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Functions, continuity, discontinuity, linear system of equation, homogenous system of linear equation, taylor's series, macluran's series, eigen values and eigen vectors
Subjects
functions
continuity
discontinuity
gamma and beta functions
maxima and minima
eigen values and
linear system of equations
homogenous system of linear equations
taylors series
maclurens series
Connected book
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M S Ramaiah University of Applied Sciences
Engineering Mathematics-1
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mathematics
Engineering -
I
, Module -
1
Functions
A function f is a rule that
weighs to each element n in a
element called fca )
ut D
exactly one in a set E .
* The domain of the function f is the set of all
possible inputs
of fecal .
The of the function f is the set of all
possible values
*
range
domain
of fecal as a varies
throughout the .
Independent variable ( symbol
A that represent
→
, y = f Ca ) arbitrary number
an the in domain )
a
variable ( that
Dependent A
symbol represent a number
in the
range of f)
x ¥ t
f! s teal
( output)
( input)
function f
machine diagram for a
f- cist
-
fi
- -
→
#
-
.. . - - - - - - - - - -
, :
!
}
rang
the '
!
ffa -
÷
.
s
,
yes ,
!
.
i
I 2 N i
,
i
-
:
domain
, Eixample : find the domain of each function .
⑨ feel ⑨ '
at a
gcais
=
n' -
a
solution The domain of f consists of all values
: ⑨
of a such that at 230
a > -
2
r
'
. The domain is E- 2. n )
⑧ the domain of gcn , is
{ KER n to ,
a # I
}
which can be written in internal notation as
1- A
,
o
) U ( O
, 1) U ( i
,
D)
The vertical line test A the
ay plane
: curve in - is the
of a function of iff vertical line intersects
graph a no
the curve more than once .
Y 't
n
wa
n
c.a.cl
~ is
fan : !
)
I
"
Ca b Ca b I
.
, ,
° ° am
. . this curve can 't
but
represent a function
represent a function
( became function auign two
different values to a ) .
, piecewise defined -
functions
-
The functions which are defined
by different
formulas in different parts of their domains .
Example .
"
{
N it
'
as a
#
' -
il fear .
'
n if a > 2
21
{
I R E C- A. o )
f- Cal =
2 N Elo ,
N )
s) Absolute value function .
{
N it a > o
f- Cnt lay =
-
n if u so
feel = - K C -
0,0 )
flat -
- a o
,
H )
ry
Int
y
-
-
o
7k
, is
4)
I
.
O
'
S n
l
{ If l
*
flat -
-
n
l 2h31
=
-
Even
-
function of odd
-
function
-
A function Fca ) is said to be even if
FC -
al -
-
flag for all n in its domain .
A function Fca ) is said to be odd if
f-C al
-
-
-
-
feng for all u in its domain .
E e) feat = ah - s even function
fl al-
-
-
Ext n' flag =
ii ) fecal = n
'
→ odd function
f- C al
-
=
f- ng? -
n! -
f Ca )
, graph symmetric
~
" If the is
→ Y "
with then
respect to y
- anis
we
say
that the function
is even .
If the graph is
symmetric
about the then
originthe
we
can that function
say
is odd .
Example : Determine whether each of the following
functions is even odd or neither even nor odd
, .
.
att 't -
1) flat = n 2)
glut I -
n 4 Hat 2x - N
solution :
'
i ) t C al -
=
faf -
a = - a - se -
-
-
Latta )
fl -
a) = -
Fca )
i . flu ) is odd function
2) C n' 't
gcn )
offal I i n
-
=
-
-
= - -
gl al
gcn,
-
- -
function
gcn
I . , even
, -
3) h1n1 = ga -
a
ht -
a ) = -
2x -
C - NII -
2x -
RE -
( satay
ht - n ) th Ca ) or ht self
- - heal
.
'
.
hla ) is neither even nor odd .
and function
Increasing decreeing
A function f called interval
increasing
is on an
I if
f- Cm ) s the 4 whenever Ncaa in I
function f called interval
A
I if
is
decreasing on an
f-Cm ) > flxz) whenever me na in I
The function flat = are
→ y
=p
- is
decreasingand on the
interval C -
H
,
)
o
the Eo )
increasing on A
.
, polynomials :
A function play is called a
polynomial if
'
ant an an
-
plat = an . , t .
. - - -
ta , n t Ao
and numbers
where n is non -
negative integer
constant called the coefficient
are ,
a. .
. .
an are
.
of polynomial peal .
any polynomial
* For the domain is A- twig
* If the
leading co -
efficient an to then the
degree of the
polynomial is n .
't
Ex : .
peak 3kt -
2N tf n 't 2
this is the
polynomial of degree 5 .
* A
polynomial tb
of degree
whie is called
of the form i
linear function
is
play an
-
-
*
polynomial
A of degree 2
,
function
plain -
an 't bat e
and called is
quadratic
power function
Attu of the form flat = sea where a
,
power function
is a constant is called a .
"
ri y a
-
n
-
y y
y
y
- .
- -
,I
, Transformation of functions .
Vertical and
Horizontal shifts
suppose c > o .
distance
of
y teal shifts the
graph y feat a
-
te
-
- -
,
c units
upward .
of distance
y teal shifts the
graph y feat a
- -
- -
c -
,
c units downward .
of distance
y ten ) shifts the
graph y feat a
-
-
- - c -
,
units the
c to
right
of distance
y ten + " shifts the
graph y feat a
- -
- -
,
e units to the left
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