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 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, 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, 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
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
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
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, 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 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 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
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
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
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, Lecture Thirty-Three
33. Multivariate Calculus, 5 of 5
In this subsection we will focus on constrained optimisation problems; namely, problems where the function f : Rn R is optimised on a proper subset D Rn dened
by imposing certain
Mathematical Methods
Exercises and Solutions for Lectures 31 and 32
Exercise 31.9.1 Let f (x, y, z) = 3x2 + 2y 2 z 2 .
(a) Find
f
f f
,
and
.
x y
z
(b) Obtain a Cartesian equation in R4 for the tangent hyperplane to the hypersurface
u = 3x2 + 2y 2 z 2 at
Mathematical Methods
Exercises and Solutions for Lectures 25 and 26
Exercise 25.4.1 Consider two bases B and C for R3 :
B = cfw_f 1 , f2 , f3
and
C = cfw_g1 , g2 , g3 .
The transition matrix PBC from B-coordinates to C-coordinates is dened by the
proper
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, 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-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-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-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 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
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 19 and 20
Exercise 19.5.1 For each of the sets Si given below, nd a basis of the vector
space Lin(Si ) and state its dimension. Provide a Cartesian description for any proper
subspace of R2 or R3 (
Mathematical Methods
Exercises and Solutions for Lectures 17 and 18
Exercise 17.4.1 Consider the set V of positive real numbers
V = vR v>0 .
Dene the operations of vector addition and scalar multiplication as follows:
For u V, v V , vector addition is den
Mathematical Methods
Exercises and Solutions for Lecture 39
Exercise 39.3.1 (a) Find the general solution of the following system of linear
dierential equations:
x (t) = x(t) + 4y(t)
y (t) = 3x(t) + 2y(t)
(b) Then nd the unique solution satisfying the ini
Mathematical Methods
Exercises and Solutions for Lectures 13 and 14
Exercise 13.6.1 Consider the vectors
u=
4
1
and
v=
2
.
2
(a) Represent the vectors u, v and their sum u + v by position vectors in R2 .
(b) Calculate the following lengths: |u|, |v|, |u +
MA100 Regarding the Final Exam
There will only be one MA100 exam. This will be held in May and will be
suitable for all candidates, including candidates retaking.
This years course is a restructured course. Although there are no big dierences in the mat