Question 1

Find the value of the marginal rate of technical substitution for the production function $Q = 300 K ^ { \frac { 2 } { 3 } } L ^ { \frac { 1 } { 2 } }$ when $K = 40 , L = 60$.

Accepted Answer

A)  $\frac { 1 } { 2 }$ 
B)  $\frac { 3 } { 2 }$ 
C)  $\frac { 3 } { 4 }$ 
D)  $\frac { 2 } { 3 }$ 
E)  2 
A)  $\frac { 1 } { 2 }$ 
B)  $\frac { 3 } { 2 }$ 
C)  $\frac { 3 } { 4 }$ 
D)  $\frac { 2 } { 3 }$ 
E)  2 A

Question 2

Evaluate the second- order partial derivative, f23, of the function $f \left( x _ { 1 } , x _ { 2 } , x _ { 3 } \right) = \frac { x _ { 3 } x _ { 2 } ^ { 3 } } { x _ { 1 } } + x _ { 2 } e ^ { x _ { 3 } }$ at the point, (3, 2, 0).

Accepted Answer

A) 1 
B) 3 
C) 4 
D) 2 
E) 5 
A) 1 
B) 3 
C) 4 
D) 2 
E) 5 E

Question 3

Which of the following could represent a function, f(x, y), with first- order partial derivatives?
$\frac { \partial f } { \partial x } = 3 x y ( x y + 2 ) \quad \frac { \partial f } { \partial y } = x ^ { 2 } ( 2 x y + 3 )$

Accepted Answer

A)  $x ^ { 3 } y ^ { 2 } + 3 x ^ { 2 } y - 6$ 
B)  $x y \left( x ^ { 2 } y + 3 \right)$ 
C)  $x ^ { 3 } y ^ { 2 } + 2 x ^ { 2 } y ^ { 3 } + 1$ 
D)  $x ^ { 2 } \left( x y ^ { 2 } + 3 \right)$ 
E)  none of these 
A)  $x ^ { 3 } y ^ { 2 } + 3 x ^ { 2 } y - 6$ 
B)  $x y \left( x ^ { 2 } y + 3 \right)$ 
C)  $x ^ { 3 } y ^ { 2 } + 2 x ^ { 2 } y ^ { 3 } + 1$ 
D)  $x ^ { 2 } \left( x y ^ { 2 } + 3 \right)$ 
E)  none of these A

Question 4

Find the value of $\frac{d y}{d x}$ at the point (- 2, 1)for the function which is defined implicitly by  $x ^ { 2 } y - \frac { x } { y } = 6$

Accepted Answer

A) 2 
B) 0.5 
C) 0.4 
D) 1.5 
E) 2.5 
A) 2 
B) 0.5 
C) 0.4 
D) 1.5 
E) 2.5

Question 5

A utility function is given by $U = x _ { 1 } ^ { \frac { 2 } { 3 } } x _ { 2 } ^ { \frac { 1 } { 2 } }$ Find the value of x2 if the points, (64, 256)and (512, x2)lie on the same indifference curve.

Accepted Answer

A) 64 
B) 4 
C) 16 
D) 256 
E) 1024 
A) 64 
B) 4 
C) 16 
D) 256 
E) 1024

Question 6

Find, and classify, the stationary points of the function 
f(x, y)= x3 + x2 - xy + y2 + 10

Accepted Answer

A) Minimum at (0, 0)and saddle point at (-0.5, -0.25) 
B) Maximum at (0, 0)and saddle point at (-0.5, -0.25) 
C) Maximum at (0, 0)and saddle point at (0.5, 0.25) 
D) Maximum at (0, 0)and saddle point at (-0.5, 0.25) 
E) Minimum at (0, 0)and saddle point at (0.5, 0.25) 
A) Minimum at (0, 0)and saddle point at (-0.5, -0.25) 
B) Maximum at (0, 0)and saddle point at (-0.5, -0.25) 
C) Maximum at (0, 0)and saddle point at (0.5, 0.25) 
D) Maximum at (0, 0)and saddle point at (-0.5, 0.25) 
E) Minimum at (0, 0)and saddle point at (0.5, 0.25)

Question 7

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.

Accepted Answer

A) $107,500 
B) $476,500 
C) $545,000 
D) $254,000 
E) $346,500 
A) $107,500 
B) $476,500 
C) $545,000 
D) $254,000 
E) $346,500

Question 8

Given the demand function Q = 450 - 2P - PA + 0.01Y2
Where P = 15, PA = 20 and Y = 100, find the income elasticity of demand.

Accepted Answer

A) 0.6 
B) 0.4 
C) 0.2 
D) 1 
E) 0.8 
A) 0.6 
B) 0.4 
C) 0.2 
D) 1 
E) 0.8

Question 9

A firm's production function is given by Q = KL
Unit capital and labour costs are $2 and $1 respectively. Find the maximum level of output if the total cost of capital and labour is $6.

Accepted Answer

A)  $\frac { 2 } { 9 }$ 
B)  $\frac { 11 } { 9 }$ 
C)  $\frac { 4 } { 3 }$ 
D)  $\frac { 9 } { 2 }$ 
E)  $\frac { 2 } { 3 }$ 
A)  $\frac { 2 } { 9 }$ 
B)  $\frac { 11 } { 9 }$ 
C)  $\frac { 4 } { 3 }$ 
D)  $\frac { 9 } { 2 }$ 
E)  $\frac { 2 } { 3 }$

Find the value of the marginal rate of technical substitution for the production function $Q = 300 K ^ { \frac { 2 } { 3 } } L ^ { \frac { 1 } { 2 } }$ when $K = 40 , L = 60$ .

Evaluate the second- order partial derivative, f₂₃, of the function $f \left( x _ { 1 } , x _ { 2 } , x _ { 3 } \right) = \frac { x _ { 3 } x _ { 2 } ^ { 3 } } { x _ { 1 } } + x _ { 2 } e ^ { x _ { 3 } }$ at the point, (3, 2, 0).

Which of the following could represent a function, f(x, y), with first- order partial derivatives? $\frac { \partial f } { \partial x } = 3 x y ( x y + 2 ) \quad \frac { \partial f } { \partial y } = x ^ { 2 } ( 2 x y + 3 )$

Find the value of $\frac{d y}{d x}$ at the point (- 2, 1)for the function which is defined implicitly by $x ^ { 2 } y - \frac { x } { y } = 6$

A utility function is given by $U = x _ { 1 } ^ { \frac { 2 } { 3 } } x _ { 2 } ^ { \frac { 1 } { 2 } }$ Find the value of x2 if the points, (64, 256)and (512, x2)lie on the same indifference curve.

Find, and classify, the stationary points of the function f(x, y)= x3 + x2 - xy + y2 + 10

Given the demand function Q = 450 - 2P - PA + 0.01Y2 Where P = 15, PA = 20 and Y = 100, find the income elasticity of demand.

A firm's production function is given by Q = KL Unit capital and labour costs are $2 and $1 respectively. Find the maximum level of output if the total cost of capital and labour is $6.

Linear Equations

Non-Linear Equations

Mathematics of Finance

Differentiation

Integration

Matrices

Linear Prgamming

Dynamics

Filters

Exam 5: Partial Differentiation

Find the value of the marginal rate of technical substitution for the production function Q=300K23L12Q = 300 K ^ { \frac { 2 } { 3 } } L ^ { \frac { 1 } { 2 } }Q=300K32​L21​ when K=40,L=60K = 40 , L = 60K=40,L=60 .

Find the value of dydx\frac{d y}{d x}dxdy​ at the point (- 2, 1)for the function which is defined implicitly by x2y−xy=6x ^ { 2 } y - \frac { x } { y } = 6x2y−yx​=6

A utility function is given by U=x123x212U = x _ { 1 } ^ { \frac { 2 } { 3 } } x _ { 2 } ^ { \frac { 1 } { 2 } }U=x132​​x221​​ Find the value of x2 if the points, (64, 256)and (512, x2)lie on the same indifference curve.

Find, and classify, the stationary points of the function f(x, y)= x3 + x2 - xy + y2 + 10

Given the demand function Q = 450 - 2P - PA + 0.01Y2 Where P = 15, PA = 20 and Y = 100, find the income elasticity of demand.

A firm's production function is given by Q = KL Unit capital and labour costs are $2 and $1 respectively. Find the maximum level of output if the total cost of capital and labour is $6.

Linear Equations

Non-Linear Equations

Mathematics of Finance

Differentiation

Integration

Matrices

Linear Prgamming

Dynamics

Filters

Find the value of the marginal rate of technical substitution for the production function $Q = 300 K ^ { \frac { 2 } { 3 } } L ^ { \frac { 1 } { 2 } }$ when $K = 40 , L = 60$ .

Find the value of $\frac{d y}{d x}$ at the point (- 2, 1)for the function which is defined implicitly by $x ^ { 2 } y - \frac { x } { y } = 6$

A utility function is given by $U = x _ { 1 } ^ { \frac { 2 } { 3 } } x _ { 2 } ^ { \frac { 1 } { 2 } }$ Find the value of x2 if the points, (64, 256)and (512, x2)lie on the same indifference curve.