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Business Mathematics II
Sample Midterm Examination 1
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. A hospital manager has records for the total number of tests performed in a laboratory during the
years 1995 - 1999. She plotted the number of tests against the number of years after 1990, and had Excel
display a trend line and its equation. For example, 1999 is nine years after 1990, so its data is plotted over 9 on
the graph.
y = 1037e0.11x
WORK LOAD
number oftests
5,000
4,000
3,000
2,000
1,000
0
4
6
8
10
12
14
years after 1990
(i) Use the equation to predict the total number of tests in the year 2002. (Note that 2002 is 12 years
after 1990.)
(ii) Use the equation to predict the total number of test in the year 2040.
(iii) In the real world of hospital management, what would the number that you computed in Part (ii) tell
you?
2. A company estimates that the demand function for its product is given by D(q) = 95  0.01q.
(i) How many units would the company expect to sell, if the product is priced at $50?
(ii) What revenue would the company expect, if the product is priced at $50?
(iii) What is the largest price at which the company could sell any of its product?
- Business Mathematics II, Sample Test 1: page 2 -
3. Graphs of the revenue and cost functions for a product are shown in the following plot.
(i) Estimate the smallest number of units at which the company would earn a positive profit.
(ii) Estimate the profit which would result from a production level which maximizes revenue.
(iii) Estimate the number of units which should be produced in order to maximize profit.
(iv) Estimate the company’s maximum possible profit.
Questions 4 and 5 refer to the production and selling of a good. Marginal revenue and marginal cost
functions are plotted below. All monetary amounts are in dollars and all quantities are in numbers of units.
R'(q)
C'(q)
MARGINAL ANALYSIS
15
10
$/unit
5
0
-5 0
20
40
60
80
100
-10
-15
q
4. (i) How many items should be produced and sold, in order to maximize revenue?
(ii) How many items should be produced and sold, in order to maximize profit?
- Business Mathematics II, Sample Test 1: page 3 -
(iii) It appears that R(80)  9.2. What does this mean in real world terms of dollars, items, and
production?
5. The plots of three functions are shown below.
FUNCTION 1
FUNCTION2 3
FUNCTION
0
20
40
60
-10
-20
80
100
0.00
-0.05 0
20
40
60
80
100
-0.10
-0.15
0
q
FUNCTION32
FUNCTION
10
8
6
4
2
0
$/unit
0
$/unit
$/unit
10
20
40
60
80
q
100
-0.20
q
(i) Function ___ could be the graph of marginal demand.
(ii) Function ___ could be the graph of marginal profit.
6. Let f(x) = 2x2  x. Use a difference quotient, with an increment of h = 0.01 to approximate f (4).
7. Fill in the boxes of the screen capture in such a way that Solver would find a value of q which gives a
maximum value for P(q), subject to the constraint that R(q) is greater than or equal to $200,000.
- Business Mathematics II, Sample Test 1: page 4 -
8. Let f(x) = x2. You are to approximate the area under the graph of f, above the x-axis, and over the
interval from 0 to 6.
40
30
x
2 20
10
0 1
2
3
4
5
6
x
(i) Find points x0, x1, x2, and x3 that subdivide [0, 6] into three subintervals of equal lengths.
x0 = ____ .
x1 = ____ .
x2 = ____ .
x3 = ____ .
(ii) Find the midpoints m1, m2, and m3 of the subintervals.
m1 = ____ .
m2 = ____ .
m3 = ____ .
(iii) Compute the midpoint sum S3(f, [0, 6]).
9. A plot of Mega Bucks demand function is shown below. Suppose that the company sells 40 units at
the demand function price. All monetary amounts are in dollars and all quantities are in numbers of units. Note
that each grid box in the plot has an area of $20.
DEMAND FUNCTION
$10
D (q )
$8
$6
$4
$2
$0
0
10 20 30 40 50 60 70 80 90 100
q
(i) Shade the region in the plot that corresponds to the revenue from the sales.
(ii) Estimate the revenue that would result from the sale.
(iii) Estimate the consumer surplus that would result from the sale.
- Business Mathematics II, Sample Test 1: page 5 -