Calculus And Analytical Geometry
MTH 101
Virtual University of Pakistan
Knowledge beyond the boundaries
,45-Planning Production Levels: Linear Programming VU
TABLE OF CONTENTS :
Lesson 1 :Coordinates, Graphs, Lines 3
Lesson 2 :Absolute Value 15
Lesson 3 :Coordinate Planes and Graphs 24
Lesson 4 :Lines 34
Lesson 5 :Distance; Circles, Quadratic Equations 45
Lesson 6 :Functions and Limits 57
Lesson 7 :Operations on Functions 63
Lesson 8 :Graphing Functions 69
Lesson 9 :Limits (Intuitive Introduction) 76
Lesson 10:Limits (Computational Techniques) 84
Lesson 11: Limits (Rigorous Approach) 93
Lesson 12 :Continuity 97
Lesson 13 :Limits and Continuity of Trigonometric Functions 104
Lesson 14 :Tangent Lines, Rates of Change 110
Lesson 15 :The Derivative 115
Lesson 16 :Techniques of Differentiation 123
Lesson 17 :Derivatives of Trigonometric Function 128
Lesson 18 :The chain Rule 132
Lesson 19 :Implicit Differentiation 136
Lesson 20 :Derivative of Logarithmic and Exponential Functions 139
Lesson 21 :Applications of Differentiation 145
Lesson 22 :Relative Extrema 151
Lesson 23 :Maximum and Minimum Values of Functions 158
Lesson 24 :Newton’s Method, Rolle’s Theorem and Mean Value Theorem 164
Lesson 25 :Integrations 169
Lesson 26 :Integration by Substitution 174
Lesson 27 :Sigma Notation 179
Lesson 28 :Area as Limit 183
Lesson 29 :Definite Integral 191
Lesson 30 :First Fundamental Theorem of Calculus 200
Lesson 31 :Evaluating Definite Integral by Subsitution 206
Lesson 32 :Second Fundamental Theorem of Calculus 210
Lesson 33 :Application of Definite Integral 214
Lesson 34 :Volume by slicing; Disks and Washers 221
Lesson 35 :Volume by Cylindrical Shells 230
Lesson 36 :Length of Plane Curves 237
Lesson 37 :Area of Surface of Revolution 240
Lesson 38:Work and Definite Integral 245
Lesson 39 :Improper Integral 252
Lesson 40 :L’Hopital’s Rule 258
Lesson 41 :Sequence 265
Lesson 42 :Infinite Series 276
Lesson 43 :Additional Convergence tests 285
Lesson 44 :Alternating Series; Conditional Convergence 290
Lesson 45 :Taylor and Maclaurin Series 296
,1-Coordinates, Graphs and Lines VU
Lecture 1
Coordinates, Graphs and Lines
What is Calculus??
Well, it is the study of the continuous rates of the change of quantities. It is the study of how various
quantities change with respect to other quantities. For example, one would like to know how distance changes
with respect to (from now onwards we will use the abbreviation w.r.t) time, or how time changes w.r.t speed,
or how water flow changes w.r.t time etc. You want to know how this happens continuously. We will see
what continuously means as well.
In this lecture, we will talk about the following topics:
-Real Numbers
-Set Theory
-Intervals
-Inequalities
-Order Properties Of Real Numbers
Let's start talking about Real Numbers. We will not talk about the COMPLEX or IMANGINARY
numbers, although your text has something about them which you can read on your own. We will go through
the history of REAL numbers and how they popped into the realm of human intellect. We will look at the
various types of REALS - as we will now call them. SO Let's START.
The simplest numbers are the natural numbers
Natural Numbers
1, 2, 3, 4, 5,…
They are called the natural numbers because they are the first to have crossed paths with human intellect.
Think about it: these are the numbers we count things with. So our ancestors used these numbers first to
count, and they came to us naturally! Hence the name
NATURAL!!!
The natural numbers form a subset of a larger class of numbers called the integers. I have used the word
SUBSET. From now onwards we will just think of SET as a COLLECTION OF THINGS.
This could be a collection of oranges, apples, cars, or politicians. For example, if I have the SET of politicians
then a SUBSET will be just a part of the COLLECTION. In mathematical notation we say A is subset of B if
∀x ∈ A ⇒ x ∈ B .Then we write A ⊆ B .
Set
The collection of well defined objects is called a set. For example
© Copyright Virtual University of Pakistan 3
, 1-Coordinates, Graphs and Lines VU
{George Bush, Toney Blair, Ronald Reagoan}
Subset
A portion of a set B is a subset of A iff every member of B is a member of A. e.g. one subset of above set is
{George Bush, Tony Blair}
The curly brackets are always used for denoting SETS. We will get into the basic notations and ideas of sets
later. Going back to the Integers. These are
…, -4, -3, -2, -1, 0, 1, 2, 3, 4,…
So these are just the natural numbers, plus a 0, and the NEGATIVES of the natural numbers.
The reason we didn’t have 0 in the natural numbers is that this number itself has an interesting
story, from being labeled as the concept of the DEVIL in ancient Greece, to being easily accepted in the
Indian philosophy, to being promoted in the use of commerce and science by the Arabs and the Europeans.
But here, we accept it with an open heart into the SET of INTEGERS.
What about these NEGATIVE Naturals??? Well, they are an artificial construction. They also have a history
of their own. For a long time, they would creep up in the solutions of simple equations like
x+2 = 0. The solution is x = - 2
So now we have the Integers plus the naturals giving us things we will call REAL numbers. But that's not all.
There is more. The integers in turn are a subset of a still larger class of numbers called the rational numbers.
With the exception that division by zero is ruled out, the rational numbers are formed by taking ratios of
integers.
Examples are
2/3, 7/5, 6/1, -5/2
Observe that every integer is also a rational number because an integer p can be written as a ratio. So
every integer is also a rational. Why not divide by 0? Well here is why:
If x is different from zero, this equation is contradictory; and if x is equal to zero, this equation is satisfied by
any number y, so the ratio does not have a unique value a situation that is mathematically unsatisfactory .
x/0 =y⇒ x=
0. y ⇒ x =
0
For these reasons such symbols are not assigned a value; they are said to be undefined.
So we have some logical inconsistencies that we would like to avoid. I hope you see that!! Hence, no
division by 0 allowed! Now we come to a very interesting story in the history of the development of Real
numbers. The discovery of IRRATIONAL numbers.
Pythagoras was an ancient Greek philosopher and mathematician. He studied the properties of numbers for
its own sake, not necessarily for any applied problems. This was a major change in mathematical thinking as
© Copyright Virtual University of Pakistan 4
MTH 101
Virtual University of Pakistan
Knowledge beyond the boundaries
,45-Planning Production Levels: Linear Programming VU
TABLE OF CONTENTS :
Lesson 1 :Coordinates, Graphs, Lines 3
Lesson 2 :Absolute Value 15
Lesson 3 :Coordinate Planes and Graphs 24
Lesson 4 :Lines 34
Lesson 5 :Distance; Circles, Quadratic Equations 45
Lesson 6 :Functions and Limits 57
Lesson 7 :Operations on Functions 63
Lesson 8 :Graphing Functions 69
Lesson 9 :Limits (Intuitive Introduction) 76
Lesson 10:Limits (Computational Techniques) 84
Lesson 11: Limits (Rigorous Approach) 93
Lesson 12 :Continuity 97
Lesson 13 :Limits and Continuity of Trigonometric Functions 104
Lesson 14 :Tangent Lines, Rates of Change 110
Lesson 15 :The Derivative 115
Lesson 16 :Techniques of Differentiation 123
Lesson 17 :Derivatives of Trigonometric Function 128
Lesson 18 :The chain Rule 132
Lesson 19 :Implicit Differentiation 136
Lesson 20 :Derivative of Logarithmic and Exponential Functions 139
Lesson 21 :Applications of Differentiation 145
Lesson 22 :Relative Extrema 151
Lesson 23 :Maximum and Minimum Values of Functions 158
Lesson 24 :Newton’s Method, Rolle’s Theorem and Mean Value Theorem 164
Lesson 25 :Integrations 169
Lesson 26 :Integration by Substitution 174
Lesson 27 :Sigma Notation 179
Lesson 28 :Area as Limit 183
Lesson 29 :Definite Integral 191
Lesson 30 :First Fundamental Theorem of Calculus 200
Lesson 31 :Evaluating Definite Integral by Subsitution 206
Lesson 32 :Second Fundamental Theorem of Calculus 210
Lesson 33 :Application of Definite Integral 214
Lesson 34 :Volume by slicing; Disks and Washers 221
Lesson 35 :Volume by Cylindrical Shells 230
Lesson 36 :Length of Plane Curves 237
Lesson 37 :Area of Surface of Revolution 240
Lesson 38:Work and Definite Integral 245
Lesson 39 :Improper Integral 252
Lesson 40 :L’Hopital’s Rule 258
Lesson 41 :Sequence 265
Lesson 42 :Infinite Series 276
Lesson 43 :Additional Convergence tests 285
Lesson 44 :Alternating Series; Conditional Convergence 290
Lesson 45 :Taylor and Maclaurin Series 296
,1-Coordinates, Graphs and Lines VU
Lecture 1
Coordinates, Graphs and Lines
What is Calculus??
Well, it is the study of the continuous rates of the change of quantities. It is the study of how various
quantities change with respect to other quantities. For example, one would like to know how distance changes
with respect to (from now onwards we will use the abbreviation w.r.t) time, or how time changes w.r.t speed,
or how water flow changes w.r.t time etc. You want to know how this happens continuously. We will see
what continuously means as well.
In this lecture, we will talk about the following topics:
-Real Numbers
-Set Theory
-Intervals
-Inequalities
-Order Properties Of Real Numbers
Let's start talking about Real Numbers. We will not talk about the COMPLEX or IMANGINARY
numbers, although your text has something about them which you can read on your own. We will go through
the history of REAL numbers and how they popped into the realm of human intellect. We will look at the
various types of REALS - as we will now call them. SO Let's START.
The simplest numbers are the natural numbers
Natural Numbers
1, 2, 3, 4, 5,…
They are called the natural numbers because they are the first to have crossed paths with human intellect.
Think about it: these are the numbers we count things with. So our ancestors used these numbers first to
count, and they came to us naturally! Hence the name
NATURAL!!!
The natural numbers form a subset of a larger class of numbers called the integers. I have used the word
SUBSET. From now onwards we will just think of SET as a COLLECTION OF THINGS.
This could be a collection of oranges, apples, cars, or politicians. For example, if I have the SET of politicians
then a SUBSET will be just a part of the COLLECTION. In mathematical notation we say A is subset of B if
∀x ∈ A ⇒ x ∈ B .Then we write A ⊆ B .
Set
The collection of well defined objects is called a set. For example
© Copyright Virtual University of Pakistan 3
, 1-Coordinates, Graphs and Lines VU
{George Bush, Toney Blair, Ronald Reagoan}
Subset
A portion of a set B is a subset of A iff every member of B is a member of A. e.g. one subset of above set is
{George Bush, Tony Blair}
The curly brackets are always used for denoting SETS. We will get into the basic notations and ideas of sets
later. Going back to the Integers. These are
…, -4, -3, -2, -1, 0, 1, 2, 3, 4,…
So these are just the natural numbers, plus a 0, and the NEGATIVES of the natural numbers.
The reason we didn’t have 0 in the natural numbers is that this number itself has an interesting
story, from being labeled as the concept of the DEVIL in ancient Greece, to being easily accepted in the
Indian philosophy, to being promoted in the use of commerce and science by the Arabs and the Europeans.
But here, we accept it with an open heart into the SET of INTEGERS.
What about these NEGATIVE Naturals??? Well, they are an artificial construction. They also have a history
of their own. For a long time, they would creep up in the solutions of simple equations like
x+2 = 0. The solution is x = - 2
So now we have the Integers plus the naturals giving us things we will call REAL numbers. But that's not all.
There is more. The integers in turn are a subset of a still larger class of numbers called the rational numbers.
With the exception that division by zero is ruled out, the rational numbers are formed by taking ratios of
integers.
Examples are
2/3, 7/5, 6/1, -5/2
Observe that every integer is also a rational number because an integer p can be written as a ratio. So
every integer is also a rational. Why not divide by 0? Well here is why:
If x is different from zero, this equation is contradictory; and if x is equal to zero, this equation is satisfied by
any number y, so the ratio does not have a unique value a situation that is mathematically unsatisfactory .
x/0 =y⇒ x=
0. y ⇒ x =
0
For these reasons such symbols are not assigned a value; they are said to be undefined.
So we have some logical inconsistencies that we would like to avoid. I hope you see that!! Hence, no
division by 0 allowed! Now we come to a very interesting story in the history of the development of Real
numbers. The discovery of IRRATIONAL numbers.
Pythagoras was an ancient Greek philosopher and mathematician. He studied the properties of numbers for
its own sake, not necessarily for any applied problems. This was a major change in mathematical thinking as
© Copyright Virtual University of Pakistan 4
