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Basic Concept of Differential and Integral Calculus
CPT Section D Quantitative Aptitude Chapter 9
Dr. Atul Kumar Srivastava
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Learning Objectives
Understand the use of this Branch of mathematics in various branches of science and Humanities
Understand the basics of differentiation and integration
Know how to compute derivative of a function by the first principal, derivative of a function by the use of various formulae and higher order differentiation
Make familiar with various techniques of integration
Understand the concept of definite integrals of functions and its properties.
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Differential calculus-Outlines
What is Differential Calculus – An Introduction
Derivative or Deferential Coefficient (First Principal Definition)
Basic Formulas
Laws for Differentiation (Algebra of Derivative of Functions)
Derivative of A Function of Function (Chain Rule)
.Derivative of Implicit Function
.Derivative of Function In Parametric Form
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What is differential calculus – an introduction
One of the most fundamental operations in calculus is that of differentiation. In the study of mathematics, there are many problems containing two quantities such that the value of one quantity depends upon the other. A variation in the value of any ones produces a variation in the value of the other. For example the area of a square depends upon it's side. The area of circle and volume of sphere depend upon their radius etc.
Differential calculus is the Branch of Mathematics which studies changes
To express the rate of change of any function we introduce concept of derivative. The concept involves a very small change in the dependent variable with reference to a very small change in the independent variable
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Continued
y=f(x)
• x - Independent variable • y - Dependent Variable
Thus differentiation is a process of finding the derivative of a continuous function. It is defined as the limiting value of the ratio of the change in the function corresponding to small change in the independent variable as the later tends to zero.
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Derivative or differential coefficient (First principal definition) Derivative of is defined as
where
is a function
is small increment in x
corresponding increment in y or f(x)
(1) Is denoted as is also The derivative of f(x)
known as differential coefficient of w.r.t. x
The above process of differentiation is called the first principal definition.
(1)
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Examples of differentiation from first principal:
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Example :1
We have
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Example-2 f(x)=a where a is fixed real number
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Example-3
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Basic Formulas Following are some of the standard derivative:-
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Laws for differentiation (Algebra of derivative of functions)
Let f(x) and g(x) be two functions such that their derivatives are defined in a common domain. Then
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1.Sum Rule
2.Difference Rule
3.Product Rule
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4.(Quotient
5.
6.
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Rule) Continued
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Example-1
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Find
EXAMPLES
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Example-2
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Example-3
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Derivative of implicit function
Until now we have been differentiating various function given in the form y=f(x)
But it is not necessary that functions are always expressed in this form. For example consider one of the following relationship between x and y
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NEXT SLIDE….
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In second case it does not seem easy to solve for Y.
for implicit function
In the first case we can solve y and rewrite the relationship as
When it is easy to express the relation as y=f(x) we say that y is given as an explicit function of x, otherwise it is an implicit function of x
Now we will attempt to find
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Continued….
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Examples
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NEXT SLIDE….
Find for Example 1
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Continued 19
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Example-2
Differentiating on both sides …..(1)
NEXT SLIDE….
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Continued 21
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Derivative of functions in parametric forms
If relation between two variables is expressed via third variable. The third variable is called parameter. More precisely a relation expressed between two variables x and y in the form x=f(t),y=g(t) is said to be parametric form with t is a parameter.
In order to find derivative of function in such form, we have by chain rule.
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Continued
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Example 1
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Find
Given That
So
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Example-2
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Logarithmic differentiation
and
.
We differentiate such functions by taking logarithm on both sides. This process in called logarithmic differentiate.
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Examples
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Taking logarithms on both sides
Differentiate
Differentiate both sides w.r.t x
Example
1
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Example-2
28 Differentiate
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Higher Order differentiation
If is differentiable, we may differentiate it w.r.t. x. The LHS becomes which is called the second order derivative of and is denoted by . It is also denoted by . If we remark that higher order derivatives may be defined similarly.
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or
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EXAMPLES
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Example-1
Given that
Here
=
-
-
- =
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Example-2
Differentiate again
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Find
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Gradient or slope of the curve
Let y=f(x)be a curve. The derivative of f(x) at a point x represents the slope of the tangent to the curve y=f(x)at the point x.
Sometime the derivative is called gradient of the curve.
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Example-1 Find the gradient of the curve
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The gradient of the curve at point X=0 is -12
Given
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Example-2 Find the slope of the tangent to the curve
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Miscellaneous Examples
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Example:1 Differentiate
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Example-2 Differentiate
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Let
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Example-3 Find derivative of
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Example-4: Find dy /dx
differentiate implicitly w.r.t. x ., we get
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Example-5 If
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Example-6 Differentiate
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41 Example 7
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Example 8
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By cross multiplication
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Continued
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Integral Calculus What Is Integration (Definition) • Basic Formulas
Method Of Substitution (Change Of Variable)
Integration By Parts
Method Of Partial Fraction
Definite Integration
Important Properties
Miscellaneous Examples
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Integration is inverse process of differentiation. Integral calculus deals with integration and its application. It was invented in attempt to solve the problems of finding areas under curves and volumes of solids of revolution.
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Also we can define integration as the inverse process of differentiation .
What is Integration (definition)
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is obtained by giving different Evidently the integral of
values to C . Here 'C' is called constant of integration The process of finding the integral is called integration. The function which is integrated is called the integrand.
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Constant of integration
and c is an arbitrary constant we also have
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Basic Formulas
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Since integration and differentiation are inverse process we have
Example-1
1.
2.
48 Two Simple Theorem .
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Example-2
Example-3
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Example-4
Example-5
Evaluate 50
or
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Example-6 Evaluate
Example-7 Evaluate
51 Examples
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By simple division
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Example-8
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can be transformed into another form by changing the independent variable x to t by substituting
Consider
The given integral
Usually we make a substitution for a function whose derivation also occur in the integrand.
53 Integration By Substitution
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Example1 Examples
adx = dt
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or
.
Evaluate
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Example 2
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dt
56
Example-3: Evaluate
or
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2.
3.
1.
4.
Important standard formulas
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.
5.
6.
7.
8.
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Continued
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Example
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Integration of product of two function
It is useful method to find integration of product of function.
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Integration by Parts
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Example:1
Examples
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Evaluate
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Example-2
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Example-3
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(Solve first integral only)
64
Example 4
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Methods of Partial Fractions
65
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Find the partial fraction of
Example-1
we put x=2
and get
we put x=3
66
Type-1
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Comparing coefficients of and constant term on both sides
Solving we get
Therefore
67 Type-2
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68
Example
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Now consider
Here a = lower limit of integration b = upper limit of integration
is called definite integral of f(x) from a to b
Consider indefinite integral
69 Definite Integration
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70 Properties
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Example-1:
71
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or
72
Example-2
![Page 73: Basic concept of differential and integral · PDF fileBasic Concept of Differential and Integral Calculus CPT Section D Quantitative Aptitude Chapter 9 . Dr. Atul Kumar Srivastava](https://reader038.vdocument.in/reader038/viewer/2022103106/5a723d287f8b9a93538d8947/html5/thumbnails/73.jpg)
let
find
73
Miscellaneous examples
![Page 74: Basic concept of differential and integral · PDF fileBasic Concept of Differential and Integral Calculus CPT Section D Quantitative Aptitude Chapter 9 . Dr. Atul Kumar Srivastava](https://reader038.vdocument.in/reader038/viewer/2022103106/5a723d287f8b9a93538d8947/html5/thumbnails/74.jpg)
Let
74
Example-2
![Page 75: Basic concept of differential and integral · PDF fileBasic Concept of Differential and Integral Calculus CPT Section D Quantitative Aptitude Chapter 9 . Dr. Atul Kumar Srivastava](https://reader038.vdocument.in/reader038/viewer/2022103106/5a723d287f8b9a93538d8947/html5/thumbnails/75.jpg)
integration by parts
consider
75
Example-3
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solve
where
Example 4.
76
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by simple division
77
Example:5
![Page 78: Basic concept of differential and integral · PDF fileBasic Concept of Differential and Integral Calculus CPT Section D Quantitative Aptitude Chapter 9 . Dr. Atul Kumar Srivastava](https://reader038.vdocument.in/reader038/viewer/2022103106/5a723d287f8b9a93538d8947/html5/thumbnails/78.jpg)
simplify integrand
78
Example-6
First
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Example 7.
79
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80
Example 8.
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81
Example 9.
![Page 82: Basic concept of differential and integral · PDF fileBasic Concept of Differential and Integral Calculus CPT Section D Quantitative Aptitude Chapter 9 . Dr. Atul Kumar Srivastava](https://reader038.vdocument.in/reader038/viewer/2022103106/5a723d287f8b9a93538d8947/html5/thumbnails/82.jpg)
I II
82 Example 10.
dx
![Page 83: Basic concept of differential and integral · PDF fileBasic Concept of Differential and Integral Calculus CPT Section D Quantitative Aptitude Chapter 9 . Dr. Atul Kumar Srivastava](https://reader038.vdocument.in/reader038/viewer/2022103106/5a723d287f8b9a93538d8947/html5/thumbnails/83.jpg)
since it passes through the origin (o,o)
Then
Given
or
83
Example11. Find the equation of the curve where slope at (x,y) is 9x which passes through origin
![Page 84: Basic concept of differential and integral · PDF fileBasic Concept of Differential and Integral Calculus CPT Section D Quantitative Aptitude Chapter 9 . Dr. Atul Kumar Srivastava](https://reader038.vdocument.in/reader038/viewer/2022103106/5a723d287f8b9a93538d8947/html5/thumbnails/84.jpg)
Let
84
Example 12.
![Page 85: Basic concept of differential and integral · PDF fileBasic Concept of Differential and Integral Calculus CPT Section D Quantitative Aptitude Chapter 9 . Dr. Atul Kumar Srivastava](https://reader038.vdocument.in/reader038/viewer/2022103106/5a723d287f8b9a93538d8947/html5/thumbnails/85.jpg)
85
Example 13.
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SUMMARY OF THE CHAPTER Differential calculus
86
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87 Continued…….
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Integral Calculus
88
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=
=
=
89
Continued……
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=
90 Continued
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91
Continued
(a < b < c)
,
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92
Continued
= 0
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Question Time MCQ’s
93
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HINT-Logarithmic Differentiation
94 Question:1
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95
Question.2
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HINT- Logarithmic differentiation then Apply product rule
96
Question.3
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97
Question.4
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HINT- (Apply Quotient rule)
98
Question.5
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HINT - (Apply chain rule)
99
Question-6
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HINT - (Apply product rule and chain rule)
100
Question.7
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HINT- (Quotient rule)
101
Question 8
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HINT- (Take log both sides then apply quotient rule)
102
Question.9
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HINT- (Differentiation of implicit function)
103
Question.10
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HINT- (Logarithmic differentiation)
104
Question.11
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HINT- (Implicit function)
105
Question:12
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HINT- (Logarithmic differentiation)
(d)
106
Question.13
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HINT- (Quotient rule)
107
Question.14
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The value of p and q are.
HINT-
108
Question.15
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HINT-
109
Question.16
![Page 110: Basic concept of differential and integral · PDF fileBasic Concept of Differential and Integral Calculus CPT Section D Quantitative Aptitude Chapter 9 . Dr. Atul Kumar Srivastava](https://reader038.vdocument.in/reader038/viewer/2022103106/5a723d287f8b9a93538d8947/html5/thumbnails/110.jpg)
HINT- Integrate By Parts
110
Question.17
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111
Question.18
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HINT- Integration by substitution let
112
Question.19
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HINT- [Integration by substitution, let t
113
Question.20
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HINT- [Integration by substitution let
114
Question.21
![Page 115: Basic concept of differential and integral · PDF fileBasic Concept of Differential and Integral Calculus CPT Section D Quantitative Aptitude Chapter 9 . Dr. Atul Kumar Srivastava](https://reader038.vdocument.in/reader038/viewer/2022103106/5a723d287f8b9a93538d8947/html5/thumbnails/115.jpg)
HINT-
115
Question.22
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HINT-
116
Question.23
![Page 117: Basic concept of differential and integral · PDF fileBasic Concept of Differential and Integral Calculus CPT Section D Quantitative Aptitude Chapter 9 . Dr. Atul Kumar Srivastava](https://reader038.vdocument.in/reader038/viewer/2022103106/5a723d287f8b9a93538d8947/html5/thumbnails/117.jpg)
HINT- (Integration by substitution let 7x + = t)
117
Question.24
5
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118
HINT- Divide Numerator by Denominator then Integrate
Question.25
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HINT-
119
Question.26
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HINT-
120
Question.27
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HINT-
121
Question.28
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HINT-
122
Question.29
![Page 123: Basic concept of differential and integral · PDF fileBasic Concept of Differential and Integral Calculus CPT Section D Quantitative Aptitude Chapter 9 . Dr. Atul Kumar Srivastava](https://reader038.vdocument.in/reader038/viewer/2022103106/5a723d287f8b9a93538d8947/html5/thumbnails/123.jpg)
HINT-
123
Question.30
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Practice makes a man perfect and that’s what mathematics demands. .
So, students ‘all the best’ for your upcoming examinations . keep practicing.
Thank you
124