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

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Môn: Math for business

Trường: Trường Đại học Quốc tế, Đại học Quốc gia Thành phố Hồ Chí Minh

Thông tin:

28 trang 1 tháng trước

Tác giả:

Profile Mai Nguyệt
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Email: vuthithutrang2003@gmail.com
1
1
MATH FOR BUSINESS TA: THU TRANG VŨ THỊ
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
2
MATH FOR BUSINESS TA: THU TRANG VŨ THỊ
P = −4Q
1
3
𝑄
𝑠
---
P = −4Q + 120
1
3
𝑄
Hence, −4Q + 120 =
1
3
𝑄
1
3
– 𝑄𝑠
𝑄𝑠
P = −4Q + 120
𝑄
Hence, −4Q + 120 = 13𝑄
• Customer pay = –
• Supplier pay = –
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3
3
MATH FOR BUSINESS TA: THU TRANG VŨ THỊ
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
4
MATH FOR BUSINESS TA: THU TRANG VŨ THỊ
Chapter 2:
Non-Linear Equations
1. Sketch the graph
Đỉ −
𝑏
2𝑎
;−
∆
4𝑎
) ∆= 𝑏
2
− 4𝑎𝑐
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5
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MATH FOR BUSINESS TA: THU TRANG VŨ THỊ
2. Revenue, cost and profit
ổ
ổ
𝑻𝑪
𝑸
𝑭𝑪+𝑽𝑪(𝑸)
𝑸
𝑭𝑪
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 − =−24−𝑄 𝑄 𝑄−
𝜋 − 𝑄 𝑄−
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MATH FOR BUSINESS TA: THU TRANG VŨ THỊ
𝑄
1
2
𝜋′=0 𝑎𝑛𝑑 𝜋′′
𝜋 𝑄 𝑎𝑛𝑑 𝜋′= − ′′= −
𝜋 𝑄 𝜋 𝑥′= −
9
4
𝑥
9
4
−
49
8
3. Indices and logarithms
𝑓 𝐾 𝐿(𝜆𝐾,𝜆𝐿 𝜆)=
𝑛
𝑓( , )
• 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
7
MATH FOR BUSINESS TA: THU TRANG VŨ THỊ
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
8
MATH FOR BUSINESS TA: THU TRANG VŨ THỊ
x
13
100
𝑥
6.5
13
100
124
1+
63
100
𝑥
100
𝑥
– – ếu đáp
ỏ hơn 1 thì lấ ừ đi đáp án để tính đượ ả
–
2. Application
2.1. Index numbers
𝑟𝑒𝑎𝑙 𝑣𝑎𝑙𝑢𝑒 𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛 𝑦𝑒𝑎𝑟𝑖𝑛 𝑜𝑛
𝑣𝑎𝑙𝑢𝑒 𝑏𝑎𝑠𝑒 𝑦𝑒𝑎𝑟𝑖𝑛
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9
9
MATH FOR BUSINESS TA: THU TRANG VŨ THỊ
𝐼𝑛𝑑𝑒𝑥 𝑥
𝐼𝑛𝑑𝑒𝑥 𝑦
𝑆𝐹 𝑥
SF
y
𝑣𝑎𝑙𝑢𝑒 𝑥
value y
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10
10
MATH FOR BUSINESS TA: THU TRANG VŨ THỊ
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
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MATH FOR BUSINESS TA: THU TRANG VŨ THỊ
QUY ƯỚ
•
•
•
•
•
o
o
o
o
o
( “interest on the interest” – ề ỗ ẽ
đư ộ ồợ ốn để ế ụ
𝒓%
𝒌
n: tính theo năm
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12
12
MATH FOR BUSINESS TA: THU TRANG VŨ THỊ
EIR/AER
EIR/AER
EIR/AER
EIR/AER EIR/AER
= (1 +
= (1 +
= (1 +
= (1 += (1 +
𝒓
𝒌
)
)
)
))
k
k
k
kk
–
–
–
––
1
1
1
1 1
= e
= e
= e
= e= e
rn
rn
rn
rnrn
–
–
–
––
1
1
1
1 1 (continuos compound interest)
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13
13
MATH FOR BUSINESS TA: THU TRANG VŨ THỊ
ấ ố ộ
ấ ố
ộ
1,...→ Ratio:
a, ar, ar2, ar3,... → Ratio: r
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14
14
MATH FOR BUSINESS TA: THU TRANG VŨ THỊ
𝑎(
𝑟
𝑛
− 1
𝑟 − 1
)
ế ệ → sinking fund
+ At the beginning of the periods (thườ ử ụ ầ ế ậ
𝑉 A(1 +
𝑛
= 𝑖)
(1+ ) −1𝑖
𝑛
𝑖
Or =
𝐴(1+
𝑟%
𝑘
)
(1+
𝑟%
𝑘
)
𝑘𝑛
−1
𝑟%
𝑘
𝑉
𝑛
= A
(1+ −1𝑖)
𝑛
𝑖
Or =
𝐴
(1+
𝑟%
𝑘
)
𝑘𝑛
−1
𝑟%
𝑘
ế ề ửi vô đầ ỗi năm nhưng compounded khác annually thì phả
ồ ớ ử ụ ứ
𝑉 𝐴
𝑛
= (1+
𝑟%
𝑘
)
(1+
𝑟%
𝑘
)
𝑘𝑛
−1
𝑟%
𝑘
1000(1+
8
100
)
(1+
8
100
)
10
−1
8
100
=
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15
15
MATH FOR BUSINESS TA: THU TRANG VŨ THỊ
𝑉𝑛 𝐴(1+
𝑟%
𝑘
)
(1+
𝑟%
𝑘
)
𝑘𝑛
−1
𝑟%
𝑘
1000(1+
8
100
)
(1+
8
100
)
𝑥
−1
8
100
=
𝑛 𝑦𝑒𝑎𝑟𝑠
ợ → annuity
(thườ ử ụ ầ ế ậ
𝑉
0
= A
1−(1+ )𝑖
−𝑛
𝑖
Or =
𝐴
1− 1+(
𝑟%
𝑘
)
−𝑘𝑛
𝑟%
𝑘
𝑉
0
= A(1 + i)
1−(1+ )𝑖
−𝑛
𝑖
Or =
𝐴(1+
𝑟%
𝑘
)
1− 1+(
𝑟%
𝑘
)
−𝑘𝑛
𝑟%
𝑘
𝑉
0
=𝐴
1−
(
1+𝑟%
)
−𝑛
𝑟%
𝐴
1−(1+
1
100
)
−12
1
100
𝐴
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16
16
MATH FOR BUSINESS TA: THU TRANG VŨ THỊ
ộ ố ứ
Interest tháng đầ ố ề mượ ấ
ố
–
ổ ổ – ố ền ban đầ
(sau m năm vẫ ợ
L’= − A
L(1 + r%)
m
(1 + 𝑟%)
𝑚
− 1
r%
ếu đề compounded annually nhưng hỏ
ớ
IV. Investment Appraisal
ị
𝑁𝑃𝑉
𝑁𝑃𝑉
𝑁𝑃𝑉
𝑁𝑃𝑉𝑁𝑃𝑉
= PV- P
= PV- P
= PV- P
= PV- P= PV- P
0
0
0
00
𝑺(𝟏 +
𝒓
𝟏𝟎𝟎
)
−𝒕
−
𝑖𝑛𝑖𝑡𝑖𝑎𝑙
𝑖𝑛𝑖𝑡𝑖𝑎𝑙
𝑖𝑛𝑖𝑡𝑖𝑎𝑙
𝑖𝑛𝑖𝑡𝑖𝑎𝑙𝑖𝑛𝑖𝑡𝑖𝑎𝑙
𝑖𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡
𝑖𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡
𝑖𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡
𝑖𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡𝑖𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡
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17
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MATH FOR BUSINESS TA: THU TRANG VŨ THỊ
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 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑜𝑢𝑡𝑙𝑎𝑦
−
𝑡
−
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18
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MATH FOR BUSINESS TA: THU TRANG VŨ THỊ
− − − −
−
− −
𝑁𝑃𝑉 𝑆 𝑟
−
𝑡
−𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑜𝑢𝑡𝑙𝑎𝑦
− − − −
−
− −
ỉ ấ ờ ộ ộ
𝑁𝑃𝑉
𝑁𝑃𝑉
𝑁𝑃𝑉
𝑁𝑃𝑉𝑁𝑃𝑉
𝑆
𝑆
𝑆
𝑆𝑆
𝑰𝑹𝑹
𝟏𝟎𝟎
−
𝑡
𝑡
𝑡
𝑡𝑡
−
𝑖𝑛𝑖𝑡𝑖𝑎𝑙
𝑖𝑛𝑖𝑡𝑖𝑎𝑙
𝑖𝑛𝑖𝑡𝑖𝑎𝑙
𝑖𝑛𝑖𝑡𝑖𝑎𝑙𝑖𝑛𝑖𝑡𝑖𝑎𝑙
𝑖𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡
𝑖𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡
𝑖𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡
𝑖𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡𝑖𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡
⋯
⋯
⋯
⋯⋯
𝐼𝑅𝑅
100
−
𝐼𝑅𝑅
100
−
𝐼𝑅𝑅
100
−
𝐼𝑅𝑅
100
−
−
𝐼𝑅𝑅
Chapter 4:
Differentation
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19
19
MATH FOR BUSINESS TA: THU TRANG VŨ THỊ
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
20
MATH FOR BUSINESS TA: THU TRANG VŨ THỊ
𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑐ℎ𝑎𝑛𝑔𝑒 𝑑𝑒𝑚𝑎𝑛𝑑𝑖𝑛
𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑐ℎ𝑎𝑛𝑔𝑒 𝑝𝑟𝑖𝑐𝑒
𝑖𝑛
𝑃
𝑄
𝑑𝑄
𝑑𝑃
𝑃
𝑄
∆𝑄
∆𝑃
𝑃1+𝑃2
2
𝑄1+𝑄2
2
∆𝑄
∆𝑃
𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑐ℎ𝑎𝑛𝑔𝑒 𝑠𝑢𝑝𝑝𝑙𝑦𝑖𝑛
𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑐ℎ𝑎𝑛𝑔𝑒 𝑝𝑟𝑖𝑐𝑒
𝑖𝑛
𝑃
𝑄
𝑑𝑄
𝑑𝑃
𝑃
𝑄
∆𝑄
∆𝑃
𝑃1+𝑃2
2
𝑄1+𝑄2
2
∆𝑄
∆𝑃
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MATH FOR BUSINESS TA: THU TRANG VŨ THỊ
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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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Email: vuthithutrang2003@gmail.com 2
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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