UNIVERSITY OF GLASGOW
CALCULUS
1S
S C H O O L O F M AT H E M AT I C S A N D S TAT I S T I C S
Copyright 2015 AJW, after JJCN, TAW and others
published by university of glasgow
Last updated, March 2015
Course Contents
1
2
3
4
5
6
7
Maclaurin Series. . . .
UNIVERSITY OF GLASGOW
ALGEBRA
1S
S C H O O L O F M AT H E M AT I C S A N D S TAT I S T I C S
Copyright 2015 AJW, after JJCN, TAW and others
published by university of glasgow
Last updated, March 2015
Course Contents
1
2
3
4
5
Introduction to Vectors . .
C
Tuesday, 24th April, 2012
9.30 a.m. to 11.30 a.m.
EXAMINATION FOR THE DEGREES OF
M.A. AND B.Sc.
Mathematics 1S
An electronic calculator may be used provided that it does not have
a facility for either textual storage or display or for graphical display.
C
2011-12 Mathematics 1S Degree Exam - Marking Scheme
1. The angle between the vectors is
(1, 2, 1).(2, 1, 1)
|(1, 2, 1)|(2, 1, 1)|
cos1
3
= cos1
6 6
1
= cos1 = .
2
3
y 1 for formula, 1 for calculation
x nal answer
3
2. Integrate by parts twice
x2 sin x dx
Tuesday, 29th April, 2014
9.30 a.m. to 11.30 a.m.
EXAMINATION FOR THE DEGREES OF
M.A. AND B.Sc.
Mathematics 1S
An electronic calculator may be used provided that it does not have
a facility for either textual storage or display, or for graphical display.
2013-14 Mathematics 1S Degree Exam
1. The cosine of the angle, , between the given vectors is given by:
2+40
(2, 2, 0) (1, 2, 1)
3
=
=
.
cos() =
|(2, 2, 0)|(1, 2, 1)|
2
8 6
Hence the angle is = /6.
2. Integrate by parts
x3
x3 1
ln x
dx
3
3 x
x3
x2
=
ln
EXERCISES
1S
WEEK 2
Hand-in Exercises
Algebra
H 2.1 Suppose that u and v are unit vectors the angle between
which is /4. Let a = u + 3 2 v. By considering a a, nd |a|.
Solution We know that the dot product denes magnitude, that is,
a a = | a |2 .
Thus we
EXERCISES
1S
WEEK 1
Hand-in Exercises
Algebra
H 1.1 Let u be a vector of magnitude 4. What are
a) the vector with magnitude 3 and the same direction as u,
b) the vector with magnitude 20 and opposite direction to u,
c) the two unit vectors parallel to u?
1S
EXERCISES
WEEK 1
Hand-in Exercises
Algebra
H 1.1 Let u be a vector of magnitude 4. What are
a) the vector with magnitude 3 and the same direction as u,
b) the vector with magnitude 20 and opposite direction to u,
c) the two unit vectors parallel to u?
Mathematics 1S
Sample Workshop Questions 1
1. Consider a vector u with magnitude 3 and a perpendicular vector v with magnitude 4.
What is
(a) the unit vector with the opposite direction to u,
(b) the vector of length 9 with the same direction as v,
(c) th
2D degree examination May 2013 Solutions
1.(i)
t8
5
A (t) =
6 t + 3
= (t 8)(t + 3) + 30
= t2 5t + 6
= (t 2)(t 3).
So 2 and 3 are the eigenvalues of A.
" # " #
"
#" # " #
x
0
6 5
x
0
(2I A)
=
=
6x 5y = 0.
y
0
6 5
y
0
Therefore
"
"
(3I A)
x
y
#
5
#
6
"
CHAPTER 2
ORTHOGONAL DIAGONALISATION
See Anton 7.2.
Complex Conjugate
Let A be a complex matrix. The matrix A obtained from A by replacing each entry by its complex
conjugate is called the complex conjugate of A. For example,
"
#
"
#
2 + i 1 4i
2 i 1 + 4i
Ex Sheet
2
2D Topics in Linear Algebra and Calculus 2014
T1
Let
"
A=
4
2
2
7
#
.
Find an orthogonal matrix Q and a diagonal matrix D such that
QTAQ = D.
Solution
2
= (t 4)(t 7) 4 = t2 11t + 24 = (t 3)(t 8).
t7
t4
A (t) =
2
So 8 and 3 are the eigenv
Ex Sheet
1
2D Topics in Linear Algebra and Calculus 2014
Tutorial Exercises
T1
Solve the system of linear differential equations
x 1 = 5x1 2x2 ,
x 2 = x1 + 2x2 .
Solution
The system is
x = Ax,
where
"
A=
5
2
1
2
#
"
x=
and
x1
#
.
x2
= (t 5)(t 2) + 2 = t2
Ex Sheet
3
2D Topics in Linear Algebra and Calculus 2014
.
T1
Suppose that the quadratic form q in n variables can be defined
by both
n
q ( x1 , x2 , . . . , x n ) =
n
aij xi x j
i =1 j =1
and
n
q ( x1 , x2 , . . . , x n ) =
n
bij xi x j ,
i =1 j =1
whe
Ex Sheet
4
2D Topics in Linear Algebra and Calculus 2014
.
T1
Let be an eigenvalue of the real symmetric n n matrix A.
In Chapter 4, we took Eig A () to be the real -eigenspace of A. Now
take Eig A () to be the complex -eigenspace of A. In other words, le
Ex Sheet
6
2D Topics in Linear Algebra and Calculus 2014
T1
.
(i) Show that the Fourier Series of f defined on [, ] by
(
f (x) =
is
0
if 6 x < 0,
1
if 0 6 x 6
1
2
sin 3x sin 5x
+
+
+ .
sin x +
2
3
5
(ii) Sketch the graph of the sum function of this seri
University of Glasgow
School of Mathematics and Statistics
Mathematics 1S 20142015
Contents, Aims and Objectives
General aims of the course Mathematics 1S is intended to build on Mathematics 1R and to provide
a further half-years mathematics course which
EXERCISES
1S
WEEK 2
Hand-in Exercises
Algebra
H 2.1 Suppose that u and v are unit vectors the angle between
which is /4. Let a = u + 3 2 v. By considering a a, nd |a|.
H 2.2 Prove carefully that
( p + q ) ( p q ) = 2( q p )
using the properties of the vec