Question 1

A firm's short- run production function is given by Q = L3 e- 0.02L
Find the value of L which maximises the average product of labour.

Accepted Answer

A) 100 
B) 200 
C) 150 
D) 250 
E) 50 
A) 100 
B) 200 
C) 150 
D) 250 
E) 50 A

Question 2

If the consumption function is C = 0.02Y2 + 0.1Y + 25
Find the value of Y when MPS = 0.38.

Accepted Answer

A) 26 
B) 7 
C) 14 
D) 13 
E) 32 
A) 26 
B) 7 
C) 14 
D) 13 
E) 32 D

Question 3

Consider the supply equation P = 7 + 2Q2
By evaluating the price elasticity of supply at the point P = 105, estimate the percentage increase in supply when the price rises by 7%.

Accepted Answer

A) 3.75% 
B) 3% 
C) 4% 
D) 3.25% 
E) 3.5% 
A) 3.75% 
B) 3% 
C) 4% 
D) 3.25% 
E) 3.5% A

Question 4

A firm's average cost function takes the form $$A C = 4 Q + a + \frac { 6 } { Q }$$ and it is known that MC = 35 when Q = 3. Find the value of AC when Q = 6.

Accepted Answer

A) 11 
B) 25 
C) 59 
D) 36 
E) 216 
A) 11 
B) 25 
C) 59 
D) 36 
E) 216

Question 5

Find the value of the first- order derivative of the function 
$y = 3 \sqrt { x } - \frac { 81 } { x } + 13$ when x = 9.

Accepted Answer

A) 1 
B) 1.5 
C) - 0.5 
D) 0 
E) 0.5 
A) 1 
B) 1.5 
C) - 0.5 
D) 0 
E) 0.5

Question 6

Find the elasticity for the demand function Q = 80 - 2P - 0.5P2
Averaged along an arc joining P = 6 to P = 8. Give your answer to two decimal places.

Accepted Answer

A) 1.30 
B) 0.67 
C) 1 
D) 0.65 
E) 1.54 
A) 1.30 
B) 0.67 
C) 1 
D) 0.65 
E) 1.54

Question 7

A firm's demand function is P = 60 - 0.5Q
If fixed costs are 10 and variable costs are Q + 3 per unit, find the maximum profit.

Accepted Answer

A) 420 
B) 651.5 
C) 21 
D) 531.5 
E) 19 
A) 420 
B) 651.5 
C) 21 
D) 531.5 
E) 19

Question 8

Which of the following expressions represent the second- order derivative of the function
 $y = \frac { x } { 2 x + 1 } ?$

Accepted Answer

A)  = $-\frac { 2 } { ( 2 x + 1 ) }$ 
B)  $- \frac { 4 } { ( 2 x + 1 ) }$ 
C)  $- \frac { 2 x } { ( 2 x + 1 ) }$ 
D)  $\frac { 1 } { ( 2 x + 1 ) ^ { 2 } }$ 
E)  $\frac { 4 } { ( 2 x + 1 ) ^ { 2 } }$ 
A)  = $-\frac { 2 } { ( 2 x + 1 ) }$ 
B)  $- \frac { 4 } { ( 2 x + 1 ) }$ 
C)  $- \frac { 2 x } { ( 2 x + 1 ) }$ 
D)  $\frac { 1 } { ( 2 x + 1 ) ^ { 2 } }$ 
E)  $\frac { 4 } { ( 2 x + 1 ) ^ { 2 } }$

Question 9

Given that the cubic function, f(x)= x3 + ax2 + bx + c has a stationary point at (2, 5), which of the following statements must be true?

Accepted Answer

A) b + 2c = 8 
B) 2b - 3c = 12 
C) b - c = 5 
D) 2b - c = 6 
E) b + c = 9 
A) b + 2c = 8 
B) 2b - 3c = 12 
C) b - c = 5 
D) 2b - c = 6 
E) b + c = 9

Question 10

Which one of the following statements describes the shape of the graph of the function, f(x)= 2x5 - 3x4 + 2x2 - 17x + 31, at the point, x = - 1?

Accepted Answer

A) Tangent slopes upwards and graph bends downwards 
B) Tangent slopes downwards and graph bends downwards 
C) Tangent slopes downwards and graph bends upwards 
D) Tangent slopes upwards and graph bends upwards 
E) Tangent is horizontal 
A) Tangent slopes upwards and graph bends downwards 
B) Tangent slopes downwards and graph bends downwards 
C) Tangent slopes downwards and graph bends upwards 
D) Tangent slopes upwards and graph bends upwards 
E) Tangent is horizontal

A firm's short- run production function is given by Q = L3 e- 0.02L Find the value of L which maximises the average product of labour.

If the consumption function is C = 0.02Y² + 0.1Y + 25 Find the value of Y when MPS = 0.38.

Consider the supply equation P = 7 + 2Q² By evaluating the price elasticity of supply at the point P = 105, estimate the percentage increase in supply when the price rises by 7%.

A firm's average cost function takes the form $A C = 4 Q + a + \frac { 6 } { Q }$ and it is known that MC = 35 when Q = 3. Find the value of AC when Q = 6.

Find the value of the first- order derivative of the function $y = 3 \sqrt { x } - \frac { 81 } { x } + 13$ when x = 9.

Find the elasticity for the demand function Q = 80 - 2P - 0.5P2 Averaged along an arc joining P = 6 to P = 8. Give your answer to two decimal places.

A firm's demand function is P = 60 - 0.5Q If fixed costs are 10 and variable costs are Q + 3 per unit, find the maximum profit.

Which of the following expressions represent the second- order derivative of the function $y = \frac { x } { 2 x + 1 } ?$

Given that the cubic function, f(x)= x³ + ax² + bx + c has a stationary point at (2, 5), which of the following statements must be true?

Which one of the following statements describes the shape of the graph of the function, f(x)= 2x5 - 3x4 + 2x2 - 17x + 31, at the point, x = - 1?

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Exam 4: Differentiation

A firm's short- run production function is given by Q = L3 e- 0.02L Find the value of L which maximises the average product of labour.

If the consumption function is C = 0.02Y2 + 0.1Y + 25 Find the value of Y when MPS = 0.38.

Consider the supply equation P = 7 + 2Q2 By evaluating the price elasticity of supply at the point P = 105, estimate the percentage increase in supply when the price rises by 7%.

A firm's average cost function takes the form AC=4Q+a+6QA C = 4 Q + a + \frac { 6 } { Q }AC=4Q+a+Q6​ and it is known that MC = 35 when Q = 3. Find the value of AC when Q = 6.

Find the value of the first- order derivative of the function y=3x−81x+13y = 3 \sqrt { x } - \frac { 81 } { x } + 13y=3x​−x81​+13 when x = 9.

Find the elasticity for the demand function Q = 80 - 2P - 0.5P2 Averaged along an arc joining P = 6 to P = 8. Give your answer to two decimal places.

A firm's demand function is P = 60 - 0.5Q If fixed costs are 10 and variable costs are Q + 3 per unit, find the maximum profit.

Which of the following expressions represent the second- order derivative of the function y=x2x+1?y = \frac { x } { 2 x + 1 } ?y=2x+1x​?

Given that the cubic function, f(x)= x3 + ax2 + bx + c has a stationary point at (2, 5), which of the following statements must be true?

Which one of the following statements describes the shape of the graph of the function, f(x)= 2x5 - 3x4 + 2x2 - 17x + 31, at the point, x = - 1?

Linear Equations

Non-Linear Equations

Mathematics of Finance

Partial Differentiation

Integration

Matrices

Linear Prgamming

Dynamics

Filters

If the consumption function is C = 0.02Y² + 0.1Y + 25 Find the value of Y when MPS = 0.38.

Consider the supply equation P = 7 + 2Q² By evaluating the price elasticity of supply at the point P = 105, estimate the percentage increase in supply when the price rises by 7%.

A firm's average cost function takes the form $A C = 4 Q + a + \frac { 6 } { Q }$ and it is known that MC = 35 when Q = 3. Find the value of AC when Q = 6.

Find the value of the first- order derivative of the function $y = 3 \sqrt { x } - \frac { 81 } { x } + 13$ when x = 9.

Which of the following expressions represent the second- order derivative of the function $y = \frac { x } { 2 x + 1 } ?$

Given that the cubic function, f(x)= x³ + ax² + bx + c has a stationary point at (2, 5), which of the following statements must be true?