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First Semester Examination– June 2019 
MATHEMATICAL TECHNIQUES FOR FINANCE AND 
ECONOMICS 
EMET 7001 
Reading Time: 15 Minutes 
Writing Time: THREE Hours 
Permitted Materials: A Non-programmable Calculator; One Double-Sided A4 Sheet 
Page 1 of 8 – MATHEMATICAL TECHNIQUES FOR FINANCE AND ECONOMICS (EMET 7001) 
Answer all questions in this section using the answer booklet(s) provided. An- 
swers are expected to be succinct but complete. Answers that are too long and 
irrelevant will be penalized. 
Question 1 [10 marks] For each integral, determine whether it is proper and if so, 
compute it. 
 
Page 2 of 8 – MATHEMATICAL TECHNIQUES FOR FINANCE AND ECONOMICS (EMET 7001) 
Question 2 [20 marks] Each part of this question attempts to determine a function f 
on a given set A. For each part, first determine whether it indeeds defines a function and 
if it does, determine whether the function is increasing or decreasing or neither on A. 
1. [4 marks] A = [0, 3.18], f(x) = x3 − 8x2 + x− 8.13. 
2. [4 marks] A = [−5, 5], f(x) = ∫ x−10 ((sin t)2 + t4 exp(t)− log(t2 + 1) + log(30)) dt. 
3. [4 marks] A = {−1, pi}, f(x) = −x2. 
4. [4 marks] A = [0.2, 10], f(x) solves the equation x2 + 
√ 
7x+ 4y2 = 1 (with unknown 
y). 
5. [4 marks] A = [0.2, 10], f(x) solves the equation x2 + 
√ 
7x + 4y3 + log y = 1 (with 
unknown y). 
Page 3 of 8 – MATHEMATICAL TECHNIQUES FOR FINANCE AND ECONOMICS (EMET 7001) 
Question 3 [10 marks] An open rectangle in R2 is a set of the form {(x, y) ∈ R2 : 
x ∈ A and y ∈ B} for some (perhaps empty) open intervals A and B, and is denoted by 
A×B. Note that the empty set is an open rectangle by definition. 
1. [5 marks] Show that the intersection of two open rectangles is an open rectangle. 
2. [5 marks] Answer only ONE of the following two questions. If you attempt both, 
your answer to (a) will be marked. 
(a) Is the union of two open rectangles necessarily an open rectangle? Prove that 
it is or write down two open rectangles whose union is not an open rectangle. 
(b) Is the set difference between two open rectangles necessarily an open rectangle? 
Prove that it is or write down two open rectangles whose set difference is not 
an open rectangle. 
Page 4 of 8 – MATHEMATICAL TECHNIQUES FOR FINANCE AND ECONOMICS (EMET 7001) 
Question 4 [15 marks] Each part of this question contains a claim, determine whether 
the claim is true and briefly explain. If the claim contains the phrase “if and only if”, 
evaluate the “if” part and “only if” part separately. As an example, consider the following 
claim: a differentiable function on an open interval is strictly increasing if and only if its 
derivative is positive everywhere. 
The “if” part asserts that if the derivative of the function is indeed positive everywhere, 
then the function must be strictly increasing. This is true by the Mean Value Theorem. 
The “only if” part asserts that if the function is strictly increasing, then its derivative 
must be positive everywhere; in other words, it asserts it that as soon as the derivative of 
the function fails to be positive at one point, the function cannot be strictly increasing. 
This assertion is false, as f(x) = x3 is differentiable on R and strictly increasing, but its 
derivative is zero when x = 0. The conclusion is that the “if” part of the claim is true 
while the “only if” part is false. 
1. [3 marks] Let f be a function on [0, 1]. Claim: f is strictly increasing if and only if 
1 is the unique maximum of f and 0 is the unique minimum. 
2. [3 marks] Consider two bonds, Bond 1 and Bond 2, with the same maturity date. 
Claim: Bond 1’s yield-to-maturity is higher than Bond 2’s if Bond 1’s market (dirty) 
price is lower than Bond 2’s. 
3. [3 marks] Claim: the geometric series limn→∞ 
∑n 
j=0 a 
j converges (which means that 
the limit as the positive integer n approaches infinity exists and is finite) if −1 ≤ 
a < 1. 
4. [3 marks] Let f be a continuous function on [0, 1]. Claim: f has a root in (0, 1) if 
and only if f(0)f(1) < 0. 
5. [3 marks] Eating too much salt increases the risk of hypertension (high blood pres- 
sure). Scientists recommend that daily intake of sodium (the chemical element Na) 
should be no more than 2.3 grams. Calculation (known to be correct) shows that 
5.93 grams of table salt contains 2.3 grams of sodium. Claim: an adult’s daily intake 
of sodium is at or below the recommended maximum if and only if he eats no more 
than 5.93 grams of table salt every day. 
Page 5 of 8 – MATHEMATICAL TECHNIQUES FOR FINANCE AND ECONOMICS (EMET 7001) 
Question 5 [5 marks] If the value of a macroeconomic variable at period t (say Year t) 
is zt and zt is always positive, then the growth rate of the variable is defined as 
zt+1−zt 
zt 
Part of the Solow model of economic growth postulates that 
Yt = AK 
α 
t L 
1−α 
t , (1) 
where Yt is the output of the economy, Kt is the capital stock, and Lt is the labour force. 
The equation is a hypothesis, but Yt, Kt and Lt are observables. For the purpose of this 
question, assume that α is a known constant between 0 and 1. The Solow residual in 
Period (t+ 1) is defined as 
Yt+1 − Yt 
Yt 
− αKt+1 −Kt 
Kt 
− (1− α)Lt+1 − Lt 
Lt 
If we assume that α is known, then the above expression only involves observables. In 
words, the Solow residual is the growth rate of output minus α times the growth rate of 
capital stock and the (1− α) times the growth rate of labour force. 
1. [5 marks] Show that if Eq. (1) is valid with a time-independent A, then the Solow 
residual should be approximately zero in every period. 
Page 6 of 8 – MATHEMATICAL TECHNIQUES FOR FINANCE AND ECONOMICS (EMET 7001) 
Question 6 [25 marks] Each part defines an interval A and a function f on A. Find all 
the maxima of f on A for each part, if any. 
1. [5 marks] A = [−3, 3], f(x) = x− exp(x). 
2. [5 marks] A = [−10, 10], f(x) = x4 − 8x2 + 2000. 
3. [5 marks] A = R, f(x) = x4 exp(−x2). 
4. [5 marks] A = (0,∞), f(x) = x−7 
x2+x 
5. [5 marks] A = R, 
f(x) = 
 
− exp(x), if x < 0; 
1, if x = 0; 
2 exp(−x), if x > 0. 
Page 7 of 8 – MATHEMATICAL TECHNIQUES FOR FINANCE AND ECONOMICS (EMET 7001) 
Question 7 [15 marks] In some applications, it is useful to consider matrices whose 
entries depend on a parameter, which means that each matrix entry is a function. Consider 
the following example: 
A(t) = 
cos θ(t) sin θ(t) 
− sin θ(t) cos θ(t) 
, for t ∈ R. (2) 
where θ : R→ R is a differentiable function and θ(0) = 0. We can differentiate A(t) with 
respect to t by differentiating each of its matrix entries, so A′(t) is a 2× 2 matrix whose 
(1, 1) entry is −θ′(t) sin θ(t), and so on. 
1. [5 marks] Show that (A(t))TA(t) = I for every t ∈ R and A′(0) + (A′(0))T = 0, 
where the “0” on the right hand side is the zero 2× 2 matrix. (Hint: the following 
formula from trigonometry might be useful: (sinx)2 + (cosx)2 = 1 for every x ∈ R.) 
2. [10 marks] Now consider a 3× 3 matrix B(t) which also depends on the parameter 
t ∈ R. Each of the nine matrix entries of B(t) is a differentiable function of t and 
B(0) = I. Assume that (B(t))TB(t) = I for every t ∈ R. Show that B′(0) + 
(B′(0))T = 0. (Hint: trigonometry is of little help here.) 
——— End of Examination ——— 
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Page 8 of 8 – MATHEMATICAL TECHNIQUES FOR FINANCE AND ECONOMICS (EMET 7001) 
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