So, new section, new chapter, some cool stuff. We get to talk about finally some multivariable functions, how these things work, how they look, and especially how the calculus relates to them. Because when you start dealing with more than one variable, what I mean by that is more than one independent variable, things get kind of screwy, like what's a derivative when you've got more than one independent variable? That's what we're trying to answer, and we start that with just an introduction about what these things do. The domain is and how they work.
So we're used to talking about functions, but we're used to talking about functions in just one variable. Please get this straight. When I say one variable, I mean one independent variable.
So like with x. The dependent variable y, there's always only one dependent variable. All right?
But now we're going to start saying, okay, what happens when we have more than one independent variable? Well, There's good news and bad news. The good news is that the same stuff basically works.
You still have a domain, you still have a range, you still get pretty pictures out of it. That's the good news. The bad news is there's a little bit more aspects to it.
Especially when we start talking about limits of more than one variable. That's the next section. Don't worry about it right now.
But what happens to these things? Here's some ideas. Here's some things that you need to know before we go any further.
So number one. In order to graph a function, these are universal, in order to graph a function, you have to have one more dimension than you have independent variables. We're always going to revert back to independent variables when we talk about those things. You always have to have one more dimension than you have independent variables. Thank you.
By the way, the reason that we talk about independent variables and not total variables is because when we do function notation, the only thing you really can count really easily is independent variables. That's what we talk about. Let me give you some examples here real quick. So how about that one? Very simple stuff, but how about that one?
How many independent variables do we have here? One independent variable. What is that independent variable?
x. By the way, you can always do this no matter what. We can always go from function notation to dependent variable. So there's one independent variable, there's one dependent variable. How many dimensions would you need to graph this graph?
Notice it's one more dimension than the number of independent variables you have. So we have a direction for the x, we have a direction for the y, that's two dimensions, two directions. So this would be graphed in 2D.
Now we step it up. A function with two variables listed like this, by the way, whenever you see your function notation, we have another one in a second, this always lists out for you, always, what your independent variables are, no matter what. So here we got...
one independent variable, it's only x. Here, we got two independent variables. They're x and y.
We're going to talk a lot about what that means. In order to get an output, you no longer need just one number. You have to have two numbers.
You literally have to have a point. on the xy plane, and when you plug in that point, it gives you out a height. So given a point xy on the xy plane, you plug it in, you get a height now. That's why we have the 3D coordinate system laid out the way we do, where we have this flat surface, where you plug in points, and at every point, this function gives you a height. If you take at every point, you find a different height, it's going to give you this surface.
Well, let's think about that. If we have two independent variables, verify two independent variables. function based on those two independent variables.
In order to do this, in order to take those two independent variables, like, okay, on my xy plane that's flat and take a point out here somewhere, and I plug in that point, two values, two numbers. And I get this height. I get a different height than every single point.
It's going to create this surface in what dimension? Two independent? What dimension do I need to graph it? 3D. 3D.
I need 3D to graph not only the X and Y motion, those points, but also the height above it. So that's a 3D. A 3D, we need 3D to graph that surface. Also, you can always do this, just go one variable different than what you have for a dependent variable. If we have g of xy equals x squared plus y squared, we can represent that z.
plus x squared plus y squared. I'd like you to get used to knowing these things mean the same thing. You know from a long time ago that f of x we can replace with y.
But now any function we can replace with one dependent variable. Notice, if we follow this, you will have at... Most how many dependent variables? One. One.
One dependent variable. Because we have these functions based on independent things, we say, hey, let it equal a variable. You can only do that one time. There's only at most one dependent variable.
Head on if you're okay with the idea so far. Let's step it up one more time. This is going to be kind of crazy. So first thing, recognize the function notation.
Hey, ladies and gentlemen, left-hand side, how many independent variables do we have? Can you tell me what the independent variables are? This always gives you the independent variables every single time. So this one said just x. Hey, there's only x's.
This said x and y. Hey, there's x and y's. This is x, y, z.
There's three independent variables. Could I write it not in function notation but base it on, well, how many dependent variables could we maximally have? How many? One.
So the dependent variable, call it something like w, just not x, y, or z. It says the same thing. Now this is weird, alright? Let's picture this for a second.
Picture what this is going to be. I know you can't see what the graph is, that's okay, neither can I. But I want you to think about what's going on with this.
Just track with me here for a second. If I plug in one number, that one number is on the x-axis, correct? I'm going to get a height. That's what the y is.
So I go over. Okay, so that's one dimension, two dimension. That's all that happened. If I plug in two numbers, I'm plugging in a quarter and a pair. It's a point on the xy plane.
That's why we have xy. It's on the xy plane. I plug in that point.
I use it in a function. I'm going to be getting out a height. So at every point, I get a height.
That's the 2d combined with a height. That would be three dimensions. That's why two independent variables gives you three dimensions. What would three independent variables give you? Four dimensions.
Four dimensions. What? Think about it. This is an order of triple.
It means you give me an x, a y, and a z. You plug it in, you get something else. You get a fourth dimension. How we graph it?
I don't know. It's really hard to think in 4D. How do we represent 4D? We live in a 3D world.
What would that 4D be? Some people use time to represent the changing of the 3D system over time as that fourth dimension. But you can't draw it. You can't draw it.
We need one dimension lower to draw it. Well, you can, but it's got to be able to move. You need one dimension lower than the graph to be able to represent it. For instance, you can represent 3D on a plane, but 4D... You need a 3D model to do that.
You need to draw it on 3D. So do you see the point, though? No pun intended, but do you see the point?
If I give you an ordered triple and plug it in, it's going to give me something greater than that, some dimension more. So for one independent variable, we need 2D to graph it. For two independent variables, we need 3D.
We need one more dimension than however many independent variables we have. And this is like, I'm sorry, okay? Or what? Yeah, 40 is weird to think about. What we typically do for graphing, well, you know what, we'll get to it later.
We'll get to it when we actually start graphing these things. Right now, I want to know if you understand the concept. of one independent variable, two-dimensional picture. Two independent variables, three-dimensional picture.
That's what I wanted to know if you understand the concept. That the two independent variables is the point, height above the point gives you something in 3D, the surface. Now, let's talk about the domain, because the domain, we spent a lot of time talking about domain here.
To graph the domain. To graph the domain of the function, you've got to have a dimension equal to the number of independent variables. It's one lower than the graph itself. So, well, we'll talk about it in a second. Write this out.
To graph the domain, you have to have the same dimension as the number of independent variables. We'll always revert back to independent variables. Thanks for watching So let's see, let's see. Everyone in class right now should be at this point.
How many independent variables do I have right here? One. How many independent variables do I have right here?
Two. How about here? Three. So if we're graphing the domain, we're literally graphing that. Variable.
Well, how many dimensions do you need to represent numbers on an x-axis? One. One.
You just need the x-axis. So in order to graph the domain of a function, just the domain, we need the dimension equal to the number of variables here. Here, for the domain, we need just 1D, just the x-axis. That's it. How about this one?
What if you wanted to, man, this should make a lot of sense, okay? How many dimensions do we need to graph the domain with two independent variables, 2D? Well, think about why. If the domain, look, if the domain are xy points, do you guys get it?
We put in the point, we get a height. If the domain is the xy points. We need something on the xy plane to graph those points.
The domain here would be intuitive. Just something on xy. This is a weird one. What if we had a function with three independent variables? We talked about that.
It's got an x. It's got a y. It's got a z. It's in three space.
What do I need to describe that domain? I need three. I need to be able to represent x, y, and z. So that's why we have this statement here. If you have one independent variable, you need one dimension to describe the domain.
Two independent variables, two dimensions. Three independent variables, three dimensions. Please don't get these two ideas confused.
The picture is always one dimension more because you're plugging in two things and getting out the third. Plugging in three things, getting out a fourth. But to graph the domain, we have the equal number of dimensions for that. This will become a lot clearer when we do some examples, but right now do you understand the concept behind it of the number of independent variables, what the graph looks like, one dimension higher, and what the domain looks like, equal number of dimensions.
Show hands if you're okay with that idea. Are you sure? Any comments, questions, or anything before we continue?
Yeah? If we had four independent variables, would we not be able to graph it at all? Graph the function or graph the domain?
The domain. The domain would have to be graphed in 4D. Okay.
Four independent variables you'd graph in the domain in 4D. The shape would, the surface, it's called surface, the surface would be graphed in 5D. Sure. Go for it.
Let's try, let's try, just practice. Let's just practice one. Just to get a feel for the taste of how these things work. So, we're gonna talk about domain in just a second. I wanna see if we can just look at a function and do some stuff with it that we would normally do, like plug in a point, manipulate stuff.
What can we do? Anna, I want to make sure you're paying attention. Ladies and gentlemen, firstly, function, yes, no, what do you think? Yes. How many independent variables do we have?
Three. How many dependent variables would we have? One. How many total variables do we have?
Four. Four. How many dimensions do we need to graph this shape for? Four. One more than the number independent because you count the dependent.
So total number of variables, total dimensions for the shape. Number of independent variables, you've got to have one more than that for the shape. So this would be a 4D graph. Next question, right side only. How many dimensions do you need to graph the domain of this thing?
So we need 4D for the shape, whatever the surface is. We need 3D for the domain, or something called a contour plot. We haven't talked about contour plots yet, but we will. I'll show you exactly how they work.
They're not too hard. Do you remember traces from section 11.6? Those are... Pretty much what contour plots are.
So we take those and project them, and then we get this contour plot. Have you ever heard of topographical maps? That literally is a contour plot. That's what that is. It's taking different...
What it's doing is taking this mountain range, right, and going... up 100 feet and 100 feet and 100, or some segment that's always the same, and say, hey, make a curve that's level around that surface, and then look down as you, like, as you're projected to a plane. And that's what a contour map is. We'll talk a lot more about that, how to find it. Could you do stuff like if I asked you, find the function evaluated at.
Zero to negative 1. Could you do it? Yes. What would you plug in for the x?
Zero. And the y? Zero. And the z?
Negative 1. Yep, there's order triples. Notice how it's in order triple has to be in 3D. That's why the domain is 3D.
So if we plug that in, we get, well, let's see. Zero squared plus 2 times 2 squared, 3 times negative 1 squared equals 11, square root of 11. Notice what you're doing. You just took something that has a 3D representation in order triple, you plug it into a function, give you a different number of it, gave you something that depends on what you plug in.
Your dependent variable value would be that height in 4D over that point in 3D. That's what's going on here. Are you sure you're okay? For real. We can also do one more thing with these just for right now before we start talking about domain.
I'm just making this up. I'm just coming out of nowhere. I'm just making up this representation, but check this out. If you had a way to parameterize x, y, and z in terms of another parameter, like, I don't know, t.
Remember that? Okay, another parameter. Sometimes we can trick the problem a little bit. We say, hey, can I change all of these into values of u and make it a little bit simpler to at least work with?
Yeah, sometimes we can. So if we have this parameterization, what would you plug in for x? What would you plug in left side for y?
And what would you plug in for z? Why don't you try that right now? Try it. Distribute it.
Simplify it. See what you get. I'm going to do it on the board as you're doing it, but I'll do a step about every 10 seconds or so.
See if you can substitute this in and change this into terms of you. Hopefully that's not the screen. I don't know, it's pretty close. Thank you So just a quick and simple manipulation where you're able to get at least the first part substituting that end, that end composition.
So you guys feel okay with the idea of just what these things are? So just a little 20-second recap. We've got these things called multivariable functions now.
With one independent variable, it takes 2D. Two independent variables, 3D. Total number of variables equals what you need to graph it.
2, 2. 3, 3. 4, 4. It's always one more than the number of independent. Domain is always equal to the number independent. That makes sense.
You've got one, you've got one dimension. Two, you've got two dimensions. Three, you've got to have three dimensions.
You've got to have that, our coordinate system. So what we're going to focus on right now is finding domain. Man, as I mentioned to you, finding domain is huge as far as your ability to graph a region to integrate over. When we get to chapter 14, we're going to be integrating over these things and understanding what we're actually doing. What region are we integrating over?
That's important. That's what we use this domain for. I'll show you exactly how to do it when we get there, but it all starts right here. So let's do some domain.
Let's see exactly how it works. I'll show you some things you can do, some things you can't do, and some things you absolutely must do. So first one.
I'm also going to really focus on your ability to understand what dimension we're working with. So we're going to do a couple things. We're going to find domain and we're going to find range.
Here's the cool stuff. The same things you're looking for, like for domain of any one variable function, one independent variable function, same stuff you look for here, which are like square roots. anything that has the possibility of being undefined.
It's the same thing you look for. So denominators, they should be slapping you in the face. Those sort of things, they work universally for functions, and we're going to use them here. So when you look at the function of a function, When we look at that one, ladies and gentlemen, function, yes, no.
Yes. Everybody, how many independent variables do we have? Two.
Two. How many total? Three.
Three. This would be graphed. The surface itself would be graphed in?
Three. Three. And the domain would be graphed in?
Two. Two. I'm going to show you how to graph domain in the next few sections, sorry, examples.
Right now, I just want to get a feel for what the domain is. So, let's take a look at it. Does the numerator give us some things that have a possibility of not being defined?
If I multiply any two numbers, even if I get zero, I'm good to go. Not a problem. How about the denominator? What do you know about denominators? Like universally any fraction, what do you know about them?
Okay. So what we know is that... x minus y, in no way can we ever let that equal 0. Does that make sense? Yes.
Okay. Solve it. Just solve it.
Okay, well, if I solve that, then what that means is that x can never equal 1. As long as we don't have x... SQL and Y, we're good. We can plug in anything.
Do you recognize that? I gave you an easy one to start with. Would you recognize that? As long as these two numbers aren't the same, we're fine.
That's exactly what this says. Now, all we've got to do is put that in domain. It looks a little strange. But it's the same statement as having one independent variable. We go, okay, well, the domain would be a set of numbers.
We use set notation generally. It's pretty hard. Well, it's not super, but it's difficult to describe other than this because this is, well, this is the nicest way I know of. We can go, well, what's our inputs? What is our independent variables?
What are our independent variables here? So every time we plug something in, it's an ordered pair. So the domain says, hey, you've got ordered pairs, x, y. What can you let happen? So such that.
What can you let happen? I can let anything happen, any ordered pair. The only exception I have right now is that x cannot equal y.
That right there is the domain. In English it says this. It says the stuff you can plug in are any ordered pairs, but not such that x does not equal y, or but x can't equal y.
That's all there is to it. So I think you're okay with that one. Now, the range still works the exact same way as it always does, but you're going to call it the output.
So if we have inputs x and y, what traditionally do we call the output variable? Z. Okay, so range works the same way. But for range, range will be the output, call it z.
So for the range. A lot of people struggle with the range. It sometimes requires a lot of thinking, like what could we possibly have?
So is it true that we can get anything out of this? And the answer is, yeah, it is true. You can plug in a lot of differences. As long as x doesn't equal y, we can get infinity.
We can go down to negative infinity. We can get zero, no problem. Just let x equal zero when y is not zero, and you get zero.
Yeah, you get 0. So 0 times y over 0 minus something not 0, you get 0. So we can get the whole spectrum of outputs here. And how do we represent that? How do you represent that? What would you do?
How do you represent z? The range can be anything. Anything. Yeah, same thing here. So z such that z is between these two infinities.
Just make a little side note, just in your head. Just in your head. How many independent variables do you have? Two.
The domain has to say something about all of them. So your domain must include all the independent variables. You've got to make a statement about all of them. The range should have only how many? One.
One. Just the dependent. So domain's got to say everything about. all these variables, you've got to have a statement about that, the range would include just your one dependent variable.
So if hands feel okay with that one. I know I'm talking it over a lot, but honestly, when you guys first see functions, you're in a class mixed with a lot of other people. who aren't going on in math, and so sometimes it's not taught very well. I know that I kind of skim through it myself when I'm teaching to people who aren't going into calculus, but you need to be familiar with it, which is why we're taking a little bit of time here, to make sure you actually understand what it is that we're doing.
Does that make sense? What we're going to do especially that when we get to graphing. Let's try one more just talking about domain and range, then we'll start graphing the domain. Then we'll graph the range, which is surface.
So as I mentioned, man, you get the same sort of problems with... with functions of two variables that you have with one variable. So we look for things like square root. We look for denominators.
We look for LN. We look for those things that have possibilities of not being defined. So when we look at this, first thing, everybody, how many independent variables do we have?
Two. How many independent variables do we have? How many dependent variables do we have? Four.
What would you call that dependent variable? C to the D. In what dimension would you graph the surface here? One, two, or three?
Three. In what dimension would you graph the domain? Two. Perfect. So, let's talk about the domain.
Which side? I'll go right side, give you guys a chance. Tell me something about what you know about this function. What do you know?
It can't be negative. What can't be negative? It can't be negative. Okay, the negative, the output can't be negative. I know that.
Tell me something about the domain. What about the inputs? What do you know about square roots? More specifically, what do you know about the inside of square roots? They can't be, so use that.
Man, use the one thing you know. What we know here is that the radicand, the inside of our radical here. For square roots, it has to be greater than zero.
Question, can it be equal to zero? Yes. What if it was on a denominator? Okay, so use that. So this can be equal to zero since I don't have this on the bottom of the fraction, we're fine.
And now if you're okay with that one. Now, all you got to do, I promise, this is it. All you got to do, solve it for x and y.
Solve it so that you have some expression. You're literally done. Okay, this is it.
All we got to do is make sure that we have x and y. Okay, well, let's add these. Done.
Practically done. Practically done. Head on if you're okay with how to get that. So what I know is that any ordered pair, any two numbers, when I square them and add them, they have to be less than 4. There's not a better way to say that.
You can't say that a better way. Putting this in terms of y less than equal, well we have minuses and pluses there, but also we have this. Can you ever get this to be a negative?
No. So we also know this is greater than or equal to 0. x squared plus y squared is always positive. You cannot add two numbers after squaring them and have them be negative. That's what our domain is.
So our domain says, okay, I have a set of ordered pairs where my coordinates are x and y, and I need these ordered pairs to do this. I need them to be bigger than zero. That's not right.
But I also need them to be less than four. So would it be incorrect if you forgot to put the zero in there? No, no. I think that's how I first had it. I'm just going to use this for the next step.
So if you have this, that's true. That's correct too. I'm just going to use that zero thing for the next part.
So that's a good question. Any other questions? Do you guys feel okay with how to get that?
Now some of you guys are looking at me like, okay, let's go through it one more time. Do you understand that square roots can't have negatives inside of them? That's what that says. Then just solve for some expression that relates x and y to some number. Here it was pretty easy because we could solve for a variable.
Here it's not that easy because when I start solving for variables like that, first I lose the picture of the circle cylinder that we're going to get. So it's really nice to keep them in terms of things that we know. We can actually graph in, hey, 2D.
What dimension would it take to graph this? 2D. It would take two independent variables, 2D.
You can graph that in 2D. That's not hard to graph in two dimensions. It's a circle, radius of four, and everything included in that.
That's all nasty. When we start trying to solve for y, we start losing that. You start getting pluses and minuses because you get the square root in there.
Do you guys see what I'm talking about? That's nasty. So we just relate these variables to your number.
And that's it. Now, let's talk about the range. Are you better at this? Are you okay with that?
Let's talk about the range. What variable would you use for the range? This takes some thought. You've got to really think. What can you get out of this?
What can you get out of this thing? What's the maximum? What's the maximum this could be? Given this scenario, what's the maximum this could be? Well, the maximum this could be is because this...
has to be positive. Verify that. X squared plus Y squared has got to be greater than or equal to zero.
You can't add two numbers together to have it squared and get a negative. It's not possible. So the largest this can be and smallest this can be.
is going to be based on the fact that the largest this can be is 4, but the smallest that can be is 0. What that means is that if we add these two together and get 4, if you don't see it, I'll do this. It might be easier for some of you to see, factor the negative. If we add these together and get 4 here, what's the smallest this could possibly be?
Zero. So our minimum for our range is zero. What's the maximum that this could be?
Notice. Notice, the maximum this could be, the smallest this can get is 0. So if the smallest this can get is 0, what's the maximum this range could be? This L could be. Not 4. What is it?
2. Did you follow that? Most of you did. Some of you didn't.
I can only let this get, and this is why I wanted to put the 0. This is why. If you let that happen, check it out. This number has to be between 0 and 4. You get it? Which means that if I plug in 0, I'm getting out square root of 4, 2. If I plug in 4, I'm getting out 0. So the range here, the range of outputs is 0 to 2, not 0 to 4, square root of 4. So many guys are just writing down stuff and you have questions. What questions?
Come on. I know there's got to be something out there. Do you understand why it's 2 and not 4? Do you understand why it's not more than 4?
Or more than 2? Do you understand that? What's the lowest? Did you guys get this?
That factor in the negative? The lowest this can... b is 0. It can't get smaller than that.
4 minus 0 is 4. Square root of 4 is 2. The biggest this can be is 4. If it's more than 4, we get something that's undivided. We get imaginary numbers. We don't want that to happen.
So the most that can be is 4. 4 minus 4 is 0. We're somewhere between 0 and 2. That's it. That's all we're doing. That's all we're about. For real, show of hands, feel okay with the idea. We're going to move on.
We're going to start graphing these things. I'm going to start really slow with graphing regular old domain, like from one variable functions. We'll do one of those, and then we'll start stepping it up a little bit. We'll end up getting to how you graph domain. with three independent variables.
By the way, don't block the section, even though it's boring and you don't like graphing, because graphing the domain here lets you solve double integrals and triple integrals later. It's important for you. If you've had this class before, I know there's some of you, you know that what I'm saying is true because you go, oh, how do you find the region of integration?
You graph it. You go, oh. Crap.
Shoot. I forgot crap. This is going to be fun next semester for you.
Just kidding. Oops, did that come out? See you next year.
Just remember this. To graph the domain, what domain? Okay, so if you have one independent variable, how many dimensions?
For the domain, two independent variables, you need a... an axis for each independent variable, a domain, or sorry, a dimension for each independent variable. So to graph domain...
To grab domain, you have to have an axis for each independent variable. This will work all the time for your domain. Okay, quickly, everybody quickly, how many independent variables do we have? What is that independent variable?
You need one axis. You need the x-axis. That's enough to graph the domain.
So for our domain, you go, okay, what do you mean? How can I graph that? Look, here's the x-axis.
If you let this, you know this, x has to be greater than 0, correct? It can't be equal to 0, but the square root is going to be greater than 0. It must be strictly greater than 0. Can you graph that on just the x-axis? Yeah, right? Here's 0. Here's this way. You have parentheses saying I can't include 0, and it's everything positive in that.
That's a simple number line. Oh my gosh, yeah, you're right. You can graph domain on just a number line if you have one independent variable.
That's literally all that we are doing. Tripp, I guess you're okay with that one. Now, that's basic domain.
You should have learned it a long time ago, but we run with that. We do the same thing here. So let's try this one.
Everyone, right now, everyone, right now, how many independent variables do you have? That was supposed to come out. How many independent variables do you have? That sounds like I'm drunk slurring words up here or something.
My bad. How many do you have? Two.
Okay. What's one? What one specifically do you have?
So we're going to need two axes. What axes? To graph the domain with two independent variables, you're going to have two axes. So first thing we're going to do, just like we stated domain here, we're going to state the domain here.
Same stuff we did. We're just now going one more step. We're just going to be graphing this now.
So can we figure out what needs to happen with this thing? Well, it has no square roots, but it's got denominators. What do you know about denominators?
Can you do it? So I know x squared minus y squared cannot equal 0. For sure, yeah, it's a denominator. Can you solve this for one of the variables?
How do you know what to do when? Well, if you have 0, there's not a constant, then it's not giving you a picture that's very nice. It's not giving you a hyperbola. It's not giving you a circle.
It's not giving you an ellipse. It's giving you, oh, I could probably just solve this for y if I wanted to. If I solve this for y, I get y squared. Equal x squared.
Does that make sense? Let's go a step further. Can you literally solve it for y? You see the idea of trying to get something we can actually graph that you recognize that you can graph it.
You should recognize that you can graph that as a circle. Actually, it's a. It's a disk.
It's the inside of a circle. Do you see it? It's a whole bunch of circles put together. It's the inside of a circle from the origin out to a radius of 2. That's all we're doing.
Well, let's figure this one out, because y squared equals x squared doesn't look so hot to me, but if I take a square root on both sides, And you need to know this, when you take a square root, it's not on your paper, but then you put it on your paper, what must you have all the time? So y cannot equal plus or minus x. Use that. I want a set of points, x, y, such that y can't equal x and y cannot equal negative x. Here's the question to you.
Don't overthink it. Don't overthink it. Can you graph y equals x?
Quickly, what's the y-intercept? What's the slope? Can you graph y equals negative x? What's the intercept? What's the slope?
They're two diagonal lines. Look at that. That's all there is to it. So since we have two independent variables, you need two axes. You just need the x and the y axis.
It's not that hard. You just need whatever independent variable you got. Put an axis there. We need the x and the y.
If we grab y equals x. Y equals negative x. Now, you do have to understand what we're doing, okay?
So if you didn't pick it up the first time, here's what we're doing. Here's what we're doing. We're saying this. Can I draw it and then look up here so I know you're done so I can talk at you, okay? Go ahead.
Make sure you get it down. I don't want to talk over your writing because then you're going to miss two things. Here's the deal. What I'm saying here, what we're saying, what this is saying is I can take any point, any point on the plane, any ordered pair.
Ordered pairs are on the plane. They're on the x, y. That's what we're talking about. Any point on the plane, provided these things don't happen, provided I can't have y equal x. Do you understand what I'm talking about?
It just can't be on that line. Provided y can't equal negative x. I can have anything but that. So what we're talking about here for the domain is literally the entire plane, literally the entire floor, the entire xy plane besides these two diagonal lines. So we're going to shade everything but that.
Five second recap if you will. Every single time you're graphing domain you need the number of axes equal to and the same axes as your number of independent variables. One x, one x axis.
X and y, xy axis. Xyz, xyz axis. In order to graph it, we solve domain like normal, man. Just the problems that are slapping the face. Denometers can't be zero.
Great. Solve for it. Graph the lines. Graph the circles.
Graph whatever you have. And then shade the pieces that actually work here. So this says any point works unless you're on the line.
That's why they're dotted. It says the dotted lines, you cannot be there. Any other point, fine. So I guess you're okay with that.
For real. Yes, no? You guys over here? You guys over here?
We've got one more to do with these two independent variables, and we're going to step it up a little bit. Did you guys take notes when I say test question? Do you take notes of that?
This one might, you know. That looks nasty, but man, don't overthink it. It's just talking about domain. You know what has to happen with your functions. You are familiar enough with these things.
They're not going to get any crazy new functions that happen. There's just some certain rules we can't... We can't let denominators equal zero. We can't put negatives inside of even-powered radicals.
We can't go negative with ln. We can't even go to zero with ln. We can't do those things. If you mind the rules, just write it.
out, you can be fine. Let's be real careful about this. So first thing, what dimension would you need to graph this surface? What dimension for the whole surface?
What is it? Three. How about the domain? What dimension?
Is it pretty clear right now just looking at it? Oh, two variables. Domain's got to be graphed on an xy plane. It's the floor. All right?
You can put it on the up here, but it's the floor. So we're going to graph this on an xy plane. I'll just put it right here. Our goal is to get a good picture for the domain there. Now let's start working with it.
What do you know? What do you know about ln? I kind of ruined the suspense, if you will, but what do you know about ln? It can't be greater than zero.
Strictly. The argument of the natural algorithm has got to be greater than zero. The ln looks like this for you guys.
So there's an asymptote right along the y-axis. You can't even get to x equals zero. So we're going to go the inside, the argument can't equal, sorry. It's got to be strictly greater than zero. Don't put not equal, all right?
You've got to show me some inequalities if these things involve inequalities. So we don't have to be, oh, yeah, can't equal zero. Yeah, you're right, but it's also got to be, can't equal negatives.
So if you just put doesn't equal zero, that implies you can plug in negatives. Does that make sense? You've got to show inequalities if they apply. Could you solve that? Solve it for a variable that allows you to graph it.
So if we have y minus x equals zero, that's not so much fun to graph. I don't know how to graph it very well. I can't do cover-up method or anything, as you learn in Math C, Section 9.4, I believe.
Math C if you're not good at graphing any qualities. Watch Chapter 9.3, 9.4. I don't know, Math C, Intermediate Algebra, Chapter 9, shows you how to graph inequalities pretty easily, pretty well.
So you can watch it, brush up on it. We're going to graph that in just a second, but that's only part one. That's this little one here.
Are there any other pieces that allow me to have some possible undefinedness? Tell me left-siders, just you guys over there, what has to happen with that particular radican? What do you know?
Is that it? Is that it? Come on, don't look at me with those blank eyes and tell me. I don't know.
There's dead eyes out there. Goodness. I'll get you some more candy. Not right now.
It made a crinkly noise on the last video. Pissed me off. You're going to get it during your break.
It's gross, all right? I don't want to hear you chew. It's nasty. No one wants to hear someone chew. You ever hear someone...
No. Hearing someone chew and swallow is probably the worst sound. Not as bad as a fart in church, but it's close. All right.
You can't move. You're sitting there the whole time. Just a little kid.
Whatever. This is not it, because I know, yeah, I know it can't equal zero, but also we know that square roots have to have positive radicands. That's another inequality. Now, can you solve for that inequality?
Could you solve that for why? Do it. If you put just not equals all the time, you're going to not equal the right answer. I don't want that.
I want you to get the actual inequalities involved with these domains. In this case, instead of subtracting, subtracting, I'd probably just add and flip. So we get y is less than x plus 1. I don't know if you're okay with that. Now, can you graph them?
Recall that graph inequality says graph a line and then shade a half plane above or below. If you have it y equals, it's really easy. Because if we graph this, if we graph this thing, by the way, should I graph this with a solid line or a dotted line?
What do you think? Dotted. Why?
Because it's a line. If we had equals, we would have a solid line. This is the graph y equals x. That's y equals. X right there, or not equal to X.
That's what that is. Now, what this says is I want to shade. If you have this Y equals or Y and then inequality all the time, you can always just shade above or below the line. It's pretty nice.
So this says I want to shade above that line. I want to shade the half plane. Now, don't do it now.
Don't do it now. Just keep it in your insookabesa. Okay, just keep it right in your head that you're going to shade above that line. With me? Let's graph the other one.
This one. This one has a y-intercept of 1. It's got a slope of 1, same slope, they're parallel. Explain to me why I'm graphing this with a dotted line again. Again, if I have this y and then inequality, it'll tell you whether there's shade above or below. This one says shade y less than that, shade less than that, below that.
Does that make sense to you? Are you sure? Can you graph just y equals x? That's all we're doing. Can you graph y equals x plus 1?
Then put them together. The inequalities say you're going to shade a half plane. But we also want both of these things to happen. We've got to satisfy both of them.
We need this to happen, and we need this to happen. We need the overlap. So if I'm doing this, this is graph the line, got it, dotted line, no equals.
Graph the line, got it, dotted line, no equals. But I need these half planes. This is above this one, but also below. below this one. Can you see the region?
It's a strip of points. That's all that's happening. So this, our domain, is every ordered pair in that strip, in that segment.
And that's the best way we can graph our domain right there. This really, when we write our domain out and say, okay, well. The domain is every ordered pair such that y is greater than x and y is less than x plus 1. That's great.
That's technically the domain, but it doesn't give us a good picture of what's going on. This is a good picture of what's going on. It shows you, literally, it shows you every single point of what we can. Now, what are we doing? Is this the...
graph of that thing? No. But this is really interesting.
This is the graph of the points you can, I hope that you're paying attention because this, I'm trying to teach you like four things at once, all right? What the graphs look like, how we're going to do double integrals. later on, what we're going to do about all this stuff.
If that's the domain, that's the strip of points I can plug in, okay? Now picture this axis flat, the X and the Y. Can you picture it? That's the strip of points doing this.
That surface is just going, this is just the domain, just the crap we can plug in. The surface is going to be that, whatever it is, whatever shape it is, but it's only going to be above that region. Do you guys get that?
It's just above the region that I just drew. It's going to be this weird circle. surface strip that's going through space. It's kind of cool. That's what's going on.
That's what I need you to understand. Why is it important to graph domain? Because in doing so, we can better understand how the surface is behaving, where it's going to be over, and if we know where it's going to be over, integration becomes easier. That's one of the major points here.
You understand the concept. When we come back from a break, we're going to talk about how to graph domain with three independent variables. We'll graph just a couple of them.
Then we'll talk about of these multi-variable functions more than two. More than two independent variables. Verify.
More than two. How many? Three.
We can still do the same thing. We still look for exactly the same problems. It's just this. So, we're going to do How many dimensions do we need to graph this surface, the whole thing?
Four. Yeah, three independent, that's one dependent, four variables, four dimensions, or one more than independent. To graph the domain, we'll need three. So we'll need.
We'll talk about what it is. I'm not going to specifically graph it, but the shape will be readily identifiable. What do you know about that function?
Come on, quickly. What do you know about it? Or at least the domain.
What do you know about the domain? What do you know? It can't be graded.
So the inside of this square root, and if it helps you right away to do this, to go, oh, well, factor the negative. You can go ahead and do that. Question, should I remove that equals?
We did sometimes before. Should I remove that? No. No, it's not a denominator.
So all I need is for this instance. That part to be positive. Show of hands if you're okay with that one. Now, let's group our X, Y's, and Z's on one side, and this is typically how we do it for 3D. Do you remember 3D graphing at all?
Do you remember section 11.6? Do you remember surfaces and all this stuff? Try to get one of those.
So if we group all of our X's and Y's and Z's, basically add this. We get x squared plus y squared plus z squared is less than or equal to 9. So you're basically okay with that. You've literally just found your domain.
That's all there is to it. All I know is that these three, the combination of the squares of these, the sum of the squares of this order triple has to be greater than or equal to 9. I also know this. If you take three numbers, any order triple, and you square them and add them, will it be positive? The least it could be would be 0. So it's greater than or equal to 0. Now, please don't misconstrue this.
I had a question during the break. It was a good question. Well, wait a minute.
Can't you plug in a negative for x? Yes, you can. And you can plug in a negative for y.
And see, I'm not saying anything that this is individually. Each of those is greater than zero. That's not what I'm saying.
What I'm saying is that when you take a point and you square those values and add them, the combination of the sum of the squares of that point has to be greater than zero and less than nine. That's what I'm saying. Does that make sense to you?
That's what I'm saying, not that each individual one, yeah, you can plug in negatives here, no problem. But altogether, when you plug it in and square them, it won't be negative, and it can't be more than 9. So that right there, that's our domain. It's an ordered triple such that, and you don't specifically need that. I'm just trying to tell you what it is.
It's such that ordered triple, I'm sorry, such that x squared plus y squared plus a squared, it's got to be less than or equal to 9. So if Hansfield came with that one. Now again, what dimensions are you going to take? going to take to graph this function.
So it's a 4D graph. It's a 3D domain now specifically. Come on, put this together. Have you seen x squared plus y squared plus z squared equals 9 before? Yes.
What is it? A sphere. Good, it's a sphere. It's technically an ellipsoid, but it's the same in all directions, so we call that a sphere.
So this is a ball with a radius of what? And it's the? Inside of that, so it's a solid ball.
It's every order triple inside a ball radius 3. Can you have some negative x, y, and z values? Yes. There's some points in that ball that are all negative. But when you square them, they become positive. And you add them, it's still positive.
That's why it's greater than zero. So don't confuse the domain of the function with the domain of each individual value. That is not what we're doing here.
Does that make sense? So this would be a. The inside, inside, it's all the points less than that, so between 0 and 9. Inside of a sphere with radius 3 centered at the origin.
That's what this domain looks like. Hey, hey, hey, get it straight here. Is this the picture of the function?
Is a sphere the picture of the function? No. No, the sphere, inside the sphere is the picture of the domain of the function. It's given us a representation of all the order of triples I can even plug in. So it's just what I can plug in there.
Make sense? Zero, zero, zero. Sure.
Sure. 010. Sure. Negative 301. Yeah, sure.
As long as it's in that bubble, that's what I can plug in. That's what we're saying. Should have answered okay with that, for real. Okay, let's do only one more.
This one. I'm going to give you 30 seconds right now. I want you to write out two things you know about the domain.
They should be pretty obvious at this point. Write out two things you know about the domain. Right, senators, you got one of them. What's one thing you know about this function, the domain of this function?
Give me something. Okay, left-siders, tell me some... Come on, guys. I can't... Z can't be...
Okay. So, Z minus 3 can't be 0. Therefore, Z cannot be equal to 3. Left-siders, you got another one. Is there something else up here that's just like, Boom! Domain issue!
Come on. Greater than or greater than or equal to zero. If I add this, we get x squared plus y squared is less than or equal to four. It's, I think we've seen that or something so similar to that in a previous example.
Question. How many independent variables do you have? Come on people, how many independent variables do you have?
Three. What would this graph, what dimension would it take to graph what's called a surface? To graph the surface, what dimension? How about the domain? What for the domain?
Okay, so we need three dimensions for the domain. Also, any time you have a statement about the domain... You have to have every single independent variable stated in there somewhere. So, with this domain, we have ordered triples, x, y, z, such that these two things just squash them together.
I know that x squared plus y squared has to be less than or equal to 4, and I also know that z can't be equal to 3. That's fine. That's all that we're saying here. So take all the exceptions from this function, all the domain issues, squash them together. As long as you said something about every one of those independent variables, that's what we're talking about.
And I don't feel okay with that one. Now. Now.
I know I'm being redundant, but I want it to stick. What dimension do I need to graph this domain? What dimension? The domain.
What dimension for the domain? Three. I need x, y, z. I need three dimensions.
Now, here's my question to you. If this is in three dimensions, think three dimensionally. What is x squared plus y squared equal to 4 in 3D? Don't tell me a circle.
What is it? Cylinder. Cylinder.
Along the, oh my gosh, what's the radius of? Cylinder. Perfect.
This is the inside of a cylinder. Less than that, inside of a cylinder along the z axis. So basically it looks like this. Sorry about the sloppiness of my cylinder. It's nasty.
Did you guys get the idea of the cylinder though? So it's this cylinder, the inside of the cylinder going through the z-axis, going along the z-axis. But wait a minute. Now think three-dimensionally still.
What's z equal to 3? What's that look like? Z equals 0 is the xy plane.
So what's z equal to 3? The xy plane at 3. What that would do when it intersects this, it's cutting out a disc at z equals 3. Does that make sense? So this right here in 3D, man, that's inside of a cylinder. With a radius of 2 going along the z-axis, that's a plane.
So I have this exception, z cannot equal 3. So if this is at 3, it's missing this disc. So it's all the points inside that cylinder. Notice it's 3D.
It's order triples, man. It's inside that cylinder except for that disk. That's the idea here. Is it hard? Not if you understand just basic domain and how to graph in 3D.
It's not super bad. You know what planes are. You know what cylinders are.
You just need to think 3D for 3D domain, for three independent variables. That's graphing domain. So you can just be okay with that idea.
Now. Let's move on to actually graphing functions of two variables. When I say two variables, I mean two independent variables. We're not going to go on to graphing functions of three independent variables.
Why wouldn't we do that? Because we don't have 4D. I don't know how to graph 4D by hand. Can't do that, okay, because we have a 2D system here, and the most I can graph is 3D.
So here's how. So the how to graph functions of two variables. How to graph. Number one, the first thing I want you to do, please get rid of the function notation.
If we're graphing only two independent variables, they're always going to be x, y. So if we have a function in terms of x and y, my two independent variables, what would I set that function equal to? Yeah, let's choose that. Number two.
Number two, and this is a big word, try. Try to manipulate this until you get a surface that you know, verify this. If I have two independent variables, that's a point, and when I plug that point in, I get a height.
So we will have a surface out of these things. You recognize what I'm talking about. It's 3D. Two independent, 3D.
Try to manipulate. Until you get a surface that you can identify. Try to get a surface you know. Like spheres, like, I don't know, hyperbolas, like ellipsoids, like all that stuff.
The stuff that we, goodness, we had it in 11.6. That's why we did it, because we're going to get a lot of the same stuff right here. So try to get a surface you know. If you can't.
Here's your computer. Is it pretty much done? Because we can't do it by hand. It's too hard. I'm going to show you that hopefully at the very end.
There are a little extra here. We'll show you some on the computer. They're pretty cool looking. We'll talk a lot about it. So, some quick examples.
Please notice these are going to be very quick. I'm not going to. a whole lot of time on them, you're like, this is the first time we've done it.
No, it's not. No, it's not. This is like the third time we've graphed surfaces, the third time.
And the stuff we're going to get here are things you're going to identify. It's the only ones that we know how to graph by hand. So we're going to do that. I just want to give you the technique of it.
So we'll walk through, but it's going to be fast. Well, Leonard fast. That means. Ladies and gentlemen, function, yes, no. How many independent variables, how many total variables, this graph is going to be in 2D or 3D?
The domain would be in 2D. Perfect. I'm not asking about domain, but domain's all real numbers, well, all real ordered pairs.
Can you see it? What I do want, first thing, set that equal to z. So instead of f of x, I don't want that.
I want z. It says the same stuff. But it yields something that you can identify.
So, for instance, you go, okay, do you know? Man, I'm hoping you do. Do you know what that is? If you have three variables, all power ones, not power twos, nothing fancy, what is it? What if I did this?
Got all my variables on one side, which is pretty much how almost all of our surfaces look. What if I did that? Now you can't miss this. Come on, you can't miss this. What is that?
That's a plane. Could you tell me the normal to this plane? What's the normal?
Perfect. Could you graph it? I haven't showed you this yet because we haven't got to graphing, but I'm going to show you the easiest way to graph planes ever. Check it out. We can always graph planes provided there is an actual number, not zero, by doing this.
By doing this. Finding out the x, the y, and the z intercepts. How you do that, check it out just so you get the logic. If I want to figure out where this thing crossed the x-axis, every y and z coordinate on the x-axis is 0, correct?
So if I plug in 0, 0, or just cover it up, I'm going to get the x-intercept. So cover up all the x-intercepts. All the variables but x, you get 2x equals 6, x equals 3. Did you catch that? I know for a fact that this plane will intersect the x-axis at x equals 3. Kind of neat, right? Just kept it up, man. 0, 0 for y and z means you're on the x.
Solve for x. We get x equals 3. If I say I want to be on the y-axis, then x and z have to both be 0. If x and z are both 0, y is 0. Would equal negative 2. Did you catch the negative 2? Cover up method.
Cover it up. Negative 2. Now you do have to know where that is. Last one.
Can you tell me the z intercept everyone in class right now? What's a z intercept? 6. Yep, cover it up, 6. If we're on the z axis, x and y are both 0. Cover it up, z equals 6. So this plane, now we're going to get a triangle, but that triangle if we extend it would represent our plane.
So when we graph our triangle here, That triangle is representing our plane that's crossing all three of those points. That's the idea. It's just a plane.
That's graphing surfaces. That's graphing multivariable functions right there. At least a very simple one.
So, I think that's really okay with that one. Okay, just two more. Maybe two more.
We'll actually graph one of them. We'll just talk about another one. Okay, ladies and gentlemen, I said we're moving quick. We're going to move quick.
How many independent variables do we got? Two. If you graph the domain, are you going to have any issues with it at all?
No. Nope. Okay, so we have all real numbers for that one, not all real ordered pairs for that one.
What dimension do we need to graph the surface? Two. The surface, not the domain, the surface. Three. Three, just like our Calc 3 symbol.
What? Calc 3. Awesome. Do your first step, please.
Now, if we can graph it by hand, it's going to look like something we've done before from 11.6. It's going to look like something like that. Do you see all three variables? Go revert back. Hey, three variables means we're graphing in.
3D. When we include our dependent, that's true. That's exactly what you learn in 11-6.
Excuse me a little bit, slip up. 11-6, that's what we learn. Three variables, 3D. So, get this into a form that looks right to you. How many squares do you have?
How many power 1's do you have? So that's got to be one of two things. It's got to be a, do you remember at all?
It's got to be a paraboloid or hyperbolic paraboloid. It's going to be some sort of paraboloid. How do we tell? Well, you get in the correct form. What we do is we get our squares on one side.
We get our single power variable on the other. And then we go, hey, you know what? In order for these things to be hyperbolic paraboloids, that would have to be a minus, and it's not. This thing is just a typical paraboloid.
That's what this is. Along what axis? This is the end.
Opening towards the positive z or negative z. You should know that. Opening towards the negative. So opening downward towards the negative z. Opening towards the negative z.
It's going to be shifted. Do you see the shift? Get your shift straight, alright?
Where's it going to be shifted? Is it shifted up or down? It's only opposite of what you think if it's in parentheses. Is it shifted up or down? Up.
It's shifted up nine. Do you remember how to find a trace? How you find a trace is if you plug in for this variable zero.
What trace is that? Just plug it in, it's covered up. What trace is that? Come on people, shout it out to me, I can't hear you. Radius of?
Three. So at the xy plane, which is z equals zero, that is the xy plane, we got this. Now I'm going to do a lot more space than that. Let's make that pretty.
Ah, crap. Stupid small circles. We get this hyperboloid, it does this.
Sorry, paraboloid, it does this. It starts... At that point, it's opening toward the negative z, shift it up, and it's got this trace of a circle with a radius of 3 right on the xy plane.
That's what that surface looks like. Have you done it before? Yes. Yeah, we're just calling multivariable functions now. It's the same stuff, same stuff.
Let's try one more. We'll talk about level curves. We'll call it good. You know what, do you guys want to try one on your own?
Do you want to see if you can do it? I think you can. Why don't you try it?
I didn't let you answer that. Do you want to try one on your own? Sure.
If I answer that question, it's always yes anyway. So, yeah, let's try it. It'll be fine.
No matter, not square roots. I hate you. That's alright, some of you hated me anyway.
It's okay. Maybe more. Before you get going, before you get going, if I asked you to find the domain, could you do it?
I hope so. You have this thing's got to be positive. It's got to be positive.
It looks like it's going to be a circle. That's what it's going to look like. So for that thing, though, I want you to verify the domain would be in 2D. The surface is in 3D. Find me the surface.
You don't need to graph it. I'm not going to graph this thing. I just want you to identify it.
That's all I'm asking for right now. So work it out until you can identify it. I'll give you about a minute. Hey, what's the first thing you did, right, Sanders?
What would you do? That doesn't look so hot to me because right now I got a square root up there. I know that none of my surfaces look like square roots.
Okay, all the things I know have like lots of squares in places. Left-siders, what would you do next? I wouldn't do the square root first.
The 2. I would get rid of the 2. So 2z... Looks like that. Middle people, how about you? Now what would you do? Being careful that when you square both sides, you're literally squaring both sides.
When you square both sides, it's including that 2. It's starting to look better. If I gave that to you on a test, last test, you'd probably get it right. Because what are you going to do? Oh, your variables. Yeah.
Man, we're close. Everyone, what's the last thing you'd probably do? Standard form.
So even though it might not look like something you're familiar with, it is. This is all squares, all added. What is that?
That's an ellipsoid. That's an ellipsoid. That's exactly right.
And you can find your x-intercepts, your y-intercepts, your z-intercepts, draw your football shape. That's practically all that we're doing. One thing, though.
This domain, notice this, that inside of the square root, you cannot get negative. Does that make sense? That means that if you're taking square roots of all positive numbers, you're getting positives out.
That means that our z cannot be negative. This is the upper, upper part of that football, upper part of that. You guys clear what I'm talking about?
Otherwise, we'd fail our domain. So that's the idea. So fans, if you understand the concept of graphing these, you should look familiar. We'll talk about level curves right now.
They're honestly probably the easiest thing that we do for graphing. They don't sound easy. They sound very confusing.
But they're honestly probably the easiest because they're all going to be shapes that you know already, all of them. So here's what level curves do. You see, a lot of times, it's really nice to know what happens. Are you listening? Are you guys writing?
Are you writing? Don't write. Okay, just listen for now. It's really nice to know what happens with our surface when we consider what it's doing at different levels along the axis of our dependent, sorry, dependent variable. So, like the z-axis.
Like, hey, what's the graph look like if I climb the mountain and I say I'm at 100 feet and I travel around it, what's it actually look like? When I go up 100 feet higher, what's it actually look like? We're creating this topographical map.
Now, that's what level curves are. If I take a series of level curves and I squash them, we get what's called a contour plot. So, we're going to, I'm going to, that's the overall summary of what we're doing.
What a level curve is, it says the shape, this is the shape that we get, this outline. It doesn't have to be an outline. It can be in 3D if we're taking a level curve in a four-dimensional figure.
But it's this outline, the shape that we get when a plane. Intersects our surface at different levels along our dependent variable, which we consider to be Z for 3D. Does that make sense to you?
So it's that shape, it's that trace, that outline at different levels. Obviously, if you've got a surface, right, you've got this thing that's changing in 3D, and I say, cut it, cut it right here, it's going to create some sort of outline. It's going to be just a trace on a plane.
I go, now go higher, it's going to be a different shape. And let's look at a cylinder. It's going to be the same. All the time. And so we use that.
Remember the cylinders are all the same shape. But if we have surfaces, like imagine this. Imagine just a ball. Imagine a ball. Are you imagining?
You're not imagining hard enough. Imagine harder. Imagine this ball and cut it with a plane. Which shape is it going to make on the plane? It's conic section.
That's exactly what we have here. So we'd have a circle. And then if we went higher, we'd have a bigger circle or smaller circle? Smaller.
Smaller. And then smaller, smaller, until we get a dot. That's what these level curves are.
Now, if we take and we let the level curves be set a fixed amount apart, like one unit or ten units, and we take them all, so all the level curves that are equidistant from each other, and we squash them onto a plane. So with a ball, you'd have a big circle, then a little circle, so we get to a dot, right? And you took those equidistant from each other.
We looked from the top, or we squashed them down. That's what's called a contour plot. So a series of level curves.
curves projected on, this is very mathy, it's a very simple idea, think topographical map, but a series of level curves projected on an xy plane where the distance between the heights of the level curves is equidistant, that creates a contouring plot. So I'm going to say it very simply, a map of level curves is a contouring plot. Probably the best way that I can describe it in less than 10 words.
It was less than 10. Have you ever been hiking? You ever been hiking? You go up to a mountain, right? Those mountains, hopefully you hike in mountains. It's not really fun to hike just down roads.
It sucks. It's called walking. I should try. I don't know.
But you know that some paths are steeper than others. You know that. So if you went up to a certain level, 100 feet in elevation, you went around this mountain, you're going to trace a certain trace. You go up a little higher, 100 feet higher, you're going to trace a different trace. We can often get these things that look like...
Let's say we went up 10 feet and that was the look. And we go up another 10 feet. It doesn't have to follow this exactly, but let's say we did this. And we go, okay, we went up another 10 feet and then we go again.
And we go again. Again, again, that right there would give you a topographical map. That is a contour map provided these distances between, so if you, if you, this is from the bird's eye, okay?
If you went up like 100 feet and 100 feet and 100 feet. And you're getting this map of the mountain, then a 100-feet map of the mountain, a 100-feet map of the mountain. And the closer these things are together, the steeper your surface is.
The further apart, the less steep it is. So the best way to get from here to here quickest would be from here to here. That's the quickest way.
Right at the steepest side of the mountain. That'd be the fastest. The easiest way would be to go slowly. Half of...
Gradual climb. We're going to use that steepest ascent idea when we start talking about gradients, when we get there in a few chapters. But this idea of a contour plot is really important for us.
Do you understand how to get the contour plot from a level curve? So each one of these would be a particular level curve at a certain value. If they have equal values and we smash them all together and look from the top, that's called a contour plot.
So eventually you're okay with that one. Now how you find them. So just like a topographical map, how you find them. Set your function equal to k. Set your function equal to k.
What I mean by that, k's got to be a value. It's got to be a value along your, please focus. If f, if the function f represents our dependent variable and you set that equal to a number, you're basically restricting the axis.
Hey, I want you to figure out the shape of the curve at z. Let's call it z for a second. Z equals 5. Z equals 16. Z equals negative 7. That's what the k is. It's a constant that gives you a level.
It's a level along the dependent axis. That is what this level curve is. Does that make sense? Don't just nod your head.
I don't want, yes, shut up. I want, yes, I understand. Do you understand the idea? If you restrict the dependent variable to a number, you're saying, I want it at that plane.
I want it just right here, where that plane intersects my surface is exactly what we were talking about. That's a level curve. Let me show you just a few of them.
We'll talk through most of them. They're not hard. They're fast. You just have to understand the idea. You'll get it.
You will get it. You guys are pretty bright, all right? So I know that you've had it before. I know that you can do it. It's just I've seen a lot of people get confused with it.
Like, level curvers are crazy hard. They're not crazy hard. You just have to get comfortable that K is a number. It's just a number.
By the way, if you graphed that right now, would it be graphed in 2D or 3D? 3D. Perfect.
You'd have two independent variables, 3D. That right there is just like the last graph that I erased. That's an upper ellipsoid.
Can you picture it? You square both sides, add this stuff over, you have a constant up. That's what it's giving you.
That's it, upper ellipsoid. Now, to find the level curves, if it's an upper ellipsoid, ladies and gentlemen, what pictures should you get as you're traveling up the football? Either circles or ellipses.
That's what you should get. Does that make sense? That's all you should. Now, let's see that. If I want to find level curves, the idea is set k equal to this thing.
And solve it, oh, very similar to how you'd solve it with a z in there, except don't include the k where the z would. Get the k with the constants. Why? Because k is supposed to be just a number. So if I did that, okay, I'm going to square it.
K squared. 16 minus x squared minus y squared. I want to add these over here.
I want to get my variables on one side. But here's the whole point of doing this. Please, please listen.
If you restrict this to be an actual number, that k is a constant. It's not a variable any longer. Say I'm restricting this z.
You said 3d, right? I'm restricting the z-axis to a number, k. It could be 4. It could be 2, whatever it is.
I'm restricting it. It's going to intersect my ellipsoid with a plane at z equals that value, z equals k. Get that constant with the other constants.
So in other words, I have 16 minus k squared. Now, in order to just get a feel for what these shapes are, start plugging in some easy numbers, man. Start plugging in stuff like 0 for K.
Zero. Not stuff here. Zero for that. If I plug in 0 for K, what shape is that? Circle.
Circle. Circle, radius. Four.
Four. At k equals zero, that's z equals zero. That's what we're doing.
We're along the z. So at the xy plane, we have a circle radius of four. That's what our ellipsoid would look like anyway. Now go up to like. What k equals 1?
k equals 2? k equals 3? k equals 4?
k equals 5? k equals 6? Can I go, what can I go past? Can I do k equals 1?
Yes. I would get, once more, that'd be 15. It's still a circle, radius of the square root of 15. Does that make sense? It's slightly smaller. Then I go, how about 2? Okay, that'd be 4, so it'd be 12. Square root of 12, that's slightly smaller.
It's still a circle. It's climbing and growing smaller. What these things are, what's the highest I could go, by the way?
R. R. Because if I went to 4, oh, hey, it's shifted up 4. Sorry, it's got a z-intercept of 4. That's the top of our ellipsoid right there. It should be making sense. It should be cohesive to you. Is it cohesive?
Do you guys see it? You picture it. It's kind of cool, man. So what these things are are circles that are getting smaller as we're climbing higher.
That's what's happening. Circles. with a maximum radius of 4, getting progressively smaller as k gets larger.
I'll tell you what, I'm going to rip through some here real quick. I'm not going to ask for a whole lot of help. I just want to get a feel for how this stuff works. I don't really have time.
Time to go through this slow, but the idea is always the same. So equal to k, try to figure out what it's looking like at different values. Are you guys okay with that?
I'm going to be moving fast. Got to get ready. So next one.
Here you go. This has two independent variables. Domain would be graphed in 2D. Function itself is 3D. It's a 3D surface.
So if I wanted to figure out what the level curves are, how the contour plot would look, what's going to happen is I would set my function equal to a constant. Okay, well, I started thinking about it. I got my variables on one side already. What would it be if I had y squared minus x squared equals k?
Well, you know what? If that's a number, just plug it in numbers. Plug in like one.
Plug in like full, plug in some numbers here. See what it looks like. If I start plugging in numbers, I go wow, that looks a lot like hyperbolas. Do you guys see it? Hyperbolas along the y-axis or the x-axis.
If I divide it by k, it's even more clear. These are hyperbolas. They're either along the x or the y, depending on the value of k.
For positive k, they're along the y. For negative k, they're along the x. So, you can tell that there's going to be a major difference between these above the xy plane and below. k is along the z, guys. It's along the dependent variable.
So, below, the hyperbolas are along the x-axis. Above, they're along the z-axis. So, these are hyperbolas. Level curves have variables.
They're along x or y, depending on the value of k. That can't be fun. I don't know what that looks like. No idea. But I do know something.
The domain's in 2D, and I know that x plus y has got to be greater than 0. That's what I know. I know that this is a 3D graph. I also know that if I set... My dependent variable equal to a constant.
I'm going to get these things called level curves, where this curve intersects some planes at whatever these levels say. If I think that that's a number, I go, man, that's like 1 or 5 or whatever it is, that's not looking so good. But I can also do some stuff with it. If you're the type of person like me who just loves crossing stuff out, I love crossing stuff out. How's it going?
Y equals... Did I do that right? That's better. Negative x plus e to the k. This is going to be wacky here, but what are these?
Do you know what those are? So lines. What's the slope of all of that, those lines?
Negative 1. Those are lines, negative 1, with the y intercept of e to the whatever the value of k is. These are just a series of lines. We can do the same exact thing with functions of three independent variables.
So if I got, hey, f of x, y, z, and you go, okay, equals this. In order to graph that, what dimension do we have to have to graph that? Fourth dimension. Fourth dimension. We can graph the contour plus of, and that's what we typically.
When we get to these functions, man, three variables, how am I supposed to graph that? You're not by hand. But we can graph the contour plot because the contour plot is just reduced a domain, or sorry, reduced a degree.
That's kind of cool. So when we have a 3D graph, the contour plot says this. If you've got a 3D graph intersected with a plane, it's now 2D.
Take a 4D graph, intersect it with a surface, it's now 3D. Does that make sense? That's kind of cool.
Cool. Do the same thing. So set this equal to a constant. Set your dependent variable equal to a constant. Group it so that it looks like something you're used to.
All your variables, get those on one side. Get your constants. Verify, k is a constant.
Yes, no. Get your constants together. So if I do that, I'll have 2x plus 4y minus 3z equals 1 minus k. k minus 1. k minus 1. So subtract the 1 and you got it.
Now, question. If I plugged in like, I don't know, 0, 1, 2, 3, 4, any constant, because k is a constant, what am I getting? Planes.
I'm getting this series of planes. That's all I'm getting. Series of planes. What's the normal to every one of those planes? 2. That's right.
So keep in mind though, this series of planes, it's weird, but this is not the graph of our surface, it's the graph of the contour plots. It's how that surface in 4D is intersecting something at these levels in 3D. That's weird, but that's what we're getting here. We're getting a contour plot out of this. So these are planes with a normal.
Last one, we're going to call it good. And then what I'm going to do later for this section is I'm going to show you some stuff on the computers. I'll probably, hopefully, if we have time, go back and graph all these on the computer and show what those things actually look like. If you have your notes handy, that would be real nice. Best Scott, how many independent variables?
Three. This would be graphed in what dimension? Four.
That means the contour plot would be graphed in the same dimension as the domain. What's that? Three.
So there's going to be 3D contour plots. What's your first step in determining level curves which gives us contour plots? What would you do?
Same. Yeah. Right there, if you had section 11.6 down, you know what that is. Plug in a number for K, like any number in your head, like 1. What's it going to give you? It's going to give you what now?
One sheet. If K is 1, you've got one sheet hyperboloid, yes? What if K is 0?
You've got a cone. What if K is negative? Two sheet, you guys are great at this. That's fantastic.
That's exactly right. So there's three cases. If K is positive, you'd have like one, right? That's all square one negative.
That would be a one sheet hyperboloid. If k equals 0, well, if k is 0, we have all squares, one negative equals 0, no constant means cone. No constant, cone.
k is less than 0. If this is like negative 1, you'd have to divide everything by a negative. It would switch all of your signs. Do you guys see it?
And then we'd get this two negatives, one positive. We'd have a two-sharp algorithm. I'm going to ask you a question, and they're all along z. It's still, it's all along z, but that doesn't change. The shape that we would get would change.
I'm going to ask you a question. Do you feel like you understand how to do probably the two most important things? Graph domain of any function that I give you right now. Graph surfaces if I give you two independent...
Perfect. Okay, we're going to talk about ones that you can't graph by hand, hopefully a little bit later. Okay, we're going to talk about some cool pictures.
What I'm going to do is I'm going to give you the images for those last, like, five graphs that we found contour plots for, that we found level curves for on your notes. So I said, hey, these are... These curves are circles. Why?
These curves are lines. Why? These curves are hyperbolas. Why?
And I'll show you some ones that you cannot grab by hand. So, here we go. This right here, that was the...
that first graph, it created the top half of an ellipsoid. You guys see the ellipsoid at work? If we talked about this and say, well, what does the contour look like for ellipsoids? We found they were circles. There's the circles.
Look how the circles are getting smaller and smaller until we reach right up there to the top. If we do this and look at it like a bird's eye, that's what a contour plot does. It creates a topographical map. Do you see it? See?
See how we're climbing fast, fast, fast, fast, fast, slow, and then we reach the top. That's a contour block. Same thing happens here. This was that f of xy equals y squared minus x squared.
It created hyperbolas as level curves. Here are those hyperbolas. Do you see them?
Above the xy plane we have along the y, below we have along the x. If we do this, that right there is the contour plot. It gives you the mapping. Pretty neat.
Do you guys like that? Pretty neat. This was that one that we said, hey, here's an outline of x plus y. What we got were lines.
What in the world? How are we getting lines from this thing? Well, check this out.
If we do the... Contour, there's the lines. That's exactly what we're getting, a whole bunch of lines climbing that thing right up there. Do you guys see it? See, that's a contour plot.
If I do this, they're really, really, really, really, really steep, and then we level off right at the top. Oh, the function itself is really, really, really, really, really steep, and then it levels off. That's what they're showing you. They're showing you the contour of this surface, how it's climbing, how it's falling, how it's changing.
This one, this is a weird one. It's cosine of the, we didn't do it in class. I'm just giving you a cool picture.
Cosine of the square root of x squared plus y squared. Isn't that a cool looking, cool looking shape? I just like that.
It's really neat. If we do the contour, what shapes? Do you think you're going to get? Look at it.
If I cross this with a z equals k planes, like z equals 4, z equals 5, what am I going to get? Circles? Whole bunch of circles. Pretty darn cool though.
Tells you how it's climbing and how it's falling. That's a neat graph. Just circles.
It should be filled in right here. It's not because it's a computer and a computer's not perfect. This other one, square root of, well, I don't know.
I just made it up off my head. Pretty interesting little picture there. Let's see what we get.
We get these really neat shapes that go out and out and out and out and out. It's not anything that we have a cool name for, but that's the sort of stuff that we're talking about when we are talking about surfaces and then... level curves. Each one of these guys is a level curve. It's going around the surface at a plane.
So we're setting z equal to 5, z equal to 6, z equal to 7, and solving for it. It's going to be a curve around that surface. You put them together.
And smash them down to the xy plane and that's what a contour plot actually does. So fancy to understand that concept. Okay, we're going to call it good.