Continuity
Md. Masum Billah
Assistant Professor of Mathematics
Department of Arts and Sciences
AUST
1
Continuity of a Function
A function f is continuous at the point x = a if
the followings are true:
i ) f (a) is defined
y
ii ) lim f ( x) exists
x a
f(a)
Md. Masum Billah
Assistant Professor of Mathematics
Department of Arts and Sciences
AUST
1
DEFINITION:
As x approaches a, the limit of
f (x) is L, written as
y f ( x)
L
lim f (x) L,
xa
a
if all values of f (x) are close to L for values of x that are suffi
Differentiation
Differentiability
A function f(x) is said to be differentiable at a point x = a if
f a hlim
0
f (a h) f (a )
exist.
h
f ( a h) f ( a )
will exist if
Again f a lim
h 0
h
f ( a h) f (a )
lim
h 0
h
and
f (a h ) f (a )
lim
h0
h
f ( a h) f ( a
Partial Differentiation
The Partial Derivative
The ordinary derivative of a function of one variable can be carried out
because everything else in the function is a constant and does not
affect the process of differentiation.
When there is more than one v
9/3/2015
In the name of Allah most gracious
most merciful
Integration
1.
2.
3.
4.
5.
6.
f ( x )
d x ln f x
f ( x)
1
1
d tan 1 ; a 0
2
2
a
a
a
1
1
a
d
ln
2
2
2a
a
a
1
1
a
d
ln
2
2
2a
a
a
1
d sin 1
a
a 2 2
1
d ln 2 a 2 s in h 1 .
a
2 a 2
1
9/3/
In the name of Allah most gracious
most merciful
Tangent and Normal:
Let P be a given point on a curve y = f(x) and Q be another point on it
and let the point Q moves along the curve nearer and nearer to the
point P then the limiting position of the sec
WELCOME
TO
the Presentation on
Maxima and Minima of a function
Md. Masum Billah
1
Issues to be discussed
Why
we need to optimize a function.
dy
What is the geometrical significance of
dx .
When a function is increasing & decreasing.
What is a stationa
In the name of Allah most gracious
most merciful
Asymptote
P
Definition:
3
x
2
y
If P be a point on a branch .
of a curve
1 extending
2a x and a straight line exists
beyond the finite region,
at a finite distance from the origin from which the
distance
9/3/2015
In the name of Allah most gracious
most merciful
Beta and Gamma Function
Beta Function:
The integral
1
m 1
x 1 x
n 1
dx, for m 0, n 0
0
is called beta function of m, n. We write as
1
m, n x m 1 1 x
n 1
dx, for m 0, n 0
0
1
9/3/2015
Gamma Fu
AHSANULLAHUNIVERSITY OF SCIENCE AND TECHNOLOGY
DEPARTMENT OF ELECTRICAL AND ELECTRONIC ENGINEERING
ROOM NUMBER: 4A03
Time
Day
Saturday
Sunday
CLASS SCHEDULE (SPRING 2016)
1.
YEAR: 1
SEMESTER: 1
SECTION: A
8:00 am 8:50 am 9:40 am 10:30 am 11:20 pm 12:10
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