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ENGINEERING MATHEMATICS-II FUNCTION,LIMIT,CONTINUITY,DIFFERENTIATION AND ITS APPLICATIONS
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1ENGINEERING MATHEMATICS-II
FUNCTION,LIMIT,CONTINUITY,DIFFERENTIATION AND ITS
APPLICATIONS
Dr A K Das, Sr. Lecturer in Mathematics
U C P Engineering School Berhampur
, 2
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)
, 3
y B ( x belongs to B)
y = f(x) can also be written as
(x,y) f
Since (x,y) represents a pair of elements we can think of these in relations
f AXB or
f can thought of as a sub set of the product of sets A and B we have earlier
referred to.
And, therefore, the elements of f are pair of elements like (x,y).
In the discussion of a function we must consider all the elements of set A
and see that no x is associated with two different values of y in the set B
What is domain of function
Since function associates elements x of A to elements y of B and function
must take care of all the elements of set A we call the set A as domain of
the function. We must take note of the fact that if the function can not be
defined for some elements of set A , the domain of the function will be a
subset of A.
Example 1
Let A={ }
B={ }
The function is given by
y = f(x) = x + 1
for x=1, y= 2
x=2, y=3
x=3,y=4
x=4,y=5
, 4
x=-1,y=0
x=0,y=1
x=-4,y=-3
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
, 5
Let y = f(x) = k =2.5
3
2
y axis
1
0
-3 -2 -1 0 1 2 3 4 5
x axis
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
4
3, 3
2 2, 2
Axis Title
1, 1
0 0, 0
-4 -3 -2 -1 -1, -1 0 1 2 3 4
-2, -2 -2
-3, -3
-4
Axis Title
, 6
Modulus Function
( ) | | { }
The rule here is: the value of y is always equals to the numerical value of x,
not taking in to consideration the sign of x.
Example
Y =f(2)=2
Y=f(0)=0
Y=f(-3)=3
This function is usually useful in dealing with values which are always
positive for example, length, area etc.
Dom = R
Range = R+ U { }
Graph of Modulus Function
5
4
y Axis
3
2
1
0
-6 -4 -2 0 2 4 6
x Axis
Signum Function
| |
( ) { } { }
This is also a very simple rule in the sense that the value of y is 1 if x is
positive , 0 when x=0, and -1 when x is negative.
, 7
Dom = R
Range = { }
Graph of Signum Function
1.5
1
0.5
0
-6 -4 -2 -0.5 0 2 4 6
-1
-1.5
Greatest Integer Function
( ) [ ]
For Example [ ] [ ] [ ] [ ]
Dom = R
Range=Z(set of all Integers)
Graph of The function
6
4
2
0
-6 -4 -2 0 2 4 6
-2
-4
-6
Exponential Function
( )
Dom = R
, 8
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
20
15 y=2x
y axis
10
5
0
-2 -1 0 1 2 3 4 5
x Axis
20
15
y=(0.5)x
y Axis
10
5
0
-5 -4 -3 -2 -1 0 1 2
x Axis
Logarithmic Function
( )
Dom = R+
Range =
Graph of Logarthmic Function
4
y = log x
2
y Axis
0
0 1 2 3 4 5 6
-2
-4
x Axis
, 9
4
3
y = log x
2
1
0
-1 0 1 2 3 4 5 6
-2
-3
, 10
LIMIT OF A FUNCTION
Consider the function
y =2 x + 1
lets see what happens to value of y as the value of x changes.
Lets take the values of x close to the value of, say, 2. Now when we say
value of x close 2. It can be a value like 2.1 or 1.9. in one case it is close to 2
but greater than 2 and in other it is close to 2 but less than 2.Now consider
a sequence of such numbers slightly greater than 2 and slightly less than 2
and accordingly calculate the value of y in each case.
Look at the table
x y=2x+1
1.9 4.8
1.91 4.82
1.92 4.84
1.93 4.86
1.94 4.88
1.95 4.9
1.96 4.92
1.97 4.94
1.98 4.96
1.99 4.98
2.01 5.02
2.02 5.04
2.03 5.06
2.04 5.08
2.05 5.1
2.06 5.12
2.07 5.14
2.08 5.16
2.09 5.18
2.1 5.2
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