q = 200.
$ 40
D(q)
$ 20
(ii) Use the graph to estimate the revenue, R(200).
(iii) Use the picture to estimate the Consumer Surplus,
when q = 200.
0
100
200
300
400 500
(iv) Give a formula, part of which will involve an integral, for the Consumer Surplus, when q = 200.
5. The revenue function for a good is R(q) =  0.3q2 + 12q. Use a difference quotient, with an
increment of h = 0.01 to approximate the marginal revenue at q = 10, R (10).
6. This question refers to a good whose
graphs of revenue, R(q) and cost, C(q), are shown at
the right.
Revenue
Cost
Revenue & Cost Function
R(q) & C(q)
$4,000
$3,000
$2,000
$1,000
$0
0
100
200
300
400
500
q
(i) Use the above graphs to estimate the value of q at which R (q) = C (q).
Marginal Function 2
Marginal Function 3
60
0
100 200 300
q
400 500
-0.05 0
100 200 300 400 500
$/unit
0.00
$/unit
$/unit
Marginal Function 1
10
8
6
4
2
0
30
-0.15
0
-30 0
-0.20
-60
-0.10
q
(ii) Marginal function number ___ could be the derivative of R(q).
100 200 300 400 500
q
- Business Mathematics II, Sample Final Examination: page 3 -
(iii) Marginal function number ___ could be the derivative of C(q).
7. Fill in the boxes of the screen capture in such a way that Solver would find a value in the Cell A2
which gives Cell E2 the value of $5,000, subject to the constraint that Cell B2 is less than or equal to $27.
Questions 8 and 9 refer to the following data on test markets and costs for a good. All monetary
amounts are in dollars and all quantities are single units.
- Business Mathematics II, Sample Final Examination: page 4 -
Potential national market: 5,000
Test Markets
Market Number Market Size
Price
Projected Yearly Sales
1
200
$89.95
30
2
300
$84.95
54
3
150
$79.95
32
4
350
$74.95
84
5
200
$69.95
54
6
400
$64.95
120
Cost Data
Fixed Cost: $1,500
Variable Costs
Cost per unit
Quantity
$5
First 1,000 units:
$2
Next 1,000 units:
$4
Further:
8. From the data in Test Market 2, compute the number of units that you would expect to sell in the
entire national market, at a price of $84.95.
9. Sketch a graph of the marginal cost function, MC(q) for 0  q  3,000.
MC(q)
6
5
4
3
2
1
q
0
500
1000
1500
10. Let f(x) = x3. You are to approximate the
area under the graph of f, above the x-axis, and over
the interval from 1 to 5.
x0 = ____ .
x1 = ____ .
x2 = ____ .
2500 3000
250
200
f (x )
(i) Find points x0, x1, and x2 that subdivide [1,
5] into two subintervals of equal length.
2000
150
100
50
0
0
1
2
3
x
4
5
6
- Business Mathematics II, Sample Final Examination: page 5 -
(ii) Find the midpoints m1, and m2 of the subintervals.
m1 = ____ .
m2 = ____ .
(iii) Compute the midpoint sum S2(f, [1, 5]).
Questions 11 - 13 refer to a continuous random variable X, whose p.d.f., fX(x), and c.d.f., FX(x), are
plotted below.
f X( x )
2.5
1
2
0.8
1.5
F X( x )
1
0.6
0.4
0.2
0.5
0
0.2
0.4
0.6
0.8
0
1
0.2
0.4
0.6
0.8
1
x
x
11. (i) Show how you can use the graph of fX(x) to estimate P(X  0.6).
(ii) Show how you can use the graph of FX(x) to estimate P(X  0.6).
12. (i) Which of the following is most likely to be the mean of X, X? Circle your choice.
a. X = 0.75
b. X = 0.65
c. X = 0.5
(ii) Which of the following is most likely to be the standard deviation of X, X? Circle your choice.
a. X = 1.0
b. X = 0.2
c. X = 0.05
13. (i) Is X a normal random variable? Why or why not?
(ii) Let m be the random variable that is the sample mean for a random sample of size 100 from X. Is
S
m  X
X
approximately normal?
10
14. X is a finite random variable whose p.m.f. is given below.
x
fX(x)
1
0.4
2
0.2
3
0.4
- Business Mathematics II, Sample Final Examination: page 6 -
(i) Compute the mean, X, of X.
(ii) Compute the variance, V(X), of X.
(iii) Compute the standard deviation, X. of X.
15. The following sample of size n = 4 was taken for the values of a random variable, X.
7, 4, 4, 9
(i) Compute the sample mean, x .
(ii) Compute the sample standard deviation, s.
16. Let T be the random variable that gives the time, in hours, for a rush order of parts to arrive at your
business. The parameters for T are T = 12 and T = 3. Let t be the random variable that is the mean of a
random sample of n = 100 delivery times.
(i)   ?
t
(ii)   ?
t
(iii) Give a formula for the random variable, S, that is the standardization of t .
(iv) What is the approximate distribution of S?
17. Graphs of the probability density functions for random variables X and Y, with the same means, are
shown below.
0.4
0.4
0.3
0.3
FfXX(x)
( x ) 0.2
F fYY(y)
( y ) 0.2
0.1
0.1
x
y
(i) Which is likely to give you a better estimate of the mean, a sample of size 4 from X, or a sample of
size 4 from Y? Explain your answer.
(ii) Could X be a normal random variable?
(iii) Could X be a standard normal random variable?
- Business Mathematics II, Sample Final Examination: page 7 -
18. Let X be a normal random variable with X = 100 and X = 8. Fill in the information that would be
needed to have the Excel function Random Number Generation create random values of X in Cells A1:A50.
19. Let X be a normal random variable with X = 100 and X = 8. Fill in the information that would be
needed to have the Excel function NORMINV create a random value of X.
20. It is expected the 20 companies will bid in oil lease auctions. Let R be the random variable giving
the error, in millions of dollars, of randomly selected geologist's estimate for the fair value of a lease. Historical
data leads us to assume that X = 0 and X = 10.
- Business Mathematics II, Sample Final Examination: page 8 -
(i) Use R and other random variables to explain the winner's curse, and indicate how it could be
approximated?
(ii) If X was increased to 15 million dollars, would the winner's curse increase or decrease? Justify
your answer.
(iii) If X remained at 10 million dollars, but the number of bidding companies was lowered to 12,
would the winner's curse increase or decrease? Justify your answer.
Formulas
f Z ( z) 
V(X ) 
E( X ) 
2
1
 e  0.5 z
2 

i 1

V(X ) 
 x   X 
2
 f X ( x ) dx

xi  a  i  x
n
s2 
xi
1

n 1
mi 
 xi  x 
n
2
i 1
xi 1  xi
2
X  X
X
P(q) = R(q)  C(q)
R(q) = qD(q)
q
0
 D(q) dq  q0  D(q0 )
0
 x  f X ( x) dx

 x   X 2  f X ( x)
ba
x 
n
1

n
E( X ) 
all x
all x
x

 x  f X ( x)
 X2  V ( X )
S n  f , [a, b] 

V x 
n
 f (mi )  x
i 1
V (X )
n
MP(q) = MR(q)  MC(q)
f ( x ) 
f ( x  h)  f ( x  h)
2h