Question 1

If the consumption function is C = 0.02Y2 + 0.1Y + 25&#10;Find the value of Y when MPS = 0.38.&#10;A)26&#10;B)7&#10;C)14&#10;D)13&#10;E)32

Accepted Answer

D

Question 2

Which of the following expressions represent the second- order derivative of the function

Accepted Answer

B

Question 3

A firm's average cost function takes the form   and it is known that MC = 35 when Q = 3. Find the value of AC when Q = 6.&#10;A)11&#10;B)25&#10;C)59&#10;D)36&#10;E)216

Accepted Answer

D

Question 4

A firm's demand function is P = 60 - 0.5Q&#10;If fixed costs are 10 and variable costs are Q + 3 per unit, find the maximum profit.&#10;A)420&#10;B)651.5&#10;C)21&#10;D)531.5&#10;E)19

Accepted Answer

To find the maximum profit, we need to first find the total revenue and total cost functions. Total Revenue =Price x Quantity = P x Q
Substituting the demand function, we get TR = (60 - 0.5Q)Q = 60Q - 0.5Q^2
Total cost = Fixed cost + Variable cost = 10 + (Q+3)Q = Q^2 + 3Q + 10
Profit = Total Revenue - Total cost = 60Q - 0.5Q^2 - Q^2 - 3Q - 10
= -1.5Q^2 + 57Q - 10
To find the maximum profit, we take the derivative and set it equal to 0:
d(Profit)/dQ = -3Q + 57 = 0
Q = 19
Substituting Q back into the profit function, we get the maximum profit:
Profit = -1.5(19)^2 + 57(19) - 10 = 531.5
Therefore, the best choice is D.

Question 5

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?&#10;A)Tangent slopes upwards and graph bends downwards&#10;B)Tangent slopes downwards and graph bends downwards&#10;C)Tangent slopes downwards and graph bends upwards&#10;D)Tangent slopes upwards and graph bends upwards&#10;E)Tangent is horizontal

Accepted Answer

The derivative of the function is f'(x) = 10x^4 - 12x^3 + 4x - 17. At x = -1, f'(-1) = 10(-1)^4 - 12(-1)^3 + 4(-1) - 17 = 10 - 12 - 4 - 17 = -19. Since f'(-1) < 0, the tangent line slopes downwards. The second derivative of the function is f''(x) = 40x^3 - 36x^2 + 4. At x = -1, f''(-1) = 40(-1)^3 - 36(-1)^2 + 4 = -40 - 36 + 4 = -72. Since f''(-1) < 0, the graph bends downwards.

Question 6

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?&#10;A)b + 2c = 8&#10;B)2b - 3c = 12&#10;C)b - c = 5&#10;D)2b - c = 6&#10;E)b + c = 9

Accepted Answer

The answer of Given that the cubic function, f(x)= x3...

Question 7

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

Accepted Answer

The price elasticity of supply is given by the formula:```Ed = (dQ/dP) * (P/Q)```where:* Ed is the price elasticity of supply* dQ/dP is the change in quantity supplied divided by the change in price* P is the price* Q is the quantity suppliedAt the point P = 105, we have:```Ed = (dQ/dP) * (P/Q) = (2Q * 2P) / (7 + 2Q^2) = (4PQ) / (7 + 2Q^2)```Substituting P = 105 and solving for Ed, we get:```Ed = (4 * 105 * Q) / (7 + 2Q^2) = (420Q) / (7 + 2Q^2)```To estimate the percentage increase in supply when the price rises by 7%, we can use the formula:```Percentage increase in supply = Ed * (Percentage increase in price)```Substituting Ed = (420Q) / (7 + 2Q^2) and Percentage increase in price = 7%, we get:```Percentage increase in supply = (420Q) / (7 + 2Q^2) * 7% = (2940Q) / (7 + 2Q^2)```At the point P = 105, we have:```Percentage increase in supply = (2940Q) / (7 + 2Q^2) = (2940 * 105) / (7 + 2 * 105^2) = 3.75%```Therefore, the correct answer is A) 3.75%.

Question 8

A firm's short- run production function is given by Q = L3 e- 0.02L&#10;Find the value of L which maximises the average product of labour.&#10;A)100&#10;B)200&#10;C)150&#10;D)250&#10;E)50

Accepted Answer

To maximize the average product of labor (APL), we set the derivative of APL with respect to L to zero. The APL is given by Q/L = L^2 e^(-0.02L). Differentiating and solving for L gives L = 100.

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

Accepted Answer

The answer of Find the elasticity for the demand function...

Question 10

Find the value of the first- order derivative of the function   when x = 9.&#10;A)1&#10;B)1.5&#10;C)- 0.5&#10;D)0&#10;E)0.5

Accepted Answer

We need to find the first order derivative of the given function with respect to x, which is given by:
$y' = \frac { 3 } { 2\sqrt { x } } + \frac { 81 } { x^{ 2 } }$
Substituting x=9:
$y' = \frac { 3 } { 2\sqrt { 9 } } + \frac { 81 } { 9^{ 2 } } = \frac { 3 } { 6 } + \frac { 81 } { 81 } = \frac { 1 } { 2 } + 1 = \frac { 3 } { 2 }$
Therefore, the value of the first order derivative of the function when x=9 is 1.5. Hence, the answer is B.

Deck 4: Differentiation