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

1 MATH FOR BUSINESS TA: VŨ THỊ THU TRANG Chapter 1: Linear Equations
1. Supply and Demand analysis – 
• Customer pay = P (at new equi’ point – old equi’ point • Supplier pay = tax – • Q’ ’ Khi đề ả ẩ ớ ủ ỗ ả ẩ
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2 MATH FOR BUSINESS TA: VŨ THỊ THU TRANG P = −4Q 1𝑄 3 𝑠 --- P = −4Q + 120 1𝑄 3 1 Hence, −4Q + 120 = 𝑄 3 1 3 – 𝑄𝑠  𝑄𝑠 P = −4Q + 120 𝑄 Hence, −4Q + 120 = 13𝑄 • Customer pay = – • Supplier pay = –
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3 MATH FOR BUSINESS TA: VŨ THỊ THU TRANG ậ ả ị ấ ấ ị ỗi phương trong đó ấ ị ạ để ễ ế ế ẽ ần lượ ệ ệ đườ
ẳng tương ứng đã vẽ ấy giao điể ạ
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4 MATH FOR BUSINESS TA: VŨ THỊ THU TRANG Chapter 2: Non-Linear Equations 1. Sketch the graph Đỉ − 𝑏; − ∆ 2𝑎 ) ∆= 𝑏2 − 4𝑎𝑐 4𝑎
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5 MATH FOR BUSINESS TA: VŨ THỊ THU TRANG
2. Revenue, cost and profit ổ ổ
𝑻𝑪 𝑭𝑪+𝑽𝑪(𝑸) 𝑭𝑪 𝑸 𝑸 𝑸 – π = TR − TC
o Make a profit of a units: π = a units: π = o Breaks even: π = 0 π
𝜋′=0 𝑎𝑛𝑑 𝜋′′<0. function is P = 10 − 2Q
obtain an expression for π in terms of Q. – 𝜋 𝑇𝑅−𝑇𝐶
𝑄− 𝑄 −4−𝑄=−2𝑄 𝑄− 𝜋 − 𝑄 𝑄−
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6 MATH FOR BUSINESS TA: VŨ THỊ THU TRANG 𝑄 1
2 𝜋′=0 𝑎𝑛𝑑 𝜋′ 𝜋′= − 𝑄 𝑎𝑛𝑑 𝜋′′= − 𝜋′= 𝑄 𝜋 − 𝑥 9 𝑥 9− 49 4 4 8
3. Indices and logarithms
𝑓(𝜆𝐾,𝜆𝐿)=𝜆𝑛𝑓(𝐾,𝐿)
• n < 1, the function is said to display decreasing returns to scale.
• n = 1, the function is said to display constant returns to scale.
• n > 1, the function is said to display increasing returns to scale. – α β → homogeneous of α + β α + β 𝑓 𝜆𝐾 𝜆𝐿 𝜆𝐾 𝜆𝐿
𝜆𝐾𝜆 𝐿 𝜆 𝐾𝐿 𝜆 𝑓 𝐾 𝐿
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7 MATH FOR BUSINESS TA: VŨ THỊ THU TRANG 𝑓 𝜆𝐾 𝜆𝐿 𝜆𝐾 𝜆𝐿 𝜆 𝐾 𝐿 𝜆𝑓 𝜆𝐾 𝜆𝐿 Chapter 3: Mathematics of Finance I. Percentage 1. Percentages |𝑥2−𝑥1| 𝑥1 • x – > 0 → Increase • x – < 0 → Decrease ệ ố ỉ ệ
𝑁𝑒𝑤 𝑣𝑎𝑙𝑢𝑒 Old value Đề ắ ị trướ Đề Đề ị Đề Đề
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8 MATH FOR BUSINESS TA: VŨ THỊ THU TRANG x 13 100 𝑥 13 6.5  100 124 1+ 63 100 𝑥 𝑥 100 – – ếu đáp ỏ hơn 1 thì lấ
ừ đi đáp án để tính đượ ả – 2. Application 2.1. Index numbers
𝑟𝑒𝑎𝑙 𝑣𝑎𝑙𝑢𝑒 𝑖𝑛 𝑜𝑛 𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛 𝑦𝑒𝑎𝑟
𝑣𝑎𝑙𝑢𝑒 𝑖𝑛 𝑏𝑎𝑠𝑒 𝑦𝑒𝑎𝑟
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9 MATH FOR BUSINESS TA: VŨ THỊ THU TRANG 𝐼𝑛𝑑𝑒𝑥 𝑥 𝑆𝐹 𝑥 𝑣𝑎𝑙𝑢𝑒 𝑥 𝐼𝑛𝑑𝑒𝑥 𝑦 SF y value y
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10 MATH FOR BUSINESS TA: VŨ THỊ THU TRANG
ếu đề cho bài toán ngượ ầ ngượ ạ Năm bắt đầ ạ ạ Trướ ạ
real values of the employees’ salaries at constant ‘2001 prices’.
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11 MATH FOR BUSINESS TA: VŨ THỊ THU TRANG QUY ƯỚ • • • • • o o o o o
( “interest on the interest” – ề ỗ ẽ đượ ộ ồ ốn để ế ụ 𝒓% 𝒌 n: tính theo năm
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12 MATH FOR BUSINESS TA: VŨ THỊ THU TRANG EIR/AER = (1 +𝒓)k – 1 1 𝒌 = ern r n– 1
1 (continuos compound interest)
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13 MATH FOR BUSINESS TA: VŨ THỊ THU TRANG ấ ố ộ ấ ố ộ 1,...→ Ratio:
a, ar, ar2, ar3,... → Ratio: r
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14 MATH FOR BUSINESS TA: VŨ THỊ THU TRANG 𝑟𝑛 − 1 𝑎( 𝑟 − 1 ) ế ệ → sinking fund
+ At the beginning of the periods (thườ ử ụ ầ ế ậ 𝑉 𝑖 𝑛 𝑛=A(1 + 𝑖) (1+ ) −1 𝑖 𝑘𝑛 ) −1 Or =𝐴(1+𝑟%) (1+𝑟%𝑘 𝑘 𝑟% 𝑘 𝑉 𝑖)𝑛 𝑛= A (1+ −1 𝑖 (1+𝑟%)𝑘𝑛−1 Or =𝐴 𝑘𝑟% 𝑘 ế ề ửi vô đầ
ỗi năm nhưng compounded khác annually thì phả ồ ớ ử ụ ứ 𝑘𝑛 ) −1 )10−1
𝑉𝑛=𝐴(1+𝑟%) (1+𝑟% 𝑘 ) (1+ 8100 = 𝑘 𝑟% 1000(1+ 8 100 8 𝑘 100
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15 MATH FOR BUSINESS TA: VŨ THỊ THU TRANG 𝑘𝑛 ) −1 )𝑥−1
𝑉𝑛 𝐴(1+𝑟%) (1+𝑟%𝑘 1000(1+ 8 ) (1+ 8100 = 𝑘 𝑟% 100 8 𝑘 100 𝑛 𝑦𝑒𝑎𝑟𝑠 ợ → annuity (thườ ử ụ ầ ế ậ 𝑉 𝑖 −𝑛 0= A 1−(1+ ) 𝑖 −𝑘𝑛 1−(1+𝑟% Or =𝐴 𝑘 ) 𝑟% 𝑘 𝑉 𝑖 −𝑛 0= A(1 + i) 1−(1+ ) 𝑖 −𝑘𝑛 1−(1+𝑟%) Or = 𝐴(1+𝑟%) 𝑘 𝑘 𝑟% 𝑘 −12 𝑉 1−(1+ 1 0=𝐴 1−(1+𝑟%)−𝑛 100) 𝑟% 𝐴 1 𝐴 100
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16 MATH FOR BUSINESS TA: VŨ THỊ THU TRANG ộ ố ứ Interest tháng đầ ố ề mượ ấ ố – ổ ổ – ố ền ban đầ (sau m năm vẫ ợ
L’= L(1 + r%)m − A(1 + 𝑟%)𝑚− 1 r% ếu đề
compounded annually nhưng hỏ ớ
IV. Investment Appraisal ị
𝑁𝑃𝑉= PV- P0 𝑺(𝟏 + 𝒓 )−𝒕 𝟏𝟎𝟎
− 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑖𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡
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17 MATH FOR BUSINESS TA: VŨ THỊ THU TRANG
NPV > 0: worth to invest (PV > P0) => acceptable NPV = 0: neutral
NPV < 0: not worth to invest (PV < P0)
involves an initial outlay of $13 500 and yields $18 000 in 2 years’ time. 𝑁𝑃𝑉 𝑆
𝑟 −𝑡 −𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑜𝑢𝑡𝑙𝑎𝑦 − − 𝑁𝑃𝑉 𝑆
𝑟 −𝑡 −𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑜𝑢𝑡𝑙𝑎𝑦 − − 𝑁𝑃𝑉 𝑆 𝑟
−𝑡−𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑜𝑢𝑡𝑙𝑎𝑦
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18 MATH FOR BUSINESS TA: VŨ THỊ THU TRANG − − − − − − − 𝑁𝑃𝑉 𝑆
𝑟 −𝑡 −𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑜𝑢𝑡𝑙𝑎𝑦 − − − − − − − ỉ ấ ờ ộ ộ 𝑁𝑃 𝑁 𝑉 𝑃 𝑆 𝑰𝑹𝑹 −𝑡 𝟏𝟎𝟎
− 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑖𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡 ⋯ 𝐼𝑅𝑅− 𝐼𝑅𝑅− 𝐼𝑅𝑅− 𝐼𝑅𝑅 − 100 100 100 − 100 𝐼𝑅𝑅 Chapter 4: Differentation
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19 MATH FOR BUSINESS TA: VŨ THỊ THU TRANG
Slope (gradient) = Δy Δx= y2−y1 x2−x1 ⇒ − ax − ax 𝑑(𝑇𝑅) 𝑄
∆(𝑇𝑅) ≅ 𝑀𝑅 × ∆𝑄 𝑑(𝑇𝐶) 𝑄
∆(𝑇𝐶) ≅ 𝑀𝐶 × ∆𝑄
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20 MATH FOR BUSINESS TA: VŨ THỊ THU TRANG
𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑑𝑒𝑚𝑎𝑛𝑑 𝑃1+𝑃2 𝑃 𝑑𝑄 𝑃 ∆𝑄 2 ∆𝑄
𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑝𝑟𝑖𝑐𝑒 𝑄 𝑑𝑃 𝑄 𝑄1+𝑄2 ∆𝑃 2 ∆𝑃
𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑠𝑢𝑝𝑝𝑙𝑦 𝑃1+𝑃2 𝑃 𝑑𝑄 𝑃 ∆𝑄 2 ∆𝑄
𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑝𝑟𝑖𝑐𝑒 𝑄 𝑑𝑃 𝑄 𝑄1+𝑄2 ∆𝑃 2 ∆𝑃
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21 MATH FOR BUSINESS TA: VŨ THỊ THU TRANG
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