CALCULUS
CALCULUS 1
,1 PRELIMINARIES FOR CALCULUS 1
1.1 Elementary Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Notation and Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.2 Some Special Number Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.3 Fundamental Operations on Sets . . . . . . . . . . . . . . . . . . . . . . . 4
2 LIMITS AND CONTINUITY OF FUNCTIONS 5
2.1 Functions and their Graphs .............................. 5
2.1.1 Domain and Range of Functions . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1.2 Some Elementary Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Limits of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.1 Informal Definition of a Limit . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.2 Formal ( ) Definition of a Limit . . . . . . . . . . . . . . . . . . . . .. 9
2.3 Properties of Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3.1 Right-Hand Limits and Left-Hand Limits . . . . . . . . . . . . . . . . . . . 14
2.3.2 Limits at Infinity and Infinite Limits . . . . . . . . . . . . . . . . . . . . . 16
2.3.3 Limits at Infinity for Rational Functions . . . . . . . . . . . . . . . . . . . 18
2.3.4 The Sandwich or Squeeze Theorem . . . . . . . . . . . . . . . . . . . . . . 20
2.3.5 Techniques of Evaluating Indeterminate Limits . . . . . . . . . . . . . . . . 21
2.4 Continuous Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.4.1 Continuity Test and Points of Discontinuity ................. 22
2.4.2 Formal ( ) Definition of Continuity . . . . . . . . . . . . . . . . . . . . 22
2.4.3 Types of Discontinuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4.4 Right Continuity and Left Continuity . . . . . . . . . . . . . . . . . . . . . 24
2.4.5 Algebraic Properties of Continuous Functions ................ 25
2.5 Exercises and Some Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . 25
3 DIFFERENTIATION OF FUNCTIONS OF A SINGLE VARIABLE 29
3.1 Tangent Lines and Their Slopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.1.1 Tangent Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.1.2 The Slope of a Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2 The Derivative ..................................... 32
3.2.1 Right Derivative and Left Derivative ..................... 32
iii
iv CONTENTS
3.2.2
Leibniz Notation ................................ 33
3.2.3 Differentiability and Continuity . . . . . . . . . . . . . . . . . . . . . . . . 33
3.3 Differentiation Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.4 Derivatives of Trigonometric and Inverse Trigonometric Functions . . . . . . . . . 39
3.4.1 Derivative of Sine and Cosine . . . . . . . . . . . . . . . . . . . . . . . . . 39
, 3.4.2 Derivatives of other Basic Trigonometric Functions ............. 40
3.5 Derivatives of Exponential and Logarithmic Functions . . . . . . . . . . . . . . . . 41
3.6 Derivatives of Hyperbolic and Inverse Hyperbolic Functions . . . . . . . . . . . . . 42
3.7 Derivatives and Indeterminate limits of type and
3.7.1 L’Hˆopital’s First Rule for the Indeterminate form
3.7.2 L’Hˆopital’s Second Rule for the Indeterminate form
3.7.3 Indeterminate forms 0.
3.8 Differentiation of Implicit and Parametric Functions . . . . . . . . . . . . . . . . . 46
3.8.1 Implicit Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.8.2 Logarithmic Differentiation .......................... 47
3.8.3 Parametric Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.8.4 Differentiating Inverse Functions using Implicit Differentiation . . . . . . . 49
3.9 Linear Approximations and Differentials ....................... 49
3.9.1 Differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.9.2 Linearizations are Linear Replacement Formulas . . . . . . . . . . . . . . . 51
3.9.3 Average and Instantaneous Rates of Change . . . . . . . . . . . . . . . . . 52
3.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4 ANTI-DERIVATIVES OF
FUNCTIONS AND APPLICATIONS TO AREAS 57
4.1 Anti-Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.1.1 Indefinite Integral of a Function . . . . . . . . . . . . . . . . . . . . . . . . 57
4.1.2 Integration Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.1.3 Rules for Indefinite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.2 Techniques of Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.2.1 Integration by Substitution .......................... 59
4.2.2 Integration by Parts .............................. 60
4.2.3 Integration of Rational Functions . . . . . . . . . . . . . . . . . . . . . . . 63
4.2.4 Inverse Substitutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.2.5 Inverse Hyperbolic Substitutions ....................... 69
4.2.6 Other Inverse Substitutions .......................... 70
4.3 The Definite Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.3.1 Properties of the Definite Integral . . . . . . . . . . . . . . . . . . . . . . . 72
4.4 Applications of the Definite Integral . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.4.1 Area and the Definite Integral . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.5 Exercises and Some Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . 76
CONTENTS
, Preface
This volume aims to present calculus in an intuitive yet intellectually satisfying way and to
illustrate the many applications of calculus to the pure sciences and management sciences. The
only co-requisite for mastering the material in the book are SMA 101: Basic Mathematics and an
interest in mathematics and a willingness occasionally to suspend disbelief when a familiar idea
occurs in an unfamiliar guise. But only an exceptional student would profit from reading the book
unless he/she has previously acquired a fair working knowledge of elementary set theory, algebra
and geometry. This book is a development of various courses designed for first year students of
science at the University of Nairobi, whose preparation has been some rudimentary knowledge of
set theory, algebra and geometry.
What is Calculus?
Algebra and geometry are useful tools for describing relationships between static quantities.
However, they do not involve concepts appropriate for describing how quantity changes. For this
we need new mathematical operations that transcend or go beyond algebra and geometry. We
require operations that measure the way related quantities change.
Calculus provides the tools for describing motion quantitatively. It introduces two new operations
called differentiation and integration which are inverses of each other: what differentiation does,
integration undoes. The process of differentiation is closely tied to the geometric problem of
finding tangent lines. Integration is related to the geometric problem of finding areas of regions
with curved boundaries. These two concepts defined in terms of the concept of a limit. This will be
developed in Chapter 2, and marks the beginning of calculus.
Origins of Calculus
Calculus was invented independently by two 17th-century mathematicians: Isaac Newton and
Gottfried Wilhelm Leibniz.
Objectives
At the end of this course unit the learner will be able to:
Understand a the concept of a limit of a function and how to compute the same.
Appreciate the concept of a continuous function.
Appreciate and apply the concepts of derivatives and antiderivatives.
vii
viii CONTENTS
CALCULUS 1
,1 PRELIMINARIES FOR CALCULUS 1
1.1 Elementary Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Notation and Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.2 Some Special Number Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.3 Fundamental Operations on Sets . . . . . . . . . . . . . . . . . . . . . . . 4
2 LIMITS AND CONTINUITY OF FUNCTIONS 5
2.1 Functions and their Graphs .............................. 5
2.1.1 Domain and Range of Functions . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1.2 Some Elementary Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Limits of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.1 Informal Definition of a Limit . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.2 Formal ( ) Definition of a Limit . . . . . . . . . . . . . . . . . . . . .. 9
2.3 Properties of Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3.1 Right-Hand Limits and Left-Hand Limits . . . . . . . . . . . . . . . . . . . 14
2.3.2 Limits at Infinity and Infinite Limits . . . . . . . . . . . . . . . . . . . . . 16
2.3.3 Limits at Infinity for Rational Functions . . . . . . . . . . . . . . . . . . . 18
2.3.4 The Sandwich or Squeeze Theorem . . . . . . . . . . . . . . . . . . . . . . 20
2.3.5 Techniques of Evaluating Indeterminate Limits . . . . . . . . . . . . . . . . 21
2.4 Continuous Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.4.1 Continuity Test and Points of Discontinuity ................. 22
2.4.2 Formal ( ) Definition of Continuity . . . . . . . . . . . . . . . . . . . . 22
2.4.3 Types of Discontinuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4.4 Right Continuity and Left Continuity . . . . . . . . . . . . . . . . . . . . . 24
2.4.5 Algebraic Properties of Continuous Functions ................ 25
2.5 Exercises and Some Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . 25
3 DIFFERENTIATION OF FUNCTIONS OF A SINGLE VARIABLE 29
3.1 Tangent Lines and Their Slopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.1.1 Tangent Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.1.2 The Slope of a Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2 The Derivative ..................................... 32
3.2.1 Right Derivative and Left Derivative ..................... 32
iii
iv CONTENTS
3.2.2
Leibniz Notation ................................ 33
3.2.3 Differentiability and Continuity . . . . . . . . . . . . . . . . . . . . . . . . 33
3.3 Differentiation Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.4 Derivatives of Trigonometric and Inverse Trigonometric Functions . . . . . . . . . 39
3.4.1 Derivative of Sine and Cosine . . . . . . . . . . . . . . . . . . . . . . . . . 39
, 3.4.2 Derivatives of other Basic Trigonometric Functions ............. 40
3.5 Derivatives of Exponential and Logarithmic Functions . . . . . . . . . . . . . . . . 41
3.6 Derivatives of Hyperbolic and Inverse Hyperbolic Functions . . . . . . . . . . . . . 42
3.7 Derivatives and Indeterminate limits of type and
3.7.1 L’Hˆopital’s First Rule for the Indeterminate form
3.7.2 L’Hˆopital’s Second Rule for the Indeterminate form
3.7.3 Indeterminate forms 0.
3.8 Differentiation of Implicit and Parametric Functions . . . . . . . . . . . . . . . . . 46
3.8.1 Implicit Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.8.2 Logarithmic Differentiation .......................... 47
3.8.3 Parametric Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.8.4 Differentiating Inverse Functions using Implicit Differentiation . . . . . . . 49
3.9 Linear Approximations and Differentials ....................... 49
3.9.1 Differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.9.2 Linearizations are Linear Replacement Formulas . . . . . . . . . . . . . . . 51
3.9.3 Average and Instantaneous Rates of Change . . . . . . . . . . . . . . . . . 52
3.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4 ANTI-DERIVATIVES OF
FUNCTIONS AND APPLICATIONS TO AREAS 57
4.1 Anti-Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.1.1 Indefinite Integral of a Function . . . . . . . . . . . . . . . . . . . . . . . . 57
4.1.2 Integration Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.1.3 Rules for Indefinite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.2 Techniques of Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.2.1 Integration by Substitution .......................... 59
4.2.2 Integration by Parts .............................. 60
4.2.3 Integration of Rational Functions . . . . . . . . . . . . . . . . . . . . . . . 63
4.2.4 Inverse Substitutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.2.5 Inverse Hyperbolic Substitutions ....................... 69
4.2.6 Other Inverse Substitutions .......................... 70
4.3 The Definite Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.3.1 Properties of the Definite Integral . . . . . . . . . . . . . . . . . . . . . . . 72
4.4 Applications of the Definite Integral . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.4.1 Area and the Definite Integral . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.5 Exercises and Some Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . 76
CONTENTS
, Preface
This volume aims to present calculus in an intuitive yet intellectually satisfying way and to
illustrate the many applications of calculus to the pure sciences and management sciences. The
only co-requisite for mastering the material in the book are SMA 101: Basic Mathematics and an
interest in mathematics and a willingness occasionally to suspend disbelief when a familiar idea
occurs in an unfamiliar guise. But only an exceptional student would profit from reading the book
unless he/she has previously acquired a fair working knowledge of elementary set theory, algebra
and geometry. This book is a development of various courses designed for first year students of
science at the University of Nairobi, whose preparation has been some rudimentary knowledge of
set theory, algebra and geometry.
What is Calculus?
Algebra and geometry are useful tools for describing relationships between static quantities.
However, they do not involve concepts appropriate for describing how quantity changes. For this
we need new mathematical operations that transcend or go beyond algebra and geometry. We
require operations that measure the way related quantities change.
Calculus provides the tools for describing motion quantitatively. It introduces two new operations
called differentiation and integration which are inverses of each other: what differentiation does,
integration undoes. The process of differentiation is closely tied to the geometric problem of
finding tangent lines. Integration is related to the geometric problem of finding areas of regions
with curved boundaries. These two concepts defined in terms of the concept of a limit. This will be
developed in Chapter 2, and marks the beginning of calculus.
Origins of Calculus
Calculus was invented independently by two 17th-century mathematicians: Isaac Newton and
Gottfried Wilhelm Leibniz.
Objectives
At the end of this course unit the learner will be able to:
Understand a the concept of a limit of a function and how to compute the same.
Appreciate the concept of a continuous function.
Appreciate and apply the concepts of derivatives and antiderivatives.
vii
viii CONTENTS
