s
BRIGHTON BUSINESS SCHOOL
Final Examination
ML211
Professional Practice
Student No..................................................
Time allowed:
3 hours
Answer:
All questions
Items permitted:
Calculator
Instructions:
For Sections A and B write all answers on the
exam paper, together with all your
calculations. Write Section C answers in the
answer booklet provided.
Marks for Sections A and B
1
2
3
4
ML211 Final Examination 2013-14
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6
1
7
8
9
10
Total
Section A: Calculus
Answer ALL questions in this section.
Question 1 (8 marks)
Differentiate the following with respect to x :
(a)
(b)
𝑦 = 𝑥 7 + 3𝑥 5 + 1
𝑦 = (𝑥)
3⁄
2
(2 marks)
4
+ 𝑥 + √𝑥
(3 marks)
(c)
𝑦 = (2𝑥 − 1)4
(3 marks)
Question 2 (7 marks)
(a) If 𝑓(𝑥) = 2𝑥 2 (𝑥 − 3) find
𝑑𝑦
𝑑𝑥
(2 marks)
Question 2 continued overleaf ...
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Question 2 continued
(b)
Hence, find the coordinates of the stationary points.
(2 marks)
(c)
Determine which point is a maximum and which is a minimum.
(3 marks)
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Question 3
(14 marks)
A laptop retailer has established that for medium to high end laptops the
revenue function and cost function in pounds are
𝑅(𝑥) = 2𝑥 3 + 40𝑥 2 + 8𝑥
𝐶(𝑥) = 3𝑥 3 + 19𝑥 2 + 80𝑥 − 800.
respectively.
Find the price per unit that will maximize the company’s profit.
(14 marks)
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Question 4
(8 marks)
EcoPower, a producer of solar panels finds that monthly demand for their 100W
12v solar panels is related to the selling price by the equation
𝑝 = 300 − 0.25𝑥
(a)
Find the elasticity of demand 𝐸(𝑝) at price 𝑝.
(3 marks)
(b)
Find the elasticity of demand at price 𝑝 = £200 and interpret the result.
(2 marks)
(c)
Contrast the situation for 𝐸(200) in part (b) with that for a price of
𝑝 = £100.
(3 marks)
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Question 5
(4 marks)
Find the anti-derivative (𝐹(𝑥)) of the function
𝑓(𝑥) = −5𝑒−5𝑥 + 5𝑥 4 −
For which 𝐹(1) = 1 +
5
𝑥
1
𝑒5
(4 marks)
Question 6 (9 marks)
(a)
Anuska’s grandfather would like to support her undergraduate education
and has established an annuity for this purpose. The annuity will provide
5000√𝑡 pounds per year for four years.
What is the total nominal value of the annuity?
(4 marks)
Question 6 continued overleaf …
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Question 6 continued
(b)
Anuska’s grandfather, who is now 50 years old, also needs to think
about his own future and wants to fund an annuity that will pay him
£15,000 per year for 10 years, beginning at age 70 (he has thus far
lived a healthy life and intends to live a long time). Assuming a
prevailing interest rate of 5% for the entire 30 year period (age 50-80),
what should he expect to pay as the single premium?
(5 marks)
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Section B: Linear Algebra
Answer ALL questions in this section.
Question 7
(a)
(10 marks)
Compute the products of the following pairs of matrices:
(i)
(2 marks)
(ii)
(3 marks)
b)
Find the inverse of the following matrices (if they exist):
(3 marks)
(i)
(ii)
Question 7 continued overleaf …
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Question 7 continued
(c)
If A and B, (n x n) invertible square matrices confirm using matrix
algebra that 𝐵−1 𝐴−1 is the inverse of the product 𝐴𝐵.
(2 marks)
Question 8
(6 marks)
The Weekly Guardia has online book shops in both the UK and the US. In
January, the new autobiography ‘Put a sock in it’ by the well-known footballer
Christian Penaldo, sold 700 hardback, 1300 paperback and 2000 e-copies from
the UK site, whereas from the US site 400 hardback, 300 paperback and 500 ecopies were sold.
The company (unlike some of their competitors) sell books at the same price on
both sites. Penaldo’s book was $30 for a hardback copy, $20 for paperback and
$10 for an e-copy. Customers with a loyalty card, however, get a 10% discount.
Use matrix algebra to compute the total revenue in dollars for each site (UK
and US) if the same amount of books were sold to each customer type.
(6 marks)
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Question 9
(6 marks)
Put the following system of linear equations into an augmented matrix and
solve using Gauss-Jordan elimination:
𝑥 + 2𝑦 + 3𝑧 = 4
2𝑦 + 3𝑧 = −2
𝑥
+ 𝑧 = 3
(6 marks)
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Question 10
(13 marks)
Two sectors of the Japanese economy are (1) communications equipment
(Comms) and (2) electronic components (Elecs). In 2001 the input-output table
involving these two sectors was as follows (in millions of US dollars):
To
From
(a)
Comms
Elecs
6000
500
Elecs
24,000
30,000
Total Output
90,000
140,000
Comms
Determine the external demand for the two sectors. (You do not need to
use matrix algebra for this part.)
(2 marks)
(b)
Suppose that external demand for communications equipment rises to
$100,000 million and for electronic components the external demand
rises to $90,000 million. How do the production levels in the two sectors
have to change to accommodate these rises?
(8 marks)
(More space for calculation overleaf)
Question 10 continues overleaf …
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Question 10 continued
(c)
Interpret the coefficients of the technology matrix you found in part (b).
(3 marks)
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Section C:
Regression
Answer SIX from EIGHT questions in this section in the answer
booklet provided. Answers on the question paper will not be
marked.
All questions carry equal marks
Question 1(C)
Assuming that you would like to investigate the effect of income and interest
rates on money demand.
(a)
What would you choose as the dependent variable?
(b)
Write out the full theoretical model as an equation and make an
initial assessment of the way that the explanatory variables will
influence the dependent variable.
(c)
What other factors could be important for money demand? How
could they be incorporated into the model?
Question 2(C)
Ordinary Least Squares estimators are said to be BLUE.
(a)
What does BLUE stand for?
(b)
Why is BLUE important?
(c)
What conditions must be in place for OLS estimates of coefficient
to be BLUE?
Question 3(C)
In an initial visual observation of data, what are you looking for and why?
Question 4(C)
Explain the following and how they are used in OLS
(a)
𝑅2
(b)
The Durbin-Watson statistic
Question 5(C)
Identify one problem that you may face with the structural form of the
regression model. How would you deal with that?
Section C continued overleaf …
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Section C continued
Question 6(C)
Why is an absolute value of 2 for the t-statistic important for assessing the
regression coefficients?
Question 7(C)
What are the main things to look for in the regression residuals and why?
Question 8(C)
What is likely to be the problem if you see a high 𝑅 2 and a low Durbin-Watson
statistic? How can you make an adjustment that will allow you to use OLS?
End of Exam
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Formula sheet overleaf
ML211 Formula sheet: Calculus and Linear Algebra
Derivatives:
𝒚 = 𝒇(𝒙)
𝒅𝒚
= 𝒇′(𝒙)
𝒅𝒙
𝑘𝑥 𝑛
𝑛𝑘𝑥 𝑛−1
𝑒𝑛𝑥
𝑛𝑒𝑛𝑥
(𝑎𝑥 + 𝑏)𝑛
𝑛𝑎(𝑎𝑥 + 𝑏)𝑛−1
Integrals: (add constant of integration unless limits of integration known)
𝑥 𝑛+1
𝑛+1
∫ 𝑥 𝑛 𝑑𝑥
1 𝑛𝑥
𝑒
𝑛
∫ 𝑒𝑛𝑥 𝑑𝑥
∫
(𝑛 ≠ 1)
1
𝑑𝑥
𝑥
ln|𝑥|
Definite integral:
𝑏
∫ 𝑓(𝑥) 𝑑𝑥 = 𝐹(𝑏) − 𝐹(𝑎)
𝑎
Profit, Revenue, Cost, 𝑝 = price, 𝑥 = no. of items:
𝑃(𝑥) = 𝑅(𝑥) − 𝐶(𝑥)
𝑅(𝑥) = 𝑥𝑝
𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑐𝑜𝑠𝑡:
𝑐(𝑥) =
𝐶(𝑥)
𝑥
Elasticity of Demand:
𝐸(𝑝) =
−𝑝𝑄′(𝑝)
𝑄(𝑝)
Revenue Streams:
𝑏
𝑇𝑜𝑡𝑎𝑙 𝑉𝑎𝑙𝑢𝑒
𝑇𝑉 = ∫ 𝑅(𝑡) 𝑑𝑡
𝑎
𝑏
𝑃𝑟𝑒𝑠𝑒𝑛𝑡 𝑉𝑎𝑙𝑢𝑒
𝑃𝑉 = ∫ 𝑅(𝑡)𝑒−𝑟𝑡 𝑑𝑡
𝑎
Input-output model: A = technology matrix, X = production vector, D = external
demand
𝑋 = 𝐴𝑋 = 𝐷
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