TEXT APPENDIX C – Differential Calculus Techniques in Management
MULTIPLE CHOICE
1. Differentiate the following TC function: TC = 150 + 200 Q - 4 Q 2 + .6 Q3
a. dTC/dQ = 200 - 8Q + 1.8 Q2
b. dTC/dQ =-8 + 1.8 Q2
c. dTC/dQ = 200
d. dTC/dQ = 200 - 4Q + .6Q2
a. dTC/dQ = 1.8 Q2
ANS: A PTS: 1
2. The total revenue function (where Q = output), is: TR = 400 Q - 4 Q2
a. TR is maximized at Q = 20
b. TR is maximized at Q = 30
c. TR is maximized at Q = 40
d. TR is maximized at Q = 50
e. TR is maximized at Q = 60
ANS: DPTS: 1
3. The following is a cubic demand function in P. Find the derivative dQ/dP of: Q= 4+3P-.5P2
+ .02P3.
a. dQ/dP = 4 + 3P – P + .06P2
b. dQ/dP = 3
c. dQ/dP = 3 – P + .06P2
d. dQ/dP = .06P2
e. dQ/dP = .06
ANS: C PTS: 1
4. If the first derivative of Y with respect to X is: dY/dX = -4·X2, then the second derivative is:
a. -4
b. -8•X
c. -4•X
d. -8·X2
e. -8
ANS: B PTS: 1
5. The second derivative of the function (d2Y/dX2 ) is negative at the optimal solution of X=22.
Therefore, we know that the solution X=22, where the first derivative equals zero…
a. must be a minimum.
, b. must be a maximum.
c. may be either a maximum or a minimum.
d. would be nothing, because the second derivative is negative.
ANS: A PTS: 1
6. Differentiate the following function with respect to Q: TC = 50 + 100Q -6Q 2 +.5Q3
a. dTC/dQ = 50 + 100 -6Q + .5Q2
b. dTC/dQ = 100 -12Q + 1.5Q2
c. dTC/dQ = 50 + 100 -2Q + 3Q2
d. dTC/dQ = 100
ANS: B PTS: 1
,Web Appendix C—Linear Programming Applications
MULTIPLE CHOICE
1. The fixed per-unit profit contribution coefficients of the objective function in a linear
programming problem imply the following economic assumptions except:
a. selling prices per unit of the products (outputs) are constant
b. constant returns to scale in the production process
c. buying prices per unit of the resources (inputs) are proportional to the amount purchased
d. both b and c
e. both a and c
ANS: C PTS: 1
2. Which of the following statements concerning dual variables is (are) true?
a. Dual variables are obtained automatically in an algebraic solution of a linear programming
problem.
b. Dual variables are similar to the artificial variables used in the LaGrange Multiplier
technique.
c. A dual variable indicates how much the objective function will change if one additional
unit of a given resource is made available, provided that the increase in the resource does
not shift the optimal solution to another corner of the feasible solution space.
d. a and c
e. a, b, and c
ANS: E PTS: 1
3. If the primal linear programming problem has two variables and four constraints (excluding the
non-negativity constraints), the corresponding dual linear programming problem will have ____.
a. two variables and four constraints
b. four variables and two constraints
c. two variables and two constraints
d. four variables and four constraints
e. none of the above
ANS: B PTS: 1
4. A dual variable equal to zero in the optimal solution to a profit-maximization linear programming
problem indicates that the objective function will not increase if an additional unit of the given
resources is made available.
a. true
b. false
ANS: A PTS: 1
, 5. An optimal solution of a linear programming problem always lies on the boundary of the feasible
solution space.
a. true
b. false
ANS: A PTS: 1
6. A primal linear programming problem has multiple optimal solutions if it contains two or more
variables.
a. true
b. false
ANS: B PTS: 1
7. An optimal solution of a linear programming problem always occurs at one (or more) of the
corner points of the feasible solution space.
a. true
b. false
ANS: A PTS: 1
8. Slack variables are given coefficients of ____ in the objective function.
a. +1
b. 0
c. 1
d. +.00001
e. none of the above
ANS: B PTS: 1
9. In a maximization linear programming problem, the ____ variables represent the difference
between the right-hand sides and left-hand sides of less than or equal to () inequality
constraints.
a. dual
b. slack
c. primal
d. both a and b
e. none of the above
ANS: B PTS: 1
10. In a minimization linear programming problem, the ____ variables are subtracted from the
greater than or equal to () inequality constraints in order to convert these constraints to
equalities.
a. dual
b. primal
MULTIPLE CHOICE
1. Differentiate the following TC function: TC = 150 + 200 Q - 4 Q 2 + .6 Q3
a. dTC/dQ = 200 - 8Q + 1.8 Q2
b. dTC/dQ =-8 + 1.8 Q2
c. dTC/dQ = 200
d. dTC/dQ = 200 - 4Q + .6Q2
a. dTC/dQ = 1.8 Q2
ANS: A PTS: 1
2. The total revenue function (where Q = output), is: TR = 400 Q - 4 Q2
a. TR is maximized at Q = 20
b. TR is maximized at Q = 30
c. TR is maximized at Q = 40
d. TR is maximized at Q = 50
e. TR is maximized at Q = 60
ANS: DPTS: 1
3. The following is a cubic demand function in P. Find the derivative dQ/dP of: Q= 4+3P-.5P2
+ .02P3.
a. dQ/dP = 4 + 3P – P + .06P2
b. dQ/dP = 3
c. dQ/dP = 3 – P + .06P2
d. dQ/dP = .06P2
e. dQ/dP = .06
ANS: C PTS: 1
4. If the first derivative of Y with respect to X is: dY/dX = -4·X2, then the second derivative is:
a. -4
b. -8•X
c. -4•X
d. -8·X2
e. -8
ANS: B PTS: 1
5. The second derivative of the function (d2Y/dX2 ) is negative at the optimal solution of X=22.
Therefore, we know that the solution X=22, where the first derivative equals zero…
a. must be a minimum.
, b. must be a maximum.
c. may be either a maximum or a minimum.
d. would be nothing, because the second derivative is negative.
ANS: A PTS: 1
6. Differentiate the following function with respect to Q: TC = 50 + 100Q -6Q 2 +.5Q3
a. dTC/dQ = 50 + 100 -6Q + .5Q2
b. dTC/dQ = 100 -12Q + 1.5Q2
c. dTC/dQ = 50 + 100 -2Q + 3Q2
d. dTC/dQ = 100
ANS: B PTS: 1
,Web Appendix C—Linear Programming Applications
MULTIPLE CHOICE
1. The fixed per-unit profit contribution coefficients of the objective function in a linear
programming problem imply the following economic assumptions except:
a. selling prices per unit of the products (outputs) are constant
b. constant returns to scale in the production process
c. buying prices per unit of the resources (inputs) are proportional to the amount purchased
d. both b and c
e. both a and c
ANS: C PTS: 1
2. Which of the following statements concerning dual variables is (are) true?
a. Dual variables are obtained automatically in an algebraic solution of a linear programming
problem.
b. Dual variables are similar to the artificial variables used in the LaGrange Multiplier
technique.
c. A dual variable indicates how much the objective function will change if one additional
unit of a given resource is made available, provided that the increase in the resource does
not shift the optimal solution to another corner of the feasible solution space.
d. a and c
e. a, b, and c
ANS: E PTS: 1
3. If the primal linear programming problem has two variables and four constraints (excluding the
non-negativity constraints), the corresponding dual linear programming problem will have ____.
a. two variables and four constraints
b. four variables and two constraints
c. two variables and two constraints
d. four variables and four constraints
e. none of the above
ANS: B PTS: 1
4. A dual variable equal to zero in the optimal solution to a profit-maximization linear programming
problem indicates that the objective function will not increase if an additional unit of the given
resources is made available.
a. true
b. false
ANS: A PTS: 1
, 5. An optimal solution of a linear programming problem always lies on the boundary of the feasible
solution space.
a. true
b. false
ANS: A PTS: 1
6. A primal linear programming problem has multiple optimal solutions if it contains two or more
variables.
a. true
b. false
ANS: B PTS: 1
7. An optimal solution of a linear programming problem always occurs at one (or more) of the
corner points of the feasible solution space.
a. true
b. false
ANS: A PTS: 1
8. Slack variables are given coefficients of ____ in the objective function.
a. +1
b. 0
c. 1
d. +.00001
e. none of the above
ANS: B PTS: 1
9. In a maximization linear programming problem, the ____ variables represent the difference
between the right-hand sides and left-hand sides of less than or equal to () inequality
constraints.
a. dual
b. slack
c. primal
d. both a and b
e. none of the above
ANS: B PTS: 1
10. In a minimization linear programming problem, the ____ variables are subtracted from the
greater than or equal to () inequality constraints in order to convert these constraints to
equalities.
a. dual
b. primal
