MATH101 Problem sheet 5
Tutorial problems discuss in Week 6 tutorial.
1. For each sequence (an ) below and function f , decide whether the sequence (f (an ) converges
and, if it does, find the limit.
(a) an = 1/n and f (x) = x2 + 2;
(b) an = (2n
MATH101 Problem sheet 4
Tutorial problems discuss in Week 5 tutorial.
1. Which of the sequences with the following tems is increasing, which decreasing, which are
bounded below and which bounded above?
(a) xn = 3n for n N;
(b) xn = 1/n! for n 1;
MATH101 Problem sheet 3
Tutorial problems discuss in Week 4 tutorial.
1. Solve the following inequalities
(a) 5 2/x > 7;
(b) x2 4 < 3.
2. Find all solutions to the following equations. Give your answers exactly, in radians.
(a) sin x = 3/2;
MATH101 Problem sheet 9
Tutorial problems discuss in Week 10 tutorial.
1. Evaluate the following integrals.
(a) 2 1/x2 dx.
(b) 0 exp( x) dx. [Hint: You may assume that exp( x) ! 0 as x ! 1.]
(c) 0 tan x dx. [Hint: Use the substitution
MATH101 Problem sheet 7
Tutorial problems discuss in Week 8 tutorial.
1. Find the equation of the tangent line to the curve x3 y 2 + xy + 3x
2. Find and classify the stationary points of the function f (x) = x4
5 = 0 at (1, 1).
4x3 + 2.
MATH101 Problem sheet 2
Tutorial problems discuss in Week 3 tutorial.
1. For each pair of functions f and g below compute the sum f + g, the product f g and the
composites f g and g f .
(a) f (x) = x2 + 3, g(x) = x2 ;
(b) f (x) = x3
MATH101 Problem sheet 1
The questions on this sheet (as on all subsequent sheets for this module, and on the exam)
are divided into two sections, A and B. Those in A test your ability to perform standard
calculations, and to recall and apply definitions a
MATH101 Problem sheet 10
Tutorial problems discuss in Week 11 tutorial.
1. In each case below use the Ratio Test (second version) to decide if the series
(a) an = n/3n ;
(b) an = 2n n/3n ;
(c) an = n!/5n ;
(d) an = ( 1)n n3
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Solution to MATH267 Homework 11
1. We are given F = C = 100, r = 10%, n = 8, i = 8%.
F r(Ia) n|i + n C v n
F ra n|i + C v 8
0.1(Ia) 8|0.08 + 8(1.08)8
0.1a 8|0.08 + 1.088
P = 1000v + 2000v 2 + 3000v 3 at i = 25%
Solution to MATH267 Homework 6
1. Time diagram
This is a 4-year deferred perpetuity immediate or 5-year deferred perpetuity due. We
deal with this annuity as a 4-year deferred perpetuity immediate. The present value is
4096 = 1.
Solution to Homework 10
1. (a) The at-par yield rat r for a three-year bond can be calculated by following formula:
1 (1 + s3 )3
r = P3
t=1 (1 + st )
s1 = 0.06 + 0.01 1 = 0.07
s2 = 0.06 + 0.01 2 = 0.08
s3 = 0.06 + 0.01 3 = 0.09
1 (1 + 0.09)3
(1 + 0
MATH267 Homework Sheet 10
Please hand in your solutions by 2pm, Friday December 12, 2014.
1. (a) The current term structure is defined by st = 0.06+0.01t for t = 1, 2, 3. Calculate
the at-par yield rate for three-year bond.
(b) You are given a
Solution to MATH267 Homework 3
Denote by X the annual retirement income the woman will receive according to her
proposal, then the present value of her income is
(1 + 5%)5 X a 20|5%
The present value of the inheritance is 50000
= 1, 000,
Solution to MATH267 Homework 4
1. We are given i(2) = 7%, which is convertible semiannually. The payment is made every
other year. Then the payment period is longer than the interest conversion period. We
can find the answer by either interest conversion
MATH267 Homework Sheet 1
Please hand in your solutions to all questions by 2:00pm, Friday, 9th of October 2014.
1. Consider the amount function A(t) = t2 + 2t + 3.
a) Find the corresponding accumulation function a(t).
b) Find In .
2. It is know
Solution to MATH267 Homework 9
1. We are given P = 115.84, r(2) = 7% r = 3.5%, i(2) = 6% i = 3% per half-year,
n = 12 2 = 24, and F = 100. By using the Basic Formula, we have
P = F ra n|i + C v n
115.84 = 100 3.5% a 24|3% + C (1.03)24
C = 115.
2. Let i