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Calculus of One Variable

2003–2021

Revised February 2021

Table of contents

Acknowledgements 1

Introduction 2

1 Real and Complex Numbers 5

1.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Number Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 The real number line – Intervals . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4 Complex numbers - Cartesian form . . . . . . . . . . . . . . . . . . . . . . . 11

1.5 Arithmetic in Cartesian form . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.6 The set of complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2 Polar Forms of Complex Numbers 26

2.1 Standard Polar form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.2 Polar exponential form - Euler’s formula . . . . . . . . . . . . . . . . . . . . 33

2.3 Arithmetic in polar form . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.4 Roots of complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.5 Roots of polynomial equations . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.6 Sine and cosine in terms of exponentials . . . . . . . . . . . . . . . . . . . . 45

2.7 Complex exponential function . . . . . . . . . . . . . . . . . . . . . . . . . 46

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3 Functions 52

3.1 Functions – definitions and examples . . . . . . . . . . . . . . . . . . . . . . 52

3.2 Combining functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.3 Injective and inverse functions . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.4 Inverse trigonometric functions . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.5 Hyperbolic functions and their inverses . . . . . . . . . . . . . . . . . . . . 63

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4 Limits and Continuity 70

4.1 Informal definition of limit . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.2 One-sided limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.3 The basic limit laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.4 Limits at infinity – Horizontal asymptotes . . . . . . . . . . . . . . . . . . . 76

4.5 Infinite limits – Vertical asymptotes . . . . . . . . . . . . . . . . . . . . . . 78

iii

4.6 The squeeze law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.7 Continuous and discontinuous functions . . . . . . . . . . . . . . . . . . . . 82

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5 Differentiation 90

5.1 The derivative at a point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.2 The derivative as a function . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.3 Basic rules of differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.4 The chain rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.5 Implicit differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.6 The Mean Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6 Applications of Differentiation 103

6.1 Optimizing functions of one variable . . . . . . . . . . . . . . . . . . . . . . 103

6.2 Increasing and decreasing functions . . . . . . . . . . . . . . . . . . . . . . 107

6.3 Concavity and points of inflection . . . . . . . . . . . . . . . . . . . . . . . 111

6.4 Curve sketching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6.5 L’H?pital’s rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

7 Taylor Polynomials 123

7.1 An approximation for ex . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

7.2 Taylor polynomials about x= 0 . . . . . . . . . . . . . . . . . . . . . . . . . 126

7.3 Taylor polynomials about x= a . . . . . . . . . . . . . . . . . . . . . . . . . 129

7.4 Taylor’s formula – The remainder term . . . . . . . . . . . . . . . . . . . . . 131

7.5 How good is the Taylor polynomial approximation? . . . . . . . . . . . . . . 132

7.6 Proof of the remainder formula . . . . . . . . . . . . . . . . . . . . . . . . . 134

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

8 Taylor Series 137

8.1 Infinite series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

8.2 Taylor series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

8.3 Euler’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

8.4 The binomial series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

8.5 A series for the inverse tan function . . . . . . . . . . . . . . . . . . . . . . 148

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

9 The Riemann Integral 151

9.1 Riemann sums – The area problem . . . . . . . . . . . . . . . . . . . . . . . 151

9.2 The Riemann integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

9.3 Calculating Riemann sums . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

9.4 Properties of the Riemann integral . . . . . . . . . . . . . . . . . . . . . . . 159

iv

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

10 Fundamental Theorem of Calculus 164

10.1 Integrals as functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

10.2 The Fundamental Theorem of Calculus I . . . . . . . . . . . . . . . . . . . . 166

10.3 The Fundamental Theorem of Calculus II . . . . . . . . . . . . . . . . . . . 167

10.4 Leibniz Integral Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

10.5 The natural logarithm and exponential functions . . . . . . . . . . . . . . . . 171

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

11 Integration Techniques 178

11.1 Basic rules of integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

11.2 Integration by substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

11.3 Integration by parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

11.4 Partial fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

12 Applications of Integration 193

12.1 Further integration techniques . . . . . . . . . . . . . . . . . . . . . . . . . 193

12.2 Length of a curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

12.3 Area between two curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

12.4 Solids of revolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

A Formal Definition of Limits 208

B Geometric proof that limx→0 sinx/x= 1 212

C Linear approximations and differentials 214

D The Distance Problem 219

E Growth Rates 223

F Table of Standard Integrals 225

G Answers to Selected Exercises 226

v

Acknowledgements

The material in these notes has been developed over many years by the following members

of the School of Mathematics and Statistics:

Eduardo Altmann Mary Myerscough

Sandra Britton Nigel O’Brian

Chris Durrant Sharon Stephen

Dave Galloway Fernando Viera

Jenny Henderson Haotian Wu

Andrew Mathas

1

Introduction

Calculus is one of the major achievements of the 17th century. It plays a key role in almost

every instance where mathematics is applied in the sciences, engineering, or in economics.

Without calculus, we would not have cars, computers, televisions or mobile phones; Einstein

would never have penned his theory of relativity; we would not know of the existence of

DNA; we would never have landed on the moon. The list goes on.

Whether you end up continuing in mathematics or majoring in another field it will be im-

portant for you to learn and understand the meaning of calculus. The reason for this is quite

simple. In high school you can progress simply by memorizing formulas; in university there

will be times when you need to develop formulas for yourself, and this is where a proper

understanding of calculus will be a definite asset.

These notes are intended to supplement the lectures ofMATH1021. Your lecturers will almost

certainly use different examples and they will also explain some of the material in the course

slightly differently from these notes. In some places these notes go into more detail than your

lectures; at other times your lecturer will go into more detail.

Reading mathematics is not like reading a novel; we have to think and struggle with every

sentence. We are professional mathematicians and we are not ashamed to say that in our

research there have been times when we have spent more than a day trying to understand a

single line of mathematics! You will be pleased to know that in this course you should not

have to spend this long on a single sentence; however, there will be times when you do have

to think quite hard to understand what is going on. If you do get stuck then go and ask your

lecturer or tutor to explain it to you!

In addition to thinking when you read mathematics you should also work through the calcu-

lations yourself using pen and paper.

At some places in the notes and in the Appendices we have included material which is more

“advanced” than we expect you to know or understand. You are free to either read these

sections or skip over them, as you wish.

Tutorial problems and Exercise sheets

There are plenty of problems with full solutions for you to practice.

a) Worked examples with full solutions have been included in these lecture notes

throughout all chapters.

b) Exercises are available at the end of the chapters in these notes and answers to Selected

Exercises can be found in Appendix G.

2

Introduction 3

c) Exercise sheets containing problems to be solved before the tutorial session are avail-

able on the MATH1021 web page. Full solutions will be available online at the end of

the corresponding week.

A detailed list of mathematical objectives (knowledge, understanding and skills) for a

given chapter is provided in the weekly Exercise Sheets.

d) Board tutorial sheets will be handed out during tutorials with problems to be solved

during the tutorial class. Full solutions will be available online at the end of the corre-

sponding week.

e) Solutions will be provided to assignments 1 and 2.

f) Questions and solutions to selected past exam papers will be made available near the

end of semester.

Why study mathematics

The study of mathematics enhances your ability to think logically and an-

alytically, move from the particular to the general, work quantitatively and

improve problem-solving skills. By reading and working carefully through

the material in these notes you will develop the following additional generic

skills:

Generalise simple and familiar ideas to more complex settings.

Use geometric/visual techniques to help understand new concepts.

Apply simple techniques in unfamiliar situations.

Estimate values by using suitable approximation techniques.

Recognise that bounds on the error are an important part of any good

approximation.

A note about definitions

A mathematical definition is a precise description of some mathematical con-

cept. Historically, many concepts in mathematics have been used extensively

before a precise definition of the concept has been formulated.

While precision in definitions is certainly important, learning a definition off

by heart, without an understanding of the concept, is unlikely to be helpful.

It is important to spend some time thinking about a definition in order to gain

this understanding.

4 MATH 1021 Calculus of One Variable

C H A P T E R 1

Real and Complex Numbers

Mathematics includes not only the study of logic, structure and geometry, but also ideas about

numbers. Real numbers in particular, are fundamental to calculus and many other branches

of mathematics. In this chapter we review the concepts of sets and extend previous work on

numbers, particularly the real numbers, before introducing the set of complex numbers.

1.1 Sets

Set notation is a convenient and precise way to write about collections of numbers. We start

by talking about general sets.

Definition

A "set" is a collection of objects which are called "members" or "elements"

of the set.

Example 1.1a A set can be written as a list, for example, A= {a,b,c,d}, where

A is the name of the set,

a,b,c,d are the elements of the set enclosed in braces and separated by commas.

If the list of elements is large, three dots may be used to mean ’and so on’. For example,

the set of natural numbers may be denoted by N= {0,1,2,3, . . .}.

1.2 Number Sets

Our understanding of numbers, what they are and how they work, develops from simple

counting through fractions and negative numbers to an appreciation of irrational numbers

and real numbers. Mathematically, different types of numbers belong to different sets.

5

6 MATH 1021 Calculus of One Variable

The set of "natural numbers" {0,1,2,3,4, . . .}, is denoted by the symbol N. It is closed

under the operations of addition and multiplication. That is, adding two natural numbers

gives another natural number, as does multiplying them together.

The set of "integers" {. . . ,?4,?3,?2,?1,0,1,2,3,4, . . .}, denoted by Z, is the set of

whole numbers, including both positive whole numbers, negative whole numbers and zero.

The set of integers is closed under the operations of addition, subtraction and multiplication.

The set of "rational numbers", denoted by Q, is the set of all numbers of the form n/m

where n and m are integers and m 6= 0. Some examples are 1

. Rational numbers

include decimals which either terminate or repeat. Note that the integers are a subset of the

rational numbers, since they are of the form n/m where m = 1. The set of rational numbers

is closed under the operations of addition, subtraction, multiplication and division, provided

that division by zero is excluded.

The set of "real numbers", denoted by R, includes all rational numbers and all irrational

numbers. Irrational numbers cannot be expressed as n/m, where m and n are integers, al-

though some may be interpreted geometrically. For example,

√

2 is the length of a diagonal

of a unit square. The irrational number pi is the ratio of the circumference of a circle to the

circle’s diameter.

The set of "complex numbers", denoted by C, contains all the other number sets mentioned

above and all the imaginary numbers to be introduced in Section 1.4.

In fact we can summarise these numbers sets diagrammatically as shown in Figure 1.1.

Natural Numbers, N

0, 1, 2, 3, . . .

Integers, Z

. . . ,?2,?1,

Rational Numbers, Q

eg. 1

2

,?4

3

, 0.1,?7.2, 0.3˙

Real Numbers, R

eg. pi, e,?√5, 0.1010010001 . . .

Complex Numbers, C

eg. 1+2 i, 3 i . . . , with i2 =?1

Figure 1.1: Number Sets

Chapter 1: Real and Complex Numbers 7

Set notation

Element of a set – The symbol ∈ means “is an element of”. For example, ?3 ∈ Z is

read as “3 is an element of the set of integers”; y ∈ B is read as “y is an element of

the set B” or “y is a member of the set B”.

Subset of a set – The symbol? should be read as “is a subset of”. For example, N?Z

is read as “the set of natural numbers is a subset of the set of integers” or “the set of

natural numbers is contained in the set of integers”.

Strictly a subset – Sometimes you may see the symbol?which means that the smaller

set is strictly a subset of the larger; the two cannot be equal. For example, it is most

precise to write N? Z as the two sets are not the same.

Contains a set – The reversed symbol?means “contains”. For example, R?Q reads

“the set of real numbers contains the set of rational numbers”. (There is also a symbol

which means “contains, but is not equal to”.) If A? B then B? A.

Not an element of a set – A forward slash through any of these symbols above means

“not”. For example, ?1 6∈ N is read as “?1 is not an element of the set of natural

numbers”.

Not a subset – Another example, R 6? Z, is read as “the set of real numbers is not a

subset of the set of integers.”

There are other symbols which describe sets formed from other sets:

Union of sets – The expression A∪B denotes the union of set Awith set B. The "union"

of two sets is the set of elements which are members of either one or both of the sets.

If an element occurs in both sets, it is only listed once in the union. For example

{1,2,3,4,}∪{3,4,5,6}= {1,2,3,4,5,6}

Intersection of sets – The intersection of sets A and B is written A∩B. The "inter-

section" of two sets is the set of elements which are members of both of the sets. For

example

{1,2,3,4,}∩{3,4,5,6}= {3,4}

Subtraction of sets – The symbol \ which is read “minus” or “without”, is used to

indicate the set of elements which are in one set but not in another. That is, A\B is the

set of all elements which are in A but not in B. So for example,

{1,2,3,4,}\{3,4,5,6}= {1,2}.

8 MATH 1021 Calculus of One Variable

Venn Diagrams

A set can be represented in a simple, graphical way by a "Venn diagram". Each set is drawn

as a circle, a square or some other closed shape. Shapes representing sets may overlap one

another if sets have elements in common. Sometimes, the elements of the sets are written on

the Venn diagram but often they are not. Different parts of a Venn diagram can be shaded to

illustrate different parts of the set.

Venn diagrams are a useful way to represent relations between sets. Note that A\B is not the

same as B\A.

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