Mathematics & Data Analysis for E&BE
(University of Groningen)
Summary 2022-2023
Stuvia: marcellaschrijver
Mathematics
Ch 7. Derivatives in Use ................................................................................................................ 2
Ch 9. Integration .......................................................................................................................... 3
Ch 10. Topics in Financial Mathematics ......................................................................................... 5
Data Analysis
Ch 1. Examining Distributions ....................................................................................................... 7
Ch 2. Examining Relationships .................................................................................................... 11
Ch 3. Producing Data .................................................................................................................. 16
,Tangent line
Just touches the curve at a given point
𝑦 − 𝑓(𝑎) = 𝑓 ′ (𝑎)(𝑥 − 𝑎)
Ch 7. Derivatives in Use
Implicit differentiation
Differentiable function where one variable is a function of the other (𝑦 = 𝑦(𝑥))
- Differentiate each side with respect to 𝑥, using the chain rule
d
[𝑥 2 + 2𝑦 2 = 6]
d𝑥
d[𝑦(𝑥)]2
= 2𝑥 + 2 ⋅ d𝑥
d𝑦 2 d𝑦
= 2𝑥 + 2 ⋅ d𝑦 ⋅ d𝑥
′
= 2𝑥 + 2 ⋅ 2𝑦 ⋅ 𝑦
- Solve for 𝑦’
Differentiating the inverse
If 𝑓 is differentiable and strictly increasing/decreasing in 𝐼, then 𝑓 has an inverse function 𝑔, which is
strictly increasing/decreasing in 𝑓(𝐼)
- Find 𝑥0 , an interior point of 𝐼, at which 𝑓(𝑥0 ) = 𝑦0
- Compute 𝑓 ′ (𝑥) and find 𝑓 ′ (𝑥0 )
1
- If 𝑓 ′ (𝑥0 ) ≠ 0, then 𝑔 has a derivative at 𝑦0 , given by 𝑔′ (𝑦0 ) =
𝑓′ (𝑥0 )
1
If the slope at 𝑃 is 𝑎, then the slope at 𝑄 is 𝑎
, Ch 9. Integration
Indefinite integrals
∫ 𝑓(𝑥)𝑑𝑥 = 𝐹(𝑥) + 𝐶 when 𝐹 ′ (𝑥) = 𝑓(𝑥)
Not one definite function, but a whole class of functions with the same derivative 𝑓
∫ 𝑎𝑓(𝑥)𝑑𝑥 = 𝑎 ∫ 𝑓(𝑥)𝑑𝑥, 𝑎 ≠ 0
∫[𝑓(𝑥) + 𝑔(𝑥)]𝑑𝑥 = ∫ 𝑓(𝑥)𝑑𝑥 + ∫ 𝑔(𝑥)𝑑𝑥
Some important integrals
1
∫ 𝑥 𝑎 𝑑𝑥 = 𝑎+1 𝑥 𝑎+1 + 𝐶, 𝑎 ≠ −1
1
∫ 𝑥 𝑑𝑥 = 𝑙𝑛|𝑥| + 𝐶
1
∫ 𝑒 𝑎𝑥 𝑑𝑥 = 𝑎 𝑒 𝑎𝑥 + 𝐶, 𝑎 ≠ 0
1
∫ 𝑎 𝑥 𝑑𝑥 = 𝑙𝑛𝑎 𝑎 𝑥 + 𝐶, 𝑎 > 0 and 𝑎 ≠ 1
Definite integrals
𝑏
The area under the curve 𝑦 = 𝑓(𝑥), in between 𝑥 = 𝑎 and 𝑥 = 𝑏, is ∫𝑎 𝑓(𝑥)𝑑𝑥 = 𝐹(𝑏) − 𝐹(𝑎)
𝑏
If 𝑓(𝑥) ≤ 0 for all 𝑥 ∈ [𝑎, 𝑏], the area is − ∫𝑎 𝑓(𝑥)𝑑𝑥
𝑏 𝑎
∫𝑎 𝑓(𝑥)𝑑𝑥 = − ∫𝑏 𝑓(𝑥)𝑑𝑥
𝑎
∫𝑎 𝑓(𝑥)𝑑𝑥 = 0
𝑏 𝑏
∫𝑎 𝑎𝑓(𝑥)𝑑𝑥 = 𝑎 ∫𝑎 𝑓(𝑥)𝑑𝑥
𝑏 𝑐 𝑏
∫𝑎 𝑓(𝑥)𝑑𝑥 = ∫𝑎 𝑓(𝑥)𝑑𝑥 + ∫𝑐 𝑓(𝑥)𝑑𝑥
Derivative of the definite integral with respect to the upper limit of integration
d 𝑡
∫ 𝑓(𝑥)d𝑥 = 𝐹 ′ (𝑡) = 𝑓(𝑡)
d𝑡 𝑎
Derivative of the definite integral with respect to the lower limit of integration
d 𝑏
∫ 𝑓(𝑥)d𝑥
d𝑡 𝑡
= −𝐹 ′ (𝑡) = −𝑓(𝑡)
Derivative of the definite integral in general
d 𝑏(𝑡)
∫ 𝑓(𝑥)𝑑𝑥 = 𝑓(𝑏(𝑡))𝑏 ′ (𝑡) − 𝑓(𝑎(𝑡))𝑎′ ((𝑡))
d𝑡 𝑎(𝑡)
Riemann integral
𝑏
∫𝑎 𝑓(𝑥)𝑑𝑥
Subdivide [𝑎, 𝑏] into 𝑛 parts by choosing points 𝑎 = 𝑥0 < 𝑥1 < 𝑥2 < ⋯ < 𝑥𝑛−1 < 𝑥𝑛 = 𝑏
Let Δ𝑥𝑖 = 𝑥𝑖+1 − 𝑥𝑖 , 𝑖 = 0, 1, … , 𝑛 − 1 and choose an arbitrary number 𝜉𝑖 in each interval [𝑥𝑖 , 𝑥𝑖+1 ]
𝑙𝑖𝑚 ∑𝑛−1
𝑖=0 𝑓(𝜉𝑖 )Δ𝑥𝑖 = 𝑓(𝜉0 )Δ𝑥0 + 𝑓(𝜉1 )Δ𝑥1 +. . . +𝑓(𝜉𝑛−1 )Δ𝑥𝑛−1
Number of individuals in [𝒂, 𝒃]
𝑏
𝑁 = 𝑛 ∫𝑎 𝑓(𝑟)𝑑𝑟
𝑟 Income
(University of Groningen)
Summary 2022-2023
Stuvia: marcellaschrijver
Mathematics
Ch 7. Derivatives in Use ................................................................................................................ 2
Ch 9. Integration .......................................................................................................................... 3
Ch 10. Topics in Financial Mathematics ......................................................................................... 5
Data Analysis
Ch 1. Examining Distributions ....................................................................................................... 7
Ch 2. Examining Relationships .................................................................................................... 11
Ch 3. Producing Data .................................................................................................................. 16
,Tangent line
Just touches the curve at a given point
𝑦 − 𝑓(𝑎) = 𝑓 ′ (𝑎)(𝑥 − 𝑎)
Ch 7. Derivatives in Use
Implicit differentiation
Differentiable function where one variable is a function of the other (𝑦 = 𝑦(𝑥))
- Differentiate each side with respect to 𝑥, using the chain rule
d
[𝑥 2 + 2𝑦 2 = 6]
d𝑥
d[𝑦(𝑥)]2
= 2𝑥 + 2 ⋅ d𝑥
d𝑦 2 d𝑦
= 2𝑥 + 2 ⋅ d𝑦 ⋅ d𝑥
′
= 2𝑥 + 2 ⋅ 2𝑦 ⋅ 𝑦
- Solve for 𝑦’
Differentiating the inverse
If 𝑓 is differentiable and strictly increasing/decreasing in 𝐼, then 𝑓 has an inverse function 𝑔, which is
strictly increasing/decreasing in 𝑓(𝐼)
- Find 𝑥0 , an interior point of 𝐼, at which 𝑓(𝑥0 ) = 𝑦0
- Compute 𝑓 ′ (𝑥) and find 𝑓 ′ (𝑥0 )
1
- If 𝑓 ′ (𝑥0 ) ≠ 0, then 𝑔 has a derivative at 𝑦0 , given by 𝑔′ (𝑦0 ) =
𝑓′ (𝑥0 )
1
If the slope at 𝑃 is 𝑎, then the slope at 𝑄 is 𝑎
, Ch 9. Integration
Indefinite integrals
∫ 𝑓(𝑥)𝑑𝑥 = 𝐹(𝑥) + 𝐶 when 𝐹 ′ (𝑥) = 𝑓(𝑥)
Not one definite function, but a whole class of functions with the same derivative 𝑓
∫ 𝑎𝑓(𝑥)𝑑𝑥 = 𝑎 ∫ 𝑓(𝑥)𝑑𝑥, 𝑎 ≠ 0
∫[𝑓(𝑥) + 𝑔(𝑥)]𝑑𝑥 = ∫ 𝑓(𝑥)𝑑𝑥 + ∫ 𝑔(𝑥)𝑑𝑥
Some important integrals
1
∫ 𝑥 𝑎 𝑑𝑥 = 𝑎+1 𝑥 𝑎+1 + 𝐶, 𝑎 ≠ −1
1
∫ 𝑥 𝑑𝑥 = 𝑙𝑛|𝑥| + 𝐶
1
∫ 𝑒 𝑎𝑥 𝑑𝑥 = 𝑎 𝑒 𝑎𝑥 + 𝐶, 𝑎 ≠ 0
1
∫ 𝑎 𝑥 𝑑𝑥 = 𝑙𝑛𝑎 𝑎 𝑥 + 𝐶, 𝑎 > 0 and 𝑎 ≠ 1
Definite integrals
𝑏
The area under the curve 𝑦 = 𝑓(𝑥), in between 𝑥 = 𝑎 and 𝑥 = 𝑏, is ∫𝑎 𝑓(𝑥)𝑑𝑥 = 𝐹(𝑏) − 𝐹(𝑎)
𝑏
If 𝑓(𝑥) ≤ 0 for all 𝑥 ∈ [𝑎, 𝑏], the area is − ∫𝑎 𝑓(𝑥)𝑑𝑥
𝑏 𝑎
∫𝑎 𝑓(𝑥)𝑑𝑥 = − ∫𝑏 𝑓(𝑥)𝑑𝑥
𝑎
∫𝑎 𝑓(𝑥)𝑑𝑥 = 0
𝑏 𝑏
∫𝑎 𝑎𝑓(𝑥)𝑑𝑥 = 𝑎 ∫𝑎 𝑓(𝑥)𝑑𝑥
𝑏 𝑐 𝑏
∫𝑎 𝑓(𝑥)𝑑𝑥 = ∫𝑎 𝑓(𝑥)𝑑𝑥 + ∫𝑐 𝑓(𝑥)𝑑𝑥
Derivative of the definite integral with respect to the upper limit of integration
d 𝑡
∫ 𝑓(𝑥)d𝑥 = 𝐹 ′ (𝑡) = 𝑓(𝑡)
d𝑡 𝑎
Derivative of the definite integral with respect to the lower limit of integration
d 𝑏
∫ 𝑓(𝑥)d𝑥
d𝑡 𝑡
= −𝐹 ′ (𝑡) = −𝑓(𝑡)
Derivative of the definite integral in general
d 𝑏(𝑡)
∫ 𝑓(𝑥)𝑑𝑥 = 𝑓(𝑏(𝑡))𝑏 ′ (𝑡) − 𝑓(𝑎(𝑡))𝑎′ ((𝑡))
d𝑡 𝑎(𝑡)
Riemann integral
𝑏
∫𝑎 𝑓(𝑥)𝑑𝑥
Subdivide [𝑎, 𝑏] into 𝑛 parts by choosing points 𝑎 = 𝑥0 < 𝑥1 < 𝑥2 < ⋯ < 𝑥𝑛−1 < 𝑥𝑛 = 𝑏
Let Δ𝑥𝑖 = 𝑥𝑖+1 − 𝑥𝑖 , 𝑖 = 0, 1, … , 𝑛 − 1 and choose an arbitrary number 𝜉𝑖 in each interval [𝑥𝑖 , 𝑥𝑖+1 ]
𝑙𝑖𝑚 ∑𝑛−1
𝑖=0 𝑓(𝜉𝑖 )Δ𝑥𝑖 = 𝑓(𝜉0 )Δ𝑥0 + 𝑓(𝜉1 )Δ𝑥1 +. . . +𝑓(𝜉𝑛−1 )Δ𝑥𝑛−1
Number of individuals in [𝒂, 𝒃]
𝑏
𝑁 = 𝑛 ∫𝑎 𝑓(𝑟)𝑑𝑟
𝑟 Income
