Calculus Introduction - Differentiability Video Lecture - Class 12

hello friends this video on continuity
and differentiability part 1 is brought
to you by exam viacom no more fear from
exam so before we go ahead since this is
the first introduction to calculus let
me give you a fair idea on what the
calculus person in the world of calculus
we have to be important term the first
term is called differentiation and the
second term is called integration
please note calculus is all about
differentiation and integration so what
does differentiation and what is
integration differentiation is nothing
but is the process of finding slope of a
curve slope of a curve if you have a
curve and if you want to find slope you
can use differentiation and what is
slope slope is nothing but change in Y
with respect to change in X that is
change in Y with respect to change in X
this is called slope so example if you
have this car moving so generally you
know you have a car you go out somewhere
the car moves and if you plot this
distance with time graphs example
suppose this is my distance my Y is
distance covered and X is the time at T
is equal to 0 you're at 0 I know and you
keep accelerating and somehow you get
this kinda graph at T is good to let
suppose 1 2 3 4 5 seconds your hair and
now you want to find different thing for
example at this point T is equal to 3
you want to find the speed speed is
nothing but change in distance by change
in time that is Delta X by delta T I'm
talking about an instantaneous speed at
this point of time you want to find the
instantaneous speed well what you loop
if you have this distance time graph
then need to just have to find slope of
this point so at this point
slope of the graph is nothing but the
instant we speak correct and that is
determined using differentiation just
you tell you what is the difference
between instantaneous speed and average
speed for example you are traveling from
let us suppose c d e u c DB and the CDA
to c DB
are the distance the distance is there
suppose hundred kilometer and you
covered the whole distance in let's
suppose to are correct so you have read
speed is 50 kilo Interborough correct
because total distance metal in time
your average speed is septic lamentable
are but your speed was not constant
somewhere you found the traffic was more
your vehicle was moving in a lesser
speed and then you know you got some
highway and you were speeding up and
then again you got some traffic area and
the vehicle was moving at a slower speed
so you try this you speed is not
constant and somewhere in the let's
suppose in the highway this is a highway
area a police guy catch you but and he
tells you're over speeding but if you
tell that i cover 100 kilometer in to or
my speed is 50 limit the power that is
not a valid answer because that police
guy is not interested in your average
speed but he is more interested in your
instantaneous speed what is your speed
at that particular point of time so in
your highway your speed was hundred
kilometer per hour let's suppose and
here you have to live with lesser speed
maybe ten kilometer per hour the traffic
was more so average came out to be
particular a bar but finding
instantaneous B is difficult how to find
that so if we have this graph at any
point of time you want to find the
instantaneous speed you just take that
point and find the slope and to find the
scope what you do you just find the
differentiation of this equation at this
point of time so you learn more when we
learn this definition more but just
understand the basic concept this
magician is not about is just about
finding the slope of a curve that is dy
by DX change in Y with respect to X in
this case change in distance with
respect to time this is just one example
and one example what I told you was you
are traveling from CDE to cdb and Perez
like a shoe in a point let's suppose at
this point see and he wanted to find
what was your speed at that particular
point of time and that particular time
your speed was finally able to per are
but the average speed is you take him
into power
so to find average speed how you got
this you use the differentiation if you
have the slope if you have a graph you
just differentiate it you get in this
case differentiation will give you
change in Y by change in X correct here
Y is nothing but distance that is Delta
distance by X is time.deltatime and this
is nothing but is then taneous speed
please note it is not average speed is
in instantaneous speed so that is one
application of differentiation the next
topic is integration integration is used
to find area between graph of a curve
and next census the whole point is you
have this graph and then what you do is
you find the area for example this is my
graph and I have my curve like this
right and I now want to find the area
this area this area the area between the
curve and the x-axis what can I do I can
use the process of integration and what
integration does is it just cut this
area into small sections small area it
will cut all this area into small small
areas correct small small areas and then
it will add
this area's so if you see integration
and differentiation are totally
different in differentiation we find
slope dy by DX in integration what we do
is find area and we add up you cut area
into smaller and then add up so
integration is more about adding for
example for the same graph and you want
to find differentiation you divide Y by
X correct but integration is nothing but
you multiply Y into X because that's
what you get area you say why you say X
you multiply you get area so
differentiation integration you can say
that both deals with a curve both DS
with a curve which has X&Y in case of
differentiation we say Delta Y by Delta
X in case of integration we say Delta X
into Delta Y and we add all the small
small areas we get and get a combined
area so that is all about
differentiation and integration to start
with I think it should be enough for you
to understand what is differentiation
and integration just in the layman term
and then we can now proceed with the
difference in each other
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