Math for Economists - Lecture 9

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Math for Economists - Lecture 9
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Statistical hypothesis testing Functional (mathematics) Equaliser (mathematics) Mereology Parabola Fraction (mathematics) Derivation (linguistics) Goodness of fit Mathematics Sign (mathematics) Root Lecture/Conference Different (Kate Ryan album) Square number Negative number Energy level Circle Genetic programming Social class Alpha (investment) Area Pairwise comparison Multiplication Model theory Calculus Linear algebra Unit circle Variable (mathematics) Cartesian coordinate system Flow separation Mathematics
Point (geometry) Greatest element Functional (mathematics) Graph (mathematics) Multiplication sign Graph (mathematics) Two-dimensional space Variable (mathematics) Dimensional analysis Root Lecture/Conference Square number Circle Genetic programming Figurate number Nichtlineares Gleichungssystem Spacetime Maß <Mathematik>
Point (geometry) Functional (mathematics) Multiplication sign Graph coloring Dimensional analysis Prime ideal Derivation (linguistics) Mathematics Plane (geometry) Positional notation Lecture/Conference Saddle point Mathematical optimization Social class Addition Graph (mathematics) Model theory Graph (mathematics) Maxima and minima Line (geometry) Calculus String theory Variable (mathematics) Cartesian coordinate system Equivalence relation Tangent Curvature Figurate number
Point (geometry) Derivation (linguistics) Goodness of fit Functional (mathematics) Lecture/Conference Calculus Variable (mathematics)
Group action Functional (mathematics) Sequel Differential (mechanical device) Multiplication sign Direction (geometry) 1 (number) Mereology Rule of inference Power (physics) Product (business) Derivation (linguistics) Mathematics Plane (geometry) Positional notation Lecture/Conference Well-formed formula Oval Arrow of time Partial derivative Mathematical optimization Social class Predictability Logarithm Multiplication Process (computing) Radon transform Curve Calculus Variable (mathematics) Tangent Network topology Quotient Universe (mathematics) Right angle
Addition Compact space Functional (mathematics) Routing Calculus Graph coloring Power (physics) Product (business) Carry (arithmetic) Expected value Derivation (linguistics) Goodness of fit Root Lecture/Conference Term (mathematics) Negative number Marginal distribution Partial derivative
Derivation (linguistics) Lecture/Conference Meeting/Interview Different (Kate Ryan album) Open set
Far-right side So that's the starts the 2nd part of the class of what we we did linear algebra for the 1st portion of the class and the 2nd portion is multi variable calculus and you just took the test of linear algebra And I'm warn you that the multi variable step as is part so it doesn't mean that test is necessarily artery It's just that the material as you should expect the work a little bit harder to get the same level of understanding for the multi variable calculus you did for the linear algebra I actually enjoy teaching the calculus are a little bit better because it's it's more well it's less arithmetic I think that's probably the reason I like linear algebra staff's nite Nunavut happen to a lottery ticket maggots But hopefully you who had gotten a little sharper with your arithmetic skills as having them that you're out OK So calculus as several variables it's helpful if you're kind expiry calculus of a single A single variables which he did in math two-way so or can start off with Some comparison always try take you back to matter to relevant example there when I can and then will lift you up into the air The multi variable setting so map to a that's a single variable OK so what would we have but before we get into the calculus far less is deal with the single variable the calculus is going be derivatives that single variable part if this talk about that than I thought about a function have to start with that so what's a typical function that we have in calculus there several on I want I want go back in and just make sure that when you see this symbolism you have a picture that goes in your mind of this one's easy and that's the parabola But then also by rude X is squared banks that This is a talked about this 1 a little bit before about how in economics functions that have a fractional powers like this 1 has a fractional power this is really at the one-half so when I say functions in Mali fractional powers I mean this 1 but Alpha me next to the one-third an extra 1 The next 1 better than all the others Next to the 2 fractional powers this is the only 1 really having your mind you I have a picture for if I say what's X to the 2 that's not pops in your head A lot of this is the general shape for fractional Expo movements in the reason they show the nite economics lot because it models read a lot diminishing Richard it's 1 of the things they can model that OK I am so that that's These are standard functions and calculates that 1 over acts is a picture pop in your mind an irate that symbolism of the In there may not but it's helpful That the 1 over X looks like there's there's this part can here see this this is an interesting thing that happens when you start studying calculus with the goal of learning economics so 1 might status in calculus this it's this whole function there's the positive are the negative are but in economics usually ignore the negative because you thought about that amount of goods Amount of utilities or something it's always a positive numbers to usually and that his squadron when you're talking about economics fully talk about Matthew the whole there's a little bit of Sakhalin on where we have the math versus the economics in all tried it make those connected whenever I can another 1 here Still get away from these effort Axis I could write them X squared plus Y squared 1 of our you talked about that a little bit better circle that's the unit circle when you're calculus at the unit circle and can actually could write add salt for YYZ plus and minus square root of 1 minus sex where I want write as a function like the way I did those I would have the top half of the circle is the squared in the bottom half the negative square E Now will make the lead math for math is multi variable calculus of weird single variable calculus Sears areas are variable X in all those cases here have equal something with X so I had if it acts as variable and now I have multi variable calculus then and then had different kinds of functions and they have several variables at least to ready if it was only 1 maybe in that case we have to have something
Like an X and Y now have 2 variables x and Y Look at some of these guys what would be analogous function here it would be I summer plus the villagers 1 variable assist X squared where y squared so know that looks like now all of a sudden we get to the variables How do you even start to grab this year picture what you have a few functions that you can look at the symbolism and have the image up in your head in the same way you could do that there I have to do that for you to get you started so 1st of all what I grab over here I've got my variable X and helpful is the 2nd dimension They that next dimension here I've got really 3 dimensions I've got X and Y those of the inputs And then so let's graphic couple points here if I have X While and So let's put in the city 0 4 X and 1 for Halliday indicate that here I go and go out this way 0 units and stay there And I got 1 unit for why that's the point 0 and then I put it in the Function and 0 plus 1 is 1 of the function as 1 there and that guy sitting floating in space Nicky graph another point like Don 1 If X is wondering why is 1 that corresponds to there's wives whining years X is 1 phenomena go out there's 1 1 the in the floor there that represents is 1 And why is 1 now I plug that in an To further function Now I have to go up to A plot that point Right away he should be realizing with me here that this is very tedious and its is not an award even if I sat there and did 50 years I might not had any idea what that thing looks like whereas if I did 50 points here I'd start to get a feel for the curtain I could really sketch it out so what they but I could just tell you what this is rather than sitting applauding the point but that's how you would do it just wanna get individual points the real Graf here is well If it's that travel at it's gotta be similar trade with just squared Be travel at I at the Y squared the question is what does it what does that do adding the extra variable and the answer is you get what's called a travel and so we want from Corabel us to paraboloid Supporters of Another due This 1 here The analogous equation would be that would see it I just that square foot wide squared equals 1 than that's the circle that I have a fair enough variable here I could stall for a Z Z will be wary of plus or minus My sex with minors wife square fed assault By bringing these over the other side and then take the square root so now I have the analogous situation here as compared to this 1 but But what's going on here we now instead of having a two-dimensional circle we have a three-dimensional spirit And the top half of their services equals the positive version in the bottom half Calls minus version features start getting nervous because it's it's much more complicated use go from 1 very adjusted to it gets really complicated because you can't really just plot points and figure out what the graphs are Luckily in this day and age we have all this technology we have Wolfram Alpha you can just go away from our typing these things in plot for you that's fine but you that we please don't bring year electronic devices to the final exam but
For spitting at home and learning about it that time let's just take this 1 year and modified slightly and to show you how wild things can get if I take care of xxx warlord Equals
X squared minus white surges taking that simple function his putting a mine is in the middle of a plus it if I did just the minus in this situation images for Graf will be really similarly is essentially the same graph but uh with my has performed very different misses But had to draw now I have all put color in here so could see the 2nd op so this is a That's a saddle on him dead Yellow with Graf 1st the exit Grew so I out That's what's called a saddle Like Saddleback now really at Saddleback you could look out there and see it let's called saddle they have names So a paraboloid severe probably don't need to put that on there was your 1st graphs akin to variable is to get get a feel for it now we wanted you calculus rocket spend the whole lot time graphing stuff because it's complicated but I just wanna show you a few functions see lower dealing and also scare you a little bit and realizing you get a half half-day will over time and that's OK Now what do we do when it comes to calculus political go back said math two-way again and what was that what was 1 of the key issues Math to a single variable so the video anybody's watching the video when I say Matthew empire about single very out single variable calculus here UC Irvine who OK well Here's a here's a picture that was common in this class we have a function The acts and then we had had a single point here say X Nyah just X legal appear there's the function value at that And was the issue The issue was what the slope Of the because that told you what the function was doing if the slope was positive the functions increasing the slope with Nadia then if we had a slow was 0 land that might have been a special point have a minimum or maximum so that the issue here was what is the slope of the injured OK Who would have more variables so 1 thing you can do here just a kind teacher brain ready 1st more variables is Weather What is the picture do highly change from this picture to picture when you have to bury so 1 think it is Think of each variable of adding dimension so here had 2 dimensions at the single variable plus the function than over here I have 2 variables plus the function so I'd really just added dimension bad a variable added dimension sold with a word like for you guys do with your brain here is just grab the picture and poured out from the wall but so did you get so really this Two-dimensional on the bipolar out from the wall that gives me the extra dimension of the sea which you see in your head and I have that I would say what I say Barak half of it to you get all pull that out from from the wall there so the picture that you may or may not get a bite if I pull it out from the wall then beat the line will you pull that out that's like getting a piece of paper if it was a line in the new pulled out from the wall It would give you a plane Draw that ends the axis say pull out an extra dimension that this 1 here X and Y the function isn't as Cornelia Street now it's now than be so the generic function into 2 dimensions of 1 variable is like a like a stirring so added dimension it's like taking a string of pointed out of Waldman which you get is like 2 people shaking a sheet is floating cheaper a floating at Whatever you wanna thinker Also let's Something like that something floating in there that I won a figure that is flatter anything I give it some kind of curvature Put died there something like that day so is my general function and over here I took a single point and then I went up and checked the function value so here I'll do that too too But it's more complicated my single point is X not why not I get to variables to consider that puts me at a single point down here and the plane around the floor And then I go and see what the function is at that point guards and that's not why not just like I had acted not over there I go when I check that value And then what's the equivalent of the Tianjin line well That's be attends plank We need to get to where we're doing calculus that's appropriate for economics and so we can escape over this idea can play 1 mistake because what's appropriate for economic who wanna know about taxes is that that's what that that optimization that appropriate economic models so it is in this particular class we can escape this concept but the idea is still there I take a line that standard and I pull out from the wall I get a plane that had a single point
Like 2 Up here the slope of the tangent line his wife It's F prime that's not that's the notation we used that's the derivative at that point that gives you the slope of the Tianjin Soto you here
Here I have team isn't playing And I had to variables What have to do is on the map Dad to add derivatives of what's good picture But
A single variable calculus wonder that it is only 1 variable 1 slope at 2 variables so you have to do so I care so there is my generic function I go to a single point here
And what we're gonna do it because it's complicated and you can go along in any direction along this time on this floating she would restrict ourselves to basically Doing that now to a problem in both directions both direct the variable so I mean by that I'm going to get a derivative a slope of the tangent lying in the x direction and McGinnis slope of the tangent line in the wider action At the end But someone is right at a few things here so I had 2 variables that implies to slopes Suwannee taste for this because over here added At prime of that that was satisfactory that that was everything we needed for derivative but now it by being the derivative with respect to x 90 notation for that and derivative with respect to wire notation for that as well 1st of all what we really doing or finding of this curve here this this she has if I just walk in the X direction here I just focus walking along the extraction and it's got a little slow And I'm detecting the slope of the extraction slowed in weak acts direction and that also that the slope The wide action in these slopes of and be the same kind of its slope We found that 2 is more complicated So it's even I does little arrow here there's a little Kirvin on this finding little Tianjin along the right now just at the stage where were just getting the concept that obviously have to learn how to calculate these things in the same way he learned at a calculated OK so high let's have some notation here that set X This is going replace at times it's just that The little setbacks replaces the primary tells me what the derivative with respect to x and then I I can't play favorites is neither the variables is more important than the others so whatever I do for X ever got at the same thing for why derivative with respect Why touched E a K How do you take these derivatives filled with what's happening you checking the slope The extra direction soap if I'm if I'm only looking at the X direction and that means that why is being held I look appears see this When I look along this little curve right Yeah and picture That is why is being fixed along that that plane there and I'm just checking the slope along that occurs when a right that out now words when I'm going to compute f setbacks the derivative of X A derivative of f with respect to them what I'm gonna do a gun that are sold can't stay at stake Yeah whatever I do X did the same thing for why so what I duets of Why is the opposite of what or hold Called stood up and secret you bowl add with respect to X WRT that with respect to its save time by that we are That is what it is that what now we have that some examples of lawyers were spinning their wheels here OK so let's go back to math legacy of how we actually took derivatives if I had look that sequel to eggs in the end it we have a formula for this heady take a derivative you take whatever power you have era matter what it is multiplied in front by subtract the matter to a Prime that's was born in X latest 1 If not for the matter for a quality this is again I have 2 variables of a function of X and Y and let's put together this kind of thing you sad at City M Y city hit by each is X is a wise have their own power plants not specifying that the same let to which
The The same is that they're back and say OK So now let's go calculate these partial derivatives 1st partial derivative of this get the formula OK so f sub backs this is going to be What is this interview over there says trees that hold the y constant cases You can't just ignore it just pretend like it And it's just it's just a constant woody do when you have a constant and Germany just ignore you bring along for the ride he take a derivative of the heart and soul when I'm doing the derivative with respect to x I'm holding this constant can I take the derivative with respect dead that'll be a N C minus 1 the same as it would be in single variable but I get a little extra Goody lighted and it comes along for the ride because it's like a constant You write that out is That's the end of World War I. M. didn't change much banality duet of why that's why I will be the derivative with respect to Y and that means a little pretend like the ex parte is a constant so it's just coming along for the ride may take a derivative of widening em so that part X and come along with me because being thought of as a constant here and then the derivative of wider is m wide to him but it did Oliver all of this these types in the universe that actually These are the ones that show up all the time so that were focused and I'm Were there were a lot of how say cool and this isn't class good Taking derivatives Multi variable function to years of variable it's a class that involves this again I'd do it but if were real class on that and we spent a long time we be doing things like product rule and quotient rule we be doing logarithm functions cosine functions candidates all that stuff so since we're this is not really a class on on partial derivatives that These are Week For our markets and a whole lot of time I wanna get the ones that we need for this class that we can start studying how we use these derivatives answer that relevant question like what the matter man But Other calculus glish optimization pocket psilosis have a few examples here just to practice The will head of bed why she will let why he said so There's 2 derivatives to take At 7 X Here's Right now I'm thinking X is the thing and taking the derivative and everything else is is considered a constant so if you want you can write this as just for the time being but he can isolate the the exits like to y I that so this is just a constant what's the derivative of the exits just 1 1 time the minus a text of his era Other derivative here As to what I just think of these 2 things as a constant and take derivative that Which is 1 of the applied to like a same thing with F Y and now think of the 2 X as a constant pledges bring that along with me was a derivative of why it's 1 then attached to relax nicer coupled with whatever they are Detached would have the axes wires are attached to each other like that I still go through the same process but is Axel that easier going here At the next is is the derivative with respect to acts Well I'm supposed to think of why Wiese constant pesos Now when I take derivative here there is no Wyatt at the exodus take the derivative of that with respect that they get to wet but now this is thought of as a constant so what's the derivative of a constant 0 0 2 derivative of 4 0 if they're thinking of this is a constant I have to think of that derivative of why when I'm thinking of it as a constant is 0 Luckily we get to do it again the if it is to practice it still f sub now on me X is the constant so when I take a derivative of this fired at 0 and then the derivative of why am I am taking the derivative with respect to so I have differentiate that normally just to let you don't have to rent out 0 1 has putting that it for the 1st few they were doing
Get used to it A fur coat on OK so X over
Now that the disguised version of this very year this ecstasy and M package that could have disrupt like that that's why 1 but that's not how we write these things It helps you to ride out the way in terms of just the powers that be Alice duets banks when I do setbacks and take a derivative of that Which is 1 and the white Carter's comes along for the ride is it's considered a constant that'll be just wide the derivative with respect to x The language to our get into this wanted to do a few examples by I a cop partial derivatives in economics so their cult that marginal products who heard the term marginal product that's what I at wireless is due this folly 1 more example of 3 per cent of the that OK so at Why I'm now thinking of X is the constant Bring it with us The list is focused on the wipers 2nd derivative here at the multiplying by the negative 1 and subtract 1 flight get negative Slate X comes along for the ride and I get negative ones such as this Negative while the multiplier front but by the negative 1 that gives me this negative and that subtract 1 that is me the negative too I can write this negative ads life's where we see that the derivatives of can be very different they don't have to be that this a similar here I get to to exit here I get to exit to wire that you're starting think always maybe they're always gonna look the same they don't also Said you have to be careful when they start to get more complicated 1 more get that square root going case That a typical because colors It's like a utility function because X and Y is different goods I've got the square root so I've Inc. The Love diminishing returns and so if I want quick the derivatives but maybe it helps to write like exit one-half collided with it helps you do that do whatever it takes to put it in situation directed by so when I X that means I'm thinking of the can't lose their bring them along with me and I do derivative of that they get 1 half had negative 1 half of that's equal to prove why over to rude and saw it is that if if you're good year derivatives rusty derivative of rude Exs won over to X and carry along the route y with with few and Stalinism address OK had that's up Why Now I'm thinking of the X is a constant the bring them along for the ride in a different that that What about war The negative one-half so Do you have what Talk and that's another villager that's in the new compact is that CA is that discussion up to that on Monday yes For I Yet they still Go back to the original calculus if I just to hear
Then the derivative a 0 but if I have to X now attached now derivative comes along the constant come along with me that's the difference