REVIEW OF MATHEMATICAL CONCEPTS USED IN
MANAGERIAL ECONOMICS
(*NOTE: THIS CHAPTER IS NOT IN THE TEXT. IT CAN BE FOUND ON THE
COMPANION WEBSITE WWW.PRENHALL.COM/KEAT)
QUESTIONS
1. Function: The relationship between a dependent variable and one (or more) independent variables.
Variable: An entity that can assume different values
2. Assume the following demand equation: QD = 250 – 10P. Given this linear equation, we can express
demand tabular form as follows:
P Q
$25 0
20 50
Figure 1
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0
50
100
150
200
250
0 5 10 15 20 25
P
Q
Q = 250 – 10P
129
Review of Mathematical Concepts 130
130
Figure 2
3. The following are suggested ways to answer this question. Instructors may wish to devise their own
equations.
a. QD = f(A) where A = dollar amount spent on advertising
4. A function is said to be continuous over a given interval if it is continuous at every point on that
interval. In real world situations, it may be impractical to consider the value of an economic
5. The slope of a line measures the rate of change of the dependent variable (measured along the
vertical axis) with respect to some independent variable (measured along the horizontal axis). It is
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0
5
10
15
20
25
0 50 100 150 200 250
Q
P
Review of Mathematical Concepts 131
131
6. Marginal analysis is the consideration of small changes around some given point. If discrete
changes are considered, then a unit change is usually considered. (e.g., marginal revenue is equal to
the change in total revenue relative to a unit change in quantity). Sometimes, it may not be practical
7. The first derivative of a function is important because it determines the rate of change of a
8. By setting the first derivative equal to zero and solving for the value of the independent variable
9. An analysis of the second derivative of a function enables us to distinguish between a function’s
maximum and minimum value, in the event that it has both. If at the point where the first derivative
10. ∆Y/∆X indicates discrete changes in Y relative to X. dY/dX indicates the change in Y relative to a
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Review of Mathematical Concepts 132
132
PROBLEMS
1. a. Q = 75 – .5P
Figure 3
Figure 4
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0
10
20
30
40
50
60
70
80
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160
P
Q
0
20
40
60
80
100
120
140
160
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75
Q
P
Review of Mathematical Concepts 133
133
2. (a, b, and c are answered together for each equation)
Q = 450 – 16P
Figure 5
Figure 6
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0
5
10
15
20
25
30
0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450
Q
P, MR
28.125
14.06
MR
D
0
500
1000
1500
2000
2500
3000
3500
0 50 100 150 200 250 300 350 400 450
Q
TR
3164.0625
Review of Mathematical Concepts 134
134
Figure 7
Figure 8
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0
1
2
3
4
5
0 90 180 270 360
Q
P, MR
4.50
2.25
MR
D
0
100
200
300
400
500
0 90 180 270 360
Q
TR
405
Review of Mathematical Concepts 135
135
Figure 9
Figure 10
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0
1
1.5
2.5
3
3.5
0 100 200 300 400 500 600 700 800 900 100011001200 1300140015001600
Q
P, MR
MR
D
2
0.5
0
200
400
600
800
1000
1200
0 375 750 1125 1500
Q
TR
1125
Review of Mathematical Concepts 136
136
3. a.
TC = 1500 + 300Q – 25Q2 + 1.5Q3
1500
b. For graphs of these equations, please refer to the next page.
c. In order to determine the quantity that minimizes the marginal cost in the cubic equation, take
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Review of Mathematical Concepts 137
137
Figure 11
Figure 12
Figure 13
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0
200
400
600
800
1000
1200
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Q
$
Av. Cost
Av. Var. Cost
Marg. Cost
0
200
400
600
800
1000
1200
1400
1600
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Q
$
Av. Cost
Av. Var. Cost
Marg. Cost
0
200
400
600
800
1000
1200
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Q
$
Av. Cost
MC = AVC
Review of Mathematical Concepts 138
138
4. Problems a. and b. are answered together for each equation.
Q = 10 – .004P
Figure 14
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–4
–3
–2
–1
0
1
2
3
4
0 1 2 3 4 5 6 7 8 9 10
Q
$ (Thousands)
Review of Mathematical Concepts 139
139
Figure 15
Figure 16
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–8
–7
–6
–5
–4
–3
–2
–1
0
1
2
3
4
0 1 2 3 4 5 6 7 8 9 10
Q
$ (Thousands)
–6
–5
–4
–3
–2
–1
0
1
2
3
4
0 1 2 3 4 5 6 7 8 9 10
Q
$ (Thousands)