WEBVTT
Kind: captions
Language: en
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Okay so we start the third module today This
module will be mainly devoted to discussion
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on maxima minima for functions taking value
in real I mean to discuss about maxima minima
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we need to compare the values of F define
on RN and you can compare only if they are
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defined their domain is R
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So let us start again the motivation from
function of 1 value so let it be a differentiable
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function differentiable on entire A B open
interval A B
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So you know that there one beautiful theorem
is called Taylor’s formula and I mean one
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can motivate this maxima and minima problem
in many ways but explanation of this comes
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more beautifully if you understand the Taylor’s
formula
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So today let us concentrate on this beautiful
theorem called Taylor’s formula So we will
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discuss a several variable part of it but
let us recall what it says I told you in the
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beginning while you were discussing about
the motivation for differentiability that…
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Suppose I have X not the point and X not and
X two point in A B maybe they are closed enough
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Then this F X so let us say X in some interval
X not minus delta to X not plus delta then
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in this interval F X is approximated right
by F X F prime at X not plus F at some point
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Z X minus X not this is…
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Actually if you say MVT then you can write
it as exactly this form but Z and X not are
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close enough then you actually kind of approximate
F X by this right Z and X not ‘close enough’
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Close enough have to be justified with the
other thing and that comes in the statement
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of the theorem we will do it do not worry
What it says that that F is differentiable
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then around X not in some interval X not F
X linearly approximated this is a linear function
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X going to F X not plus F prime X not X minus
X not this is linear function
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So F X is linearly approximated around F X
not now F is suppose F is nice enough F is
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more nice in the sense that F is N times differentiable
that is F prime X exists for 1 to N Then this
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Taylor’s formula tells you that F X is approximated
around F X not we have a better approximation
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there is a first degree approximation but
here we can have end degree approximation
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by uppernomial and it is much better approximation
than the derivative part here goes faster
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than derivative part here goes faster to 0
as X goes to X not than here
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And actually if you want to write exact expression
then I have to write here then replace here
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by some Z Z in X to X not so for ‘close
enough’ X not this is so called Taylor’s
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formula And this part is called as reminder
term right So up to N minus 1 degree find
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this is all coefficient F prime X not and
X minus X not and N at coefficient is Z So
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the previous term here was F N minus 1 X not
X minus X not power N minus 1 divided by N
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minus 1
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And this gives to so called Taylor’s series
also that if F is differentiable of all order
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that is F N exists for all N then to F X I
can associate this series which goes on and
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if you put conditions like for all F N in
for some delta FNX is bounded for all X in
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X not minus delta X not plus delta then this
series is actually converged to FX and in
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that case we will write equal to that is the
Taylor’s theorem
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But in general given F X which is differentiable
of all order we can associate with this series
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Taylor’s series it converges or not depends
on certain nice conditions like this but there
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is a one way to express if it does if it does
there is a one way to F X as power expression
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So called linearity function So far so good
for one variable
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Now how do you generalize it to F from U in
RN to RM U open set Now how do you do it once
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again the idea is that as you have done it
before that is F is from U to RM so you have
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M components so do it component wise so we
will assume M to be 1 and if we can write
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down the generalize form of Taylor’s formula
here we can write down for any F taking values
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in RM component wise
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Well to do this I have to talk about this
higher derivative F prime F double prime we
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have already discussion this but then I have
to talk about any derivatives and as you have
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seen that F N in the function for real variable
so second derivative is already a matrix So
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third derivative onwards you cannot write
it do nicely and to put so what we try to
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do in mathematics that if you cannot write
things nicely we try to take help of notations
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We put some nice notations so that the final
formula looks nice
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But putting it notation does not mean that
the calculation becomes easy but any way to
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put it or to look it nicely we will fix up
some notations well So suppose I have a T
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TN vector in RN so this F prime at so at some
X in RN X in U X T just means Grad F X dot
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T so this is summation I equal to 1 to N del
X I at X dot TI
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Similarly F double prime X capital T equal
to I write it in the summation for again then
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you try to recognize what it is summation
I equal to 1 to N summation J equal to 1 to
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N del square F del XI del XJ (I am assuming
everything exists of course) TJ TI There is
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notation for this thing and if you recognize
this is actually T prime Hession of F we have
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already introduced as FX
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Similarly F three prime X T summation I equal
to 1 to N summation J equal to 1 to N summation
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K equal to 1 to N del cube F del XI XJ XK
TK TJ TI if you want to write it in a compact
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nicer looking form this will be actually T
prime H FX T dot T So this is TI Do not worry
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about this part; it becomes more difficult
when you write in general for any M or N So
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you can write it in the summation formula
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There is a small hint that actually I have
written I equal to 1 to N XI XJ TJ TI but
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this is real numbers all Ts are real numbers
TIs I could have written TI TJ here also I
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could have written TI TJ TK but I have written
it in this order because when you talk about
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complex function then certain things happen
then you have to put bars complex conjugate
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so therefore this is the standard notation
we can easily generalize to complex
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Now with this I can state the Taylor’s formula
for several variables which will of course
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match so I will write Taylor’s formula in
this notation So here is the theorem let F
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from an open set U in RN to R be such that
all its partial derivatives exists up to order
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M up to order say less than M and differentiable
so partial derivative up to order M exists
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and they are also differentiable I do not
say F is differentiable up to order M or not
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partial derivative up to order less than M
exists and also partial derivative of M exists
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So I can put it in one maybe not to confuse
you too much partial derivative are differentiable
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so it automatically assumes that they exist
differentiable up to order less than M for
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some M Let X and Y be two points in U such
that the line joining X Y we have used this
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notation before this is set of all point T
X plus 1 minus TY T
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in 0 1 this is also in U So if you use a convex
set there is no problem here
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Then there exists a Z in the line X Y such
that F of Y let me write it in this form F
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X equal to how do I write it okay F prime
X at Y minus X (I am following that rotation)
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half of Y minus half means 1 by 2 factorial
1 by minus 1 factorial plus F M Z at so this
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is a eroterm divided by M This is the generalization
of Taylor’s formula so written more compactly
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This will look more nice you will agree so
this is the writing this approximation part
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plus ero part Well if I prove it the same
proof will follow for N equal to 1 what we
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started and proof is again very easy Proof
is just using first variable for Taylor’s
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formula (18:04) variable function And we have
already done it before in case of last week
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what we are doing Mean Value Theorem
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So let’s say we proof here as you can guess
I will start with U is an open set so what
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we will do I will find a delta greater than
0 such that TX plus 1 minus TY this belongs
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to U for all Tin this is again as I said to
avoid one sided derivativeness at N point
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So here in X and Y since this line is there
and there is the ball like I can also extend
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little bit I can extend here little bit so
I have extended this line this line is there
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little bit extended like will also be there
because U is open
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And we look at the same kind of function here
that we did for MVT So take GT equal to F
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of X plus T Y minus X okay Then F 1 equal
to G sorry GFY G1 equal to FY and G 0 equal
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to FX and as you can and you see 0 and 1 both
are in this interval so I will apply the Taylor’s
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formula for G so what I will apply one variable
Taylor’s formula for G and what we get G1
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minus G0 equal to summation K equal to 1 to
M minus 1 If F is differentiable this is the
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composition of two functions F and a linear
function
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So just observe GT is composition of two functions
if PT where F is as according to our assumption
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U to RM R and P T is simply X plus T Y minus
X This is a linear function so differentiable
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of all order and F is assumed to be all partial
derivative are differentiable up to order
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less than M G will have the same property
by Chain rule
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So I will write it 1 by K factorial K derivative
of G at 0 1 minus 0 1 minus 0 is 1 so I should
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have 1 minus 0 1 minus 0 is 1 so this part
plus the part 1 by M factorial for some T
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not
in 0 1 Correct Now what I have to do to get
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this formula I have to see what are these
fellows are Let us see we will apply Chain
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rule here
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Well what will happen P is from R2 to RN so
P can be written as P1 P2 PN components where
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each P I T so if my X is X1 X2 XN Y is Y1
Y2 YN then each P I T is X I plus T YI minus
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XI so PI PI prime T is just YI minus XI so
that will be me my Chain rule G prime of T
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is how much F prime PT and following that
notation at derivative of PI that is YI minus
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XI sorry which is by your notation F prime
PT Y minus X
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Now you apply G double prime T
this is what will be there again apply Chain
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rule now there will be summation I equal to
1 to N J equal to 1 to N del square F del
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XI del XJ if at PT YJ minis XJ YI minus XI
which is according to our notation is F double
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prime PT Y minus X so on you continue you
will see GMT equal to FM ET Y minus X
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Now when I put it back I will have G K at
0 and now observe P 0 P T here P 0 is simply
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where is P here P 0 is simply X So put it
back you will get the formula I have written
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for Taylor’s formula I have written in the
theorem so put them back altogether So this
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is the straight forwards proof of Taylor’s
formula So next time we start from the Taylor’
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formula and talk about maxima minima
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Thank you!