MA212: Further Mathematical Methods (Linear Algebra) 201617
Solutions to Exercises 10: Generalised inverses and Fourier series
This document contains answers to the exercises. There may well be some errors; please do let me
know if you spot any.
1.
If we
MG211.1: Solutions for Chapter 4
4.1 Setting up a new TV channel for TV station CCB involves a number of activities.
These are identified below, together with the cost per day for each activity (in 000) and
its the normal time to completion (in days). How
MG211.1: Solutions for Chapter 7
7.1. Revisit Exercise 1.4: What is the expected number of times you have to roll a die
to get a 6? Solve the problem by using Markov chains.
Solution. Let us first define the problem as a Markov chain. We have two states:
MG211.1 OPERATIONAL
RESEARCH TECHNIQUES
Lecture 1
Introduction
Course logistics & content
History of OR
Depts of Mathematics & Management
Applications of OR today
Office hours: Fri 14:30-16:30pm
Dr Lszl Vgh
Office: NAB3.05
[email protected]
Teaching
MG211
MG211.1: Solutions for Chapter 5
5.1. A consumer who purchases one of two brands of soap powder every week is influenced
by her choice of the previous week but not by earlier experience. If she purchased brand
A the previous week, her current purchase wou
MG211.1: Solutions for Chapter 3
In the solutions we provide the full description of Dijkstras algorithm in every step
using tables as in the lecture notes. You do not need to use this format on the exam or in
your submissions, and can use any other reaso
MG211.1: Solutions for Chapter 2
1. As in the notes we use n to denote the number of men (and women).
For part a), each woman will receive only one proposal in the first iteration, since
the most preferred woman is different for every man. Thus every woma
MA107: Quantitative Methods (Mathematics) 201617
Solutions for Exercises 5: Optimisation
Exercise 1. This question is similar to worked examples 8.1 and 8.2 of the Anthony & Biggs book.
(a) We are given the function f (x) = x3 9x2 + 26x 24 and we note tha
MA107: Quantitative Methods (Mathematics) 201617
Solutions for Exercises 3: Recurrence equations and limits
Exercise 1. Excise tax is discussed in section 1.4 of the Anthony & Biggs book. See also worked
examples 1.3, 1.4 and 1.5.
The supply and demand se
MA107: Quantitative Methods (Mathematics) 201617
Solutions for Exercises 4: The cobweb model and derivatives
Exercise 1. See worked example 4.3 of the Anthony & Biggs book. Given a principal, P , invested
at r% per annum, we know that after 1 year the bal
MA107: Quantitative Methods (Mathematics) 201617
Solutions for Exercises 6: Partial derivatives
Exercise 1. By now, the rules of differentiation should be more or less known. If not, see sections
6.2 and 6.4 of the Anthony & Biggs book. See also worked ex
MA107: Quantitative Methods (Mathematics) 201617
Solutions for Exercises 1: Revision
Exercise 1. We manipulate inequalities just like equations with the exception that multiplying (or
dividing) by a negative quantity reverses the direction of the inequali
MA212: Further Mathematical Methods (Calculus) 201617
Solutions to Exercises 3
1. (a) This requires evaluating successive derivatives of f (x) = ln x at x = 1. We see that
f (n) (x) = (1)n+1
(n 1)!
,
xn
for n 1. This means that the Taylor series is
X
f (j
MA212: Further Mathematical Methods (Calculus) 201617
Solutions to Exercises 4
1. This question is mostly about the Fundamental Theorem of Calculus, which says that
Z x
d
f (z) dz = f (x),
dx z=c
for any constant c, provided f is continuous at x.
This que
MA212: Further Mathematical Methods (Calculus) 201617
Solutions to Exercises 1
I think these answers are an important part of the course: do have a look through them even if you
and your class teacher agree that your answers were perfect.
The answer given
MA212: Further Mathematical Methods (Calculus) 201617
Solutions to Exercises 5
1. The first part of this question, which tests your understanding of the idea that an integral is an
antiderivative, is more-or-less exactly Question 2 from Exercises 4.
2
Wha
MA212: Further Mathematical Methods (Calculus) 201617
Solutions to Exercises 2
1. Plugging in a few values should suggest to you that the limit is likely to be 1. Sure, the evidence
could equally support the suggestion that the limit is 0.999999999567, bu
MA212: Further Mathematical Methods (Calculus) 201617
Exercises 6: Improper integrals
For these and all other exercises on this course, you must show all your working.
1. Sketch a graph of f (x) = sin2 ( x) for x 0. (Or use Maple.)
Z
f (x) dx converges.
MA212: Further Mathematical Methods (Calculus) 201617
Exercises 7: More improper integrals and the manipulation of proper integrals
For these and all other exercises on this course, you must show all your working.
1.
Determine whether each of the followin
MA212: Further Mathematical Methods (Calculus) 201617
Exercises 1: Assumed background
The lectures do not cover practical techniques for integration of functions of a single variable as
students on this course are supposed to be skilled in this already. H
1 Revision of Integration
Compute the following inde nite integrals
1. Quickies:
Z
Z
Z p
2 + ln x
x
cos x
p
dx;
dx;
dx;
x
(1 + x2 )2
1 + sin x
Z
Z
Z
sin x
sin2 x cos xdx;
tan xdx =
dx;
cos x
Z 0
Z
f (x)
f 0 (x)
p
dx;
dx
f (x)
f (x)
Note these are all of t
MA212: Further Mathematical Methods (Calculus) 201617
Exercises 2: Approximate behaviour and convergence
For these and all other exercises on this course, you must show all your working.
1
1. For x > 0, let f (x) = x ln 1 +
.
x
Evaluate f (x) for some lar
MA212: Further Mathematical Methods (Calculus) 201617
Exercises 3: Taylor series, more limits and the definition of the Riemann integral
For these and all other exercises on this course, you must show all your working.
1.
(a) Find the Taylor series expans
MA212: Further Mathematical Methods (Calculus) 201617
Exercises 5: FTC (again), transformations and more double integrals
For these and all other exercises on this course, you must show all your working.
1.
The function, p(x, y), of two variables is defin
MA212: Further Mathematical Methods (Calculus) 201617
Exercises 4: The fundamental theorem of calculus and double integrals
For these and all other exercises on this course, you must show all your working.
1.
Let f be a continuous function taking positive
A Quick Tour of Game
Theory
Paul Dutting
1
Non-cooperative game theory
Players motivated by individual incentives
Interactions resulting in payoffs
Explains:
Selfish but collectively damaging behavior
How to think strategically
More than one possible