Calculus for Business - Endterm Topic Summary
Ngày 31 tháng 5 năm 2023
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QUESTIONS
Lecture 1. Integral: Topic 1 to Topic 2
Topic 1. Antiderivative - Netchange: Economic problem.
Exercise 1. (Hoffmann, Section 5.1, Question 43) A manufacturer estimates that the marginal cost
0
of producing q units of a certain commodity is C (q) = 3q 2 − 24q + 48 dollars per unit. If the cost of
producing 10 units is 5000 dollars, what is the cost of producing 30 units?
Exercise 2. (Self practise) A manufacturer estimates that the marginal cost of producing q units of
q2
7
a certain commodity is
− q + dollars per unit. If the cost of producing 2 units is 240 dollars, what
4
2
is the cost of producing 6 units?
0
Exercise 3. (Hoffmann, Section 5.1, Question 50) A biomass is growing at the rate of M (t) = 0.5e0.2t .
By how much does the mass change during the second hour?
Exercise 4. (Self practise - Hoffmann, Section 5.3, Question 53) Bejax Corporation has set up aproductionline to manufacture
a new type of cellular telephone. The rate of production of the telephones
t
is 1500 2 −
units/month. How many telephones are produced during the third month? The
2t + 5
final numerical result is asked to be rounded to the hundred decimal place.
Exercise 5. A manufacturer estimates that the marginal cost of producing q units of a certain
commodity is 6q 2 − 20q + 500 dollars per unit.
(a) By how much will the total manufacturing cost increase if the level of production is raised from
2 units to 5 units?
(b) Let C(q) be the total cost for producing q units. Knowing the total cost for producing the first
Z 10
3 units is 1500 dollars, evaluate
C(q)dq?
0
Exercise 6. (Self practise) A manufacturer estimates that the marginal cost of producing q units of
q
q2
− + 50 dollars per unit. All next final numerical results are asked to be
a certain commodity is
5
3
rounded to the hundred decimal place.
(a) By how much will the total manufacturing cost increase if the level of production is raised from
3 units to 7 units?
(b) Let C(q) be the total cost for producing q units. Knowing the total cost for producing the first
Z 8
2 units is 450 dollars, evaluate
C(q)dq?
0
1
1
QUESTIONS
Topic 2. Consumer surplus - Producer surplus.
√
Exercise 7. (Hoffmann, Section 5.5, Question 18) Given the demand function D(q) = 245 − 2q
and the supply function S(q) = 5 + q, find the consumers’ surplus and the producers’ surplus at
equilibrium.
Exercise 8. (Self - practice, Section 5.5, Question 17) Given the demand function D(q) = −0.3q 2 +50
and the supply function S(q) = 0.1q 2 + q + 20, find the consumers’ surplus and the producers’ surplus
at equilibrium.
Exercise 9. (Hoffmann, Section 5.5, Question 33) A manufacturer of machinery parts determines
that q units of a particular piece will be sold when the price is p = 110 − q dollars per unit. The total
cost of producing those q units is C(q) dollars, where
C(q) = q 3 − 25q 2 + 2q + 3, 000
(a) How much profit is derived from the sale of the q units at p dollars per unit?
(b) For what value of q is profit maximized?
(c) Find the consumers’ surplus for the level of production q0 that corresponds to maximum profit.
Exercise 10. (Self - practice, Hoffmann, Section 5.5, Question 34) Repeat the Exercise 9 for C(q) =
2q 3 − 59q 2 + 4q + 7600 and p = 124 − 2q.
Lecture 2. Differential Equations: Topic 3 to Topic 5
Topic 3. Resale Model
Exercise 11. The resale value of a certain industrial machine decreases over a 15-year period at a
rate that depends on the age of the machine. When the machine is x years old, the rate at which its
value is changing is 80(x − 15) dollars per year. Express the value of the machine as a function of its
age and initial value. If the machine was originally worth 12000 dollars, how much will it be worth
when it is 15 years old?
Topic 4. Price Adjustment Model
Exercise 12. Suppose the price p(t) of a particular commodity varies in such a way that its rate of
dp
D−S
change with respect to time is proportional to the shortage D − S :
=
, where D = 6 − 2p
dt
3
and S = −8 + 4p are the demand and supply functions respectively. Given p(0) = 3, find p(t).
Topic 5. Other practical model
Exercise 13. (New, 31/05/2023) Some populations are subject to seasonal fluctuations. The population in a vacation resort serves as one example. A model for describing such situations is the
dP
= (kcost)P , where k = 0.2 and t is measured in months. Find the solution
differential equation
dt
of the differential equation subject to P (0) = 400.
Exercise 14. (New, 31/05/2023) The rate of change of the number of birds N (t) in a flock is directly
proportional to 650 − N (t), where t is the time in years. When t = 0, the flock is 300, and when t = 2,
the flock has increased to 500. Find the number of birds when t = 3.
Lecture 3. Function of Several variables: Topic 6 to Topic 8
Topic 6. Small Incremental Approximation
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ANSWER
Exercise 15. (Hoffmann, Section 7.2, Question 46) A manufacturer estimates that the annual output
at a certain factory is given by Q(K, L) = 30K 0.3 L0.7 units, where K is the capital expenditure in
units of 1000 dollars and L is the size of the labor force in worker-hours.
(a) Find the marginal productivity of capital QK and the marginal productivity of labor QL when
the capital expenditure is 630000 dollars and the labor level is 830 worker-hours.
(b) Should the manufacturer consider adding a unit of capital or a unit of labor in order to increase
output more rapidly?
Exercise 16. (Hoffmann, Section 7.2, Question 75) The output at a certain plant is Q(x, y) = 0.08x2 +
0.12xy + 0.03y 2 units per day, where x is the number of hours of skilled labor used and y is the number
of hours of unskilled labor used. Currently, 80 hours of skilled labor and 200 hours of unskilled labor
are used each day. Use calculus to estimate the change in output that will result if an additional 0.5
hour of skilled labor is used each day, along with an additional 2 hours of unskilled labor.
Topic 7. Free Extrema
Exercise 17. (Hoffmann, Section 7.3, Question 1 - 22) Find the critical points of the given functions
and classify each as a relative maximum, a relative minimum, or a saddle point.
(a) (Question 4) f (x, y) = x2 + 2y 2 − xy + 14y
(b) (Question 7) f (x, y) = 2x3 + y 3 + 3x2 − 3y − 12x − 4
(c) (Question 9) f (x, y) = x3 + y 2 − 6xy + 9x + 5y + 2
Exercise 18. (Hoffmann, Section 7.3, Question 23) A T-shirt shop carries two competing shirts, one
endorsed by L. Messi and the other by C. Ronaldo. The owner of the store can obtain both types at a
cost of 2 dollars per shirt and estimates that if Messi shirts are sold for x dollars apiece and Ronaldo
shirts for y dollars apiece, consumers will buy 40 − 50x + 40y Messi shirts and 20 + 60x − 70y Ronaldo
shirts each day. How should the owner price the shirts in order to generate the largest possible profit?
Exercise 19. (Hoffmann, Section 7.3, Question 27) A company produces x units of commodity A and
y units of commodity B. All the units can be sold for p = 100 − x dollars per unit of A and q = 100 − y
dollars per unit of B. The cost (in dollars) of producing these units is given by the joint-cost function
C(x, y) = x2 + xy + y 2 . What should x and y be to maximize profit?
Topic 8. Constrained Optimization
Exercise 20. (Hoffmann, Section 7.5, Question 9) Let f (x, y) = 2x2 + 4y 2 − 3xy − 2x − 23y + 3. Find
the minimum value of the function f (x, y) subject to the constraint x + y = 15.
Exercise 21. (Hoffmann, Section 7.5, Question 17) A manufacturer of television sets makes two
models, the Deluxe and the Standard. The manager estimates that when x hundred Deluxe sets and
y hundred Standard sets are produced each year, the annual profit will be thousand dollars, where
P (x, y) = −0.3x2 − 0.5xy − 0.4y 2 + 85x + 125y − 2500. The company can produce exactly 30000 sets
each year. How many Deluxe and how many Standard sets should be produced each year in order to
maximize annual profit?.
Exercise 22. (Hoffmann, Section 7.5, Question 27) When x thousand dollars are spent on labor and
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y thousand on equipment, the output of a certain factory is Q(x, y) = 60x 3 y 3 units. Suppose 120000
dollars is available for labor and equipment. How should the money be allocated between labor and
equipment to generate the largest possible output?
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ANSWER
Z
Answer 1.
First solution: Net change. C(30) = C(10) +
Z 30
3q 2 − 24q + 48 dq = 22360.
10
3
30
10
C 0 (q)dq
= 5000 +
2
Z
Second solution: C(q) =
C 0 (q)dq =
Z
ANSWER
3q 2 − 24q + 48 dq = q 2 − 12q 2 + 48q + C. The cost is
5000 dollars for producing 10 units: C(10) = 5000 ⇒ C = 4720. When 30 units are produced, the cost
is C(30) = 22360.
766
dollars.
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Answer 2.
Z
2
Answer 3. Net change theorem: M (2) − M (1) =
0
Z
2
0.5e0.2t dt ≈ 0.6761. Note
M (t)dt =
1
1
that the initial condition is not
Z given.
0.5 0.2t
5 0.4
Second solution. M (t) = M 0 (t)dt =
e
+ C ⇒ M (2) − M (1) =
e − e0.2 ≈ 0.6761.
0.2
2
Conclusion: The mass increases by 0.6761 grams during the second hour.
Answer 4. 2626, 26 telephones.
Answer 5.
Z
(a) Net change theorem: C(5) − C(2) =
5
Z
6q 2 − 20q + 500 dq = 1524
M C(q)dq =
2
2
(b) C(q) =
5
Z
M C(q)dq = 2q 3 − 10q 2 + 500q + K.C(3) = 1500 ⇒ K = 36 ⇒
Z
10
C(q)dq ≈ 27026.67.
0
Answer 6. (a) 214.4 dollars (b) 4440.89
Answer 7.
(a) The
√ equilibrium price corresponds to the 2point where supply equals demand: D(q) = S(q) ⇔
245 − 2q = 5 + q ⇔ 245 − 2q = (5 + q) , q + 5 > 0 ⇒ qe = 10 ⇒ pe = 15.
Z 10 p
Z qe
D(q)dq − pe qe =
(b) The consumer’s surplus is CS =
245 − 2qdq − 150 = 3.29 or 3290
0
0
Z 10
Z qe
S(q)dq = 150 −
(5 + q)dq = 50 or 50000
dollars. The producer’s surplus is PS = pe qe −
0
dollars.
0
Answer 8 . Equilibrium: qe = 10. Consumer surplus: 200 dollars. Producer surplus ≈ 116.67 dollars.
Answer 9.
(a) The profit function is P (q) = (110 − q)q − q 3 − 25q 2 + 2q + 3000 = −q 3 + 24q 2 + 108q − 3000
(b) P 0 (q) = −3q 2 + 48q + 108 = 0 ⇒ q = 18(q > 0). Table of sign of P 0 (q) gives q = 18 corresponds
to the maximum profit.
Z 18
(c) When q = 18: p = 92. The consumer’s surplus is CS =
(110 − q)dq − 18 × 92 = 162
0
−2q 3
57q 2
Answer 10. (a) P (q) =
+
+ 120q − 7600 (b) Profit is maximized when q = 20 (c) The
corresponding consumer’s surplus is 400 dollars
Answer 11. The car value y(x) = 40x2 − 1200x + 12000. Initial value: y(0) = 12000 ⇒ C = 3000.
When the car is 15 years old y(15) = 3000.
Answer 12. The requested price p(t) =
14 5 −x
− e .
3
3
Answer 13. P (t) = 400e0.2 sin t .
Answer 14. N (3) ≈ 552 birds.
Answer 15. (Hoffmann, Section 7.2, Question 46). Result:
Answer 16. (Hoffmann, Section 7.2, Question 75). Result:
Answer 17. (Hoffmann, Section 7.3, Question 1 - 22)
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2
ANSWER
(a) (Question 4)
(b) (Question 7)
(c) (Question 9)
Answer 18. (Hoffmann, Section 7.3, Question 23). Result:
Answer 19. (Hoffmann, Section 7.3, Question 27). Result:
Answer 20.
f (8; 7) = −18
(Hoffmann, Section 7.5, Question 9). Critical point: (8; 7), λ = 9. Minimum value
Answer 21. (Hoffmann, Section 7.5, Question 17). 12500 Deluxe sets and 17500 Standard sets
should be produced.
Answer 22. (Hoffmann, Section 7.5, Question 27). 40000 dollars should be spent on labor and
80000 dollars should be spent on equipment.
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TÀI LIỆU
TÀI LIỆU
Tài liệu
[1] Hoffmann, Calculus for business, economics and social life, Brief edition.
[2] James Stewart, Calculus Early Transcendentals, Brooks/Cole, 2012.
[3] Ian Jacques, Mathematics for Economics and business, Fifth editions, Prentice Hall, 2006.
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