Mathematical Methods
Exercises and Solutions for Lectures 9 and 10
Exercise 9.5.1 Use the method of partial fractions to calculate the integral
x3 x2 + x + 1
dx
(x2 + 1)2
and the method of integration by parts to calculate the integral
1
arcsin(x) dx.
0
S
Mathematical Methods
Exercises and Solutions for Lectures 21 and 22
Exercise 21.4.1 Consider the set of vectors X = cfw_v1 , v2 , v3 , v4 where
1
v1 = 2
4
2
1
v2 =
3
1
v3 = 7
29
9
6 .
v4 =
8
(a) Find the reduced row echelon form of the matrix A = (
Mathematical Methods
Exercises and Solutions for Lectures 23 and 24
Exercise 23.6.1 For the following linear transformation T , nd a basis for the
kernel of T , ker(T ), and the range of T , R(T ). Obtain a Cartesian description and
a vector parametric de
Mathematical Methods, Lecture Twenty-Seven
27. Linear Transformations, 5 of 6
27.1 Diagonalisation
In the last lecture we saw examples of linear transformations T : R2 R2 where a
basis B of R2 can be found such that the matrix ABB representing T is diagon
Mathematical Methods
Exercises and Solutions for Lectures 27 and 28
3 1 2
Exercise 27.4.1 Consider the matrix AT = 5 3 5 representing a linear
1 1 2
transformation T : R3 R3 by T (x) = AT x.
(a) Find a basis B = cfw_f1 , f2 , f3 for R3 such that
T (f1 )
Mathematical Methods, Lecture Twenty-Eight
28. Linear Transformations, 6 of 6
We now focus on symmetric matrices and a special form of diagonalisation applicable
to symmetric matrices, known as orthogonal diagonalisation. We begin by introducing orthogona
Mathematical Methods
Exam-Style Questionsa.nd Solutions for Lectures
32 and 33
Exam-style question 32.9
R3 R dened by
(a) Find the stationary points of the function h :
h(x, y, z) = 9z + 6xy 8x3 3y 2 3z 3 .
Classify each of them as a local maximum, a loca
Mathematical Methods
Exercises and Solutions for Lectures 33 and 34
Exercise 33.7.1 The production function P for a particular manufacturer has the
Cobb-Douglas form
P (x, y) = 100x3/5 y 2/5
where the variables x and y represent labour and capital, respec
Mathematical Methods, Lecture Three
3. One-Variable Calculus, Part 1 of 7
3.1 Sets, subsets and intervals
A set is a collection of distinct objects. These objects are called the elements of
the set. A set can be dened either by listing its elements or by
Mathematical Methods
Exercises and Solutions for Lectures 3 and 4
Exercise 3.8.1
Consider the function
f : R [3, ) dened by f (x) = x2 + 7.
(i) Find the domain, codomain and range of f .
(ii) Sketch the graph of f on the Cartesian plane and write down a C
Mathematical Methods
Exercises and Solutions for Lectures 11 and 12
Exercise 11.7.1 (a) Use elementary row operations to solve the linear system
2x + 5y + z = 4
6x + 15y + 2z = 9
z = 3.
(b) Then use the algorithm based on elementary row operations to inve
Mathematical Methods
Exercises and Solutions for Lectures 35 and 36
Exercise 35.6.1 (a) Show that the sequence
yt = mt + mt
where m = 1 +
3i, satises the dierence equation
yt+2 2yt+1 + 4yt = 0
for arbitrary complex constants and .
(b) Show that yt can be
Mathematical Methods
Exercises and Solutions for Lectures 37 and 38
Exercise 37.5.1 (a) Find the particular solution of the equation
d2 y
dy
2 + 10y = 4
2
dx
dx
which satises the conditions
y(0) = 1
and
6
y
= 0.
(b) Describe the behaviour of this solutio
Mathematical Methods
Exam-Style Questions and Solutions for Lectures
34 and 35
Exam-style question 34.8 (a) A principal P is invested for t years at an annual
interest rate r. Write down expressions for the sum accumulated when the interest
is compounded
Mathematical Methods
Exam-Style Questions and Solutions for Lectures
36 and 37
Exam-style question 36.6 (a) Find the general solution of the dierence equation
yx+2 + 2yx+1 + 4yx = x2 .
(b) Describe the behaviour of the general solution as x .
(c) Find y4
Mathematical Methods
Exam-Style Questions and Solutions for Lectures
38 and 39
Exam-style question 38.11 (a) Show that the ordinary dierential equation
8xy 3 + cos(y) dx + 12x2 y 2 xsin(y) dy = 0
is exact.
(b) Hence nd its general solution in the form
F (
Mathematical Methods
Exam-Style Questions and Solutions for Lectures
30 and 31
Exam-style question 30.7 Consider a function h : R2 R and a point
(a, b, h(a, b) on the graph of h in R3 .
(a) Dene the directional derivative hu (a, b) in terms of the gradien
Mathematical Methods
Exam-Style Questions and Solutions for Lectures
24 and 25
Exam-style question 24.6 Let V be the vector space of all functions f : R R
of the form f (x) = a + bx + cx2 where vector addition and scalar mutiplication are
dened in the sta
Mathematical Methods, Lecture Fourteen
14. Developing Geometric Insight, 2 of 2
14.1 Visualising vectors and operations in R3
a
By analogy with R2 , R3 is dened as the set of all 31 column matrices b where
c
a
a, b and c are real numbers: R3 = b a R,
Mathematical Methods
Exam-Style Questions and Solutions for Lectures
26 and 27
Exam-style question 26.7 (a) Give the denition of an eigenvector of a square
matrix A and that of its corresponding eigenvalue.
Let T : R2 R2 be a linear transformation and con
Mathematical Methods
Exam-Style Questions and Solutions for Lectures
22 and 23
Exam-style question 22.8 (a) Given an m n matrix A, state the Rank-Nullity
theorem associated with A.
Consider the linear system Ax = b where
1 3 1 0
A = 2 6 0 2 ,
4 12 1 3
x
Mathematical Methods
Exam-Style Questions and Solutions for Lectures
20 and 21
Exam-style question 20.9 (i) Name the three properties that need to be satised
by an inner product , on an inner product space V .
(ii) Given a vector space V equipped with an
Mathematical Methods, Lecture Thirty-Eight
Practice Questions and Solutions for Lecture 38
Practice question 38.9.1 Find the general solution of the following equations by
any appropriate method.
(a) (x2 + 6x) dy = y 2 dx 12dy
(b) 2x
dy
=2 x+y+2
dx
Soluti
Mathematical Methods, Lecture Thirty-Five
Practice Questions and Solutions for Lecture 35
3i and w = 1+i as points on the complex
z6
plane. Express z and w in polar exponential form and nd q = 10 in both polar
w
exponential and Cartesian form.
Practice qu
Mathematical Methods, Lecture Thirty-Three
Practice Questions and Solutions for Lecture 33
Practice question 33.6.1 Consider the cost function C : R2 R dened by
C(x, y) = 4x2 + 4y 2 2xy 40x 140y + 1800
for a rm producing two goods x and y.
(a) Show that C
Mathematical Methods, Lecture Thirty-Seven
Practice Questions and Solutions for Lecture 37
Practice question 37.4.1 Write down a linear dierential equation P (D)y = 0
whose solutions include the function e2x and another such equation whose solution
includ
Mathematical Methods, Lecture Thirty-Six
Practice Questions and Solutions for Lecture 36
Practice question 36.4.1 (a) Find the general solution of the dierence equation
yx+2 yx+1 2yx = 4x.
(b) Consider the initial conditions y1 = 1 and y2 = 2. Generate y3
Mathematical Methods
Exercises and Solutions for Lectures 7 and 8
Exercise 7.6.1
(i) Show carefully that the derivative of f (x) = arccos(x) is given by
f (x) =
1
.
1 x2
(ii) For a > 1, nd the derivative of g(x) = ax .
Hint: Express g as a composite func
Mathematical Methods
Exercises and Solutions for Lectures 5 and 6
Exercise 5.8.1
Consider the function f : R R given by
f (x) = x3 3x.
(i) Find the stationary points of f (x). For each stationary point, determine whether
it is a local maximum, a local min
Mathematical Methods
Exercises and Solutions for Lectures 15 and 16
Exercise 15.6.1 (a) Write down the augmented matrix for each of the following
systems of equations, where (x, y, z) R3 , and solve the system by reducing the
augmented matrix to reduced r