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.

Differential calculus-Outlines

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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

What is differential calculus – an 4 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.

Derivative or differential coefficient (First principal definition) Derivative of

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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.

The derivative of f(x) is also w.r.t. x

The above process of differentiation is called the first principal

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Examples of differentiation from first principal: 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

Basic Formulas

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Following are some of the standard derivative:-

Laws for differentiation

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(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 1.Sum Rule

2.Difference Rule

3.Product Rule

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Continued

4.(Quotient Rule)

5.

6.

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EXAMPLES

Example-1

Find

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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

NEXT SLIDE….

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Continued…. 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. 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

for implicit function

Examples Example 1

Find

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for

NEXT SLIDE….

Continued

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Example-2

20 …..(1)

Differentiating on both sides

NEXT SLIDE….

Continued

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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 Find Given That

So

Example-2

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Logarithmic differentiation

.

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and

We differentiate such functions by taking logarithm on both sides. This process in called logarithmic differentiate.

Examples Example

1

Differentiate

Taking logarithms on both sides

Differentiate both sides w.r.t x

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Example-2

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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 or and is denoted by . It is also denoted by . If we remark that higher order derivatives may be defined similarly.

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-

EXAMPLES

Example-1 Given that =

-

Here =

-

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Example-2

Differentiate again

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

Given

The gradient of the curve at point X=0 is -12

Example-2 to the curve

Find the slope of the tangent

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Miscellaneous Examples Example:1

Differentiate

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Example-2

Differentiate

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

Example 7

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Example 8

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Continued

By cross multiplication

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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What is Integration (definition) 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.

Also we can define integration as the inverse process of differentiation .

Constant of integration

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and c is an arbitrary constant we also have

Evidently the integral of

is obtained by giving different

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.

Basic Formulas

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Two Simple Theorem 1.

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.

2.

Since integration and differentiation are inverse process we have Example-1

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Example-2

Example-3

50 Evaluate

Example-4

or

Example-5

Examples Example-6 Evaluate

Example-7 Evaluate

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Example-8 By simple division

Integration By Substitution The given integral can be transformed into another form by changing the independent variable x to t by substituting Consider

Usually we make a substitution for a function whose derivation also occur in the integrand.

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Examples

Example1

adx = dt

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Example 2 .

or

Evaluate

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Example-3: Evaluate

or

dt

Important standard formulas 1.

2.

3.

4.

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Continued 5.

6.

7.

8.

.

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Example

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

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Examples Example:1

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Example-2

Evaluate

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Example-3

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Example 4

(Solve first integral only)

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Methods of Partial Fractions

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Type-1

Example-1

Find the partial fraction of

we put x=2

we put x=3 and get

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Type-2

Comparing coefficients of

and constant term on both sides

Solving we get

Therefore

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Example

Definite Integration

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Consider indefinite integral Now consider Here a = b =

lower limit of integration upper limit of integration

is called definite integral of f(x) from a to b

Properties

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Example-1:

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Example-2

or

Miscellaneous examples find

let

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Example-2 Let

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Example-3 integration by parts

consider

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Example 4. solve

where

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Example:5 by simple division

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Example-6 First simplify integrand

Example 7.

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Example 8.

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Example 9.

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Example 10. I

II

dx

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

since it passes through the origin (o,o) Then or

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Example 12.

Let

Example 13.

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SUMMARY OF THE CHAPTER Differential calculus

Continued…….

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Integral Calculus

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Continued……

=

= =

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Continued =

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Continued

,

(a < b < c)

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Continued =0

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MCQ’s

Question Time

Question:1

HINT-Logarithmic Differentiation

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Question.2

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Question.3

HINT- Logarithmic differentiation then Apply product rule

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Question.4

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Question.5

HINT- (Apply Quotient rule)

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Question-6

HINT - (Apply chain rule)

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Question.7

HINT - (Apply product rule and chain rule)

Question 8

HINT- (Quotient rule)

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Question.9

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

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Question.10

HINT- (Differentiation of implicit function)

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Question.11

HINT- (Logarithmic differentiation)

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Question:12

HINT-

(Implicit function)

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Question.13

(d)

HINT-

(Logarithmic differentiation)

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Question.14

HINT-

(Quotient rule)

Question.15 The value of p and q are.

HINT-

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Question.16

HINT-

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Question.17

HINT- Integrate By Parts

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Question.18

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Question.19

HINT- Integration by substitution let

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Question.20

HINT- [Integration by substitution, let t

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Question.21

HINT- [Integration by substitution let

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Question.22

HINT-

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Question.23

HINT-

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Question.24

HINT- (Integration by substitution let 7x + 5 = t)

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Question.25

HINT- Divide Numerator by Denominator then Integrate

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Question.26

HINT-

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Question.27

HINT-

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Question.28

HINT-

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Question.29

HINT-

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Question.30

HINT-

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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