Final Test revision - Math for business | Trường Đại học Quốc tế, Đại học Quốc gia Thành phố HCM

MATH FOR BUSINESS 1 Math for Business
Chapter 4: Differentiation
4.3. Marginal function: Marginal revenue: the effect on TR of a change in the value of Q from some existing level (Q thay đổi 1 lượng thì
ảnh hưởng TR như thế nào) MR = � ( � ����) ���� Δ(TR) ≅ MR × ΔQ
Marginal cost: derivative of total cost with respect to output (Q thay đổi 1 lượng thì
ảnh hưởng TC như thế nào) MC = � ( � ��� ) �
����Δ(TC) ≅ MC × ΔQ PRACTICE
1. A company estimates that the marginal revenue (in dollars per unit) realized by
selling x units of a product is 48 – 0.0012x. Assuming the estimate is accurate, find
the increase in revenue if sales increase from 5,000 units to 10,000 units.
3. A firm’s demand function is given by P = 100 - 4√ �� – 3Q
(a) Write down an expression for total revenue, TR, in terms of Q.
(b) Find an expression for the marginal revenue, MR, and find the value of MR when Q = 9.
(c) Use the result of part (b) to estimate the change in TR when Q increases by 0.25
units from its current level of 9 units and compare this with the exact change in TR(*) ĐINH HOÀNG TÚ November 30, 2020 2 MATH FOR BUSINESS Marginal product of Marginal propensity to Marginal propensity labour consume to save Definition Derivative of output Derivative of consumption Derivative of savings with respect to labor with respect to income with respect to income Formula MPL = ���� MPC = ���� MPS = ���� ���� ���� ���� Note Law of diminishing 1 = MPC + MPS returns when MPL’ < 0 PRACTICE
4. A firm’s production function is given by Q = 5√�� − 0.1 �� .
(a) Find an expression for the marginal product of labour, MPL.
(b) Solve the equation MP = 0 and brie L
fly explain the significance of this value of L.
(c) Show that the law of diminishing marginal productivity holds for this function.
4.1. If the consumption function is C = 0.01Y + 0.2Y 2 + 50
a) Calculate MPC and MPS when Y = 30
b) Assume that all your money will be seperated into investment, charity, consumtion and saving with I = 0.04Y + 0.5Y 2 and S = 0.03Y + 0.4Y + 2 20
Find the marginal propensity to charity when Y = 50 4.5. Elasticity : |E| < 1 inelastic 🡪 E
�� �������� ����
��ℎ�������� ���� D = % ℎ
����������
������������ % |E| = 1 unit elastic 🡪 |E| > 1 elastic 🡪 E = ��������
��������x∆��
∆��= ����x���� ����
Lưu ý: Nếu đề cho P theo Q, ta tìm ����
arc elasticity (trung bình cộng) ����
point elasticity (thế điểm có sẵn)
đảo ngược lại sẽ thành ���� ����rồi
Supply hay demand đều cùng công thức: ĐINH HOÀNG TÚ ED < 0 ; ES > 0 November 30, 2020 MATH FOR BUSINESS 3 PRACTICE
5. (a) Find the elasticity of demand in terms of Q for the demand function: P = 20 - 0.05Q
(b) For what value of Q is demand unit elastic?
(c) Find an expression for MR and verify that MR = 0 when demand is unit elastic.
6. If the supply equation is Q = 7 + 0.1P + 0.004P2
find the price elasticity of supply if the current price is 80.
(a) Is supply elastic, inelastic or unit elastic at this price?
(b) Estimate the percentage change in supply if the price rises by 5%
7. A supply function is given by Q = 40 + 0.1P2
(a) Find the price elasticity of supply averaged along an arc between P = 11 and P = 13.
Give your answer correct to 3 decimal places.
(b) Find an expression for price elasticity of supply at a general point, P. Hence:
(b’) Estimate the percentage change in supply when the price increases by 5% from its
current level of 17. Give your answer correct to 1 decimal place.
(b’’) Find the price at which
supply is unit elastic. (*) 4.6. Optimisation of economic functions: Local maximum
Stationary points
(Critical points, Tìm f′′( ) a turning points,
℘ f ″(a) > 0 pt đạt cực tiểu tại 🡪 x = ℘
a f ″(a) < 0 extrema)
🡪 pt đạt cực đại tại x = a ℘ f ″(a) = 0 🡪 không thể
tìm cực đại hay cực tiểu với dữ liệu đề cho ĐINH HOÀNG TÚ Global minimum (Minimum value) Local minimum Global maximum (Maximum value)
Stationary point of inflection
Cách tìm cực đại, cực tiểu:
Bước 1: Giải f′(x) = 0 để tìm cực trị, x = . Bước 2: a November 30, 2020 4 MATH FOR BUSINESS
Cách vẽ đồ thị bậc 3:
Bước 3: Kẻ bảng giá trị (bao gồm: giới hạn, giao
trục tung, trục hoành, cực trị) Bước 1: Tìm TXĐ
Bước 4: Vẽ đồ thị (chú ý: phần < 0 kẻ đứt khúc vì
Bước 2: Tìm sự biến thiên của hàm số ( giới hạn
trong kinh tế không có Q < 0, L < 0,…)
– cực trị – bảng biến thiên)
Average product of labour (labour productivity): measures the average output per worker �� APL = �� At the point of maximum AP L 🡪 L AP = MPL
At the point of maximum profit MR = MC and 🡪 � ( � ��� ) � �� ���� ����< ( ) ���� PRACTICE
8. The supply and demand equations of a good are given by
P = 12QS + 25 and P = -2Q + 50 respectively D . The government decides to
impose a tax, t, per unit. Find the value of t which maximises the government’s
total tax revenue on the assumption that equilibrium conditions prevail in the market
9. The demand and total cost functions of a good are
4P + Q - 16 = 0 and TC = 4 + 2Q -3��2 10+��3 20 respectively.
(a) Find expressions for TR, π, MR and MC in terms of Q.
(b) Determine the value of Q which maximises profit.
10. Daily sales, S, of a new product for the first two weeks after the launch is modelled by
S = t3- 24t2 + 180t + 60 (0 ≤ t ≤ 13)
where t is the number of days.
(a) Find and classify the stationary points of this function. (b) Sketch a graph of against S
t on the interval 0 ≤ t ≤ 13.
(c) Find the maximum and minimum daily sales during the period between t = 5 and t = 9 (*)
11. A firm’s total cost and demand functions are given by
TC = Q2 + 50Q + 10 and P = 200 - 4Q respectively.
(a) Find the level of output needed to maximise the firm’s profit.
(b) The government imposes a tax of $t per good. If the firm adds this tax to its costs ĐINH HOÀNG TÚ November 30, 2020 MATH FOR BUSINESS 5
and continues to maximise profit, show that the price of the good increases by two-fifths
of the tax, irrespective of the value of t. (*)
12. The demand functions for a firm’s domestic and foreign markets are P1 = 50 - 5Q1 P2 = 30 - 4Q2
and the total cost function is TC = 10 + 10Q
where Q = Q + 1
Q2. Determine the prices needed to maximise profit (a) with price discrimination
(b) without price discrimination . (*)
Chapter 5: Partial differentiation
5.1. Functions of several variables: �� = �� ��( ,
�� ��) = ��2��3 + 3��3 + 6� � + ��+ 4 ��� � =���� ����� � =��2�� 1 2 ���
� = 2����3 + 9��2 + �� ���� = 2��3 + 18 �� ��� � =���� ����� � =��2�� ���
� = 3��2��2 + 6 −����2 2 2��
���� = 6��2�� + �� ��3 ��� � = ����� � =��2��
Small increments formula: Implicit differentiation: if ( �� ��, ) = constant then �� ���� 1 �������
� = 6����2 − ��2 ∆z =���� ����=−���� ��� � ∆x +���� ���� ���
� ∆y = ����∆x + ����∆y
Ex: Find the derivatives of y + 2xy 3 − x = 5 2
Ý tưởng: Xem vế trái là 1 phương trình = �� ( �� ��, )
�� = 5 (constant) 🡪 dùng công thức PRACTICE
1. Find expressions for all first- and second-order partial derivatives of the following
functions. In each case verify that:��2�� ������� � = ��2�� ������� � (∗) ĐINH HOÀNG TÚ November 30, 2020 6 MATH FOR BUSINESS a) �� = ���� b) �� = c) �� = ��2 + 2 �� + �� d) 2
e) �� =���� +���� ���� �� = 16 �� ��1/4��3/4
2. Use the small increments formula to estimate the change in
z = x2y4- x6 + y 4 when
(a) x increases from 1 to 1.1 and y remains fixed at 0
(b) x remains fixed at 1 and y decreases from 0 to -0.5
(c) x increases from 1 to 1.1 and y decreases from 0 to -0.5.
3. Find the value of ����
����at the point (-2, 1) for the function which is defined implicitly by
x2y –����= 6
5.2. Partial elasticity and marginal functions:
Q = f(P, PA Y , )
Price elasticity of demand Cross-price elasticity of demand Income elasticity of demand Formula E
�� ��������
�� ��������
�� �������� P = % ℎ EPA= % ℎ EY = % ℎ ��� � �� ��� � �� ��� � �� % % ℎ
�� �������� % ℎ
�� �������� ℎ
�� �������� ���� �� ���� �� �� ���� �� = ���� = ����x���� = ����x���� ���� ���� ��x���� ������ Note |EP| < 1 : inelastic EPA< 0: complementary E� � < 0: inferior goods |EP| = 1 : unit elastic goods EPA> 0: substitute E� � > 0: normal goods |EP| > 1 : elastic goods E� � > 1: superior goods
Utility: Satisfaction of buying goods U = U (��1, ��2) Marginal utility of xi:
▪ With change: ∆U =����
������∆��� � =����
����1∆��1 +���� ����2∆��2 + ⋯ ▪ At point:���� ����1and ���� ����2
Ex: U (20, 5) = 25, U (15, 10) = 30 means that buying 15 of G1 and 10 of G2 will satisfy
customer more that 20 of G1 and 5 of G2 ĐINH HOÀNG TÚ November 30, 2020 MATH FOR BUSINESS 7
Law of diminishing utility: the higher x the lower satisfied rate of 1 unit c 🡪 hange in
x (therefore, the indifference curve commonly has downward-slope) Indifference
curve: at A, B people have the same satisfaction. U = U (x 0 , y 0 ) 0
Marginal rate of commodity substitution (MRCS):
Marginal rate of technical substitution (MRTS):
increase in G needed to maintain utility if G 2 1 decrease by 1 unit PRACTICE 4. Given the demand function
Q = 200 - 2P - PA + 0.1Y 2
where P = 10, PA = 15 and Y = 100, find
(a) the price elasticity of demand
(b) the cross-price elasticity of demand
(c) the income elasticity of demand. MRCS = −����2 ���� ���� ����1= / 1 / ���� ����2
Estimate the percentage change in demand if PA rises by 3%. Is the alternative good
substitutable or complementary? 5. Given the utility function U = �� 1/2 1 ��21/3
determine the value of the marginal utilities ĐINH HOÀNG TÚ November 30, 2020 8 MATH FOR BUSINESS ����1and ���� ���� ����2 at the point (25, 8). Hence
(a) estimate the change in utility when x and 1
x both increase by 1 unit (b) find the mar 2 ginal rate of commodity substitution at this point. 6. Evaluate MP and MP K
L for the production function Q = 2LK + √ ��
given that the current levels of K and L are 7 and 4, respectively. Hence (a) write down the value of MRTS
(b) estimate the increase in capital needed to maintain the current level of output given a 1 unit decrease in labour.
5.3. Unconstrained optimisation: = �� ��( ,
�� ��) = ��2��3 + 3��3 + 6�� +����+ 4 (thường là �� , TR, TC – quy ra 2
biến) Bước 1: Giải ����, ����, ������, ������, ������ Bước 2: Xét ��� � = 0, ���
� = 0 🡪 Stationary points
Bước 3: Xét ∆ = ������������ − ������2 ֍ Nếu > 0 ∆ o ����� � < 0, ����� � < 0 🡪 maximum o ����� � > 0, ����� � > 0 🡪 minimum ֍ Nếu < 0 ∆ saddle point 🡪 ĐINH HOÀNG TÚ November 30, 2020 MATH FOR BUSINESS 9 PRACTICE
7. A monopolist sells its product in two isolated markets with demand functions
P1 = 32 - Q1 and P2 = 40 - 2Q2
The total cost function is TC = 4Q where Q = Q1 + Q2
Find the values of Q and 1
Q which maximise profi t and calcula 2 te the value of the maximum profit.
8. A firm’s production function is given by
Q = 2L1/2 + 3K1/2
where Q, L and K denote the number of units of output, labour and capital. Labour
costs are $2 per unit, capital costs are $1 per unit and output sells at $8 per unit. Show that the profit function is
π = 16L1/2 + 24K1/2– 2L – K (*)
9. An additional cost of $50 per unit is incurred by a firm when selling to its non-EU
customers compared to its EU customers. The demand function is the same in both markets and is given by 20P + Q = 5000
and the total cost function is given by TC = 40Q + 2000
where Q is total demand.
Find the maximum profit with price discrimination. (*)
10. A firm is able to sell its product in two different markets. The corresponding demand functions are P1 + 2Q1 = 100
2P2 + Q2 = 2a
and the total cost function is TC = 500 + 10Q where Q = Q + 1 Q and 2 is a positive const a ant.
Determine, in terms of , the prices needed to maximise profit a (a) with price discrimination
(b) without price discrimination. ĐINH HOÀNG TÚ November 30, 2020 10 MATH FOR BUSINESS
Show that the profit with price discrimination is always greater than the profit without
discrimination, irrespective of the value of (*) a.
5.4. Constrained optimisation:
Objective function (��): a function subjects to constraintss
With ��(��, ��) and ��(��, ��) = �� 5.4.1. If
�� is simple – linear equations (ax + by = M) �� ���� 🡪 �� = − ��🡪 replace �� in the function ( �� ��, ) �� to become ��( ) ��
Find ��′(��)= 0 🡪 x0, y0
Find ��′′(��0) (x) to confirm that stationary point is a maximum or a minimum PRACTICE
11. A firm’s production function is given by
Q = 10K1/2L1/4
Unit capital and labour costs are $4 and $5 respectively and the firm spends a total of
$60 on these inputs. Find the values of and K
L which maximise output. (*) 5.4.2. If If
�� is not simple – nonlinear equations (we use Larrange
multiplier) Bước 1: Tìm 1 phương trình mới
g(x, y, λ) = f(x, y) + λ[ - φ( M x y , )]
Bước 2: Giải các phương trình sau đồng thời ���� ��� � = 0; ���� ��� � = 0, ���� � λ � = 0
Bước 3: Giải các biến số và tìm yêu cầu đề PRACTICE
12. An individual’s utility function is given by U = ��1√��2
where ��1and ��2 denote the monthly consumption of two goods. Unit prices of
these goods are $2 and $4 respectively, and the total monthly expenditure on these goods is ĐINH HOÀNG TÚ November 30, 2020 MATH FOR BUSINESS 11 $300.
(a) Write down the budgetary constraint.
(b) Show that if the budgetary constraint is satisfi ed the maximum value of U is 500
and write down the corresponding values of x1 and x2. Verify that the stationary point is a maximum.
13. A monopolistic producer of two goods, G and G 1
, has a joint total cost function 2 TC = 10Q + 1 Q1Q + 10 2 Q2 where Q and 1
Q denote the quantities of G 2 and G 1 respectively 2 . If P and 1 P denote 2
the corresponding prices then the demand equations are
P1 = 50 - Q1 + Q2
P2 = 30 + 2Q1 - Q2
Find the maximum profit if the firm is contracted to produce a total of 15 goods of
either type. Estimate the new optimal profit if the production quota rises by 1 unit. Chapter 6: Integration
6.1. Indefinite integration:
Integral (anti – derivative or primitive): ( �� ) if �� ��′(��) = ( �� ��): nguyên hàm ∫ ��� � d� � =1 � � + 1����+1 + �� + �� 1 ∫ ��d�� = ln + �� ∫ �� ����� � d� � PRACTICE 1 PRACTICE
= �������� + ��
1. (a) Find the total cost if the marginal cost is MC = Q + 5 and fixed costs are
20. (b) Find the total cost if the marginal cost is MC = 3e0.5Q and fixed costs are 10.
2. Find the short-run production functions corresponding to each of the following marginal product of labor functions: a) 1000 – 3L b) 2 6√��− 0.01 3. Find: 6 a) ∫ ��(��5 − 2)d
�� b) ∫ ��10 − 3√�� + ��−� � d
�� c) ∫ ��3 −5�� +2��− 4��−4��d �� ĐINH HOÀNG TÚ November 30, 2020 12 MATH FOR BUSINESS
6.2. Definite integration: ��0 CS = ∫ ( �� ����)d �� − ��0 0 ��
Definite integral: tích phân ∫ ( �� )d �� �� �� Limits of integration: a, b
Applicability of definite integration: 🡪 Consumer’s surplus 🡪 Producer’s surplus 🡪 Investment flow 🡪 Discounting ��0 ��0
PS = ��0��0 − ∫ ��(����)d �� 0
Consumer’s surplus Producer’s surplus Net Discounting: ��
investment (I): Rate of change of capital stock K �� �� �� −��� / � 100 �� =d�� 2 = ∫ e d�� d��∫ ��( )d �� �� 0 ��1 PRACTICE
4. Find the consumer’s surplus at P = 5 for the following demand functions:
(a) P = 25 – 2Q (b) P = 10 √��
5. Find the producer’s surplus at Q = 9 for the following supply functions:
(a) P = 12 + 2Q (b) P = 20√ + 15 �� ĐINH HOÀNG TÚ November 30, 2020 MATH FOR BUSINESS 13
6. If the net investment function is given by I(t) = 100e0.3t calculate
(a) the capital formation from the end of the second year to the end of the fifth
year (b) the number of years required before the capital stock exceeds $100 000. Chapter 7: Matrices
7.1. Basic matrix operations: Entry Column A = [ 1 7 3 4 5 6] Matrix order 2x3 Row
֍ Transposition: AT = B (transfer column with rows)
��12: row 1, column 2 of A 🡪 ��12= 3 5
Row vector: only 1 row C vector: only 1 column D = [ 1 5 6]B = [6 2 1 4 2 = [5 2 1 −4] Column ] A = [7 3 4 0 4 4] ֍ if C = Addition and subtraction: A + B or C = B + A then C = [7 + 6 3 + 2 4 + 1 1 + 0 5 + 4 6 + 4] = [13 5 5 1 9 10]
Note: to add or subtract, 2 matrices must have the same order (m x n)
Zero matrix: all entries are 0 ֍ tích vô hướng Scalar multiplication:
Ex: To find 12A, we multiply each entry with 12 🡪 12A = [7��12 3 12 4 �� 12 �� 1� 12 � 5� 12 � 6��12] = [84 36 48 12 60 72] ✔ ��( + �� ) = �� + ���� ����
✔ ��(����) = (����)��
֍ Matrix multiplication: just have C = AB when A: m x s and B: s x ; then C: n
m x n ĐINH HOÀNG TÚ November 30, 2020 14 MATH FOR BUSINESS
Note: Row x Column, Multiply 🡪 Plus AB ≠ BA PRACTICE
1. A firm manufactures three products, P1, P2 and P3, which it sells to two
customers, C1 and C2. The number of items of each product that are sold
to these customers is given by
The firm charges both customers same price for each product according to P1 P2 P3 B = [100 500 200]T
To make each item of type P1, P2 and P3, the firm uses four raw materials, R1, R2,
R3 and R4. The number of tonnes required per item is given by
The cost per tonnes of raw materials is In addition, let E = [1 1]
Find the following matrix products and give an interpretation of each one.
(a) AB (b) AC (c) CD (d) ACD (e) EAB (f) EACD (g) EAB – EACD
2. A firm orders 12, 30 and 25 items of goods G1, G2 and G3. The cost of each item of G1, G2
and G3 is $8, $30 and $15 respectively.
a) Write down suitable price and quantity vectors, and use matrix multiplication to work
out the total cost of the order.
b) Write down the new price vector when the cost of G1 rises by 20%, the cost of G2
falls by 10% and the cost of G3 is unaltered. Use matrix multiplication to work out
the new cost of the order and hence find the overall percentage change in total cost. ĐINH HOÀNG TÚ November 30, 2020 MATH FOR BUSINESS 15
3. Two matrices A and B are given by
and if A = B find the values of , a , b c and . (*) d 4. If
find the matrix X which satisfies the matrix equation: 2A + XT = 3B 7.2.
Matrix inversion: (just apply for square matrix)
7.2.1. 2x2 Matrix: Identity: I where I = [1 0 0 1])
✔ AI = A and IA = A or A-1A = I and AA-1 = I For A = [�� �� �
� ��] 🡪 A-1 = 1 det (A) or |A| or |�� �� ��� − � ���� [ − �� �� −� � ��] Determinant:
(A) ≠ 0 🡪 non - singular �� � |
� = ad – bc ❖ If det (A) = 0 🡪 singular ❖ If det
To find x, y through simultaneous equations: + ���� ���� = �� �� + ����
���� = �� �� ��] x = [ ��] b = �� [ ��]
can be written as Ax = b where then we can solve x = A- 7.2.1. 3x3 Matrix: 1 0 0 1b �� = [�� �� Identity: I = [ 0 1 0 0 0 1 ] [
To find cofactor: cross the vertical andThe cofactor is the determinant of the + − + − + − + − +
horizontal line through that entry ]
2 x 2 matrix not lined and the sign Ex: A = [ 3 6 2 1 5 3 4 3 2 ], find A , 23 A ? 31 A23 = − |3 6
4 3| = − (3x3 – 4x6) = 15 A31 = + |6 2 5 3| = + (6x3 – 2x5) = 8
To find determinant of a 3 x 3 matrix: multiply entries in any one row or column by their
corresponding cofactors and adding together. ĐINH HOÀNG TÚ November 30, 2020 16 MATH FOR BUSINESS
Ex: With previous A, find det (A)
det (A) = a11A11 + a21A21 + a A
31 31 = 3 x 1 + 1 x (-6) + 4 x 15 = 57 A11 = + |5 3 3 2| = + (5x2 – 3x3) = 1 A21 = − |6 2
3 2| = − (6x2 – 3x2) = −6 Adjugate matrix Transposition of cofactors For A = [ ] �� ] 🡪 A-1 = 1|� |
� [��11 ��21 ��31
11 ��12 ��13 ��21 ��22 ��13 ��23 ��33
��23 ��31 ��32 ��33 ��12 ��22 ��32
To find inversion of a matrix 3x3:
1) Find cofactors of 3 entries 2) Find det
A 🡪 ≠ 0 or = 0 3) Find 6 other cofactors PRACTICE 5. ĐINH HOÀNG TÚ 6.
4) Transpose the cofactors to have adjugate matrix
5) Finish the scalar multiplication November 30, 2020 MATH FOR BUSINESS 17 7. 8.
Use matrices to solve the following pairs of simultaneous equations:
(a) 3x + 4y = -1 (b) x + 3y = 8
5x - y = 6 4x - y = 6 5 −2 7
9. Find the cofactor of the matrix [ 6 1 −9 4 −3 8 ]
10. Find (where possible) the inverse of the
matrix Are these matrices singular or non- singular?
7.2. Cramer’s rule: (apply for any n n matrix) x ��
column of A by the right-hand-side vector b. � � = det(����) det( ) �� where Ai is the x n matrix found by replacing the n ith
Ex: Solve the system of equations
Using Cramer’s rule to find x1
A is the coefficient matrix:
A1 is constructed by replacing the first column of A by the right-hand-side vector which gives A = 1 det (A ) 1 = | 9 2 3 −9 1 6 13 7 5 7 5| – 2|−9 6
13 7| = 9(-37) – 2(–123) + 3 (–76) = – 315 | = 9|1 6 13 5| + 3|−9 1 det (A) = … = -63 ĐINH HOÀNG TÚ November 30, 2020 18 MATH FOR BUSINESS ��1 =det(����) det(� ) � =−315 −63= 5 PRACTICE 11.
Chapter 8: Linear Programming
8.1. Graphical solution of linear programming problems: Step 1: Sketch the feasible region
Step 2: Identify the corners of the feasible region and find their coordinates.
Step 3: Evaluate the objective function at the corners and choose the one which has the maximum or minimum value.
Ex: Solve the linear programming problem ĐINH HOÀNG TÚ November 30, 2020 MATH FOR BUSINESS 19
Maximize 5x +3y
Subject to 2x + 4y ≤ 8 x ≥ 0 y ≥ 0 Step 1: The non-
negativity constraints x ≥ 0 and y ≥ 0 indicate that the region is bounded by the coordinate axes in the positive quadrant.
The line 2x + 4y
= 8 passes through (0, 2) and (4, 0). Also, at the test
point (0, 0) the inequality 2x + 4y ≤ 8 is true ( 0 ≤ 8).
The feasible region is sketch below:
Step 2: The feasible region has 3 corners, (0, 0), (0, 2) and (4, 0). Step 3:
Corner Objective function (0,0) 5(0) + 3(0) = 0 Note:
The maximum value of the objective function is 20,
(0,2) 5(0) + 3(2) = 6 (4,0) 5(4) + 3(0) = 20
which occurs when x = 4, 4, y = 0
+ If, in step 3, the maximum (or minimum) occurs at corners then the problem has infinitely 2
many solutions. Any point on the line segment joining these corners, including the two corners
themselves, is also a solution
+ If, there are many corners and 1 corner is not with integer x, y ignore that value + If, there 🡪
is only 1 corner not with integer x, y find 4 points near that value to find min, max + In 🡪
unbounded feasible region, we just can find minimum value of the objective function 8.2.
Application of linear programming: ĐINH HOÀNG TÚ November 30, 2020 20 MATH FOR BUSINESS
Decision variable: the unknowns in a linear programming problem which can be
controlled. (Ex: x = number of bowls, y = number of plates)
Integer programming: a linear programming problem in which the search for solution is
restricted to points in the feasible region with wholenumber coordinates.
Shadow price: the change in the optimal value of the objective function due to a one unit
increase in one of the available resources (nearly like marginal) PRACTICE
1. A manufacturer of outdoor clothing makes wax jackets and trousers. Each jacket requires 1
hour to make, whereas each pair of trousers takes 40 minutes. The materials for a jacket cost $32
and those for a pair of trousers cost $40. The company can devote only 34 hours per week to the
production of jackets and trousers, and the firm’s total weekly cost for materials must not exceed
$1200. The company sells the jackets at a profit of $12 each and the trousers at a profi t of $14 per
pair. Market research indicates that the firm can sell all of the jacketsthat are produced, but that it
can sell at most half as many pairs of trousers as jackets.
(a) How many jackets and trousers should the firm produce each week to maximise profit? (b)
Due to the changes in demand, the company has to change its profit margin on a pair of
trousers. Assuming that the profi t margin on a jacket remains at $12 and the manufacturing
constraints are unchanged, fi nd the minimum and maximum profit margins on a pair of trousers
which the company can allow before it should change its strategy for optimum output.
2. A food producer uses two processing plants, P1 and P2, that operate 7 days a week. After
processing, beef is graded into high-, medium- and low-quality foodstuffs. High-quality beef is
sold to butchers, medium-quality beef is used in supermarket ready-meals and the low-quality
beef is used in dog food. The producer has contracted to provide 120 kg of high-, 80 kg of
medium- and 240 kg of low-quality beef each week. It costs $4000 per day to run plant P1 and
$3200 per day to run plant P2. Each day P1 processes 60 kg of high-quality beef, 20 kg of
medium-quality beef and 40 kg of low-quality beef. The corresponding quantities for P2 are 20
kg, 20 kg and 120 kg, respectively. How many days each week should the plants be operated to
fulfill the beef contract most economically? ĐINH HOÀNG TÚ November 30, 2020 MATH FOR BUSINESS 21 Cách tìm det = máy tính
MODE 6 1: nhập matrix cần tính 🡪 AC Shift 4 🡪
🡪 7 Shift 4 🡪 3 : tìm được det của A, tương tự nếu nhiều matrix thì nhập B, C
bằng cách Shift 4 🡪 2 2, 30 🡪 Cách tìm A cofactor T
ATcofactor sẽ tính = A-1x det (A) ĐINH HOÀNG TÚ November 30, 2020 22 MATH FOR BUSINESS
֍ GOOD LUCK TO YOU ֍ ĐINH HOÀNG TÚ November 30, 2020