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TEXT APPENDIX C – Differential Calculus Techniques in Management
MCQS
1. Differentiate the_ following TC function: TC = 150 + 200 Q - 4 Q2 + .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
ANSWER: 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
ANSWER: D PTS: 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
ANSWER: 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
ANSWER: 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.
ANSWER: A PTS: 1
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6. Differentiate the_ following function with respect to Q: TC = 50 + 100Q -6Q2 +.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
ANSWER: B PTS: 1
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Web Appendix C—Linear Programming Applications
MCQS
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
ANSWER: 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
ANSWER: 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
ANSWER: 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
ANSWER: A PTS: 1
5. An optimal solution of a linear programming problem always lies on the_ boundary of the_ feasible solution
space.
a. true
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b. false
ANSWER: A PTS: 1
6. A primal linear programming problem has multiple optimal solutions if it contains two or more variables.
a. true
b. false
ANSWER: 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
ANSWER: 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
ANSWER: 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
ANSWER: 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
c. surplus
d. both a and b
e. both a and c
ANSWER: C PTS: 1
11. A computer solution of large-scale linear programming problems typically employs a procedure (or variation of
the_ procedure) known as the_ ____ method.
a. least squares
b. analysis of variance
c. simplex
d. primal/dual
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TEXT APPENDIX C – Differential Calculus Techniques in Management
MCQS
1. Differentiate the_ following TC function: TC = 150 + 200 Q - 4 Q2 + .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
ANSWER: 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
ANSWER: D PTS: 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
ANSWER: 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
ANSWER: 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.
ANSWER: A PTS: 1
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6. Differentiate the_ following function with respect to Q: TC = 50 + 100Q -6Q2 +.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
ANSWER: B PTS: 1
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Web Appendix C—Linear Programming Applications
MCQS
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
ANSWER: 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
ANSWER: 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
ANSWER: 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
ANSWER: A PTS: 1
5. An optimal solution of a linear programming problem always lies on the_ boundary of the_ feasible solution
space.
a. true
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b. false
ANSWER: A PTS: 1
6. A primal linear programming problem has multiple optimal solutions if it contains two or more variables.
a. true
b. false
ANSWER: 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
ANSWER: 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
ANSWER: 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
ANSWER: 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
c. surplus
d. both a and b
e. both a and c
ANSWER: C PTS: 1
11. A computer solution of large-scale linear programming problems typically employs a procedure (or variation of
the_ procedure) known as the_ ____ method.
a. least squares
b. analysis of variance
c. simplex
d. primal/dual
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