Math for Economists - Lecture 13

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Math for Economists - Lecture 13
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Point (geometry) Statistical hypothesis testing Functional (mathematics) Existence Graph (mathematics) Poisson-Klammer Sheaf (mathematics) Maxima and minima Open set Mathematics Derivation (linguistics) Mathematics Arithmetic mean Kritischer Punkt <Mathematik> Lecture/Conference Extremwertstatistik Boundary value problem Social class
Point (geometry) Functional (mathematics) Focus (optics) Closed set Direction (geometry) Adaptive behavior Maxima and minima Mass Rectangle Variable (mathematics) Equivalence relation Derivation (linguistics) Mathematics Kritischer Punkt <Mathematik> Lecture/Conference Extremwertstatistik Boundary value problem
Point (geometry) Trail Focus (optics) Functional (mathematics) Process (computing) Multiplication sign Gradient Maxima and minima Calculus Rectangle Flow separation Vector potential Kritischer Punkt <Mathematik> Lecture/Conference Algebra Spacetime
Logical constant Derivation (linguistics) Functional (mathematics) Kritischer Punkt <Mathematik> Lecture/Conference Square number Mereology Variable (mathematics) Rectangle Directed graph
Point (geometry) Functional (mathematics) Kritischer Punkt <Mathematik> Process (computing) Sequel Lecture/Conference Multiplication sign 1 (number) Maxima and minima Calculus Vector potential
Area Derivation (linguistics) Focus (optics) Kritischer Punkt <Mathematik> Octahedron Lecture/Conference Multiplication sign Gradient Order (biology) 1 (number) Rectangle Partial derivative
Prime ideal Derivation (linguistics) Functional (mathematics) Kritischer Punkt <Mathematik> Matching (graph theory) Lecture/Conference Multiplication sign Maxima and minima Line (geometry) Mereology
Point (geometry) Frequency Derivation (linguistics) Functional (mathematics) Kritischer Punkt <Mathematik> Lecture/Conference Multiplication sign Square number Mortality rate Variable (mathematics)
Point (geometry) Logical constant Functional (mathematics) Process (computing) Multiplication sign 1 (number) Maxima and minima Water vapor Rectangle Variable (mathematics) Numerical analysis Derivation (linguistics) Formal power series Kritischer Punkt <Mathematik> Lecture/Conference Different (Kate Ryan album) Order (biology) Saddle point Square number Charge carrier Right angle Resultant Physical system Spacetime
Point (geometry) Functional (mathematics) Group action Multiplication sign Equaliser (mathematics) 1 (number) Sheaf (mathematics) Parabola Frequency Mathematics Kritischer Punkt <Mathematik> Lecture/Conference Different (Kate Ryan album) Well-formed formula Analogy Oval Negative number Energy level Circle Algebra Set theory Pairwise comparison Addition Multiplication Constraint (mathematics) Process (computing) Matching (graph theory) Concentric Uniqueness quantification Consistency Closed set Gradient Maxima and minima Mortality rate Calculus Line (geometry) Unit circle String theory Cartesian coordinate system Variable (mathematics) Rectangle Flow separation Tangent Radius Cube Mathematician Table (information) Maß <Mathematik>
Point (geometry) Logical constant Functional (mathematics) Direction (geometry) Equaliser (mathematics) Multiplication sign Mereology Graph coloring Expected value Lecture/Conference Energy level Modulform Nichtlineares Gleichungssystem Algebra Descriptive statistics Set theory Social class Multiplication Focus (optics) Constraint (mathematics) Gradient Maxima and minima Line (geometry) Category of being Vector space Histology
Point (geometry) Functional (mathematics) Constraint (mathematics) Lecture/Conference Equaliser (mathematics) Green's function Gradient Nichtlineares Gleichungssystem Variable (mathematics) Mereology Physical system
Statistical hypothesis testing Point (geometry) Lecture/Conference Multiplication sign Right angle Maxima and minima Line (geometry) Theory
Point (geometry) Functional (mathematics) Group action State of matter Multiplication sign Mereology Food energy Theory Power (physics) Mathematics Lecture/Conference Energy level Circle Set theory Addition Multiplication Constraint (mathematics) Process (computing) Weight Gradient Maxima and minima Unit circle Line (geometry) Calculus Mortality rate Variable (mathematics) Statistical hypothesis testing Order (biology) Triangle Right angle
Derivation (linguistics) Lecture/Conference Multiplication sign Gradient Mortality rate
Point (geometry) Statistical hypothesis testing Polar coordinate system Three-dimensional space Functional (mathematics) Group action Euler angles Multiplication sign 1 (number) Routing Rule of inference Plane (geometry) Lecture/Conference Square number Negative number Energy level Circle Nichtlineares Gleichungssystem Set theory Constraint (mathematics) Graph (mathematics) Gradient Maxima and minima Unit circle Line (geometry) Numerical analysis Tangent
So Finals a week from today that feel obligated to announce that surveillance each other for a while As of today with a new material Friday new material and Monday only review should get everything that I Friday infect them and even a little early to where we do agree now were wary in section ahead 12 . 3 and were
We're doing that's it's worth it to just reveal little bit so this scenario And not to aim is This if I have had the equals the say 3 minus sex squared so we can actually graph that goes on 6 But of like this And then when if I restrict the inner wall so can't a closed Case so what should make the interval let's say from 5 1 2 3 Close interval went when I say close that means there is Particular meaning in math were closed but in this case what it means is that the the end point Our included it open interval I'd have the parentheses their illustrates closed in remote has the brackets if I say half closed you got 1 bracket 1 The open side OK so that sister were closed but you can see what's going on here if I if I do 1 here when I plug-in 1 get the value to years 3 function continues on down there That's a negative 6 she conceded the extreme values of this Find the Mac's owned by the maximum and upon this interval there's the maxim there's the men but the method is what we really want exposed and so The that method is were getting used extreme value and that says if you have ever of X on a closed Interval A move The function needs to be continuous in math classes like this they usually are so doesn't come off but if you have ever Vivek summer closed The Macs and the Exists and the fur hat there is a critical point for the boundary so the reason is call the extreme value theories because You you're going to get an extreme value If I if I make this an opener will you might you you might not get an extreme value instance you wouldn't in this case if I make this an open enrollment there's a hole here GCC never actually arrived at the extreme value so it's crucial that it's a closed in Irbil that he actually have boundary points over this problem here which you do is you you just go ahead and there's 2 places where it could occur you critical point or the boundary and so we can go get the derivative the derivative is negative 2 amount said that he called a 0 that's as good Mexico 0 so we did get a critical point that 3 years we see the critical point doesn't occur in the interval that I'm interested in so I rejected this this is not that interval so you know it is a critical point in this particular context it's not relevant to what I'm trying to to do so now my only potential points for the maximum Be at the extremes are at the at the boundaries of the center what you could see that in the picture too That's where the Max's announced with minute but I just test points He didn't have a picture will you would do you just plug in the car the boundary points of 1 is equal to about 3 minus 1 of 2 3 3 minus 9 negative 6 This is the max and this is the man which we are you from the picture is good it is practiced the method so sometimes when you go get your critical point is just not is in a role that you're interested in and see Rejected and then focus on down the path
So format for illustrate an extreme value theory for mass for it's the same idea you have the expert here in math four-wheel out at We'll ourselves that more variables so you might have this is you my head of X which is it's Exeter's why so there's a function here and then what's the equivalent of the interval now in this context dinner have is a white here announced that of a closed It's a close directing so We have ever of XY just basically repeat this bill put it in this context so we have ever ex-wife on I can't say a close durable now I have to say on a closed rectangle pop and being closed means it has its edges That's it did you could have an open rectangle where 1 of the sad is dotted and it's not included in this case will have a close rectangle and its continuous muzzle just continuous has to be different because adaptability A derivatives but all these things originally given in our problem I just wanna focus on closed for so we have a close rectangle that Max For the millions exists and occur If there is a critical point August the mouth no 1 with so the 2nd targets either critical point the boundary here the boundaries just 2 points this is the edges of interval here when I got the boundary it's much more complicated because the boundary includes these edges And these corners so let's put the 3rd 1 here this 3rd 1 here represents the edges of this accessories plugging in the real extreme points which are the corners of the corners represent the edges of the interval and then the extra the thing that we have in here is we may get critical points along the edges 81 I think at the edges each Edge here Is like 1 of those problems because each edge is a little and 1 of the variables is constant unimaginable And so it you really just reverting back to this promise skidded use go do this problem and see what with the dealers OK so list extra thing that I need that I didn't have here It is I might have a critical point on the edges consumed The overall is These are the 3 possibilities we didn't have this possibility in or if you wanna think of it as the other possibilities is said to have 3 this extra OK so what what could happen here as we plug-in points from inside here what we get is that you know we might A little something that's a little flat that there but it's it's not it's not flat the flat that has some shaped to its of three-dimensional shape to you might get a little little maximum here on the edge year you might get a minimum over here on the on the edge of those or trying to discover OK so that's that
That's year the recipe for how much do Medusa 3 examples of this kind have all the little possibilities of that time you watch me do these should be pretty good at it because it really is the same thing over and over again
It said here and X Y is equal to 20 for AIX plugs Well I kind squared minus what So right away Don't you really know what that looks like so we have to rely on the calculus of the algebra to get the job done and then I wanna restricts the X and wise to a rectangle looked just say were restricted to between 0 and 1 so that helps you I mean the tardy keep track of all the singer had gone right down with the Union of the left interval the rectangle But whose X here while I 1 of this is mine This is my rectangle are And then I'm not really be interested in the interior so much evidence which I terrifying critical points of some critical points land inside their then I'll be interested in them but really the reason why I drug because I wanna label edges Scott that held 1 held to L 3 4 or you don't have to call itself 1 that's just the way I do it I stared go counterclockwise there yet to pick something start so I don't care what they call no 1 else E 1 foredge L 4 line segment focus and that the center will really have to do is just go through those 3 things are getting it Charter find the critical points in the traditional way so that's taking the gradient and setting it equal a 0 Phillips get back here the gradient of death that at the next of what I wanna do the green here get for minus Q X and then The derivative with respect to wire that 0 0 I get 6 and then minus 2 I and define the treaty is the critical points we put said equal to 0 OK and this day Equals 0 this so that tells me there for my next to exit 0 gonna say will use the space conceded X is equal to their But is ready for minors to Mexico 0 that implies that QXC for which I that too And then I have 6 miners to I equal 0 that's as to why equal 6 3 so I did get a critical point using my traditional way however 2 3 if autograph it to 3 years Days after it's out of my region yes I just think it's given the problem will hours he said Yeah that is a two-minute in include this year it's kind of a mass in on Are give you the rectangle on go find the maximum and get this is given problem OK so since it's given the problem when I going at this critical point after it has to be in his I'm only interested in things that are happening inside that rectangle so to 3 is not in off so we don't we don't include the center list of potential points for the accident could occur if it turns out you wanted check that go take the 2nd review of high-level that back from this function please see I'm not interested in the the global maximum only interested in what the maximum and is on that will wear for practice you see that is a matter of when you're OK so now what am I left without a did this berating get anything for this many questions focus under this now signed the critical points on the edges in each 1 of those edges it's item for separate problems and they're all just like the the matter to a example will go through that that's very repetitive peso on L 1 this is what I was describing due last time
Is it when you when you decide that you're on a particularly line segment 1 of the variables as constant so immediately decide what what the cases for L 1 you see that y values moving up and down but the X values constant at once so clear the defining feature of L 1 is that X equals 1 leaders go through all of them and do that for propelled to the defining feature is that why what propelled the defining feature over there is that
Was 0 for the defining feature you see the bottom segment there X is moving but That is 0 2 liking you go through each 1 of us and put down defining feature what at what happens is as soon as you said 1 of the variables equal to a constant then your functioned as becomes A functional 1 variable and you go right back to the matter to a when x equaled 1 a colony in 1 of 4 X and my functions becomes 20 plus 4 plus 6 y minus 1 miners squares that little But most speaks the 20 For the 6 minus 1 minus wise Puerto Rican simple file that 23 . 6 while minus square OK so now I've got 1 variable sources take drew severely critical points along that segment from 0 1 Motyka derivative its 6 Chiu Wai considered equal to 0 I get too wide 0 6 1 3 of the White equals 3 is not part of that rectangle so this is that this is not a member of our fund often include that even I did get a critical point it's not part of my just like over here
I I saw that nite at Mexico 0 was an inalienable values district
OK for altitude Now why is is equal 1 OF is equal to 20 clubs for X plus 6 minus X squared plus 1 that is equal to C minus 1 of 25 plus or minus X squared take the derivative alienated us for so X equals to set the 0 2 a David again are only interested in X is between 0 and 1 so X equals 2 It it is not a member of 0 what is not a memorable The reacted this disk No but so far we have no potential points for Max's so it leaves look ahead this is what going happen in each case the only 1 who in again as the quarters in the same way that the only ones we got here with That's what I'm looking ahead that's what's gonna happen let's fill in the the details here Practice so when x equals 0 this'll easier we get 20 minus 6 while minus 1 weird And then there is the same as we had every time minus 2 I said Eagles 0 I 3 this is not a member of 0 won't get any critical points that are valid for the same thing here When a wife is is equally 0 20 miners are plus 4 acts 6 where's derivative is for minors U.S. Senate equaled a 0 sequels to which is not in general it's much more complicated in this workers had to go through all this process but you see the repetition when she's would you do if you these Easter to just be able to look ahead and see what happens OK so Once again so so far we have no points and the only thing left to check the corners checked the corner piano happy doing calculus and literally plugging in the The value of 0 0 I have 0 1 1 0 have warned Solis 0 0 I get 20 The employee Back in the function 0 1 so Texas 0 but why is 1 that looks like a 25 that check that you and then taxes 1 but why is 0 so here and the year Bruce 3 20 23 and then at 1 1 5 24 30 miners to lose 20 So now we reveal what the maximum and here's the men say this is where they occur occur at 0 0 and 1 Hot if Jim questions about that book I was going to go through a couple more of these
The practice if you're able to do all ones that human lecture me really set the final exam that the place star who want challenge yourself you can do the ones suggested exercise and book but you're limited on time is getting ready for the exam think you'd be OK if you just do ones from lecture
And so there's my rectangular muzzle Drive A X 1 label that that's 1 There's X 2 X 1 is between 1 and 0 and 3 in next from 0 2 to a rectangle We must all label want held to help 3 4 will have to go through each of a case of 1st things 1st as go see if there's any critical points that happened it to occur in the room in the middle of the Rectangle otherwise them will be relegated to the edges So 1st things 1st Demand for this fine but the maximum and on Oct OK so let's get too greedy and the gradient areas my partial derivatives are bad 1 has to F 1 only 2 x 1 and then minus 2 x 2 plus 0 at the end of the derivative with respect to the 2nd variable that 0 get negative 2 6 1 and then plus 2 Back Focus so what can we get from this river the solemnly 0 This 1 down here can be solved immediately put it boasts a stood in order so as Turek 1 minus QXQ equal 0 that tells me that X 1 equal has to have a relationship between 1 and 2 from the is bring the Soviet decided that they too And then the 2nd 1 2 X 1 plus 2 equals 0 that implies that excellent equals 1 so X 1 equals 1 x 1 equals X. so that implies next to equals 1 so I have a critical point at 1 and that 1 1 is in fact even so I do have a valid critical . 51 seeking can attack at says you go to see you find appointed potentially Allegis plug-in in Athens see what it is if a duet of 1 1 half of 1 1 was 1 minutes to a negative 1 plus 2 is 1 That's the 1st
Part of it Just finding the critical points now a move on to the edges So for L 1 of the defining feature for Alwan over here is that X 2 gold miners REX won the whole time is equal to 3 x 2 varies but X 1 is stuck at 3 into the function f is going to be 9 6
6 to What do you do It's 9 minus 4 x 4 putting Annex 1 equals 3 along that line so sticker derivative that Prime is equal to negative for so what we do the drew said ego 0 it's never gonna be equal derivative is negative for them so you wanna think of a long match We've revealed something there along this edge the thing is decreasing kind because the maximum is decreasing along the edge does X 1 goes from 0 firms X to those from 0 up to decrease focus so but the bottom line is for this to get no critical L to this was slightly different because the the last 1 we got critical points every time but they didn't occur at intervals of that rejected here we just don't even get helps to have a historical differently so on held too
The X 1 variables moving but next to it is equal to to the global power plant next to the function of X 1 squared minus 4 1 plus that the new F now 180 derivative that crime stew X 1 minus and that gives us that equals 0 X 1 is equal to So you put these 2 points together at the point by June 4 x 1 and 1 4 x 2 we have a critical point that What has happened here I am He said eroded Text used to thanks to to that that's OK so here's a critical point rate I knew it had to be on the edge If you like you can you can check effort this point have to to that's equal to 4 minus is negative for plus 4 0 between the 2 that I have so far that's the Macs and that's the end so far but we might get at the age of 3 A L 3 years this little segment there the defining feature for that is that X 1 of the 0 the whole time now want plug-in X 1 equals 0 I just get at this really simple it's just x 2 0 0 plus 2 x 2 And when I take the derivative here I just get to which is now 3 0 so I get no critical just like they always say For the last 1 of the defining feature here 1 is moving but X 2 is . 0 when equity next to equal 0 I get S squares and then derivative 0 x 0 0 so I had a critical point have 0 Lakers and muzzled check at that point so this critical point off at 0 0 is 0 so rude dance step to to the original critical point we checked that gave us a value of 1 we get to other critical points along the edges of those viewers values of 0 for the function of the last step is to check the corners of quarters we have 0 0 We check 0 0 it is a little overlap just happened 1 over critical points was a corner check at that 0 At that point he knew that we have the . 3 0 this quarter over the lower rate 3 0 I get 9 minus 0 plus 0 . 9 And then I have The . 3 2 half of 3 2 out of 3 period 9
Minus 12 is native 3 Plus for his once said that Minus 12 native 3 for this 1 In the last 1 is 0 2 was the last quarter of what their 1st turn 0 9 for their best at this point and then everything I got all the critical points just check which is the biggest in which is the smallest I looks like this is gonna be straight here as equals 9 is the max and then any negative values said 2 places wary 0 so this is the new here and this is also
Images right down the formal Donnelly space sports Religious right down the final Back The minimum value is 0 and it occurs it too little station 0 0 a hand 2 0 in the maximum That occurs At 3 3 2 where they will ahead Yes to 2 And the This will occurs at 3 Our Broker But say everything all written out all strangers with Dravid carrier On the deck What is free Because I just couldn't I set the derivative equal a 0 and couldn't equal 0 let him get a variable here so it could solve anything This just says it has a slope a 2 0 Longmeadow alleged never again had a maximum but in Notice the difference here I had an exodus For myself the exit number and then I could say at a critical point here here I was there's no exercise could never equals What For the same reason as L 2 Woods when I took derivative I got something the salt The found 6 1 0 take care that up with that 1 0 0 Now The good news is when you when you do this you really feel a sense of satisfaction you're working for a while you work and now this details and you come to the conclusion it can you feel like really really did Semak there Requested system 1 more legal faster through this we've seen the process a bunch of times now culture And We have A simpler function it The do the saddle Troops subtle But that's the saddle Giving but say that X 1 is between negative wanted 1 and so is X 2 boats immediately right down the picture here negative 1 positive 1 positive 1 negative 1 Talk about this Rectangle And Will call this a very creative a disk Yukon the same 1 a 1 0 2 0 3 0 4 Look at those would go find a critical point do this these These derivatives will be quick so a worker more overall he appeared ingredients is at once she will 2 x 1 negative 2 x 2 and then we said that it will do so X 1 equals 0 2 x 2 equals that to 0 have occurred 0 . 0 0 And might need it does occur inside the rectangle so I need to use it as a candidate who realize all check effort that point check That I will Check L 1 4 L 1 of the defining feature here is that X 1 is stuck it won the ball on the water so that left was equal to Next 1 is 1 that I put 1 in their Agate one-liners sex to square than the derivative is equal to negative to add to their equal 0 6 2 0 0 0 OK so That That is between 0 and 1 so I've got the point 1 comma 0 is on 1 common 0 this came amid a put all the critical points and now Chris moves out that keep order of things down bit of garlic critical a high there's the first one 1 0 they really do just work at exactly the same way minute but 1 constant does 1 thing that we go over the other stagger sensory playing same variable and the constants of nearly the same results start to notice of patterns in the this book L to the defining feature of There is next to is equal 1 of them had physique Exxon squared minus 1 the derivative 2 x 1 said equal 0 Thank you for your next 1 equal 0 9 . 0 1 and that They say that L 3 L 3 is the defining feature for L 3 is that X 1 is equal to negative and have becomes 1 of my my next June squared Crime Negative 2 x 2 0 0 2 0 0 and . negative quarters The next day and health for the last 1 that's where X 1 movement but X to his stuck A negative ones have become X 1 squared minus 1 prime
OK so were down 0 1 we can check at that all these points here that is equal that of 1 0 Equal 1 that but 0 1 his evil ones that negative 1 0 That's little 1 and half of 0 negative want ads nice easy function apply those things in the recently I'll take that back This was negative and this wants Negative as well Be careful there is a severe rebuke Talkers so far this was my completely I would have Max's occurring here and minimums occurring here because zeros In between rates more extreme is 1 that's higher and negative ones lower so that 1 that 1 won't be counted But I needed please check the corners to make sure that I didn't get some other values that beat out the of the corners It those occur at But there's 1 1 negative 1 there's 1 1 1 so let's plug those in of mind negative 1 that's equal to 0 You put that in an effort 1 1 That 0 as well We haven't beaten a those and then got a negative 1 1 at a negative 1 1 that's also equal 0 and have a negative ones because if that's the other corner negative 1 negative 1 effort that has also so this 1 hears Nunavut Max shared the negative 1 this is a Max schism in this isn't The maximum as 1 one-handed occurs at 2 locations and the minimum is negative wanted occurred at 2 locations A when she get That I can't really mess you up because what what I could do to make it harder is just make the algebra that silly I can make the process and are it's you find the critical point if it's in the region you keep it You find the critical point it is then you get the a quarter point to collect all those things in check that the function site I can't really changed the method all I can do is Make the function is a lot harder to deal with but they just adds and necessary algebra so this is really the level that we need to demonstrate steady this you learn the method and you can expect functions that are at this caliber the lesser putting cubes in Things it is because of past OK So I mini questions about that Great value to move on to the last topic section 13 . 1 of us is this is the culmination of away this was 2 of the next section is 1 step up from the and the doesn't really have an analogous thing from math to This truly is just a unique set of variable calculus Grant multipliers the bronze was a mathematician who loses a person's name and we'll find out what these multipliers are off in a minute what I wanna do is I wanna take our standard example that I always start with the ah paraboloid and I wanna work out that really detailed example of all this before even tell you what the formulas work set up the problem and then we will come to discover what the method is a little quieter more general function OK so here's our said it will take All-Star with the paraboloid as weary know that 1 of its it's something familiar we can ride out all the pictures and we can see exactly what's going on if I start with 1 be on the fiction that I can show you what I can only do the algebra which is not as From only remind you how the picture here actually dot dot dot There's a parabola label accidents here say why This S softened Z period F schools OK so as it stands right now that there is nothing new so I have to give you a new command for you to be 11 new problem the new command find them acts And manage if it exists just Tread a fine them the maximum in 5 subject to the constraints But X Y Tu So what's different about this This comparisons we have another example of what we were doing Over here and have the word constraint but I was constrained I was put that was put into a little region we'll use that language to say on that region now I wanna find the man Maximum minimum but it's not a region that I am Interested and I'm interested in this life I'm only picking points along the line And then seeing what the maximum and is on this problem so let's see what the weather's if I if I do This is really wide equals A negative ads to Huletts graft that negative X plus 2 is that when Texas 0 wise astute 1 y 0 Texas too So that's the was put the label the eggs here there's X there's while novelists is good practice for your brain see if you can put this in the XY plane over there so that X has a matchup with VAX
Put it down in the In the floor there The match at the X and Y so what we're what's gone here I have to actually extraction have wider action and that connect those values So now you see this This line here playing the role of the rectangle appears the rectangle was like this carries a rectangle and then I have some function that I only plug-in values from there so I yell rectangle on the function and I'm only in a 1 0 at the Max is up there on little rectangle close to the same thing for this I'm only allowed the plug-in points from this line so chiral matter what what's it didn't look like I'm here and dried on here you see it in your mind when I plug-in points said plug-in this point here in this point and so 1 would end up with is a little parabola on the prowl it looks like Take Just in case he did Take this pasted on and effect a stranger 1 You know peace the strain on the vase thing and this is a sitting on your table and you got a string here unity is straining a pace that and there's below who wanted discover that none of you can also see from this picture that there is no vision This goes forever out this way so they keep and points his get points higher and higher up there so there is no maximum if there is a minimum of there's and there's no Max so we wanted to Ozona discover the method had a fine this Nowhere Where is this minute turning its it's further up on the graft in fact if a plug-in 1 1 here that's going to be the thing that's gonna be word is the 1 1 with the words cannot occur so this is going to be at To hear so you have to imagine that on this we have all these levels reached level as I go up it intersects this grass and this is the minimum 1 but you see these levels below year they never intersect we wonder let's look at his West this This topic puts everything together here so what I wanna do put the level sets here over on this scrap and let's see what happened and that they got just been reproduced again will have a bigger version of the what's gone on how any questions so far about the set OK to recruit I would With Look at reproduced this Bigger here next year's why Who won here to 2 of this guy here is X plus while I am calls to beware what where have we ever had imagine a situation we had A bunch XY steadily said equal constant work when do we do that When do we take a function said consonant Graf What's the concept would do in that case Barry get But level set What I can do is I can't protect so this is this is like having a function equal to constant so there's a bigger function going on here And this is just 1 level said that function from call G O X Y next us why And this is billed level set of G Pat G equal stew But now I'm making it more confusing by doing that but see that gives us a look a little advantage if I if I view this as just a level set of above a larger function then I can do things like a gradient of back at Icahn agree I will do that in minute but what I wanted I wanna introduced the level set of the other function so let's let's go slowly if I have half equals 0 What is the level set that just this little wants to this and white here so that the level said Ford F equals 0 Now what happens if I go up 1 so I go to the press because it is shredded at that sex what I swear done this before us I wanna go through all the levels again this Katie a single point and the next level said that equals 1 that's X squared plus Y squared equals 1 that's the unit circle like drug that So this is difficult as 1 that little of that unit circle we see I haven't intersected yet added intersected So what that is the 1st few levels there haven't gotten that That minimum point out of the question is when do we when do we had this for the very 1st time that's going be the minimum value price so this is F. equals 1 if I go out Say to that people too Than what's that coolers so articles for forgot that he goes for the that's a circle of radius too So let's dried Now I have a circle received not intercepted it had intercepted but at this level of their vehicles for I worry intersected my goal is to find the very first one that touches that so it's not 1 Kaz's doesn't even have intersected Once I get a 4 yet that intersect that sees can agree with me that the very first one Just this It is the very 1st level That touches a slight break that that's the very 1st levels This is lower this is 0 here's 1 years for so somewhere in between 1 and 4 hits it for for the very 1st time and this point right here where their tangent is looking for that will finally find the very 1st level that what I'm drying here compare that to the levels there you see those little concentric circles as This is the bottom And this is 1 unit that I go up 1 and then years 2 units that are passed over it and then here's the Magic 101 of fine I when when exactly kisses just teams
OK so how we find this This magic point here this We're use the fact that the gradient is always perpendicular to the level set I mention that when we 1st talked about level sets but we can see it again so let's let's take a look at the gradient of G here that's equal to 1 1 So that 1 1 this is the the gradient of g perpendicular to that outline of the radiant as always The money write that down the gradient is always Perpendicular to level set forth so that the gradient of G and what's gonna happen with the that right there at that point where they're just kinda like that the gradients lineup that's the magic for the gradient because if they're Tianjin and the gradient of F was drafted with the color here it also be perpendicular to its level said so right at that point These 2 guys have the same focus 30 injured character the gradient of death would be pointing in the same direction That's the magic rockets on show you what's going on now I need to show you how we find out what's the algebra and that's really what you're be interested in because you now is to build a draw the pictures you don't really know every little thing that's gone on what you need is a method for finding the magic point where the The level set matches up with the with the constraint function just That kissing point that Tianjin point pockets so so what does it mean When 2 vectors and wonder if it he is right that What is it mainly to vectors lineup try answer that in your mind if I told you I had 2 vectors and they were pointing in the same direction what could you say about 1 relative to the Here's a here's a vector and I tell you that I have another vector W. that's in the same direction as the airline Well if it W. isn't the same direction then it's just a multiple of the scalar multiple because it keeps the same direction is very has a different like Sony write that down more precisely off The OK so the Grady Exhibit F. This is your original function and the gradient of G this keeping straight is now we have 2 functions floating around this is what's called a constraint and you know that from the original description of the problem find the Maxson and subject to the constraints of the constraint is this other Jedi function and Athens your original function so These are these are there pointing in the same direction when the gradients that is equal to A scalar multiple of the other The plan is a constant junta little beyond the scope of this class but the fact that we use land is not a coincidence we also use land doing Eigen values so this is a form of an agonizing infected looks kind of similar we did die in in values before we did a X equals landed Texas a little bit like that you think of a Exs F this X is So I just want to point out that it's not An accident using land in this case is not that important you know that what we need to do is build a solve this equation so That when the vectors appointee there we have this property that then victory is a constant multiple or scalar multiple another By the way this slander places called the Lagrange Malta where the land is the branch more but you never really Salford Freeland New Canaan we might even do it in 1 of our example But the bottom line is defined as minimum value remember that's that's our goal of 1st find OK here's the The punch lawyer so solving this Equation would find The locations of the knives in the middle of a subjective So that's the bottom line OK so that What you wanna do is when she learned that you wanna go back and look at this example and go through all the details when you 1st thing the 1st time can You might Gloria head so now we need to do is we wanna see that this solving that equation 1st on learn how to solve it And then I'm telling you that when you solve this this is where these vectors are all lying that that that candid point and I'll give you your Maddox's and give the locations of the pockets of how they do that they need to learn how to deal with that equation if you go back and start over with the problem just work through it as if we didn't have all this discussion going to use this back off OK so they have effort XUI sex where supplies where and were doing this subject to the constraints acts plus Y equal stew So immediately take this this is going to become your G function the constraints on if just take all the stuff that's not part of the constant said equal g g that's what I called faxed watchful eye so that's why I say the constraint G constant fear can't function GE equals a constant mind expired the minimum value because we know in advance here there is no maximum find the OK now I'm telling you that if you solve that equation it will find the location of the minimum value because
Of this picture I do it's right where those gradients lineup so we just wanna find where the gradients lineup And will be there at that location OK Phyllis do that go from there saw that equation have to know what the different gradients are the gradient of that that's 2 X 2 Y not mystery and then the gradient of Jedi the overhearing 1 1 1 4 X and 1 for while and I've got the 2 things the 2 sectors and now I want I wanna find out the locations of X and Y where these things online that O'Grady in advance
Equals slammed O'Grady Andrew G. that means I have 2 X 2 Y equals land Timed 1 1 And that gives me he could think of that Islam that slammed a judges me equations I've got 2 x equals Lander and I've got to why equals Not only stay there for OK so this is parts her 2nd Time Warner find I wanna find XMY but look what I have here I have a system of equations whereas 3 variables that 2 equations and 3 variables do I have any hope of finding the answers to the street variables if I only have 2 equations That's a 2nd that's under it is great that under determined system so if I were really wanna get a value for XOY needed 30 equation where's that 30 equation and come from it's going to come from GE but not but this is Gee this is the 3rd equation This is my 30 is yet use this information yet what I did was ignored the 2 1st 2nd I said OK this function and gradient but I haven't used this information so that's the last piece of information that's going go with this system of equations that can allow me this Salford X Y 3 variables 3 equations and in fact 1 of the variables isn't even in this equation so that this becomes a relic of a reality that really Alberta for that crime OK Reproduce that I have to good Islam as to why will Plus wide from that any defined the value of X and Y so what I recommend you doing is more interested in Axson land is irrelevant to the sulfur lamp then plug it into the others equation so users me so since landing equals to x 1 I just take that to X and put it in from under there so slammed girls to X and to I was that implies to wide question X and that implies that white makes so from these 2 equation Jr. relationship between X and Y many Take that relationship could you put it in here and I'll tell you what X The next wide to put wife equals X put that in there bid to exit to Mexico 1 Now Therefore I equals X. than we have like this is the location where those greens The location of the Mac's or men don't know what it is yet is is the location is the location of that special point still to do a little bit more work to figure out whether it was an accident and it at 1 Look at what point 1 is equal to plug-in X were plus squared should now I just need a plug in any other points on makin constraint to see whether that's a Mac sermon
But in any other points on text to show about 0 2 0 2 wanna compare If this is a Maxtor end the sits Max them when I put that in this will be bigger if it's a million them when put this and this will not be small at 0 2 when you put that in you get 4 That means that this must so the location of the men is at 1 1 That's the picture here This is this is 1 1 right here We just found that location and then what we did that we were sure if it was a Maxtor amid yet but I knew the location was either Macs earned him so I just test another point and I find out that the original 1 was smaller so they had to be a minimum OK Cecil that tough for the 1st time through because I'll explain everything all over the theory were also doing examples of status OK so I'll that I just wanna pick any other points on this line here so anything that works I could add 2 0 I wanted 1 1 Canary had so I just pick 1 only 0 84 negative anything just to compare Saudi puff
OK let's said Do a new example The GE questions It's so always money There The extract that method which is used here and is will arrested because of what we do here here's that
Recipe for grunge so 1st look sat a generic pictures about just some group function of 2 variables and say that I'm going to Find the Macs and the millions of that that's why subject to the constraints Gee that what I was doing this on generalities years after Get caught up in 1 example OK so what was this guy in the last example that was line the rate so the line is very specific If I won I have it in general but state that lining Gravitt gathered by the end there and just go like that so rather than having a lying there packages have some general squiggle so that he can buy I hit my little constrain so what happens when I go and I'm only allowed the depict points from here and plug women appear so what I get is a little squiggle there and I wanna find potentially the minimum that That's triangle defined the Minimax they're out there and then how I do it so let's look at what we did here we took the level sets of the original function and when the level set just got ready at that constraint in just kissed it resist change to it That's where the maximum minimum occurred I level set the level set of that function put him in written here to level sets I don't know what they look like this is just generic the let's say there is no light at these levels set Well at some point you get right up engage is just 1 of and then maybe have another 1 over here They just kiss is there and those 2 points were that level said just come up interchange in this good calculus where his team engineering drawings and that's where the magic occurs at Tianjin but those 2 locations give you the Max's emissions and the way you find them so that that's this Straight up the the recipe mind Max and for F subject to GE here we have a three-step process So what was the 1st thing we did we solve where the gradient of that is lying with the gradient energy That's the way we say it Mathematically considered in English that their lined up with mathematically means that this Sector is a scalar multiple of that When we solve this here to look back this is case straight yearly and 2 x equals landed 2 white people's Lambert and From that all we got was a relationship between X and Y all we did was escape exit polls alive in their value yet and then the next step was to take that plug it into the street as women do next so John so plugged solutions to 1 inch the constraint And saw and the 3rd step was plugs solution to Chu into add to see Max And will add a little 1 example we just did had a little aware there was part of the 3rd step there was this little extra duty theory had checked it said check an extra point here is not put it on a three-day on checked Additional weight on this On the constraint if necessary will see in Somerville examples of doesn't always happen so sweats It's a little extra step you could make it step for a few wanted to really part of step 3 could answer instead 3 plug into the function not laughs case we didn't have a true answers had to check an additional point Talk it so there's very little met here now it's however nice example go through this recipe you'll see It's really not that bad is only that when you have no idea what I'm talking about which is at the beginning now I have a little bit of an idea what I'm OK so huts Avenue example pay here is that of access to add us why Can you realize what that is what I tell you it's linear sits linear a word that were supposed to be familiar with glass its linear now have any powers so this is lead the 2 variable version of a line which is a place that This is just a playing awarded is exactly We can grab a little bit and I will on a sec I'll show you what was going on here but 1st there's the function that in order for it to be this setting I need to have a constraint on find the Macs and the men subject to the constraints X squared plus Y squared equals 1 new visualize that seen this a bunch of times now and they still have the board X squared plus Y squared equals 1 that's a circle right is the unit circle so my constraint is the unit circle Get that function we will start to the little work here 1st things 1st lids Fred Astaire and this is my G Greg This is the Genie That equals a constant so let's that Dungy that's why swear puts wise
In the 1st step in my recipe here is to take agreement that considered illegal land times agreement and just get from a I can do that until I get so let's make good clean board
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Folk O'Grady in Lander rating agency was good get these things and effort that's that's a bad steps of why I wanted you the derivative with respect exodus get to 1 in the gradient G
It is equal to on 2 x 2 I Due while let let's kill picture of what's going on here to see you can't see it is so here is that the function original function is leery of the plane just picture like this is a place that's S NGO of X war equals 1 that square pose westward equals 1 that the unit circle sitting on the floor the unit circle
X squared plus Y squared equals 1 and so then I just take points long hair 20 what'll I get here I get something like that every time a plug-in appoint appear images is ill Circle up there on that plane and I'm interested in finding this is the end And that's the So what I do is I I take the level sets of this guy for years on level sets This is not the 1 that's Tianjin here in Tianjin back there That's what I'm looking for and that occurs when I saw this equation OK so Luxor Let's all but right still Gradient Islam gradient of G that means I have to warn people Islam time to add to what is a two-week regions I have to calls to the Member X. assertion the top-line just multiply through 2 equals to land 6 and I have 1 equals To Lando What I recommend you use is going a worker rules but they golly examples I Maggiore ones and lecturer and all anyone you get on a test is just sulfur landed 1 of them plug it into the equation because then what happens is your land anymore said Exon why they get a relationship between X and Y that step to wants solve this than instead to our take that relationship and I put it back into the constraint Nike is the actual numbers so for this 1 here I say that Lambda see if you agree with me land is equaled 1 operetta I saw for Linda there Nominate that This is planned equal former excimer put it in right there OK so then I have 1 eagled to land there's 1 of our acts on July So that means that if I solved 1 equals 2 while her exit that means that expert questioned what I've seen I have a relationship between X and Y now on minute scissors step 1 now it's that to I take that relationship and plug it into extra word plus Y squared equals 1 that's the constraint of take to put it for there So as to why squared plus Y squared equals 1 of 5 wife square 1 that implies that why is eagerly cluster minus 1 over 5 OK so have Y equals 1 0 Group 5 and why people want to values and X is equal the 2 so this flies over here that X equals to 5 an over here But the X equals negative attitude So what I've found years 2 points out got the the point to over a 5 1 over 5 and the point they needed to overturn negative 1 those 2 magic points those affected the recipe I just plug in the solution no 1 into the constraint and solved it And I got answers over my final answers to the point where the where that the gradients lined at the last steps as plugs solution to a into the function to see the maximum and sows do that death 3 years is Has of 2 overrule 5 1 Route 5 what's that equal called that 2 x plus why is equal to half of the year 2 times to Route 5 plus 1 5 5 over 5 feet crucially this that people would die multiplied not bottom Beirut that's the 1st point in the 2nd point is this effort of negative to over 5 negative 1 of 5 that's equal to 2 times native over 5 plus or minus 1 so that's negative 5 over Route 5 million PCs is revealed to us which is Maxim which is imminent the magic occurs at these points and then the function values Route 5 at this point in its negative But at that point This is the bank's this is there's no need to test another point because it revealed to us which 1 is the maximum in The picture that goes with this Harry have the three-dimensional picture Billups to the two-dimensional 1 So there so that Jedi begs the such X squared plus Y squared equals 1 that's the constraint And then there's the level sets us look at the facts Stewart plus why I would level sets of this is good practice anyway to a final a week so the level said there had ever equal 0 that means I have why he was negative 2 x so that This graphic here goes through here This is 1 that is negative 1 a slope of negative tube 0 And as I do these levels said her f just due to the Jewish y So why negative she went close to so that's now this graph here it's just a bunch of the level sets of this function here Mr. much apparel line And the the magic wands that we found were the ones that just went right at the constraints and that But here But what parallel there at the again here And they looked so yet this point here and we got this point over here to so right there were those of tangent this is this is where have Eagles Route 5 and this is where that equals negative but that's the level For those points and they just can't to that circle at those points they could put in the values in this Negative to overrule 5 negative 1 over 5 that This point here and this 1 to overrule by 5 1 that we just And the gradients are lined up there because the engines their candidate of the perpendicular set to be single letter We just did OK yet few more examples of this to Coliseum for next time that's enough for today and you want but