Intermediate Microeconomics
Instructor: Sandhya Patlolla
Assignment 1
1) Calculate the derivatives of following functions with respect to x.
𝑥
(𝑥+1)
(i)
𝑦=
(ii)
𝑦 = 𝑓(𝑥) = 𝑥 3 + (1 + 𝑥 2 )
(iii)
𝑦 = (𝑥 3 + 2)2 + (1 + 𝑥 2 )3
(iv)
𝑦 = 𝑓(𝑥) =
𝑦=
(v)
(1+𝑥)
𝑥2
3
1+𝑥
𝑥2
+(2𝑥 + 1)2
2) Calculate the partial derivatives of following functions with respect to 𝑥1 and with respect to 𝑥2 .
(i)
𝑦 = 𝑓(𝑥1 , 𝑥2 ) = 3𝑥1 + 5𝑥2
(ii)
𝑦 = 𝑓(𝑥1 , 𝑥2 ) = 𝑥12 + (2𝑥1 + 1)3 + 3𝑥22 + 𝑥1 . 𝑥23 .
(iii)
𝑦 = 𝑓 (𝑥1 , 𝑥2 ) =
(iv)
𝑦 = 𝑓(𝑥1 , 𝑥2 ) = 𝑥10.75 𝑥20.25
(v)
𝑦 = 𝑓(𝑥1 , 𝑥2 ) = (𝑥10.5 𝑥20.5 )2
(vi)
𝑦 = 𝑓(𝑥1 , 𝑥2 ) = 4𝑥10.5 + 𝑥2
(vii)
𝑦 = 𝑓 (𝑥1 , 𝑥2 ) = 𝑥1 𝑥2 + 𝑥22 + 𝑥13
𝑥13
(1+𝑥1 )𝑥2
3) Solve the following optimization problem and find the optimal values of x1 and x2 .
Maximize 𝑦 = 𝑓(𝑥1 , 𝑥2 ) = 100𝑥1 + 20𝑥2 − 5𝑥12 𝑥2 .
4) Solve the following optimization problem and find the optimal values of x1 and x2 .
𝟎.𝟐𝟓
Maximize 𝒚 = 𝒇(𝒙𝟏 , 𝒙𝟐 ) = 𝒙𝟎.𝟕𝟓
with the constraint of 2 x1 + x2 = 15 .
𝟏 𝒙𝟐