MATH 221 FIRST SEMESTER CALCULUS
1
MATH 221 – 1st SEMESTER CALCULUS LECTURE NOTES VERSION 2.0
Calculus is a branch of mathematics that studies change and motion. It is divided into
Differential Calculus, which deals with rates of change, and Integral Calculus, which focuses on
accumulation. Together, they form powerful tools for understanding physical phenomena,
optimizing solutions, and solving complex problems in fields like science, engineering, and
economics.
, 3
3. Exercises 64
4. Finding sign changes of a function 65
5. Increasing and decreasing functions 66
Chapter 1. Numbers and Functions 5
6. Examples 67
1. What is a number? 5
7. Maxima and Minima 69
2. Exercises 7
8. Must there always be a maximum? 71
3. Functions 8
4. Inverse functions and Implicit functions 10 9. Examples – functions with and without maxima or minima
5. Exercises 13 71
10. General method for sketching the graph of a
Chapter 2. Derivatives (1) 15
function 72
1. The tangent to a curve 15 11. Convexity, Concavity and the Second Derivative 74
2. An example – tangent to a parabola 16 12. Proofs of some of the theorems 75
3. Instantaneous velocity 17 13. Exercises 76
4. Rates of change 17 14. Optimization Problems 77
5. Examples of rates of change 18
15. Exercises 78
6. Exercises 18
Chapter 3. Limits and Continuous Functions 21 Chapter 6. Exponentials and Logarithms (naturally) 81
1. Exponents 81
1. Informal definition of limits 21
2. The formal, authoritative, definition of limit 22 2. Logarithms 82
3. Exercises 25 3. Properties of logarithms 83
4. Variations on the limit theme 25 4. Graphs of exponential functions and logarithms 83
5. Properties of the Limit 27 5. The derivative of ax and the definition of e 84
6. Examples of limit computations 27 6. Derivatives of Logarithms 85
7. When limits fail to exist 29 7. Limits involving exponentials and logarithms 86
8. What’s in a name? 32 8. Exponential growth and decay 86
9. Limits and Inequalities 33 9. Exercises 87
10. Continuity 34
11. Substitution in Limits 35 Chapter 7. The Integral 91
12. Exercises 36 1. Area under a Graph 91
13. Two Limits in Trigonometry 36 2. When f changes its sign 92
14. Exercises 38 3. The Fundamental Theorem of Calculus 93
Chapter 4. Derivatives (2) 41 4. Exercises 94
1. Derivatives Defined 41 5. The indefinite integral 95
2. Direct computation of derivatives 42 6. Properties of the Integral 97
3. Differentiable implies Continuous 43 7. The definite integral as a function of its integration bounds
4. Some non-differentiable functions 43 98
5. Exercises 44 8. Method of substitution 99
6. The Differentiation Rules 45 9. Exercises 100
7. Differentiating powers of functions 48
8. Exercises 49 Chapter 8. Applications of the integral 105
9. Higher Derivatives 50 1. Areas between graphs 105
10. Exercises 51 2. Exercises 106
11. Differentiating Trigonometric functions 51 3. Cavalieri’s principle and volumes of solids 106
12. Exercises 52 4. Examples of volumes of solids of revolution 109
13. The Chain Rule 52 5. Volumes by cylindrical shells 111
14. Exercises 57
6. Exercises 113
15. Implicit differentiation 58
7. Distance from velocity, velocity from acceleration 113
16. Exercises 60
8. The length of a curve 116
Chapter 5. Graph Sketching and Max-Min Problems 63 9. Examples of length computations 117
1. Tangent and Normal lines to a graph 63 10. Exercises 118
2. The Intermediate Value Theorem 63 11. Work done by a force 118
Contents 12. Work done by an electric current 119
Chapter 9. Answers and Hints 121
GNU Free Documentation License 125
,1. APPLICABILITY AND DEFINITIONS 125
2. VERBATIM COPYING 125
3. COPYING IN QUANTITY 125
4. MODIFICATIONS 125
5. COMBINING DOCUMENTS 126
6. COLLECTIONS OF DOCUMENTS 126
7. AGGREGATION WITH INDEPENDENT WORKS 126
8. TRANSLATION126
9. TERMINATION 126
10. FUTURE REVISIONS OF THIS LICENSE126
11. RELICENSING 126
, CHAPTER 1
Numbers and Functions
The subject of this course is “functions of one real variable” so we begin by wondering what a real
number “really” is, and then, in the next section, what a function is.
1. What is a number?
1.1. Different kinds of numbers. The simplest numbers are the positive integers
1,2,3,4,···
the number zero
0,
and the negative integers
··· ,−4,−3,−2,−1.
Together these form the integers or “whole numbers.”
Next, there are the numbers you get by dividing one whole number by another (nonzero) whole number.
These are the so called fractions or rational numbers such as
or
By definition, any whole number is a rational number (in particular zero is a rational number.)
You can add, subtract, multiply and divide any pair of rational numbers and the result will again be a
rational number (provided you don’t try to divide by zero).
One day in middle school you were told that there are other numbers besides the rational numbers, and
the first example of such a number is the square root of two. It has been known ever since the time of the
greeks that no rational number exists whose square is exactly 2, i.e. you can’t find a fraction such that
, i.e. m1 = 2n2.
x x2
1.2 1.44
1.3 1.69
1
) were then we could perhaps answer such questions. It turns out to be rather difficult to give a precise
description of what a number is, and in this course we won’t try to get anywhere near the bottom of this issue.
Instead we will think of numbers as “infinite decimal expansions” as follows.
One can represent certain fractions as decimal fractions, e.g.
.
4
1
MATH 221 – 1st SEMESTER CALCULUS LECTURE NOTES VERSION 2.0
Calculus is a branch of mathematics that studies change and motion. It is divided into
Differential Calculus, which deals with rates of change, and Integral Calculus, which focuses on
accumulation. Together, they form powerful tools for understanding physical phenomena,
optimizing solutions, and solving complex problems in fields like science, engineering, and
economics.
, 3
3. Exercises 64
4. Finding sign changes of a function 65
5. Increasing and decreasing functions 66
Chapter 1. Numbers and Functions 5
6. Examples 67
1. What is a number? 5
7. Maxima and Minima 69
2. Exercises 7
8. Must there always be a maximum? 71
3. Functions 8
4. Inverse functions and Implicit functions 10 9. Examples – functions with and without maxima or minima
5. Exercises 13 71
10. General method for sketching the graph of a
Chapter 2. Derivatives (1) 15
function 72
1. The tangent to a curve 15 11. Convexity, Concavity and the Second Derivative 74
2. An example – tangent to a parabola 16 12. Proofs of some of the theorems 75
3. Instantaneous velocity 17 13. Exercises 76
4. Rates of change 17 14. Optimization Problems 77
5. Examples of rates of change 18
15. Exercises 78
6. Exercises 18
Chapter 3. Limits and Continuous Functions 21 Chapter 6. Exponentials and Logarithms (naturally) 81
1. Exponents 81
1. Informal definition of limits 21
2. The formal, authoritative, definition of limit 22 2. Logarithms 82
3. Exercises 25 3. Properties of logarithms 83
4. Variations on the limit theme 25 4. Graphs of exponential functions and logarithms 83
5. Properties of the Limit 27 5. The derivative of ax and the definition of e 84
6. Examples of limit computations 27 6. Derivatives of Logarithms 85
7. When limits fail to exist 29 7. Limits involving exponentials and logarithms 86
8. What’s in a name? 32 8. Exponential growth and decay 86
9. Limits and Inequalities 33 9. Exercises 87
10. Continuity 34
11. Substitution in Limits 35 Chapter 7. The Integral 91
12. Exercises 36 1. Area under a Graph 91
13. Two Limits in Trigonometry 36 2. When f changes its sign 92
14. Exercises 38 3. The Fundamental Theorem of Calculus 93
Chapter 4. Derivatives (2) 41 4. Exercises 94
1. Derivatives Defined 41 5. The indefinite integral 95
2. Direct computation of derivatives 42 6. Properties of the Integral 97
3. Differentiable implies Continuous 43 7. The definite integral as a function of its integration bounds
4. Some non-differentiable functions 43 98
5. Exercises 44 8. Method of substitution 99
6. The Differentiation Rules 45 9. Exercises 100
7. Differentiating powers of functions 48
8. Exercises 49 Chapter 8. Applications of the integral 105
9. Higher Derivatives 50 1. Areas between graphs 105
10. Exercises 51 2. Exercises 106
11. Differentiating Trigonometric functions 51 3. Cavalieri’s principle and volumes of solids 106
12. Exercises 52 4. Examples of volumes of solids of revolution 109
13. The Chain Rule 52 5. Volumes by cylindrical shells 111
14. Exercises 57
6. Exercises 113
15. Implicit differentiation 58
7. Distance from velocity, velocity from acceleration 113
16. Exercises 60
8. The length of a curve 116
Chapter 5. Graph Sketching and Max-Min Problems 63 9. Examples of length computations 117
1. Tangent and Normal lines to a graph 63 10. Exercises 118
2. The Intermediate Value Theorem 63 11. Work done by a force 118
Contents 12. Work done by an electric current 119
Chapter 9. Answers and Hints 121
GNU Free Documentation License 125
,1. APPLICABILITY AND DEFINITIONS 125
2. VERBATIM COPYING 125
3. COPYING IN QUANTITY 125
4. MODIFICATIONS 125
5. COMBINING DOCUMENTS 126
6. COLLECTIONS OF DOCUMENTS 126
7. AGGREGATION WITH INDEPENDENT WORKS 126
8. TRANSLATION126
9. TERMINATION 126
10. FUTURE REVISIONS OF THIS LICENSE126
11. RELICENSING 126
, CHAPTER 1
Numbers and Functions
The subject of this course is “functions of one real variable” so we begin by wondering what a real
number “really” is, and then, in the next section, what a function is.
1. What is a number?
1.1. Different kinds of numbers. The simplest numbers are the positive integers
1,2,3,4,···
the number zero
0,
and the negative integers
··· ,−4,−3,−2,−1.
Together these form the integers or “whole numbers.”
Next, there are the numbers you get by dividing one whole number by another (nonzero) whole number.
These are the so called fractions or rational numbers such as
or
By definition, any whole number is a rational number (in particular zero is a rational number.)
You can add, subtract, multiply and divide any pair of rational numbers and the result will again be a
rational number (provided you don’t try to divide by zero).
One day in middle school you were told that there are other numbers besides the rational numbers, and
the first example of such a number is the square root of two. It has been known ever since the time of the
greeks that no rational number exists whose square is exactly 2, i.e. you can’t find a fraction such that
, i.e. m1 = 2n2.
x x2
1.2 1.44
1.3 1.69
1
) were then we could perhaps answer such questions. It turns out to be rather difficult to give a precise
description of what a number is, and in this course we won’t try to get anywhere near the bottom of this issue.
Instead we will think of numbers as “infinite decimal expansions” as follows.
One can represent certain fractions as decimal fractions, e.g.
.
4
