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Hello this is Dr. Brown
In this video we are going to look at the
derivative as
a rate of change
You've already been introduced to the concept
of the instantaneous rate of change
with respect to x at a point
it's just simply that lead to real reform
external physical to of election of course
each -12 works no
old-world each as
well each field 0
well lose
arm let's look at an example here
are as everyone knows your
circle is reloaded to
worker this equation I'm sure you
absolutely the Eagles
arch where course or
physical the armored
whose with this five times
to you to on
squared which is very cool follow
forecast where she suppose you ask a question
of plasticity area change with respect to the
*** with ruston movers well
what you do yesterday the derivative
public area with this book to the doc
just using the power rule
that would be
power for arms
power which to the
to our moms warn which uses
equal for work you
times the and so
if you want to see how s your
changing no with you
to it would be simply
far yeah that would be
are square meters
now we're going to all interviews
conserves we were told by a little bit
we have all generally talked about
yes representing position
whole body
at T
and though we're now formally
the following the world lost city you
to as the derivative
us with this book to and as you can see this
we will limit I'll to the plus
bill to the moms of to the divided by build
to you still to go 0 are
if you're familiar with the arm change into the free
each missus this absolutely use
ulam sake pros she's 0
over to force each myself
to the world by each for ages
built here we
the farm bill to just to show that there was a change
and time pass
the time the so we have
instantaneous velocity which is the
ru with respect to time all
position
now suppose you the graph like this
in which
the position was increasing you can look
this group conceive the st.
which is a slope pendulum is
more 0 so positions increasing
it has a policy slow so the
love body is movement for arm
you the position
is decreasing I'm then you can see the
slow July very negative the st. to
would be less 0 and so
the bodies Reebok now we're going to
to find what we call speed and we call that the absolute value
gloss that's really how fast your blog loaded on
forward so if you're headed north
waynesboro a hundred miles per hour
we lost the is a hundred miles per hour
if you're headed love self
Swainsboro you might be going moms
100 balls for or this the
which is an absolute you
were lost both cases your
all myself s you're traveling
here's just
you all law of the velocity
our way body from equal 0
you servants sockets just
showing what some other possible or
as you can see here they were beginning
are the velocities increasing
between
220 to one are basically the body is speeding up
in between the to
want to the velocities
artist all
body is still moving forward with this
use cards and in between two equal to
to you through you can see that we've lost
use decreasing all down 20
and so the body has been slowed by now
now
between to the 3 empty for
you conceive the body
is headed in the other direction this philosophy
are actually
absolute you at this particular point
is the is going too fast is
that has for work
looking the fears estas
it will cure its velocity
is negative but its 3 its
be is largest that has been all
during are
this time backwards moving backwards
and in between for him to 240 love
the negative velocity
heads towards
targets are smaller and smaller enough
you following the loss egos 0
so the arm the body is slowing down
and then between five and six
soccer's we lost the use
0 and so the body's basically standing still
and then unless between six
themselves accuse it speeds up again
the begins to move forward 10 things you need to be able to do
perfect you know graph like this
now
when you're going along on a call or and your
going asserts the you so you need to speed up
you do we call acceleration
but you the accelerator min creatures
and so we do farm the acceleration
as derivative of the Lost worked
love so
body's position is a sequel the
than the acceleration
is DVD which is
suck Peru public position
it and informally
there is your kermit
will be using all which is the
just of SC acceleration is changing work
time it's called your and
is Jesus the derivative you with respect to T
which resist the urge to rue all bloopers issue
split know as we were already talked about nearly
service earlier all bodies fall
the same cost acceleration this came from girl whose experiment and you
notice we've already talked about
all the position all
object on earth all
be you
given by sixty T-square
that is just one
times 32 comes to the square
word she is
exhilaration gravity
32
use 32 the for soccer for circuit and that is an
English units and you taken
physics see this express Emeka
use all for
Yessica 10 GT squared and
the next page rule who express
him actually before we
go to the next page just want to show you the relative sizes
alva some planets that you know
well obviously yours or the
rotational constant this non ***
the meters per socket
for soccer and
because a gravitational classes directly
proportional to love
my s love all
the planet you see Venus
would have a rotation cars somewhat less
non for are
Morsi the evil us you 20
which no longer a planet it would
be very small and then you can look at Jupiter and Saturn
which over or small or
their revocation cost very
for no Road
member some
love with just four everything
we're gonna work problem here
and we're going to do this one are in
metric units
arm
the figure here shows so freefall
lovely every ball bearing as it's dropped
from rest position to you hero
and here are she
is known for
and yes would be one
GATT's were which would be
4.9 T-square
so the question is how many meters does
ball fall the first to suck
and then answer the question what is velocity
the an acceleration
went to cool to suck
so just plugging in to equation for
s review 4.9 times two squared
equal like to import six meters is
for the ball falls
in the first two suckers
now to calculate the velocity
we need to take the derivative position
which is DDT for 490 square
using the power rule you're going to get to
4.9 times the to
or -1 which is known for
to so this gives the velocity
two sockets obviously is two times Nonpoint
or 19 points Barnes six meters
suck
arm which the
timetable to was absolute value
which is night 3.6 meters per second also
now the acceleration
is the derivative glossy
which is a secure review position well we knew
velocity
was not born to the so
free-for-all missus importantly
so the acceleration
mmm love
this non *** meters per second squared
and we we knew that
another example all we've got
are a an equation for the position
heavy rock are that
his mom straight up with watch Lost the
over 160 for soccer which is about a hundred
miles per hour arm
this is called fun of dynamite blast and
its reaches a high
yes equal 160 Mon 16 to the squared
after to suck so
your questions your hard as a rock go world
the all several ways to do that
you can calculate the velocity
all the rock which is two years due to the
which would be DD to you 160 -60
T-square taking that the room with you 160
months 32 the feet per second
and the rock will continue to climb until
velocity this 0 so
if we sit the Los the which is 160 moms
32t 0 solid to you
T sockets now
so it climbs for farms occurs
the sea of for its goal you plug his fav
into s equation and we go to hundreds issued
-60 I'm Times Square calculating this
review that it goes 403 Business School in store
work now next question
is for velocity ins the rock
when it is 256 the aboveground
formal way of no we know that it goes up
to 400°c and so
as its climbing you reaches 256 the
point walking no all
what the loss to use
speeded Rockies won its 256 so we have to say the position equal to 256 the
and solve for t to find out of order that
times all with him the
Rockies 256 so
solving this equation
to the we get a quadratic equation
we take 256
over to the other side and
actually love were taking
these terms over to the other side and creating
you know quadratic equation
with love 62 squares pause
yours so we good equation 16 to the square
minus for 160
for 256 hero immediately so you can factor sixty you know
and so for a quadratic equation
the square miles to empty sixty so that equals 0
from this particular one can be factored into T minus two
steven is a always check your factors
your farm in this case to equal to soccer's
to you kuwait's occurs these are the points in time
with the Rockies likely 256
above right now so the
rocks velocity these times can be
determine bar the velocity equation
remember who lost the was legal
160 moms 32
see so plugging to in
we go to 160 164
plus 96 week so it's
losses pause your own way of
calculating this at equally
when it's on the way down we good 161
32 times I the coal mines 96
free hook Norris its actual
the is exactly the same to
two circuits and to collects
and then the next question is what is its operation
rock at any time to you during
flawed well the acceleration
this DVD and recall that the
velocity was a 160 moms 22
taking that to Rubidoux we get -32
for soccer for soccer and that rudeness moms
32 for suckers were so
the acceleration of gravity as you know is always downward
was always trying to bring anything
above the surface your torture
on this is acted upon by another force
like a rocket turkey call for your throwing the ball
the last question is when does rock it brown
and of course rockets brown
are the positive time to the
for which Yessica 10 so you recall that
s was given by 162
Mon 16 square setting that equal to 0
you obviously can't factor sixty
to you all and your 62 times too much to the
because 0 the solution to this is to Google 0
obviously that's the very beginning with the ball with the
all when the Rockies
is blasted off the other time is tee
tune with its a groom so the
ball hits or
rockets a groom cancer occurs after
blast for so now we
see your own love this law first ball
position versus time Aug
the rock we see going up to a maximum
position for the you remember
to the to and
see cool the it was 256 the
we will also
applauded the losses the
of this rock which was a 160 moms
32 to which is a straight line and you can see the velocities paused
first five seconds the rock is on his way up
and the lost it becomes negative on the next lives
accuse as it comes now right now