Basic Concept of Differential and Integral Calculus

CPT Section D Quantitative Aptitude Chapter 9

Dr. Atul Kumar Srivastava

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

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

4.(Quotient

5.

6.

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Rule) Continued

Example-1

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Find

EXAMPLES

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….

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….

Examples

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NEXT SLIDE….

Find for Example 1

Continued 19

Example-2

Differentiating on both sides …..(1)

NEXT SLIDE….

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Continued 21

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

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

Example-2

28 Differentiate

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

EXAMPLES

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Example-1

Given that

Here

=

-

-

- =

Example-2

Differentiate again

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Find

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

Example-2 Find the slope of the tangent to the curve

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Miscellaneous Examples

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Example:1 Differentiate

Example-2 Differentiate

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Let

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

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

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)

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

Since integration and differentiation are inverse process we have

Example-1

1.

2.

48 Two Simple Theorem .

Example-2

Example-3

49

Example-4

Example-5

Evaluate 50

or

Example-6 Evaluate

Example-7 Evaluate

51 Examples

By simple division

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Example-8

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

or

.

Evaluate

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Example 2

dt

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Example-3: Evaluate

or

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

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

Evaluate

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Example-2

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Example-3

(Solve first integral only)

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Example 4

Methods of Partial Fractions

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Find the partial fraction of

Example-1

we put x=2

and get

we put x=3

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Type-1

Comparing coefficients of and constant term on both sides

Solving we get

Therefore

67 Type-2

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Example

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

70 Properties

Example-1:

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or

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Example-2

let

find

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Miscellaneous examples

Let

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Example-2

integration by parts

consider

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Example-3

solve

where

Example 4.

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by simple division

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Example:5

simplify integrand

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Example-6

First

Example 7.

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Example 8.

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Example 9.

I II

82 Example 10.

dx

since it passes through the origin (o,o)

Then

Given

or

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Example11. Find the equation of the curve where slope at (x,y) is 9x which passes through origin

Let

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Example 12.

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Example 13.

SUMMARY OF THE CHAPTER Differential calculus

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87 Continued…….

Integral Calculus

88

=

=

=

89

Continued……

=

90 Continued

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Continued

(a < b < c)

,

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Continued

= 0

Question Time MCQ’s

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HINT-Logarithmic Differentiation

94 Question:1

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Question.2

HINT- Logarithmic differentiation then Apply product rule

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Question.3

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Question.4

HINT- (Apply Quotient rule)

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Question.5

HINT - (Apply chain rule)

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Question-6

HINT - (Apply product rule and chain rule)

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Question.7

HINT- (Quotient rule)

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Question 8

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

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Question.9

HINT- (Differentiation of implicit function)

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Question.10

HINT- (Logarithmic differentiation)

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Question.11

HINT- (Implicit function)

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Question:12

HINT- (Logarithmic differentiation)

(d)

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Question.13

HINT- (Quotient rule)

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Question.14

The value of p and q are.

HINT-

108

Question.15

HINT-

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Question.16

HINT- Integrate By Parts

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Question.17

111

Question.18

HINT- Integration by substitution let

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Question.19

HINT- [Integration by substitution, let t

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Question.20

HINT- [Integration by substitution let

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Question.21

HINT-

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Question.22

HINT-

116

Question.23

HINT- (Integration by substitution let 7x + = t)

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Question.24

5

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HINT- Divide Numerator by Denominator then Integrate

Question.25

HINT-

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Question.26

HINT-

120

Question.27

HINT-

121

Question.28

HINT-

122

Question.29

HINT-

123

Question.30

Practice makes a man perfect and that’s what mathematics demands. .

So, students ‘all the best’ for your upcoming examinations . keep practicing.

Thank you

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