Notesale: Turn your study into money

Document Preview

Extracts from the notes are below, to see the PDF you'll receive please use the links above

---

Basic Concept of
Differential and Integral Calculus
CPT Section D Quantitative Aptitude Chapter 9

Dr
...


Differential calculus-Outlines

3

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 Function In Parametric Form

What is differential calculus – an 4
introduction
One of the most fundamental operations in calculus is that of differentiation
...
A variation in the
value of any ones produces a variation in the value of the other
...
The area of circle and volume of
sphere depend upon their radius etc
...
The concept involves a very small change in the dependent variable
with reference to a very small change in the independent variable

5

Continued

y=f(x)
•
•

x - Independent variable
y - Dependent Variable

Thus differentiation is a process of finding the derivative of a
continuous function
...


Derivative or differential coefficient
(First principal definition)
Derivative of

6

is defined as
(1)

is a function

where

is small increment in x
corresponding increment in y or f(x)
(1) Is denoted as
known as differential coefficient of

definition
...
r
...
x

The above process of differentiation is called the first principal

7

Examples of differentiation from
first principal:
Example :1
We have

8

Example-2
f(x)=a where a is fixed real number

9

Example-3

Basic Formulas

10

Following are some of the standard derivative:-

Laws for differentiation

11

(Algebra of derivative of functions)

Let f(x) and g(x) be two functions such that
their derivatives are defined in a common
domain
...
Sum Rule

2
...
Product Rule

12

Continued

4
...


6
...
For example consider one
of the following relationship between x and y

NEXT SLIDE…
...

In the first case we can solve y and rewrite the
relationship as
In second case it does not seem easy to solve for Y
...


Continued

19

Example-2

20
…
...


Continued

21

Derivative of functions in
parametric forms
If relation between two variables is expressed via
third variable
...

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
...


22

23

Continued

24

Example 1 Find
Given That

So

Example-2

25

Logarithmic differentiation


...
This process in called logarithmic differentiate
...
r
...
r
...
x
...
It is also denoted by

...


30

-

EXAMPLES

Example-1
Given that
=

-

Here
=

-

31

Example-2

Differentiate again

Find

Gradient or slope of the curve

Let y=f(x)be a curve
...

Sometime the derivative is called gradient
of the curve
...
r
...
x
...
Integral calculus
deals with integration and its application
...


Also we can define integration as the inverse process of
differentiation
...
Here 'C' is called constant of integration

The process of finding the integral is called integration
...


Basic Formulas

47

Two Simple Theorem
1
...


2
...


53

Examples

Example1

adx = dt

54

55

Example 2

...


2
...


4
...


6
...


8
...


59

Example

Integration by Parts
It is useful method to find integration
of product of function
...

solve

where

77

Example:5
by simple division

78

Example-6
First simplify integrand

Example 7
...


81

Example 9
...

I

II

dx

Example11
...


Let

Example 13
...


87

88

Integral Calculus

89

Continued……

=

=
=

90

Continued
=

91

Continued

,

(a < b < c)

92

Continued
=0

93

MCQ’s

Question Time

Question:1

HINT-Logarithmic Differentiation

94

95

Question
...
3

HINT- Logarithmic differentiation then Apply product rule

97

Question
...
5

HINT- (Apply Quotient rule)

99

Question-6

HINT - (Apply chain rule)

100

Question
...
9

HINT- (Take log both sides then apply quotient rule)

103

Question
...
11

HINT- (Logarithmic differentiation)

105

Question:12

HINT-

(Implicit function)

106

Question
...
14

HINT-

(Quotient rule)

Question
...


HINT-

108

109

Question
...
17

HINT- Integrate By Parts

111

Question
...
19

HINT- Integration by substitution let

113

Question
...
21

HINT- [Integration by substitution let

115

Question
...
23

HINT-

117

Question
...
25

HINT- Divide Numerator by Denominator then Integrate

119

Question
...
27

HINT-

121

Question
...
29

HINT-

123

Question
...


...
keep practicing

---