Contents
Preface 3
7 One-Variable Calculus, Part 5 of 7 4
7.1 The inverse function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
7.2 Some elementary bijective functions and their inverses . . . . . . . . . . . . 6
7.3 The derivative of an inverse function . . . . . . . . . . . . . . . . . . . . . 12
7.4 The local inverse of a non-bijective function . . . . . . . . . . . . . . . . . 14
7.5 Exercises for self study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
7.6 Relevant sections from the textbooks . . . . . . . . . . . . . . . . . . . . . 17
8 One-Variable Calculus, Part 6 of 7 17
8.1 Indefinite and definite integrals . . . . . . . . . . . . . . . . . . . . . . . . 17
8.2 The fundamental theorem of calculus . . . . . . . . . . . . . . . . . . . . . 18
8.3 Primitives of elementary functions . . . . . . . . . . . . . . . . . . . . . . . 19
8.4 Integration by recognition . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
8.5 Integration by change of variable . . . . . . . . . . . . . . . . . . . . . . . 23
8.6 Improper integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
8.7 Exercises for self study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
8.8 Relevant sections from the textbooks . . . . . . . . . . . . . . . . . . . . . 28
9 One-Variable Calculus, Part 7 of 7 29
9.1 Integration by partial fractions . . . . . . . . . . . . . . . . . . . . . . . . . 29
9.2 Integration by parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
9.3 Consumers’ and producers’ surplus . . . . . . . . . . . . . . . . . . . . . . 36
9.4 Exercises for self study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
9.5 Relevant sections from the textbooks . . . . . . . . . . . . . . . . . . . . . 38
10 Matrices, Part 1 of 3 38
10.1 Definitions, notation and terminology . . . . . . . . . . . . . . . . . . . . . 38
10.2 Operations on matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
10.3 The laws of matrix algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
10.4 The inverse matrix and its properties . . . . . . . . . . . . . . . . . . . . . 45
10.5 Powers of a matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
10.6 Properties of the transpose of a matrix . . . . . . . . . . . . . . . . . . . . 46
10.7 Matrix equations and their solutions . . . . . . . . . . . . . . . . . . . . . 47
10.8 Exercises for self study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
10.9 Relevant sections from the textbooks . . . . . . . . . . . . . . . . . . . . . 48
1
,11 Matrices, 2 of 3 49
11.1 Solving systems of n linear equations for n unknowns . . . . . . . . . . . . 49
11.2 Elementary row operations . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
11.3 Elementary matrices and row-equivalence . . . . . . . . . . . . . . . . . . . 54
11.4 Theorems on matrix invertibility . . . . . . . . . . . . . . . . . . . . . . . 57
11.5 Inversion algorithm based on elementary row operations . . . . . . . . . . . 59
11.6 Exercises for self study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
11.7 Relevant sections from the textbooks . . . . . . . . . . . . . . . . . . . . . 62
12 Matrices, 3 of 3 62
12.1 Minors, cofactors and the determinant . . . . . . . . . . . . . . . . . . . . 62
12.2 The properties of the determinant . . . . . . . . . . . . . . . . . . . . . . . 66
12.3 Calculating determinants using row operations . . . . . . . . . . . . . . . . 69
12.4 Inverting a matrix using cofactors and the determinant . . . . . . . . . . . 71
12.5 Cramer’s rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
12.6 Exercises for self study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
12.7 Relevant sections from the textbooks . . . . . . . . . . . . . . . . . . . . . 75
2
,Preface
These lecture notes have been designed as the main study resource for the MA100 Mathe-
matical Methods course at the LSE. At the same time, they complement the MA100 course
texts, ‘Linear Algebra, Concepts and Methods’ by Martin Anthony and Michele Harvey,
and ‘Calculus, Concepts and Methods’ by Ken Binmore and Joan Davies. I am grateful
to Martin Anthony and Michele Harvey for allowing me to use some materials from their
‘Linear Algebra, Concepts and Methods’ textbook and to Michele Harvey for commenting
on a draft of the Calculus part of these lecture notes. I am also grateful to Siri Kouletsis for
her invaluable help with typing and editing the manuscript and for various improvements
to its content.
3
, 7 One-Variable Calculus, Part 5 of 7
7.1 The inverse function
Let f : X → Y be a bijective function. The sets X and Y are assumed to be subsets of R.
There is a unique function f −1 : Y → X, called the inverse function of f , which is
defined by the property that
y = f (x) if and only if x = f −1 (y).
This definition is equivalent to saying that the composite function f −1 ◦ f : X → X is the
identity function on X and the composite function f ◦ f −1 : Y → Y is the identity function
on Y ; that is,
f −1 (f (x)) = x for all x ∈ X
and
f (f −1 (y)) = y for all y ∈ Y.
Figure 7.1.1
Example 7.1.2 Consider the function f : [0, 5) → [3, 13) defined by
f (x) = 2x + 3
and let the function g : [3, 13) → [0, 5) be defined by
x−3
g(x) = .
2
Show that g is the inverse of f .
Using the definition of an inverse function, we need to prove that the following statements
are both true:
if y = f (x), then x = f −1 (y)
and
if x = f −1 (y), then y = f (x).
4
Preface 3
7 One-Variable Calculus, Part 5 of 7 4
7.1 The inverse function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
7.2 Some elementary bijective functions and their inverses . . . . . . . . . . . . 6
7.3 The derivative of an inverse function . . . . . . . . . . . . . . . . . . . . . 12
7.4 The local inverse of a non-bijective function . . . . . . . . . . . . . . . . . 14
7.5 Exercises for self study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
7.6 Relevant sections from the textbooks . . . . . . . . . . . . . . . . . . . . . 17
8 One-Variable Calculus, Part 6 of 7 17
8.1 Indefinite and definite integrals . . . . . . . . . . . . . . . . . . . . . . . . 17
8.2 The fundamental theorem of calculus . . . . . . . . . . . . . . . . . . . . . 18
8.3 Primitives of elementary functions . . . . . . . . . . . . . . . . . . . . . . . 19
8.4 Integration by recognition . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
8.5 Integration by change of variable . . . . . . . . . . . . . . . . . . . . . . . 23
8.6 Improper integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
8.7 Exercises for self study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
8.8 Relevant sections from the textbooks . . . . . . . . . . . . . . . . . . . . . 28
9 One-Variable Calculus, Part 7 of 7 29
9.1 Integration by partial fractions . . . . . . . . . . . . . . . . . . . . . . . . . 29
9.2 Integration by parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
9.3 Consumers’ and producers’ surplus . . . . . . . . . . . . . . . . . . . . . . 36
9.4 Exercises for self study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
9.5 Relevant sections from the textbooks . . . . . . . . . . . . . . . . . . . . . 38
10 Matrices, Part 1 of 3 38
10.1 Definitions, notation and terminology . . . . . . . . . . . . . . . . . . . . . 38
10.2 Operations on matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
10.3 The laws of matrix algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
10.4 The inverse matrix and its properties . . . . . . . . . . . . . . . . . . . . . 45
10.5 Powers of a matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
10.6 Properties of the transpose of a matrix . . . . . . . . . . . . . . . . . . . . 46
10.7 Matrix equations and their solutions . . . . . . . . . . . . . . . . . . . . . 47
10.8 Exercises for self study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
10.9 Relevant sections from the textbooks . . . . . . . . . . . . . . . . . . . . . 48
1
,11 Matrices, 2 of 3 49
11.1 Solving systems of n linear equations for n unknowns . . . . . . . . . . . . 49
11.2 Elementary row operations . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
11.3 Elementary matrices and row-equivalence . . . . . . . . . . . . . . . . . . . 54
11.4 Theorems on matrix invertibility . . . . . . . . . . . . . . . . . . . . . . . 57
11.5 Inversion algorithm based on elementary row operations . . . . . . . . . . . 59
11.6 Exercises for self study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
11.7 Relevant sections from the textbooks . . . . . . . . . . . . . . . . . . . . . 62
12 Matrices, 3 of 3 62
12.1 Minors, cofactors and the determinant . . . . . . . . . . . . . . . . . . . . 62
12.2 The properties of the determinant . . . . . . . . . . . . . . . . . . . . . . . 66
12.3 Calculating determinants using row operations . . . . . . . . . . . . . . . . 69
12.4 Inverting a matrix using cofactors and the determinant . . . . . . . . . . . 71
12.5 Cramer’s rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
12.6 Exercises for self study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
12.7 Relevant sections from the textbooks . . . . . . . . . . . . . . . . . . . . . 75
2
,Preface
These lecture notes have been designed as the main study resource for the MA100 Mathe-
matical Methods course at the LSE. At the same time, they complement the MA100 course
texts, ‘Linear Algebra, Concepts and Methods’ by Martin Anthony and Michele Harvey,
and ‘Calculus, Concepts and Methods’ by Ken Binmore and Joan Davies. I am grateful
to Martin Anthony and Michele Harvey for allowing me to use some materials from their
‘Linear Algebra, Concepts and Methods’ textbook and to Michele Harvey for commenting
on a draft of the Calculus part of these lecture notes. I am also grateful to Siri Kouletsis for
her invaluable help with typing and editing the manuscript and for various improvements
to its content.
3
, 7 One-Variable Calculus, Part 5 of 7
7.1 The inverse function
Let f : X → Y be a bijective function. The sets X and Y are assumed to be subsets of R.
There is a unique function f −1 : Y → X, called the inverse function of f , which is
defined by the property that
y = f (x) if and only if x = f −1 (y).
This definition is equivalent to saying that the composite function f −1 ◦ f : X → X is the
identity function on X and the composite function f ◦ f −1 : Y → Y is the identity function
on Y ; that is,
f −1 (f (x)) = x for all x ∈ X
and
f (f −1 (y)) = y for all y ∈ Y.
Figure 7.1.1
Example 7.1.2 Consider the function f : [0, 5) → [3, 13) defined by
f (x) = 2x + 3
and let the function g : [3, 13) → [0, 5) be defined by
x−3
g(x) = .
2
Show that g is the inverse of f .
Using the definition of an inverse function, we need to prove that the following statements
are both true:
if y = f (x), then x = f −1 (y)
and
if x = f −1 (y), then y = f (x).
4
