Fundamental Theorem of Calculus PDF

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Contents
1 Introduction 1
1.1 How to Use the Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Keys to Success in Studying Mathematics . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Preparing for the Examination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Functions and Models 4
2.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Learning Outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.3 Principles of Problem Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.5 The Way Forward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3 Limits and Derivatives 7
3.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.2 Learning Outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.3 Prescribed Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.4 Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.4.1 Introduction to the Limit Concept . . . . . . . . . . . . . . . . . . . . . . . 8
3.4.2 Definition of a Limit: Left- and Right-hand Limits . . . . . . . . . . . . . . 10
3.5 Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.5.1 Limits as x → c (c ∈ R) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.5.2 Limits as x → ±∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.5.3 Limits Involving Absolute Values . . . . . . . . . . . . . . . . . . . . . . . 18
3.5.4 Left-hand and Right-hand Limits . . . . . . . . . . . . . . . . . . . . . . . 21
3.5.5 Limits Involving Trigonometric Functions . . . . . . . . . . . . . . . . . . . 24
3.5.6 The Squeeze Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.5.7 The ε-δ Definition of a Limit (Read only for other modules eg MAT2615) . 33
3.5.8 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4 Differentiation Rules 46
4.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.2 Learning Outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.3 Prescribed Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.4 The Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.4.1 Introducing the Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.4.2 Definition of the Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.5 Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.5.1 Differentiation from First Principles (derivative as a Function) . . . . . . . 51
4.5.2 Basic Differentiation Formulas . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.5.3 Derivatives of Trigonometric Functions and Inverse Trigonometric Functions 61
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4.5.4 Derivatives of Exponential and Logarithmic Functions . . . . . . . . . . . . 66
4.5.5 Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.5.6 Implicit Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.5.7 Tangents and Normal Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.5.8 The Mean Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5 Integrals 93
5.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.2 Learning Outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.3 Prescribed Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.4 Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.4.1 Antiderivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.4.2 The Definite Integral and the Fundamental Theorem of Calculus – Part II . 98
5.4.3 The Definite Integral and the Area Between the Curve and the x-axis . . . 99
5.4.4 The Definite Integral and Area Under the Curve . . . . . . . . . . . . . . . 102
5.5 The Mean Value Theorem for Definite Integrals . . . . . . . . . . . . . . . . . . . . 107
5.6 The Fundamental Theorem of Calculus – Part I . . . . . . . . . . . . . . . . . . . . 109
5.7 Integration in General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.8 Indefinite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.9 The Substitution Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.10 Integration of Exponential and Logarithmic Functions . . . . . . . . . . . . . . . . 124
5.11 Review of Formulas and Techniques of Integration . . . . . . . . . . . . . . . . . . 125
6 Differential Equation, Growth and Decay and Partial Derivatives/Chain Rule 132
6.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
6.2 Learning Outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
6.3 Prescribed Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.4 Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.4.1 Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.4.2 Growth and Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
6.4.3 Partial Derivatives/Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . 153
A Sequence and Summation Notation 169
B Mathematical Induction 174
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Contents
MAT1512 v

Study Unit 1
Introduction
Calculus is a set of formal rules and procedures. It gives you the tools you need to
measure changes both qualitatively and quantitatively. Wikipedia (www.wikipedia.org
) defines calculus as a branch of mathematics that includes the study of limits, deriva-
tives, integrals, and infinite series, which constitute a major part of modern university
education. Calculus has widespread applications in science and engineering and is used
to solve complex and expansive problems for which algebra alone is insufficient. It builds
on analytical geometry and mathematical analysis and includes two major branches –
differential calculus and integral calculus – which are related through the Fundamental
Theorem of Calculus. Differential calculus explores and analyses rates of change quanti-
tatively and qualitatively. Integral calculus deals with the analysis (quantitatively and/or
qualitatively) of how quantities or measures of values accumulate or diminish over time.
The two processes – differentiation and integration – are reciprocal.
The purpose of this module is to equip you, the student, with those basic skills in differential
and integral calculus that are essential for the physical, life and economic sciences. Most
of the time you will be dealing with functions. Basically, a function is a generalised input-
output process that defines the mapping of a set of input values to a set of output values.
It is often defined as a rule for obtaining a numerical value from another given numerical
value. You are also going to have to develop a very large repertoire of methods for depicting
functions graphically/geometrically.
This course is built around your prescribed book. The purpose of this study guide is to
guide you through those parts of the prescribed book that you must study for this module,
and to provide you with many additional worked examples. The prescribed book is:
James Stewart Calculus
Metric Version 8 Edition
Early Transcendentals
Cengage Learning
ISBN 13:978-1-305-27237-8
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1.1 How to Use the Study Guide
From now on we will refer to the prescribed book as “Stewart”. The study guide must
always be used in conjunction with the prescribed textbook, because it is not a complete set
of notes on the book. Chapters 1 to 4 of this study guide contain many additional worked
problems, taken from past examination papers and assignments. Before going through
them, study the relevant parts in Stewart, and do the examples and some of the exercises
in Stewart. Also, before going through our solution to a problem in the study guide, try
solving it yourself. Remember that reading maths often means reading the same
thing over and over again.
We have included numerous worked examples for you. These are designed to stimulate
your thinking in such a way that you will come to appreciate and master the delicate
beauty and intricacies of the subject. All you have to do is keep going! Follow all the
instructions given. Try to write down all the answers to the activities in full. This is
extremely important, as a major part of learning mathematics is thinking and writing
down what you think. By writing everything down, you will develop the essential skill
of communicating mathematics effectively. The other reason for writing down your
answers, is to prevent you from losing your train of thought about a concept or mathemat-
ical idea. If this happens, it takes a while to get your reasoning back to the same point.
If you have everything written down, you can also go back the next day and check your
reasoning. Learning mathematics is an activity, and you will only learn by doing. Because
you will be thinking about the problems, you will, in most cases, be able to determine by
yourself whether your reasoning is right, or not.
Your assignments are included in Tutorial Letter 101. Attempt these once you have com-
pleted all the related activities in your study guide. Spend a part of your time each day
doing some of the questions, and a part studying new material. Being able to do an as-
signment is proof that you have mastered the work of that particular section. In Tutorial
Letter 101 we have indicated which assignments you should submit for evaluation.
Also remember that statements, theorems and definitions are the building blocks of your
mathematical language – you cannot learn anything without knowing the basic facts. Begin
by reading the preface of your prescribed book. This should give you a good idea of the
importance of the subject you are about to study.
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