ENGINEERING MATHEMATICS-II FUNCTION,LIMIT,CONTINUITY,DIFFERENTIATION AND ITS APPLICATIONS, Exams of Engineering Mathematics

Download ENGINEERING MATHEMATICS-II FUNCTION,LIMIT,CONTINUITY,DIFFERENTIATION AND ITS APPLICATIONS and more Exams Engineering Mathematics in PDF only on Docsity!

ENGINEERING MATHEMATICS-II

FUNCTION,LIMIT,CONTINUITY,DIFFERENTIATION AND ITS

APPLICATIONS

Dr A K Das, Sr. Lecturer in Mathematics

U C P Engineering School Berhampur

1.LIMIT OF A FUNCTION

Lets discuss what a function is

A function is basically a rule which associates an element with another element.

There are different rules that govern different phenomena or happenings in our day to day life.

For example,

i. Water flows from a higher altitude to a lower altitude ii. Heat flows from higher temperature to a lower temperature. iii. External force results in change state of a body(Newton’s 1st^ Rule of motion) etc. All these rules associates an event or element to another event or element, say , x with y. Mathematically we write, y = f(x) i.e. given the value of x we can determine the value of y by applying the rule ‘f’ for example,

i.e we calculate the value of y by adding 1 to value of x. This is the rule or function we are discussing. Since we say a function associates two elements, x and y we can think of two sets A and B such that x is taken from set A and y is taken from set B. Symbolically we write x A ( x belongs to A)

x=-1,y= x=0,y= x=-4,y=-

clearly y=5 and y= -3 do not belong to set B. therefore we say the domain of this function is

the set {^ }^ which is a sub set of set A.

What is range of a function

Range of the function is the set of all y’s whose values are calculated by taking all the values of x in the domain of the function. Since the domain of the function is either is equal to A or sub set of set A, range of the function is either equal to set b or sub set of set B.

In the earlier example,

Range of function is the set { } which is a sub set of set B

SOME FUNDAMENTAL FUNCTIONS

Constant Function

Y = f(x)=K, for all x

The rule here is: the value of y is always k, irrespective of the value of x

This is a very simple rule in the sense that evaluation of the value of y is not required as it is already given as k

Domain of ‘f’ is set of all real numbers

Range of ‘f’ is the singleton set containing ‘k’ alone.

Or

Dom= R, set of all real numbers

Range= {k}

Graph of Constant Function

Let y = f(x) = k =2.

The graph is a line parallel to axis of x

Identity Function

Y = f(x)=x, for all x

The rule here is: the value of y is always equals to x

This is also a very simple rule in the sense that the value of y is identical with the value of x saving our time to calculate the value of y.

Dom = R

Range = R

i.e. Domain of the function is same as Range of the function

Graph of Identity Function

0

1

2

3

-3 -2 -1 0 1 2 3 4 5

y axis

x axis

• 3, - 3
  • 2, - 2
    • 1, - 1

0, 0 1, 1 2, 2 3, 3

0

2

4

-4 Axis Title -3 -2 -1 0 1 2 3 4

Axis Title

Dom = R

Range = { }

Graph of Signum Function

Greatest Integer Function

( ) [ ]

For Example [ ] [ ] [ ] [ ]

Dom = R

Range=Z(set of all Integers)

Graph of The function

Exponential Function

( )

Dom = R

-1.

-0.

0

1

-6 -4 -2 0 2 4 6

0

2

4

6

-6 -4 -2 0 2 4 6

Range= R+

The specialty of the function is that whatever the value of x, y can never be 0 or negative

Graph of Exponential Function

Logarithmic Function

( )

Dom = R+

Range =

Graph of Logarthmic Function

0

5

10

15

20

-2 -1 0 1 2 3 4 5

y^

axis

x Axis

y=2x

0

5

10

15

20

-5 -4 -3 -2 -1 0 1 2

y Axis

x Axis

y=(0.5)x

0

2

4

y Axis 0 1 2 3 4 5 6

x Axis

y = log x

LIMIT OF A FUNCTION

lets see what happens to value of y as the value of x changes. and accordingly calculate the value of y in each case.

• y =2 x + Consider the function
• value of x close 2. It can be a value like 2.1 or 1.9. in one case it is close to Lets take the values of x close to the value of, say, 2. Now when we say
• a sequence of such numbers slightly greater than 2 and slightly less than but greater than 2 and in other it is close to 2 but less than 2.Now consider - x y=2x+ Look at the table - 1.9 4. - 1.91 4. - 1.92 4. - 1.93 4. - 1.94 4. - 1.95 4. - 1.96 4. - 1.97 4. - 1.98 4. - 1.99 4. - 2.01 5. - 2.02 5. - 2.03 5. - 2.04 5. - 2.05 5. - 2.06 5. - 2.07 5. - 2.08 5. - 2.09 5. - 2.1 5.

We see in the tabulated value that as x is approaching the value of 2 from either side, the value of y is approaching the value of 5 in other words we say, y → 5 (y tends to 5) as x → 2(x tends to 2) or

INFINITE LIMIT

As x→ a for some finite value of a, if the value of y is greater than any positive number however large then we say

Y → ∞ (y tends to infinity)

In other words y is said have an infinite limit as x → a. And we write

Example

If

,

Then

Since x→0, is positive,

becomes very very large and is positive. Therefore the result.

Similarly,

As x→ a for some finite value of a, if the value of y is less than any negative number however large then we say

Y →- ∞ (y tends to minus infinity)

similarly

As x becomes very very large with a negative sign or in other words the value of x is less than a very large negative number , i.e. x →- ∞, if value of y is close to a finite value’ a’, then we say has a finite limit ‘a’ at infinity and write

Example

Let

As x→∞ , becomes very very small and approaches the value 0.Therefore

we write

ALGEBRA OF LIMITS

1. Limit of sum of two functions is sum of their individual limits Let ( )^ ( )^ , then ( ( ) ( ))

2.Limit of product of two functions is product of the their individual limits Let ( )^ ( )^ , then ( ( )^ ( ))

3. Limit of quotient of two functions is quotient of the their individual limits Let ( )^ ( )^ , then ( ) ( )

SOME STANDARD LIMITS

1. ( ) ( ) where P(x) is polynomial in x

Example

( )

2. where n is a rational number

Example

Example

. ( )

6. (^ )

Example

( )

Example

( ) ( ) ( )

Existence of Limits

When we say x tends to ‘a’ or write x→a it can happen in two different ways

X can approach ‘a’ through values greater than ‘a’ i.e from right side of ‘a’ on the Number Line

Or

X can approach ‘a’ through values smaller than ‘a’ i.e from left side of ‘a’ on the Number Line

The first case is called the Right Hand Limit and the later case is called the Left Hand Limit.

We, therefore conclude that Limit will exist iff the right Hand Limit and the Left Hand Limit both exist and are EQUAL

Consider the Greatest Integer Function

( ) [ ]

Consider the limit of this function as

The right hand limit of this function

[ ]

Since if the value of x is greater than 1 for example 1+h,h then the greatest integer less than equal to 1+h is 1

The left hand limit of this function

[ ]

Since if the value of x is less than 1 for example 1-h,h , then the greatest integer less than equal to 1-h is 0

In this case the right hand limit and the left hand limit are not equal

And therefore the limit of this function as does not exist

For that matter this function does not allow limit as

Since the right hand limit will be always n and the left hand limit will be n-1.

Consider the Signum Function

Consider the limit of this function as

The right hand limit of this function is 1 and the left hand limit of this function is -1 as evident from the definition of the function and concept of right and left hand limits

Therefore this function does not have a limit as

0

1

0 0.5 1 1.5 2

-1.

-0.

0

1

-1.5 -1 -0.5 0 0.5 1 1.

This function is not continuous at x=4.Since the function is not defined at x=

Consider another Function

( ) [ ]

Consider the point x=

This function does not have limit x → 2 as the Right Hand limit will be 2 and the Left Hand Limit will be 1.Hence this function is also not continuous at x = 2

Example

i.e

( ) (^) ( )

This function is therefore continuous at x=

Limiting value is same as functional value

Consider another Function

( ) {(^ )

( ) ( ) [( ) ]

i.e

limit of the function is same as value of the function at the point

therefore, the function is continuous at x=

example

consider the function

Consider the point x=

Therefore the function is continuous at x=

As,

| | | |

Taking limit as x we can conclude that