School of Mathematics, Computer Science and Engineering
BSc Economics
BSc Economics with Accounting
BSc Financial Economics
MA1202: Mathematics for Economists Post A-Level, Part One
Stage 1 Examination
January 2021
Examination duration: 24 hours
Full marks may be obtained for correct answers to ALL questions.
All necessary working must be shown
BEGIN EACH QUESTION ON A FRESH PAGE
External Examiner:
Internal Examiner:
Dr Paul Bushby
Dr Florian Eisele
Page 1 of 3
√
5 x
1. Consider the function g(x) =
with x restricted to [0, 4].
2 + 3x
(a) Find an expression for the first two derivatives of g(x) , i.e. g 0 (x) and
g 00 (x).
[4]
(b) Using the results from part (a), find the potential extreme points of g(x)
in [0, 4] (i.e. critical points and boundaries), and show which of these are
maxima or minima.
[6]
(c) Determine the location of any points of inflection of g(x) in [0, 4]. Then
sketch the graph of g(x), showing the location and type of the maxima
and minima, and the intervals of convexity and concavity.
[10]
2. Consider the function
f (x) =
2
5
+
2
x
x
e1−x .
We are interested in an approximation of f (x) about x = 1 by a polynomial of
degree three.
(a) Find the first four derivatives of the function f (x), i.e. f 0 (x), f 00 (x), f 000 (x)
and f (4) (x).
[6]
(b) Find the appropriate Taylor expansion for f (x), including an expression
for the remainder.
[9]
(c) Estimate the value of f (x) at x = 1.1 by means of the Taylor expansion
found in (b) (give five or more decimal places for all answers in this part
of the question). Calculate the minimal and maximal values of the remainder and present your estimate of f (1.1) in the form a ≤ f (1.1) ≤ b.
How does this estimate compare with the calculator value?
[5]
Page 2 of 3
3. A monopolist believes that demand of his product follows the function
D(P ) =
3
,
1 + αP3
where P is the selling price and α is an unknown constant, greater than zero
(i.e. α > 0).
(a) Calculate the marginal revenue as a function of price and the price that
maximises revenue (this will depend on α).
[8]
(b) It has been observed that revenue increased by 21 % when the selling
price was increased from P = 1 to P = 1.01. Use this information to
estimate the value of α. Once α is known, calculate the value of the price
that maximises revenue.
[12]
4. Let two 3 × 3 matrices A and B be given by


2 2 0
A = 3 5 −4
3 2 3


1 −1 −2
and B = 0 −1 1  .
0 2 −2
(a) Calculate the matrix multiplications AB and BA.
[5]
(b) Knowing that a matrix is invertible if and only if its determinant is nonzero,
does either A or B have an inverse? If either does, then calculate its
inverse by means of the cofactor matrix.
[15]
5. Consider the system of linear equations


x + 2y + z = 2
2x − 2y + 3z = 1

3 x − 2 y − 2 z = 2.
(a) Write it in the matrix form A x = b where


x
x =  y .
z
Then solve it by means of the inverse matrix A−1 . (You may calculate
this inverse matrix by any method you like as long as you show your
calculations.)
[10]
(b) Use Cramer’s rule to verify your result.
Page 3 of 3
[10]