Okay, we're going to start off, we looked at the derivatives a little bit last time, partial derivatives. I didn't even really talk about that name yet, but I talked about the concept of 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 rule, just extending the power rule from Math 2A to Math 4. Now let's look at the definition of a derivative.

Let's go back to Math 2A. What was the definition of the derivative there? It's complicated enough just doing it at this level, then when we extend it, it'll be even more complicated, so it's worth it to go back and handle this. This is stuff you've seen before, but maybe your first time through calculus, the theory

didn't mean a whole lot to you. You were just learning how to do derivatives, so you didn't really have the opportunity to look at the depth and really get fluency with this. This is another opportunity for you to just absorb some of the math theory from single variable calculus.

What was the definition of f prime of x? Let's do it by picture. It's a slope of the tangent line, right? That's the idea behind it. If I have f of x here, and then I have a point, say x naught, or let's just say x. Let's not add notation where we don't need to.

Then what we did was, we said, well, if I go to a point nearby, x plus delta x, so delta x, you're supposed to think of this as an infinitesimal amount, you're just adding a little bit, but when I expand it like this, I kind of exaggerate it to make it a bigger distance so we can see what's going on.

So I have two points here, I have x and x plus delta x, and I can go plug them in. This is f of x, and over here, this is f of x plus delta x. Since I don't have a specific function, I don't have to specify any numbers, I can just say what things are.

And then what we do is, we find the slope between those two points. So what's the slope? The slope is rise over run, it's the change in y over the change in x. What's the change in y? That's the change in the y value, that's this little bit right here. And then we have the change in x on the bottom, that's x plus delta x minus x.

So that's just change in y over change in x, and we can clean this up a little bit because on the bottom the x's cancel so there's no reason to have those there. So what I really have is that guy, that's just the slope, this is what's called

the slope of the secant line, I'm just connecting these two points and finding the slope between them. So that's not the tangent line, that's the slope of the line connecting those two points, that's what I just calculated there. So then how do I get the tangent line? What I do is I move this delta x closer and closer to x and that brings that tangent

line closer and closer and eventually, or the secant line, it brings that closer and closer and you end up with the tangent line once at the limit when delta x goes to zero. So the actual slope isn't written like this, it's written with a limit.

So f prime of x, that's equal to the limit as delta x goes to zero of that thing. And let's just for practice, let's do an example with this.

f of x, let's always go back to f of x equals x, oh no let's not do that, let's do 3x. Okay, so I know the derivative there, the derivative's 3, but let's use the definition. f prime of x, that's equal to f of x plus delta x, that means I put x plus delta

x in place here. So that's 3x plus delta x minus f of x, that's 3x over, and I forgot my limit here, limit as delta x goes to zero, and then on the bottom I have delta x.

And the reason that you can't calculate this immediately is because you have delta x equals zero on the bottom. So that's a problem point, so you have to do a little bit of manipulation first. Usually what happens is the delta x cancels and then you're able to solve the limit.

So the limit as delta, this is 3x plus 3 delta x minus 3x over delta x, and you see those 3x's cancel, and we get, and I ran out of space, but let's just go over here. So then I have f prime of x is equal to the limit as delta x goes to zero, and

all I have left is 3 delta x over delta x, those cancel, and I get 3. So that's a long way of going about it, I mean you've learned, you've taken calculus, so at this point you know, you just glance at that and you know the derivative is 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 there. Okay, so now what we're going to do is we're going to see what this looks like with two variables.

So in math 4 we now have, not this situation, we have two variables, so say maybe x and y, so there's our f of x, y, but then we have two derivatives

because we have two variables, so we have fx, that's, now let's write down what fx is, this is the, it's just like this over here except it's the slope, here's the picture here, if I have, you can kind of draw something like this to give yourself some perspective here, and then what is that

fx is the slope in the x direction, that's how you want to think of it, so what I've drawn here is it's called a slice, I've fixed y, there's a fixed y here, and so every point there is equal to that same y value, but

x is varying and I've now found the slope in the x direction, I can do the same thing for y, but that's the general idea, so you can see that's complicated, we want to get away from this, we want to have a little bit of intuition for what it's doing, but ultimately we just want to get to where we can just calculate these things.

Okay, so we have two derivatives here, okay so what should be the derivative in the x direction, well it's the slope, and the way we find the slope is I'm going to take a point here and then let x just vary a little bit just like I did there, delta x, but y isn't going to change at all,

so let's write that down in the notation, this is going to be the limit as delta x goes to zero just like it is over there, but now what's the slope, the slope is going to be f of x plus delta x comma y, so I replace the x part with x plus delta x, and I just leave y alone, and

then, what did I do over here, I let the variable change by a little bit and then I subtracted off the whole function, so I'll do that here, and then on the bottom I have delta x, so if you compare those two you can see they're really the same thing, the only thing that's

different is I just have this little extra y, but if I remove the y's you see it's the same formula, so all we're really doing is we're just finding that slope, but just in that direction, and then we're going to do the exact same thing in the y direction as well, so we can just copy this down and just say, okay, well what does it have to be, I've got here

I'm doing with respect to x, so the delta x goes to zero, here I'm doing this with respect to y, so I'm not going to have delta x going to zero, here it's going to be delta y goes to zero, the roll of x is now played by y, so everything we see in x over here is going to happen in y, okay, now here x is constant, if you want to draw a little picture for that,

so now I'm fixing x, so there's a little plane there going in there, y is changing but x is fixed, okay, so that gives me the little slope

in the y direction, that'll be f sub y, so x stays fixed and then y changes a little bit, and then I subtract off the whole original function, you can think of this as like the change in y over, the change in the function over the change in y, okay, so just take a second to compare

those the three things, we have the formula from your math, where did I put it, there it is up at the top there, from math to a, and it's really just a slight extension when you have several variables, and since we have this up here, we don't really want to go

through three variables and ten variables, but it's worth mentioning it, if I had three variables here, then I'd have three derivatives, and I would just do x plus delta x, and then I'd have y plus delta y, and then I'd have z plus delta z, that would be maybe the third variable, okay, so let's take an example, just do one, let's have f of x y equals,

let's say 3x plus 5y, so we learned how to take the partial derivatives, I can tell you right now that f sub

x, the derivative with respect to x, so let's just remember what that is, I'm holding y constant, so I'm thinking of it as a constant, the derivative of the x part will be just three, the derivative of this part will be zero, because when I'm doing the derivative with respect to x, I'm considering this a constant, so that just is zero, so this is equal to three, and the

derivative with respect to y, that means I'm thinking of x as the constant, so now when I take the derivative here, that's zero, and the derivative with respect to y here is five, so those are my answers, now let's go see if we can get that from, just like we did here, with the definition, so f sub x,

that's going to be, using that definition up there, it's the limit as delta x goes to zero, I have f of x plus delta x, so that means I put x plus delta x here, and I leave y alone, so this will be 3 x plus delta x minus 5y minus, and then what do I do

for the second part, it's minus the whole function, so I put the whole function in there over delta x, so the same kind of thing should happen, we should whittle this all down,

cancel the delta x's, and we'll end up with our answer, we know the answer's three, but let's do the algebra to see that that happens, okay, here I get 3x plus 3 delta x minus 5y minus, why do I have minus, this should be 3x plus here, and then minus 3x minus 5y

over delta x, and see, once you wade through all the notation, everything just cancels out, so that's, this is what I mean when I talked about on the first day, just gaining a little mathematical maturity, there's nothing about this that's hard except the notation,

and once you can deal with the notation, then you start, you get a different perspective on mathematics, okay, so let's cancel everything, we got the 3x cancels with that 3x, the 5y cancels there, and I just whittled it down exactly the same as this, here I ended up with this, 3 delta x over delta x, and then those cancel, so then that

that limits 3, now I, it's, you're not really going to have to do this, but it's just good for your practice, and there's some examples in the book, some homework problems that ask you to use the definition, but the bottom line is, for us, when we're in the middle of doing problems,

we're not going to want to do it this way, we're going to want to just get the derivatives and then use that information to solve the particular problem that's posed, okay, but that's generally the theory behind the derivative, okay, it's just like it was before, it's just we have slopes in two directions, or three directions, or whatever,

however many variables you have, so let's deal with the notation,

you have to be able to recognize the symbols. In math 2a, we had, if I have, we could have, let's just have an example here, f of x equals x squared, so what were the different notations we had for the derivative?

There's f prime of x, but then there was also, we could use those d's, maybe you remember that, we could have df dx, that was an equivalent notation, this is Robert, this is Bob, right, it's the same guy,

just a different name, so now when we go to math 4, it's a little bit more complicated, we don't have f of x, we now have f of xy, or even f of xyz, we'll start to introduce a third variable in a minute, but let's just say we have this example

here, so now what's, we have f of x, that's 2x, and we have f of y, that's 2y, so these two guys here are kind of like the f primes, but then we have the, we have a notation like this for this situation as well, and it's not quite a d, it's a partial d,

and that's, kind of hints at the language used, so equivalently, this is Robert, this is Bob, they're the same, this is called the partial derivative of f with respect to x,

and here you'd say the derivative of x with respect, sorry, the derivative of f with respect to x, here you would say the partial derivative of f with respect to x, and you have the partial derivative of f with respect to y, let me write a little bit of that out, so del f del x,

this is said, it's said in a couple different ways, one is the one I've already said, the partial derivative with respect to x, and then if I had a y on the bottom,

I would say it's with respect to y, and notice the language makes sense, partial derivative, right, I don't have a complete derivative because I have one for one variable and one for another variable, and so that language is appropriate, it's part of the derivative,

not the complete thing, but then it's also, because this is a little tedious, I don't want to have to say this every time I write this notation, I don't want to have to say, oh, the partial derivative with respect to x, we have a shorthand for this, you can say, del f del, del f del x, del being, this is almost a delta, if it was a delta,

a lowercase delta in the Greek language is that, it has a little tail on it,

sort of like a musical note, it looks kind of like a musical note, but when we don't include that tail, then we just use partial, it's like instead of delta, it's part of it, it's del, that's one way of thinking of it, or justifying that language, so that's just how we communicate, right, it takes me a few minutes just to talk about

how to communicate with this stuff, I'll maybe just let that sink in a little bit, I think you can appreciate, even from having taken math 2a, you prefer this, this is more satisfying, it's just faster to write, here we've got, it's just more confusing,

because it's got a fraction and everything, it's got this d in it, which is confusing, so just like that, we're probably going to prefer f sub x and f sub y, and there's a little bit more notation to introduce you to in a bit, we'll get to that eventually.

The other thing I want to mention is the language that's used in economics, so in economics they might say partial derivatives in talking about it, but usually not, usually the words that they use are marginal products, that's the equivalent, you know, that's the Robert and Bob scenario, so in econ,

these are called partial derivatives are, and actually marginal products is an abbreviation,

they're actually marginal product functions, but I'm just going to write that since I ran out of room here anyway, marginal products, I'm just going to write marginal products.

All right, we're at the point now where we want to put this stuff together, so I've got, the situation is this, I've got a function of two variables, which means got a little point here, x, y, I plug it in, that gives me a point up there, f of x, y,

and then at that point we can say what are the different slopes, what are the derivatives, so we've got one that goes in the x direction, f sub x, and we've got another slope that goes in the y direction, so it's already complicated, I've got things going in different

directions, the question is how do we compile this information, where are we going to put this, so let's have a little example here, let's say f of x, y equals x squared plus y squared, our standard example, I just want to keep using this example because one of the ideas is to just

have at least one function that you're good at doing everything with, so this one, I'm just going to keep using this example so that you just, as soon as you see this, that pops into your head and then you have some sick picture to work with. All right, so at let's say the point

one negative one, so that's, you know, if x is, here's x, there's y, so one negative one's back there, and then I go up here and I get that point, it's up at two, f x, okay, that's equal to

two x, but, and this is two y, we've done this derivative a few times now, but now I want to know what is the slope, the particular number associated to this point, so now I plug that in to get the actual value, so the notation I would use to say, okay, not just the partial derivative, but at

that point, I would indicate that, but the vertical line, and then I put the point here, that tells me I'm about to plug that in, so now I only have x, so I plug that in, that's two times one, which is two, okay, so now I've got a number, the slope is two in the x direction at that

particular point, but if I had a different point, I'd get a different slope, so now we're starting to get some numbers in here, and then f sub y evaluated at one negative one, well that's negative two, so what I was saying before is, what are we going to do with that information? Where do

we store this? I've got, okay, the slope is two in one direction is negative two in the other, so we want one thing that incorporates all that information, and that's what leads us to the

answer to my question, where do you store this information? Well, at this point in the class, you should have some idea about where you store information, where do we store numbers? In a matrix, a matrix is a vessel of information, you can put information into it, and it'll store it there for you, we don't need a giant matrix, because in this case, I've only got two pieces

of information, so really what I need is a vector, it's a matrix, but it's going to be a one column matrix, and so that's a vector. Now the idea is this, I've got, so just looking at this

example here, I've got two in this direction, and I've got negative two in that direction, so what that means is I go out two in this direction, and then I go back, you know, negative two in the y direction, and then what do I do with those two pieces of information? That tells me how far to go in the x, this tells me how far to go in the y, and so that gives me a,

let's just draw it here, and it's getting in the way here, that's the gradient, it'll be a vector that's the diagonal of the parallelogram created by the f sub x and the f sub y, it'll be the little diagonal, so how do we write that down?

So given f of x y, it's got two variables, then I can go get f sub x, and I can go get f sub y, the two partial derivatives, and then I store them in this vector called the gradient,

and that's the way you say it in English, it's the gradient of f. Now just a little

intuition for gradient, the gradient is, it's the direction that a ball would roll if you just dropped it, so in this room, the slope of this room is coming down this way, right, if I take a ball and I put it at the top of the room it rolls right down to my feet, and if I put it over there it rolls right down there, and if I put it over there it rolls

right down there, and if I change the slope of the room that changes the gradient, so you want to think of, the gradient is just downhill, it's the most downhill you can find, or the most uphill, whatever the most extreme is, so that's just an intuition for it, we're going to be dealing with mostly just how you get it, and plugging it in, and using it,

and just to give you a little preview of where we're headed, what do we do in math 2a, we go get the derivative, and then we ask the question what's the maximum or the minimum, right, and what do we do in math 2a, when you ask that question, you go and you take the derivative and you set it equal to zero, and then that gives you critical points,

and then you test those critical points, and you find out whether you have a max or a min, the gradient is going to play the role now of f prime of x, you see f prime we only had one variable, so we just had one derivative, now we have two derivatives, and so in the old setting you set the derivative equal to zero, that gives you critical points, now we're going to set the gradient equal to zero, and that's going to give you a critical point,

if you think about, if I have a maximum here, so what I just tell you that the gradient always points in the direction where it's like a maximum increase or decrease or points downhill basically, so if you're at a maximum there's no pointing downhill, so it'll be zero up there,

and then that's the equivalent of a horizontal tangent line will now become the horizontal tangent plane, and the horizontal tangent plane will just have the situation where there's no change in the x derivative and there's no change in the y, so that'll create a top for us, or a bottom, however you want to look at it, so that's just looking ahead,

just showing you that's how we're going to use this, ultimately right now we just need to just practice with it a little bit, let's just compute, okay so here's a typical situation

that you'll find yourself in, I give you a function, it's got two variables, but you have no idea what that looks like, nor should you, nor is anyone born that way, it's not like there's this monk in Tibet and he was born, you could just show him that symbolism and he just knew exactly what it looked like, you have to practice and train

yourself and you have to learn about these tools, so that's the situation we find ourselves in all the time, you just don't know what these look like, so in order to figure out what's going on you have to just practice at least taking some derivatives, all right so the gradient of this, okay that just means that I'm going to take the two partial derivatives and then put

them into a vector, f sub x, that means I take the derivative with respect to x, okay so just take it piece by piece, what's the derivative of x squared, well that's 2x, okay what's the derivative of minus xy, remember you think of y as a constant, so see if you

follow me that the derivative of that piece right there is minus y, it's like having minus 2x and the derivative would be negative 2, okay so the derivative or another way to look at it is the derivative of x is 1 and then the minus y comes along for a ride, okay now I'm taking

the derivative with respect to x so this guy is just a constant, so that's 0 right there, let's see if you believe me that that's the partial derivative with respect to x, okay now let's do y, so now for y this is thought of as a constant so it's 0, then the derivative of this piece is negative x and you should as you're learning this and

I'm explaining you should sing along in your head you know make sure that you try to guess the derivative and then I'll tell you and then that'll help you download the information, so the derivative here when I'm doing it with respect to y is negative x and then the derivative of this with respect to y is 3y squared

and then we could say what is the, you see this is the generic derivative it's got the x part and the y part but then I want to know what's the slope at the particular point so then I can plug in 1 negative 1 there that's the notation we

use and that just means I plug in this for x and negative 1 for y into those terms there, so 2 times 1 is 2 minus negative 1 I get a 3 there and then negative 1 plus 3 is 2, so there's the gradient of that function at that point pretty simple

once somebody tells you how to do it and then the only place where you're really going to be maybe stuck you got to get this right you have to learn how to take these derivatives plugging in is not going to be a problem yes yeah it is but you're taking a derivative so what's

the derivative of a constant it's 0 right so yeah that kind of see when it's by itself the derivative will be 0 but when it's attached to an x you bring it along

just like if I had 2x and I do the derivative I get 2 but if I have x plus 2 and I take the derivative I just get 1 this part's 0 yeah okay so that's important I'm glad you asked that I'm sure you're not the only one who had the same had that question

in fact it was asked last time too you know it's it's a sticking point whenever people are learning this so that's good that you ask okay another I want to add I mean as if we didn't have enough notation I want to add to

the pile okay see in economics you don't usually use variables like x and y so we have to get you ready for what you really use and so if you think of a setting that you're that you're in an economics model you might have good one and good two right and see or good seven

right good one through good seven and you label them x sub one x sub two or g sub one g sub two when we label our variables that way then we have a different slightly different notation for the derivatives so let's get to that a lot of the problems in the book are written in this form so you have to know somebody has to tell you at some point what all this notation

means so now instead of having x and y I have x sub one and x sub two little tedious to deal with let's have this example here so if I want to write down the

partial derivative I could do this I could go to the partial of f with respect to x1 that was one way of notating it with that del f this would be del f del x1 but even that's little tedious I don't like that then you might want to use the f sub x1 but that's kind of tedious

too because it was okay when we had just x but having the extra sub one that's kind of annoying so we're going to abbreviate that further and just call it f1 and that's a lot nicer and it doesn't lose any information I know exactly which variable I'm dealing with so f1 is equivalent to all these this is Robert this is Bob this is Bobby it's all the same guy okay let's calculate

that now so f1 is the derivative with respect to x1 so if you want just think of that as a constant the derivative of x1 is one and then bring the constant with you and then same

notation at del f del x2 that's the same as f sub x2 but we don't like that we'll just call it f2 the derivative with respect to the second variable okay here I'm taking the derivative of root x2 so the derivative of square root of x is one over two root x this one will be two root x2

that's the derivative of that part and then we bring the constant along with us we're thinking of it as a constant this is a good example for you just make sure that you see that these are in fact the derivatives and be able to do them yourself make sure you get those answers well this is the only new part now we can just plug in a point

so f of x1 x2 or sorry the gradient so the gradient of f at the point let's say three sorry nine four well this is a vector

and in the top part I put f sub x f sub one let's put it here f1 f2 evaluated at those points

so when I plug in four here I get a two and when I plug in four here I get four on the bottom and nine on the top so that's the gradient at that particular point that function and again it's hard to see what this is so that's why we have to learn how to just do

do this without pictures so let's this is the next concept level sets of a function so let's

think about there's also an interchangeable word here you can call level curves those are interchangeable terms level sets and level curves so let's have an example here f of x y

is equal to 2x plus y and what we want to do is you see this is a three-dimensional well it's it's it's a plane in three dimensions so it lives in three-dimensional

space it's a two-dimensional object but it lives in three dimensions and the thing is is that all your training was in two dimensions you're really good at the xy plane so what we can do is we can reduce this down a dimension and look at things in the xy plane and that's valid thing to do and you will see some economics examples where we do that so let's

just start off learning about what a level set is to see a level set so where does this word

level come from what it is is we're going to you see there's different levels of the function this is the level where down on the floor there that's the level where the function's zero then if i move up one then that's the level where we're at one and then there's the level where we're at two so to see a level set of x of f of x y choose f to be a constant

and graph the results so as with most things in math when we describe it in english

it doesn't help even if i tell you to do that it doesn't really help so we have to take it and just kind of dissect it okay so what does it say it says

choose f to be a constant and we get to choose it so it's not like we're just this is random i mean it is random but you get to choose it so let's pick easy ones like zero and one and two and things like that so let's do that let's choose f equals zero i just did this i chose f to be a constant and then graph the result so what does that mean i'm going to graph the result after

i chose to see that doesn't do me anything what is f equals zero well you see f is equal to this so that's the same as saying this equals zero so that's what it means by graph the result so if f equals zero then that implies that zero equals two x plus y which then now i can solve

for that that's y equals negative two x and i can graph that y equals negative two x that's a slope of negative two and it passes through the origin so there's the there's the level set

for when f is equal to zero and the idea of that is here we are it passes through the origin i'm going to draw this down on the floor so the level where f equals zero is the floor here that's where the the f value is zero and so what we're getting is we're getting a little

line there in the floor but see that's hard to graph so we go over here and we just graph it on our original xy plane also when you set this equal to a constant it's like getting rid of one of the variables so i don't have this side anymore it now becomes a constant then it's just a regular line to graph that's what a level set is you want to think of it like this

you've got this big function and okay you get it you're on the ground floor and you hop in the elevator and it's like in a like in a parking structure or something and you you go up a floor say the first level and you get out of the elevator and you look at the ground and you see what's painted on the ground what's the graph at that

level then you go up to the second floor and you get out of the elevator and you look around and see what's painted on the floor and you want to graph that and you can do that at every level here we just did it at the ground floor so we'll do a few more of these so let's see another

level set of f of x let's choose f to be another constant how about uh choose f equals

two okay so then that means now i have two on the left side so that implies that two equals two x plus y and then it's what does it say to do it says choose f to be a constant and graph the result so let's graph the result here that's the same as y equals negative 2x plus 2

look it's the same slope as that one so we we should expect parallel lines that's right if it has the same slope it's just it has a different y-intercept now the y-intercept is two and it's got a slope of negative two so that's the level set for where the function

is two and you want to think of that as we've gone up to the this the second floor and then i hopped out of the elevator now i'm sitting up there on the second floor

and then i see what's painted on the floor it's this graph it actually isn't that graph because it passes through the uh and it passes through um uh the y-intercept is two so it's like this i don't want to be that accurate but anyway so i go up to the second floor and i see this line painted on the ground and then

i just graph that line let's do one more let's do f equals negative two so then that means that i have negative two equals negative two x plus y and then it says

pick it to be a constant and then graph it so that means oh sorry this was two x there so y is negative two x minus two okay same slope but now the y-intercept is down here we can go on and on with this but the idea is i've just done three and what

we realized is that for this particular example each level set is a parallel line so now let's graph that all on one set of axes and then you can start to

learn how to look at this in three dimensions that's that's the goal here so all three graphs on the same axes this is what enables you to start to see what the three-dimensional

graph looks like using these level sets so the first one it was a slope of two but it passed through the origin the other one passed through two and the other one passed through negative two

okay so uh let's let's get these pictures here okay so if you use your imagination you can think of those as parallel lines but what you should see is that okay this is where you

use the level set information this line right here see if you can use your brain to do this this line is is on the chalkboard this line is two units out from the chalkboard this is this is this one here is f equals two this one is f equals zero and this is f equals negative

two so if you train yourself properly you can see this as a ramp you can see this is behind the board this is at the board and this is one unit out of the board so you want to you can actually see the tilting if you practice and train yourself to see that so the term level

set means it's it's what distance out from the board or behind the board is your level set is your is your graph so it's not three parallel lines all next to each other there's a ramp here this is further out from the board this is further in and this is right on the board and that takes practice but you can train yourself so that you start to see that

the next example it's actually even easier to see so let's take a look at the next example it's our old friend x squared plus y squared so you know just for practice do you see the picture in your head now maybe not yet but if i do this enough times you'll see it

this is the paraboloid so what about level set so let's go through this process again the level to see the level sets of f choose f to be a constant and then graph the result

okay so let's pick some easy constants let's say f equals one let's start with that so when f equals one we're supposed to think of okay we're at the level one i took the elevator up to the floor the first floor i got out and i look around what's going on at that level

well if you if you kind of cut it with a plane here what you'll get is you get a little circle here that's what happens at that level but but if we just follow that last example what do we do we set the function equal to a constant and then graph the results so let's just follow

what we did over here as soon as i pick the function to be a constant then take the rest of it that means x squared plus y squared equals one and let's go graph that that's a circle that's the unit circle circle of radius one so that's the picture that's the that's the level set

for that function at the level of one now with your mind since this is more mostly what we're intuition here now take this plane here and move it up and down

so let's take it and move it up to here what should you get there it's a circle also but maybe it's a little wider right it's got a different radius so let's let's graph that let's graph a few of these see that's what i mean by the that's why the level set language is appropriate so we just move up and down on the levels and then it changes

the graph and we're just interested in graphing each one of those then maybe putting a few of them together on one set of axes so that we can see the three-dimensional picture using two-dimensional graphs that's the goal that sounds kind of like a lofty goal but it is

possible okay so another one would be now i want to pick something convenient see i kind of know the um the equations of circles the equation of any circle is x squared plus y squared equals r squared so when i'm picking my values of f let's say i pick two well then i'm really talking

about a circle of radius root two so that's a little i'd rather not have that so let's do the next one i'll do f equals four just to make the numbers work out nicely because that'll give me a circle of radius two now okay so pick the constant for the function then i have four equals x squared plus y squared

and that is a circle of radius two and there's the graph of the level at at four so i've gone all the way up to four there and i get this circle then one of the fancier ones is actually f equals zero that that should be the first one i started with because zero is always the

easiest number but i wanted to save it because it's a little bit different than the others and you can see from the picture there what should we get at this at zero or just a single point right we should just get a single point so you can kind of use that picture to guide you but let's do f equals zero that means that i have zero equals x

squared plus y squared and then let's just think about this for a second what are the values of x x and y that give me zero well okay let's say let's say y was something positive anything point one anything positive how could i ever cancel it with this

right so i couldn't ever have anything for this other than zero because i'd never be able to cancel it right so this this means x equals zero and y equals zero so we just get one point so the level set for this one here it's a circle of radius zero that's another way to

look at it right it's it's this circle here but i've got a radius of zero so that's just a single point that point right there is the level set for f equals zero it's just a single point okay so now let's put them all on the same set of axes and then we'll do that thing

where we try to train our brain to see it in three dimensions okay so i've got one of them

is just a single point and then i've got a radius of one i've got a circle

and then at radius two i've also got a circle and let's label those this is where f now it's the circle of radius two but this was the level set for f where it's equal to four right that was the constant that we set it equal to and then this guy right here that's f equals one

and then this one here is f equals zero okay so the point here see what we're doing when we're looking at this is we're looking down the center of that paraboloid and when you look down the center you see a bunch of concentric circles so that's what you want to do is see this

as pulled out from the board it's like one of those maybe you've seen those hanging baskets where they're chain link right when they hang they give you a three-dimensional thing but when you rest them on the ground they flatten down and you just see a bunch of rings maybe you've seen those before so picture this this here this is four units out from the board

this unit is only one unit out from the board and this is right on the board so see if you can visualize something three-dimensional from that picture you're sort of putting the two together there's another reason why we look at level sets and we'll get to that but the the

first step is to just understand what they are yeah you're always looking from the top depending on what you choose out from the board yeah yep yeah so when i see this because i've labeled it that way i see a ramp coming out this way and this one here i see

this pulling out from the wall like a big cup

this is a really famous model in economics and this is this is the one model that you will see

on your final we're going to use this function and go through a lot of the different examples and do the different things that we study with this one function but so it what it is is the cob douglas and there is no you want to maybe put two s's but there isn't two s's it's one s

these are people that i think they won the nobel prize anyway the typical thing is a cob douglas production function but i added the word utility because it it works like utility too so whatever you learn when it comes to this you can apply it to your utility function theory as well and i'll make those connections as we go okay so what

what is this cob douglas production function this book has a particular notation that it uses so we'll try to stick to that but it's not always this isn't the standard notation if you try to look this up on wikipedia or something you'll find different notation for this so there's

going on here uh so i've got so y is a function it's a function of two variables now the two variables are k and l this is capital but i thought capital was spelled with a c but anyway it's okay it's it's capital in maybe in america with a k but anyway it's uh

so this is capital and labor and the idea is that if you're going to have any kind of productivity you need both right if you had a bunch of labor if i just decide okay you guys are all hired let's let's go there's nothing to produce right you need a factory or you need

somewhere to go work to go create something so labor by itself doesn't get you much production and then if i had a big factory and i didn't have anybody to work in there then i have a lot of capital but i'm still not going to have any productivity so you need the two right so that's economists notice that they're dependent on each other you you you have to have some

in order to get some productivity you need some capital and you need some labor now the idea is let's say i want to do uh let's say i want to incorporate i want to have a model that that depicts what happens in the real world in fact it allows me to predict what's going to happen in the real world well as i add labor what law comes into

effect let's say let's say i get a certain amount of product productivity with a hundred labor a labor force of a hundred what if i make it 200 when i get double the productivity well if that's so great then why don't i make it 400 wouldn't i quadruple the productivity is that how it works no what what what plays a role the law of diminishing returns right

so if i'm going to try to depict what's happening in the real world what i do is i take these variables and if i want to incorporate the law of diminishing return i put a fractional exponent there and i haven't told you those are fractions yet but that's what i'm going to do

and then we can have a constant out in front here so you see it's kind of intimidating when you just look at it and say okay there's the model but then you have to ask you as economist you have to ask the question why does that model what we see in the real world so let me tell you what alpha and beta are so we put that little stipulation on alpha and

beta now here's a great example of math notation versus math into english okay you guys all take your math to english dictionaries out and translate this in your heads say in your mind what is that how do you say that just i mean i don't need you to say just do it in your head

just practice it how would you you know if you're if you were reading this at story time to your child well how would you read that in english or if you were saying it to your grandmother so a lot of you might be saying okay zero less than alpha beta less than one something

like that right you're just reading it you're reading the symbols but that doesn't really give you an impression of what it is so let me let me help you out what if i just say alpha and beta are between zero and one then don't you know exactly what i'm talking about so you

see how math notation is intimidating it's it you have to learn to read that if you if you if it was possible i'd like for you guys to just have a picture pop into your mind for that as soon as i see that i see zero and one and i know that alpha and beta are living in there so you could have a picture the mind works much better with a picture but when you're dealing

with inequalities the message i want to give you is use the word between that's a great word when it comes to inequalities very few of you did anybody say that in your mind when you were rehearsing it did you say alpha and beta are between zero and one see nobody does it right but then when i once i say it's like oh yeah

of course now i know exactly what that means okay anyway let's do let's take it one step further what does it mean to be between zero and one what kinds of numbers are between zero and one fractions right two-thirds one-third one-half right so so if you want just give yourself some examples those are numbers that are between zero and one

and then as economist is budding economist you want to say to yourself why would we choose those fractional exponents and the answer is because we want to model the law of diminishing returns these types of these exponents give you so just i told you this before but if we

if i just go back to calculus single variable calculus this function gives you that picture so if i want to incorporate that idea the law of diminishing returns on capital and labor then i'm going to insist that the exponents are fractional and that's what that's that's the

math behind so that's that's introducing the cob douglas production function from a mathematician's point of view because i say well why why would you use those kinds of functions what's so what's so great about those functions okay yeah they do model what we see in the real world all right i haven't said anything about a there's not much to say it's just a constant so what

we allow for is you know if i just do that there might be you know a real life example might actually have an example when we take the data i mean i have a constant in front so we allow for that so that doesn't really add much it's just a constant to throw in there okay let's label it everything

The Y is the total production that you get out based on a certain amount of capital and labor.

So let's go get the, let's go do some partial derivatives and maybe some level sets. Those are the only things we know, right? We know how to do partial derivatives and level sets now. So let's go do those two things.

So F of KL, this is an example from your book. This is page 400 in your book.

Let me switch the K and the L. That's another thing. In practice, they always put the L first. I don't know why, but I'll just keep it consistent right now with what I have above. Okay. Okay, so they've got the alphas. They're less than one.

Another thing that is usually, let me add this since this might be your first time seeing this model. Usually, they add another thing in that alpha plus beta equals 1. And that just makes, as far as I can tell, I don't know whether that model is what happens in the real world. I kind of doubt that.

But it makes the computations a lot simpler. So a lot of our examples are going to have that feature so that the numbers work out better. Okay, what can we do with this? We can take our derivatives, F sub K. That's going to be, okay, so I'm doing the derivative with respect to K. So just figure this part right here.

If it helps you, you can write this out like that or like this even. They're all the same. When I do the derivative with respect to K, I get 1 over 2 root K. So I get 5 root L over root K. So I bring this along for the ride.

So I get the root L. And then I get 1 over 2 root K. And that changes the 10 to a 5. And F sub L, that's going to be the same thing. 5 root K. Now let's go look at some level sets.

Okay, so what is a level set?

I don't have it on the board anymore. It means you take the function. You set it equal to a constant. So let's say, well, what constant should we pick? How about 10, just so that you have 10 equals 10 root KL, right? So then the 10s cancel. That's why I'm picking this. If you want to add that to your notes, it's fine. So I pick F equals to 10.

And so what does that mean? That's saying I'm going to get a production level of 10. So I could do that in a bunch of different ways, right? I could have a certain amount of capital and a certain amount of labor that gives me a production of 10. Or I could have more labor and maybe less capital. There's a bunch of different combinations that give me this.

And I want to find out what they all are. That's this graph. So F equals 10. That means 10 equals 10 root K root L. So then the 10s cancel. I have 1 equals root K root L. And then it says to graph, right?

That's what we do. We graph the result after we set the function equal to a constant. So I have to decide here which one is the X and which is the Y. So let's just do it alphabetically. So let's say root L is equal to 1 over root K. And so then

that implies that L is equal to 1 over K. So if I graph that, here's the K axis, here's the L axis. What is, we can go over here and just do the math, X and Y axis. If I had Y equals 1 over X, that's this graph.

But we don't have 1 over X. We have L and K, but it's still the same shape. And this is what's called an indifference curve. Or actually, in a production function, it's called an isoquant.

Well, I'll get to these vocabulary words. But since we're talking about a production function, maybe you've heard these terms already. So let's analyze this term here, isoquant. Iso means same, and quant, quantity.

So this graph represents all the combinations of labor and capital that give you a production level of 10. This is all F equals 10. So if you think about it from a manufacturing point of view, do I care whether I had, you know, a bunch of labor and a little bit of capital if I got an output of 10,

or if I had a bunch of capital and a little bit of labor? Either way, it's an output of 10. So there might be some little sweet spot on this curve that you might find, and that's maybe you've seen these indifference curves or isoquants before. Usually, you know, you try to make it tangent to the budget constraint or something like that.

I don't know how much you guys have seen. But anyway, that's a typical isoquant, that shape. And so that's why this graph here, 1 over X, is important for economists to know. I mentioned that before. But that's, see, if you don't have those graphs at your disposal, then it's really hard to do economics,

both the fractional exponents, which are all the square roots, the law of diminishing returns, for you guys, it's like this. And then the other one that comes up all the time is this graph here, which is an isoquant or an indifference curve. If you're talking about utility functions, they're indifference curves. And that's probably the more common term.

So, so far, the tricks or the tools that we have in our tool bag are we've got our partial derivatives

and we've got our level sets. So, eventually, we're going to try to combine those two things. We also got the gradient. So, I want to put together the level sets and the gradient.

Let's put them all on one graph. So, what I want to communicate to you about this is that the gradient is always perpendicular to the level set. And we're just going to graph one example and see that, but I'm going to write that down first. So, the gradient is always perpendicular

or orthogonal to the level set. Okay, let's go see that. So, let's take our typical example, f of xy equals x squared plus y squared.

So, we already know what the level sets look like. Let's do, so at the point, let's say, 0, 1. So, at 0, 1, we've got f is equal to 0 plus 1, which is 1.

So, we're on the level set of f equals 1. When we're at that point, we're on the level set of f equals 1. So, let's graph that. It's a circle of radius 1. That's x squared plus y squared equals 1.

That particular level set. And now, let's go get the gradient. The gradient, okay, that's partial with respect to x. The partial with respect to y. This guy is 2x, 2y. And then, at that point, the gradient of f at 0, 1, okay, that means I plug in 0 for x. I get 0.

And I plug in 1 for y. I get 2. So, if I go to the point, okay, where's the point 0, 1? Here's the point 0, 1.

And at that point, I graph the gradient. What is that? That means it's 0, 2. So, it goes 0 in the x direction and 2 in the y direction. So, let's put a yellow.

That's the gradient of f at 0, 1. And you see that that's perpendicular. It points straight out. Let's do another one. The gradient at the point, let's do right here, at 1, 0.

So, that means I'm plugging in here. I get a 2 here because x is 1. So, I put that in. I get a 2. Y is 0. So, now, I get 2, 0. Okay, so let's go to this point here, 1, 0. And then, graph that. It says go 2 in the x direction.

And 0 in the y direction. That's the gradient of f at that point. And you can see here that's perpendicular. We're going to leave this topic a little bit. So, I just wanted to get that out there.

We're going to use this fact later on. I just want it to be something you've seen before, that the gradient is always perpendicular to the level set.

Okay, guess what's next? Second derivatives.

Okay, what about in math 2a, what did we use second derivatives for? What was the tool? How did it get used? What was the information that you got from the second derivative? Concavity. Yeah. Okay, so we're going to have, now, if you look at, let's just look at this guy here.

Is that concave up or concave down? You don't know, but if your life depended on it, and you had to guess. What would you say? Up, right? That's a good guess, yeah. If I have this, that's concave down.

It's going to be the same way as the parabola was concave up in calculus and then concave down. We also have an inflection point. That was usually found from the second derivative. You take the second derivative, you set it equal to zero, and that gave you another critical point where something might have happened, namely an inflection point, but it didn't always happen.

So that's what we need to do. We need to start investigating second derivatives of these guys. Well, that's a little bit complicated. So I'm going to do both notations simultaneously. This is sort of the typical. We just have two variables, x and y,

but then in the economic setting it's typical to have x1 and x2. So when we did our first derivatives here we got fx and fy, and over here we get f1 and f2.

Okay, but then when it comes to the second derivative, don't we have two derivatives we can take here? So this is a function here sitting there with x's and y's in it potentially, but when I take the derivative I have two variables, so I have to do two derivatives.

Just on that side. Now what are we going to call the notation? Well, I've got the derivative with respect to x and then I do the derivative with respect to x again. So how about fxx? That tells me that I did with respect to x both times.

Then I could come along here and I did the derivative with respect to x, but now I want to do the derivative with respect to y, and I'm going to notate it that way. So first x, then y. And then I've got the symmetric situation over here. I do the derivative with respect to y,

then I could do x next, or I could do y both times. Now think about what you're doing when you're doing the derivatives here. You're saying, okay, I got the slope in the x direction

and then I find out what the concavity is in the y direction, and then here I found the slope in the y direction and then I find the concavity in the x direction. And it turns out these are the same. It doesn't matter which order you do those. So we'll just get that out front right away. That's what's called, I'll be a little more formal about it in a second, but since I've got this up here,

I want to be clear for your notes that these are always the same for our purposes. That's called Young's Theorem. Those derivatives are the same. Those are called the mixed partials. That's the, you know, the shorthand language that we use.

Okay, let's go over here. This is the derivative with respect to x1, so now I've got two of those. I'm going to call this one f11, the derivative with respect to x1 twice, and then I have f12. And then over here I've got f21 and f22, and just

like over here, the mixed partials are going to be the same. These are the same.

I actually want to continue on here. Does anybody need these notes right here? Can I erase this board so I can continue on underneath that? Is that okay?

So then what do we do with this information? We go get these derivatives. We had, what was the thing where we compiled all the first derivatives? That was the gradient.

So for the first derivative, we had the gradient of f. That was just compiling or consolidating that information into one matrix. And then we have the second derivative.

Well, what should that be? You see I've got four pieces of information. And where should we put it? How about a matrix, right? Isn't that where we store information? So I'm going to store the information in what's called, this is called the Hessian, or the Hessian,

but I think Hessian is the right way of saying it. But I've heard it both. So this is the notation for this. It's going to be a matrix. Well, I've got four pieces. In the upper left, I'll have fxx, then fxy, fyx, fyy.

Now, it's a little bit, when you have x's and y's, you might be saying, well, okay, now how do I remember x's in the upper left-hand corner? Well, it kind of goes alphabetical order. So whatever the first variable is goes there. But it's a lot easier to deal with this in this notation. Because, look, you see this, isn't that the 1, 1 location?

So that should be in the upper left-hand corner. And this is the 1, 2 location. So that should be in the upper right. And 2, 1 and 2, 2. So those, that notation is perfect for matrices because it's already got the subscripts that are appropriate for the matrix locations. Okay, so here, the gradients for this situation is f1,

f2, and then the Hessian, this is called the Hessian.

That's named after a person. But you just think of it as the second derivative matrix, right, just like the gradient is the first derivative. The gradient plays the role of f prime

from your Math 2A class. This is playing the role of f double prime. There is a little bit of a weirdness here that you might have picked up on. Here we use a lowercase f, and here we use an uppercase F. So that's just following the book there.

But that's pretty standard. All right, there shouldn't be any questions yet because we haven't really done anything. I'm just getting all the stuff up there, all the notation. So now let's do a little example here.

So this is the level of derivatives that you're going to be having to take. Maybe slightly harder with just some more exponents, but not much. The only other one you might have to consider is where you have the roots and things, the fractional exponents.

But this is typical of the type of thing. So if you can handle this, then that's where you want to be. All right, so let's go get these different things. The command for this is find the gradient of f and the Hessian at, oh, let's just compute those things.

We'll do add at another point, another problem. Okay, so the gradient, that's fx, fy. Okay, so fx is 6x here. There's no y, so I just take the derivative like it was in math 2A. So 6x. And then the derivative here will be 2y.

And then the derivative of negative 4y squared when we're doing it with respect to x is zero. That's considered a constant, so I just get that. Now if we do f sub y, this is zero. There's no y there. Then I get 2x and then I get negative 8y.

Okay, so there's the gradient. If I wanted to plug in a particular point, we can, let's just, let's do that. Let's just do it at, let's just stick to my 1, negative 1. So the gradient at 1, negative 1 is 4, 10.

Now let's go get our second derivative matrix, the Hessian.

Okay, that's fxx, fxy, fyx, fyy. Okay, let's go get these things. So fxx, that means I take the derivative of f sub x with respect to x again. So I'm taking the 6x plus y, let's put it here.

So fx is equal to 6x plus 2y. So now when I do the derivative with respect to x again, I just get 6. And the derivative with respect to y of this, I just get 2.

And then f sub y is 2x minus 8y. So I get my two derivatives here, fyx. Now what I told you over here is that these two derivatives should be the same. That's what's called Young's Theorem,

which I'll state formally in a minute. Maybe next time. Anyway, the derivative of this with respect to x is 2. And so notice you get the same here. Fxy is the same as fyx, so that's consistent with what I was saying there. And then fyy, the derivative of this guy

with respect to y is negative 8. So now the Hessian matrix is, I've got 6 here, 2, 2, negative 8.

All right, so guess what we're going to do? When you get a 2 by 2 matrix, now ask yourself, what are all the different things? If you look at a 2 by 2 matrix, could you see a picture in your mind?

Is there a picture to be seen? Remember the little area? Remember the determinant is the area? Well, that area is going to tell you the concavity. If the area, well, the determinant is going to give us the concavity information, okay? So I've got a matrix, a 2 by 2 matrix.

We could take the determinant. In this case, this is negative 48 plus 4. That's negative 44. That's going to mean something to us. So that negative determinant is going to mean something. In this case, it will mean we have what's called a saddle point. We'll get to all that. But that's how we're going to use this information. We've got our first derivative,

and we've got our second derivative. And what we're going to do is we're going to take that and set it equal to 0, which is going to give us a system of equations to solve. So we'll get back to what we did in linear algebra a little bit. And then when we set that equal to 0, we'll get a critical point. And then we take the critical point, and we plug it in here.

And then that tells us whether we have a concave up or a concave down. And that will tell us whether we have a max or a min. All right. Now, I've been dying to let you guys go early for one day. So today is the day, okay? I've been trying to get out of here, but I always run up against the time. So I'll just stop right now, and we'll continue on with this next time.

And if you have questions about your midterm, come up and ask them at this point.