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# Math for Economists - Lecture 10

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It will start up but what we looked at the derivatives a little bit last time partial derivatives even really talk about that name yet but I talked about the concept of when you when you have more variables you have to take a derivative for each variable so we saw some examples of that last time just using the power just extending the power rule from map to a mass for Dallas look at that the definition of a derivative of go back to match to a what was the definition of the derivatives there it's complicated and not just doing it at this level them only extended it'll be even more complicated so it's worth it to go back and can handle this this is step you've seen before
But may be your 1st time through calculus the Syrian mean a lot to you or you know you're just learning do derivatives really have the opportunity to look at that depth and and really get fluency with this so so this is another opportunity for you to just absorb some of the math theory from single variable calculus
OK So what was the definition of a crime of X look withstood by pictures
The slope of the tangent line right that was That's the idea behind it so I have The next year and then I have a point say X 9 We were just just say exercise had notation what we don't need to What we did was we said we won't by Goto . nearby Next plus Dr. acts of Delta taxis boasted to think of this as an infinitesimal amount to just adding a little bit But when I expanded like this they cannot exaggerated to make it a bigger distance so we can see what's going on here so I have 2 . 0 An X plus built text Making go plug them in Tampa Bay 6 And over here this is half of X plus Delta X Since I don't have a specific function I don't have to to specify any numbers I could just say what things are and what we do is we find the slope between those 2 2 points so what's the slope of the slope was rise over overrun the change in why over the change annexed what a change in y That's change in the white valued this little bit gray here and then we have to change in X on the bottom that X plus Delta X minus acts such as changing y ever-changing week and clean this up a little bit because on the bottom the Texas cancels the Norwegian Abbas there so I really have is back at just the slope persist as is what's that call the slope of the seeking 1 and just connecting these 2 points in finding the slope between so that's not the Tianjin line that's the slope of a line connecting those 2 points that's what it is calculated the battle I get the Canton Line what I do is I move this Delta X closer and closer to X and that brings that Tianjin line closer and closer And eventually you're the seeking line it brings them closer and closer union up with the Tianjin line wants at the limit when x don't X goes to 0 so the actual slope isn't written like this it's written with a limit if so at Prime of X that's equal to the limit As Delta Texaco's 0 of that thing Now let's just For practice looks due to an example would ever next Always go back there X equals X was not to that with 2 out 3 acts as I know the derivative the derivatives 3 bullets use the definition of a F crime of That's equal to half of X plus Delta Hexamita put Expos built Texan places the 3 acts Will plus Delta My At the next that 3 x over limit their limit as Celtic text goes to 0 The bottom line had dealt X and the reason that you can calculate this immediately is because you have dealt Mexico 0 on the bottom so that the problem points you have to do a little bit manipulation 1st usually what happens is the Delta X cancels and then you're able to all 1 of the Menendez del this is 3 x plus 3 Delta X minus 3 X Over Delta When you see those 3 axes cancel We get Arianespace clueless over here so than I have that crime of acts His equal to the limit as Delta that's close to 0 and all I have left is 3 Delta X over doubt that those canceled and make 3 so it's a long way of going about it I mean you learned you taking calculus at this point you know use glance at that you know the derivatives 3 but we just went through the definition from scratch and found out that the In fact the slope of the tangent line is 3 OK so how will it do is in see what this looks like with 2 variables
So enough for we now have
Not this situation we have 2 variables estate X and Y there's RFO that's why I but whether we had to derivatives because we have to to variables X that's military Dallas sex is this is it's just like this every year except it's the slope here's the pictures here have included draw something like this to give yourself some perspective here
And then the What is that deaths of X It is slope in the x direction that they would think it would have drawn here as a scholar sliced and I fixed y There's a fixed white here and so every point there's that is equal to that same y value but taxes varying and have now found the slope and the extra I could do the same thing for wider that's the general idea that's complicated won't get away from that We wanna have a little bit and tuition for what it's doing but popular just wanna get to where we can calculate OK so we have to derivatives here so what should be the derivative In the X direction while it's the slope and the way we find the slope is undertake a point here and then let extras very little bit just like they did their Delta X but why Isn't indicated change so let's write that down in the notation is Could it be the limit as Delta next goes 0 just like it is over there but now what the slope of the slope an agreed that the Plus Delta X Carmel why so I replaced the ex parte with X plus the exercises leave while on and then said voters over here I and that the variable changed a little bit minutes attracted awful functions on their here and around the bottom I have dealt so if you compare those you can see really the same thing the only thing that's different is I just had this little extra wide Fareham the wives the same formula for all were really doing is just finding that slope in that direction and then we may do the exact same thing in the white direction as well so we could just copy this down just say cable would have to be that guy here I'm doing with respect to acts of the Delta x goes 0 here I'm doing this with respect to why not dealt X going 0 here it's gonna be dealt 0 bullet that is now played by the rules but why so everything we see Annex over here you happened and why they now here Exs constant because if you want to run a little picture for that so now I'm and fixing X a little plane they're going wise changing that Texas fixed case so that gives me the little Slocum a wide direction of what So X stays fixed and then why changes a little And then subtract . global original functions Figure this is a change in y over the change in the function over the change in what just take a 2nd to compare those 3 things we have the formula from your math Put it their the Top from to air and it's really just a slight extension you have several variables and since we have this appear normal I wanna go through 3 variables and 10 variables but it's worth mentioning it if I had 3 variables here than I have 3 derivatives and I would just 2 Expos Delta excellent of wiped the wise men and others equals the disease due to the 3rd taking example just do want house That is why holds To Greek So we learned how to take the partial derivatives but I can tell you right now that X the derivative with respect that just remember what that is I'm holding Why can't since I'm thinking of it as a constant the derivative of the export of 3 the derivative of this part of 0 because women doing the derivative with respect to accent considering this constant 0 4 this is equal the 3 and the derivative with a that means I'm thinking of X is the constant Sonoma take derivative here that's 0 and the derivative with respect to y here not so there my answers Nellis Goosey if we can get that from just like we did here with the definition of tax that's going to be found using that definition of there's the limit as text was 2 0 ad
F O X plus Delta excellent that put Expos 6 here and leave while owner of the treaty next still my describe why minus and they will do for the 2nd guards minus if the whole function but over Delta So The same kind of thing should happen we should will assault down cancel the axes 1 up with answer we know they have 3 bullets to the algebra to see that that have a Harrogate 3 x 3 Delta Air Lines Letter and my aim is to be plucked And then minus 3 Minus 5 y over Delta and see they want you wade through all the notation earnings cancels out that That's what I need to talk about our 1st date is being a little mathematical maturity is nothing about this that's hard except the notation and want you can deal with the notation than than you start you get a different perspective on mathematics focus of canceled at the 3 X cancels that 3 x five-way cancels there and I just whittled the down exactly the same as this year a but that's 3 . that's over Delta the most canceled the that limits 3 If you're not really gonna have to do this but it is good for years practice and there's some examples of the book some homework problems that ask you to use the definition but the bottom line is for us or more in the middle of doing problems we wanna do it this way get derivatives and use that information assault the particular problems posed OK let's Generally the Theory behind the derivative just like it was before
We have slopes into directly over 3 direction would ever hollow many areas Those dealers the notation that you have to build the recognize the symbols in not to we had if I have but we could have us 7 example here OPEC's equals X squared
So What were the different notations we have for the jury there's Prime attacks but then there was also could users DCA remember that we get D E F G X and that was the equivalent noted this is this is Robert Bob Wright the same guy just a different name Now only good amount for it it's a little bit more complicated we don't have effort that we now that X Y or even if that's what will start to introduce a 3rd variable minute because they have this example here So now what We have that 2 act so we have ever of why that stupid why these 2 Here How however that crimes but then we have to we haven't notation like this for this situation as well and it's It's not quite a D partial D rats Any Cunningham said that the language used by the equivalently this is Robert Bob the same this is called the partial derivative with respect to exit here you'd say the derivative of that with respect The derivative of asked with respect to x here you would say the partial derivative of that with respect and you have the the partial derivatives of that with respect Why we're a little bit of dealt dealt tags this said It said a couple different ways 1 was is 1 of our he said Partial derivatives with respect to x it by adding a Y on the bottom I would say it's with respect to wide notice the language makes sense partial derivatives trader don't have a complete derivative because I have 1 1 variable on 1 for another variable So that languages appropriate part it's part of the jury not the complete state but then it's also because this is a little tedious I want to say this every time I write this notation on to say about the partial derivative with respect X we have a shorthand for this so we can say that all that Dell tactics Dell have Dell being being this is almost a Delta if it was a Delta a lower case Delta and in the Greek language is is that has a little tail on her like a musical note that looks like a musical but when we we don't include that And used cars considered dealt As part of its Del that's 1 way of thinking Justifying the language that as how we communicate it takes me a few minutes just to talk about it communicate with the stuff committee but that's a little bit I think you can appreciate even from having taken map to aid you prefer this this is this is more satisfies faster rate here we go this is more confusing to the fraction Everything that is D unit which is confusing for just like that will probably get a prefer S setbacks and of why and there's a little bit more notation to introduce you to him and they will get to that eventually 1 mission is the language of its use in economics in economics they might say partial derivatives I'm talking about it but usually not usually the words that they use marginal products that's the equivalent of that the Robert and Bob scenario so Hindi on are called partial derivatives are and and actually marginal products is it It is an abbreviation they're actually marginal product function But I'm just gonna that surrender room anyway a marginal products A If
But But we're at the point now where we want us put this together so I've got this situation is this I've got a a function of 2 variables which means 0 . here x y
Plug it in the me point up there at of X Y And then at that point we can say where the difference saltwater the wood the derivatives so we've got 1 that goes in the x direction at that time we got another slope that goes in the wider region so tolerated complicated I've got things gone in different directions the question is how we compilers information were wary put the food but that's how little example here said that the X Y equals X squared plus Y square standard example is wanna keep using the example because money ideas is is is have at least 1 function that you're good at doing everything the sums to keep using this example so you use as you see fit that part in of your head and you have some pictured a work Are a joke at but say the point Want negative ones that's Texas here next year's while I fell 1 negative ones back there I go there make it that way It's a bit too F X OK with the sequel 2 expert And this as to why we distributed a few times but now 101 is the slope the particular number associated to this point now I put that into get the actual value The notation I would use to say OK not just Partial derivative but at that point That would indicate that both the vertical line and put the point here that tells me I'm about to plug that it now I only had put that is 2 times 1 with just 2 and now I've got a number the slope is to with the extraction at that particular point but if I had a different point I get a different sort Now states numbers in here and then have why evaluated at 1 negative 1 that's negative to so What I was saying before is what we could do with that information where where we would restore the said that OK this Lopez two-on-one 1 direction is native to the other so we we want 1 thing that incorporates all that information and that's what leads us to the next concept which is called the Great But the answer to my question where he used to store information that at this point in the class that some idea about where you store information where we store numbers in a matrix as a matrix is a vessel of information you could put information into little store it therefore don't need a giant matrix because I've only in this case only at 2 pieces of information
Give that a lot of the problems of the book a written in this form so to you have to know somebody has to tell you at some point put all this notation announced it X and Y X 1 next to a little tedious to deal with this example here Soap If I wanna right down the partial derivative I could do this they use the partial of f with respect to next 1 That was 1 way of note-taking taking it that Della Delaware well but even that's a little tedious like that then you my only used the F sub x 1 that's kind tedious to It was OK had just expert having the extra 1 that annoying so abbreviate that further Just call it at that's a lot nicer and doesn't lose any information and know exactly which variable and dealing with So F-1 physical want all businesses Robert is bonuses body all the same OK that's calculate that now so is a derivative with respect X 1 so if you want to think of that as a concept derivative of X 1 of 1 and then bring the constant with you and then same notation Bill F. Del Next to it that's the same as that next to like that was holiday 2 The derivative with respect to the 2nd variable hockey here I'm taking the derivative Ruth next 2 of the derivative of squared Exs 1 over to rejects the only to root X 2 that's a derivative that are only bring the constant along with I was thinking of it as it is a good example for you just make sure the you see that these are infected derivatives and be able to do them yourself make sure you get those with this is the only new part now we can just in a point half us so F X 1 X 2 0 sorry the gradient of the gradient of death at the point let's say a 3 9 make the water will this is sector during the top part I put S sub x Someone with fear that want to evaluated at those points us on plug-in four-year to in for here They form the bottom and 9 of the top That's the the gradient at that particular point that function again as hard as with this and that's why we have to learn how to do this without pictures off But So Of course Puts this is the next Concept level set of functions of the think about There's also an interchangeable word here call level currency is interchangeable terms level set some level So 7 example here The ex-wife and we want to do is to see this is a three-dimensional bullets it's it's a plane in three-dimensional lives in three-dimensional space at two-dimensional object but lives in 3 dimensions and the thing is that all your training was in 2 dimensions you're really good at BXY point so what we can do is we can reduce this down dimension and look at things in the XY plane and that's a valid thing to do and you will see some economics examples where we do that was to start off learning about water level said it's to see a new level said who wears a sword level come from what it is we're going to see Different levels of the function This is the level where down the Florida The level where the functions 0 But if I move up 1 then that's the level where were at 1 And then there's the level were it to go to Sealevel set next That's why Jews Effort to be a constant soft and graphs this Results Go as with most things and that when we describe it in English it doesn't help thinking if I tell you that isn't really help so tho have the Chadian discounted dissected aware the status of Jews Have to be a constant and we get to choose it's not like we're just as When a random but you get to choose so easy ones like 0 and 1 in 2 things like that so what do that but Jews have people 0 I just did this chose have to be a constant and then grabbed the results was at a monograph result after it chose see that doesn't do me highways at equal 0 busy all of this The same saying this sequel 0 so that's what it means by Graf the result so if equals 0 than that implies that 0 equals 2 x y Which then now stop that's why he was negative 2 X And I think at that a white equals native to X that's a slope of negative Tuan it passes through the origin So there's that there's the level set for while That is equal to 0 and the idea that is here we are passes through the origin of the job is done before so the level where an equals 0 is the Fourier that's where the value 0 so we're getting is getting a little lying there before but see that's hard to so we grow over here just Graf on original XY Also when you set this sequel to a constant forgetting where 1 of the variables so I don't have this side anymore now becomes a constant menace just a regular lying to graft that's what levels You wanna think of it like this you get this big functions and became you get on the ground floor you hop in the elevator and it's like a like in a parking structures in you you go up for sale the 1st level you get the elevator and you look at the ground you see what painted on the ground the graph At that level when you go up to the 2nd floor and get as military look around what painted on the floor on a graph that you can do that at every level here we just did at the ground floor so we'll do a few more
See another level set of effort excellent Tuesday to be another constant have about Jews that equals to case so then that means
Now I have 2 on the left side by the student peoples to experts y And that it was stated to choose ever to be a constant and Graf the result was grabbed the result is that the same as wide negative way to look at the same Lopez that will so
We should expect parallel lines that's if says the same slope is just as it did for a wider set out why intercepted 2 days a slope of negative to collect the level set for where the function is to be wanna think of that as we've gone out of the this the 2nd floor and a half that of the elevator announce it up there on the 2nd floor and then I see what painted on the floor it's this graph to lectures that Graf passes through
We can see the three-dimensional picture using two-dimensional graph obstacle south
But OK I guess what's next 2nd 3 So it would about What about In math to where would we use 2nd derivatives for 4 was a tool I had a really Will it is the information that you got from the 2nd period can't cavity yet they still would have now if you look at the 1st with this guy Here is that Conte leprechaun came down
You don't always get life depended on it had to guess which is a great has agreed to have this is found it is going to be the same way as the gravel was concave up In calculus Even for concave down but also have an inflection point that that was usually found from the 2nd derivative take the 2nd release a legal 0 view another critical point or something might have happened namely inflection point always happen so that so we need to deleted start investigating 2nd derivatives of these guys Well that's a little bit complicated amid both notation simultaneously this is sort of a typical always have to variables X and Y depend on economics said it had difficulty have X 1 next to What we did our 1st earlier this year we got F X F Y and over here we have 1 2 OK then when it comes to the 2nd derivative don't we have to Derivatives we can take here so this is a function here sitting there was sexism wise and potentially a 180 during the day and I have 2 variables so I have to do to just on There were leader called the notation well I've got that the derivative with respect to X and then I do the derivative with respect X again so how about you F an X exact that tells me that I did I did with respect to x both times that I could come along here and I did the derivative with respect that they now wanted to the derivative with respect to why women indicated that way The 1st X-Men walk and that symmetric situation over here I do do with respect to then I could do X next alright dude why both Times I can't think about what you do and when you doing the derivatives here Here you're saying OK I get the slope and the extra action and I find out what the caddies Amelie directions and then Here I found the slope of the wider action find cavity and texture and teachers at this as a matter which ordered to do was get that out from Reddaway that's what's called a below more formal added a 2nd but since I got set there won't be clear these for your notes that geezer are always the same for our purposes that's called Young's those derivatives of the singer's recalled the Knicks partial The shorthand language that we focus Gloria This is a derivative with respect to x once and I've got to Lotus call this half 1 1 the derivative with respect to x 1 twice and then had had 1 too Over here that hash 1 and 2 and just like over here the mixed parcels be the same How
Actually wanted continually there's anybody need these here canaries as Bush continue on underneath that OK
OK So how where we do with this information we will get these derivatives we had what was the thing where we compiled all the 1st that was the gradient so for the 1st derivative we had agreed that that was just compiling consolidating information and 1 matrix And then we had the 2nd derivative what should that be We see about 4 pieces of information the where would put it However matrix rate is now we store information store the information in what's called was called the Hessian or the has seen the passion is saying that Both So this is the notation for this is guaranteed A matrix that 4 pieces in the upper left on how their X X and that's why X Y Y now it's a little bit when you had axes and why he may be stable here however member axes in the upper left-hand corner work Penagos alphabetical order soared over the 1st goes there but it's a lot easier to deal with this This notation because look we see this is that the 1 1 location that to be in the upper left-hand corner and this is 1 2 locations that should be in the upper right and 2 1 and 2 2 of those That notation is perfect for matrices because Darty got the subscripts that are appropriate for the Matrix location so here the gradients for this situation is 1 2 and then the Hessian Mrs. called the That's named after person Peter think it is the 2nd derivative matrix legislator gradient is the 1st derivative the the gradient plays the role that Prime from your math to a glass this is playing the role of that doubled from Er There's a little bit of weirdness here that you might have picked up on that he we use a lowercase that you use and appreciates that assists following the book as by the should be questioned it because is really that anything is getting all that stuff all the notation ballots
Example here roof
Kalkül
Mathematik
Ruhmasse
Mathematisierung
Derivation <Algebra>
Schlussregel
Partielle Differentiation
Physikalische Theorie
Übergang
Arithmetisches Mittel
Variable
Vorlesung/Konferenz
Leistung <Physik>
Lineares Funktional
Kalkül
Punkt
Mathematik
Zahlenbereich
Derivation <Algebra>
Primideal
Variable
Zahlensystem
Dämpfung
Rechter Winkel
Minimum
Inverser Limes
Vorlesung/Konferenz
Abstand
Tangente <Mathematik>
Ebene
Lineares Funktional
Erweiterung
Abstimmung <Frequenz>
Punkt
Mathematik
sinc-Funktion
Derivation <Algebra>
Schlussregel
Partielle Differentiation
Ausdruck <Logik>
Richtung
Zahlensystem
Variable
Perspektive
Mereologie
Minimum
Inverser Limes
Vorlesung/Konferenz
Figurierte Zahl
Lineares Funktional
Zahlensystem
Subtraktion
Flächeninhalt
Mathematik
Perspektive
Vorlesung/Konferenz
Derivation <Algebra>
Fokalpunkt
Physikalische Theorie
Richtung
Randverteilung
Lineares Funktional
Bruchrechnung
Subtraktion
Punkt
Sterbeziffer
Derivation <Algebra>
Partielle Differentiation
Primideal
Biprodukt
Variable
Zahlensystem
Einheit <Mathematik>
Mereologie
Minimum
Vorlesung/Konferenz
Aggregatzustand
Lineares Funktional
Matrizenrechnung
Subtraktion
Gewichtete Summe
Punkt
Klasse <Mathematik>
Zahlenbereich
Fortsetzung <Mathematik>
Eins
Richtung
Zahlensystem
Negative Zahl
Sortierte Logik
Vorlesung/Konferenz
Standardabweichung
Resultante
Matrizenrechnung
Punkt
Prozess <Physik>
Extrempunkt
Natürliche Zahl
Fortsetzung <Mathematik>
Kartesische Koordinaten
Ungerichteter Graph
Richtung
Übergang
Eins
Zahlensystem
Negative Zahl
Exakter Test
Minimum
Vorlesung/Konferenz
Tangente <Mathematik>
Figurierte Zahl
Dimension 2
Lineares Funktional
Güte der Anpassung
Partielle Differentiation
Teilbarkeit
Kritischer Punkt
Menge
Rechter Winkel
Dimension 3
Diagonale <Geometrie>
Ebene
Wellenpaket
Sterbeziffer
Hausdorff-Dimension
Wasserdampftafel
Gruppenoperation
Parallelogramm
Derivation <Algebra>
Bilinearform
Äquivalenzklasse
Term
Variable
Erwartungswert
Mathematik
Graph
Vektorraum
Objekt <Kategorie>
Mereologie
Numerisches Modell
Resultante
Ebene
Lineares Funktional
Prozess <Physik>
Wellenpaket
Kreisfläche
Mathematik
Graph
Hausdorff-Dimension
Reihe
t-Test
Spieltheorie
Ungerichteter Graph
Übergang
Konstante
Menge
Einheitskreis
Vorlesung/Konferenz
Figurierte Zahl
Subtraktion
Punkt
Sterbeziffer
Ortsoperator
Hausdorff-Dimension
Wasserdampftafel
Entscheidungsmodell
Zahlenbereich
Fortsetzung <Mathematik>
Gleichungssystem
Stichprobenfehler
Übergang
Eins
F-Verteilung
Arbeit <Physik>
Variable
Zahlensystem
Unterring
Einheit <Mathematik>
Vorlesung/Konferenz
Schnitt <Graphentheorie>
Dimension 2
Einfach zusammenhängender Raum
Lineares Funktional
Kreisfläche
Graph
Biprodukt
Fokalpunkt
Konzentrizität
Menge
Rechter Winkel
Tourenplanung
Dimension 3
Faktor <Algebra>
Parabel <Mathematik>
Numerisches Modell
Resultante
Kalkül
Punkt
Gruppenkeim
Ungerichteter Graph
Gesetz <Physik>
Übergang
Zahlensystem
Arbeit <Physik>
Translation <Mathematik>
Meter
Vorlesung/Konferenz
Wurzel <Mathematik>
Funktion <Mathematik>
Bruchrechnung
Lineares Funktional
Exponent
Übergang
Partielle Differentiation
Biprodukt
Teilbarkeit
Ereignishorizont
Kugelkappe
Arithmetisches Mittel
Verbandstheorie
Forcing
Menge
Tourenplanung
Nebenbedingung
Subtraktion
Sterbeziffer
Wasserdampftafel
Gruppenoperation
Zahlenbereich
Derivation <Algebra>
Term
Erwartungswert
Multiplikation
Variable
Reelle Zahl
Jensen-Maß
Kreisfläche
Mathematik
Graph
Kurve
Zehn
Betafunktion
Orthogonale Funktionen
Kombinator
Menge
Differenzkern
Mereologie
Numerisches Modell
Lineares Funktional
Kalkül
Mathematik
Gruppenoperation
Wendepunkt
Derivation <Algebra>
Frequenz
Symmetrische Matrix
Richtung
Zahlensystem
Variable
Kritischer Punkt
Sortierte Logik
Vorlesung/Konferenz
Matrizenrechnung
Zahlensystem
Matrizenring
Mathematik
Sterbeziffer
Rechter Winkel
Vorlesung/Konferenz
Derivation <Algebra>
Primideal
Ordnung <Mathematik>
Bruchrechnung
Matrizenrechnung
Abstimmung <Frequenz>
Punkt
Betragsfläche
Determinante
Derivation <Algebra>
Gleichungssystem
Physikalisches System
Übergang
Hesse-Matrix
Kritischer Punkt
Erwartungswert
Negative Zahl
Einheit <Mathematik>
Flächeninhalt
Gleichgewichtspunkt <Spieltheorie>
Lineare Geometrie
Vorlesung/Konferenz
Kantenfärbung
Aggregatzustand