SECTION A

Question 1 (10 marks)

“Differentiate the following, simplifying the answer where appropriate:

y = 4𝑥7 + 3𝑥2 - 2𝑥0.5 – 27

(2 marks)
y = (3𝑥2 + 7)(𝑥3 – 4𝑥)
(2 marks)
y = (5𝑥4 + 2)6
(2 marks)
y = (x^2+1)/(2x^3-3x^4 )
(4 marks)”

Question 2 (8 marks)

“Find the stationary points of the following functions and determine whether each is a local maximum, a local minimum or a point of inflection:

y = 𝑥3 + 1.5𝑥2 - 18𝑥 + 7

(4 marks)
y = 1/3 x^3 – 𝑥2 + 𝑥 + 4
(4 marks)”

Question 3 (5 marks)

“For each of the following, find the partial derivatives ∂z/∂x and ∂z/∂y :

z = 4𝑥3 + 3y4 – 17

(2 marks)
z = 𝑥2y3 – 𝑥7y + 𝑥y5
(3 marks)”

Question 4 (5 marks)

“Find the following indefinite integrals:

∫(3𝑥2 – 𝑥 -2) d𝑥

(2 marks)
∫(2𝑥 + 3)(𝑥2 + 3𝑥)4 d𝑥
(3 marks)”

Question 5 (3 marks)

“Evaluate the following definite integral:”

∫_2^6▒〖(x^3 〗 + 2𝑥 – 7) d𝑥

SECTION B

Question 6 (15 marks)

“Consider the Total Cost (TC) function
TC  2.2q3  16q2  48q  150
Where q is the level of output.
Show that the Average Cost is at it’s minimum when q = 5.
(4 marks)
Find the output at which marginal cost is at its minimum.
(4 marks)
Show that MC = AC when AC is at its minimum.
(2 marks)
Sketch the graphs of the MC and AC functions, on the same axes.
(5 marks)”

Question 7 (8 marks)

“Given the demand function q = – 3p2 + 2p + 1965
Find the arc elasticity of demand when p increases
from 10 to 11
from 24 to 25.
(4 marks)
Find the point elasticity
when p = 10
when p = 24
(4 marks)”

Question 8 (8 marks)
“Find the stationary point of the following function, and determine whether it is a maximum, minimum, or saddle point:”

z = 5𝑥2 + 2y2 + 4𝑥y - 2𝑥 + 10y + 1000

Question 9 (15 marks)
“A firm’s production function is Q = K^□(1/3) L^(2/3)
Where Q is the output of a good resulting from inputs of capital (K) and labour (L).
Find the marginal products of capital and labour. Show that they can be written as:
MPK = Q/3K MPL = 2Q/3L
(4 marks)”
“Are the marginal products of capital and labour always positive? Sketch their graphs.
(3 marks)”
“Find the equations of the isoquants for
10 units of output
15 units of output.

In each case, express the isoquant both as an implicit function and also with K as an explicit function of L.
(3 marks)”
“By implicit differentiation, find the slope of any isoquant and show that this slope is given by the ratio of the marginal products of capital and labour. Is the slope always negative?
(5 marks)”

Question 10 (13 marks)

“Ben’s utility function is given by:

U = X3Y2

where X and Y are the weekly consumption levels of goods X and Y.”

The market price of good X is £2 and of good Y is £1.
“If Ben’s weekly budget is £100, use the Lagrange multiplier method to find the quantities of goods X and Y that Ben should buy each week in order to maximise his utility.”
(9 marks)

“Suppose that the price of good X rises to £3.  Find the new utility-maximising quantities.  Comment on how the rise in price of good X effects the demand for good Y.”

(4 marks)

Question 11 (10 marks)

“Given the inverse demand function:

p  9  q  0.1q2

Find the consumers' surplus when quantity purchased is q  5.

(6 marks)
How does consumers’ surplus change if q increases to q  5.5?
(4 marks)”

Sample Solution