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well if you following these presentations along in the order in
which i recording them then this is their sevens presentation intimate ask
itself by now
uh... he's talking about calculus all the time but we're basic emptiness
intently sometimes
the
materials that we have covered so far i sometimes facetiously call multi
variable pre calculus
because after all calculus is about limits derivatives integrals and we
haven't booked any of these concepts yet
uh... that does not mean that book we have done so far is is to be trivialized
or anything like that
if lays a foundation for the delay station
int without power of calculus and certainly comes from the fact that we
will be able to
uh... connected visual concepts to analytical concerts which will allow us
ten to do all sorts of really nice physics
we will start his presentation with units and continuity
although
we will mostly use derivatives integrals in calculus frightening you've done
similar book out this for about a year
and i'm talking about limits now and that that's way back hopefully in the
back of your mind that at some point the code we didn't do a little bit about
limits in a personal often
we talked about unity unity
but other than that you know it's basically
we're just
concepts that form of the underlies the reducing intercourse and and dot it's
not bad
uh... it's just that that means nimitz will also be the formal starting point
here
uh... so yeah we must start the chiapas effectively functions with limits
limits are conversant we note that because
uh... before we have no p cultural limits were really hard to compute ralph
mchugh creek that for
shortcut uh... formulas for different station
in even with all the cultural sometimes limits can be
the dvd cumbersome
thankfully a lot of the calculus of activity functions can be reduced to do
in calculus with the components
and in fact that's another
potential mantra that we're going to repeat quite a bit in this course and
that is whenever you do something with pictures
you typically do a component sup
that's first reminders to us at the old inmates are basically defined the same
way
and that is that when the input gets close to the case will be take the limit
the o put must be close to the limit itself so if you want to recall
the decision from single variable calculus was
the informal definition of the democratic point was
if he is a real number
and the function is the said to convert should've been l_a_ pete
utility for all its efficiently close to what not equal to p
we have the devil fixes costa
and analytical data function happy
an estimate l equals the middle ever picks as exco stupid
the definition format victor valued functions is going to be exactly the
same
so the informal definition of the demand for vector valued functions yes
that the victor elm
is the limit of the picture of any function sea of thier p
it's not only here for all teens that are sufficiently close to but not equal
to p
we have that sea of t is closed
and in singles who arrived at the limit of seo t s t goes to p
used instead
and so visually speaking if we just take any
trajectory of any vector valued function in its present
safe that
this place here you know is the limit author victor valued functions and
basically what happened to use that
we control a little circle of radius epson around at that point
intend for all parameters are sufficiently close to p namely for all
parameters their president even delta and people need to have the exact house
here
but for everything that is sufficiently close to p
curve must be
or that the points out
uh... are obtained by evaluating the bekaa valley function
must be within this middle circle
around the limit
now this already goes toward in in that direction off the excellent out the
definition of the damage which we are
not going to discuss
in the mainstream of these presentations
ever is because
we have been a very simple results that
replaces things computationally and that is the limit of a vector body functions
the empty can be taken component west
that is the limit as he goes to peel c_r_t_ estimate as he goes to peel off
see one of two through and
cnf d_n_a_ give notice that
we don't worry about how many components of victor asked for two dimensions
which started in situ for three-dimensional would stop at c_ three
in the main thing is that the thing that can be moved
into the compound salutes the of tea is same as
stack up a vector conf consisting often thoughts you want to use of the nazi to
to you all the way through
than of
seem in off-key
that is
how we're going to make a lot of the transitions to
multivariable calculus whenever we have a victories and output we will typically
apply at the operations of calculus
components
which means
uh... dino ke some of the equation used to be read estimate on the left side
existed for millennia
all the news on the right side exist
basically what this means is that the convergence
the victor valley calculus two single variable calculus
and is the reason for that bs well if i have basically the death of a close its
rights of the limit is about the business between c_e_o_ tedium
well that's if we try to comply by sets about there
uh... know ralph the difference between c one f t through c_n_n_ few months
l_-one through ellen
and that's a square root off the sofa squares of the differences so that nor
mister spirit of c_-one of t_-minus l_-one squared plus he too often
went out to square and so on all the way through well
it's enough to go stripping out
then fourteen close to the point where we evaluate the dentist
this left hand side must be small
and so they say must be small breakfast and a small
then none of these
uh... differences can be smaller right-side can be large on the right
side because then
the right side would be large that means all these differences must be small so
that means
the bitter about it but now defunct converges
then all the components must converge because all these differences must also
becomes final
conversely
if all the component wise
if all the components converter all these individual differences here
here's a certain you also in the thoughts
in here
if those differences become small twelve and the some will also be smaller and
that means that the overall
normal if the differences off the vectra sauce xml
so that means
we can replace closeness
of the victories with close this off the components
other phone rings i apologize and to put
okay well that pos actually came at the right link because we are now ready to
work on an example that shows that pizza computational chemist is just a
component wisc
computation
and so here we go
i want to ask you did not get the limit as to ghost is you know three months to
t squared
keys guidelines to you over to u_n_ co-sign t_ minus one over to you or
shows that it does not exist well
we would just have to compute the limited what do we remember
from computing limits well
the three months to achi square should be no problem at all
because that's what you're trying to equal zero and get free
in this one you we cannot guarantee equal zero dbe can cancel the tea so
this will just be t_ minus one
and then we can compete and and and for this one for stuff like this
we had hoped to talk to him if we take the derivative
numerator int denominator here we should end up with
uh... negative
signing off to you
uh... over one
and uh... well that's what we are getting
keep the first couple and we do algebra the second component and the u_s_
no because one of the third component that we just
take the typical point wise to get free *** neg one zero
is the limit
invested all of it
so this is nothing but a review of single variable calculus
this is just something we're uh... we
go into the spirit of what we also have to do most of your account please and
thank yous
when we needed we simply have to rely on the things
that we already have learned and no matter how deeply their store we have to
get him back
now the title of continuity indeed they were going to be
link the definition of continuity to the definition of continuity that you
already know
because again continuity is always to find the same way
the unit at the point must be equal to the value at the point
so recall for example the definition off the limit of the function if this is a
function
than if is called continuous at any different only if the following three
conditions satisfied
eppadi has to find
the limit as exposed a_t_f_ affixed exists
in limit as exposed a f affects must be perfect
indicators continues at every point eighteen percent capital a
then it was called continuous on a
uh... alright
soul
for a decade of any function will collapse in continuous acting any
differently if the phone three conditions are satisfied
seat of a must be defined
visited as t goes to pay off c_r_t_ must exists
is but the biggest single students you up to you must be equal to
see a pic so it's exactly the same definition
as in one dimension put people typically remember is
of course this equation he read it doesn't have to help
the people remember
this equation
because
uh... after all
uh... that equation implicitly assumes that the first two
our sas files
and then also there too much function of the quote continues on the set gas if
and only if
it's continues at every point in the senate s in fact
uh... leads us to the proposition that makes everything work out like it
uh... single variable calculus intensive victor body functions these continuous
acne
if it only if all its components are continuous data and that comes
directly from the fact that of course the units are taken component wise
c_e_o_ pay is also the value component wise which means all of this
reduces to conditions on the components which are just continuity conditions
inspector baseball just continue unless it s with its own earphones components
for continued assistance because if continuity i readings to continuity of
the components at a
then the same thing but with stats also
what he would come into i think the final example buddy
and we want to determine where the function seo team being natural ought
not in my list he squared went over to my knees t
in spirit if t plus one day three-dimensional bitter very function
without thingies continue as well
we have to look where the components are continuously every component
is continues on its domain
the north of nine nine ist he square discontinuous on eighty three three and
it's not defined
outside that domain right because he must have faxed
peace process articles in nine which happens exactly or negative three three
uh... one whatever two-minus team will
but can happen is that you could divide by zero so you have to exclude too so
that's continuous on
interval from negative finicky cat to union the interval from two to infinity
in security plus one
is continuous on the interval negative one
infinity because
radhika into this stuff out of the rooms
must be
greater or equal to zero
so basically determining continuity is again exactly the same as in one
variable we need to figure out where things can go over all of which are
typically
negatives were there shouldn't be such as in square roots pornographers
into divisions by zero and as soon as the excludes those we've got
case qasim
rate domains where
the components are continuous and then we just have to
intersect these things so the construction see is continues on the
intersection of these domains
which he is
interval that starts at made of one inclusive inspect knit ektu exclusive
invent union
two-three because basically
negative three three if we are sick that was negative one infinity beget negative
one two three and then sent to throw away the point
is that
is already
so this one
was hopefully refreshingly simple of course you have to take back up what you
know about limits and that me
take a minute or so because we haven't witnessed minutes for a while and then
some prosecuting its
but overall this is now a first step into calculus and know that we have
limits
into full foundation to define derivatives and integrals
as well as a computational tool
uh... or a computational insight that says basically everything ought to work
component wise
we should be able to talk about derivatives integrals and that's what
makes presentation is about
i'll see you there