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0:00
- Time for us through some practice problems involving functions of multi-variables. And we're just starting out very simple. Focusing a little bit on things like, what's our domain? What do we mean when we talk about level curves and level surfaces? Because we're kind of easing our way into our new topic of multi-variable calculus. Of course that doesn't mean we can take it easy. We gotta make sure we do the practice. The more we practice, the better we get. And we wanna be so wonderful.
0:30
We want people to be like, "Wow, look at them! They know calculus! They're awesome! I want to hang out with those people." That's our goal, right? And we get there by practice. So if you're following along with the video, make use of the pause button, see a problem, pause it, and try it yourself first and see how far you get. And if you ever do need my help, I'm there. You just un-pause me and together, our powers combined, we'll be able to figure it out. Well let's begin.
1:00
So our very first problem, this is just really a warmup, right? So, suppose you have a function F of XY equal X square plus two Y squared, find the following. So let's think about what this means. So this says, look, our function has two inputs, X and a Y.
1:23
And so in other words, we plug two values in and it says take the whatever is called X becomes X square, whatever is Y becomes Y square.
1:34
Now, be careful that you don't tie too much to the symbols.
1:40
And so we'll get into that as we go down our list. But some of these are pretty straightforward. So if I plug in zero comma two, okay, that says X equals zero, Y equals two. So it'd be zero squared plus two times two squared.
1:58
And of course we can simplify that. That becomes zero plus two times four, which is 8. Okay, that's not so bad. How bad could it be?
2:09
F of one, one. So now what does it say? Well look, it says we're plugging one in for X and one in for Y. So one squared plus two times one squared, that's one plus two. That gives us three. Okay, T one. Well what is that? Well, it says, now what we're plugging into the X slot, is we're plugging in T. And what we're plugging into the Y slot is we're plugging in one. So that says, okay, replace the X by T. So T squared and Y, we're gonna replaced by one, two times one squared, or if you like,
2:56
that becomes T squared plus two.
3:00
Okay, that's a little bit more interesting. But of course we can still get even more interesting. The next one, T plus one comma T. Okay, so what's going on here? Well, it says that we're replacing X by T plus one. So everywhere where we see X becomes a T plus one. So there's an X there. So that's gonna become a T plus one.
3:27
And here we're plugging T into the Y slot. So everywhere there's a Y is gonna become a T. So, two times T squared. Well of course you can simplify that if you want. That's really T squared plus two T plus one in the first slot, two T square in the second slot. So it becomes three T squared plus two T plus one. If you eXpand and combine.
3:56
Now, oftentimes it really makes sense to sort of have this intermediate step. If we just wrote down three T squared plus two T plus one, we'd be like, "Ah! Where did that come from?" But because we had the intermediate step, we're like, "Oh, okay, I see what's going on." Now this next one might throw you off.
4:17
Here we have F of X comma Y. It's X square plus two Y square. What is F of Y comma X? You're saying it's the same thing, right? We have variables, it's the same set of variables, shouldn't be the same function? And no, it's not. So you gotta be careful here. When we talk about a function of X and Y, we're just saying, look, these are convenient names here. We could of course done other things. So for example, I could have thought this as really
4:45
a function of, I'll put a square here and a triangle.
4:51
You're saying there's something that goes into there, there's something that goes into there. And the rule is whatever goes into there, you're going to square it and plus two, whatever goes into that slot, you're going to square it. Regardless of what the names are. That's what you do. That's what the process says to do. So F of Y coma X is not the same as F of X comma Y, unless you happen to have a really nice function. But you know, we're probably not gonna have that right away.
5:27
So what does it say? Well this is, look, it says the first slot is becoming Y. Now what we plug into the first slot, it says to square that, so this would be Y square plus two.
5:41
And see now that's the second slot. Second slot is X. X squared. So you get Y squared plus two X squared.
5:51
All right? Now the last one, what is F of cat comma dog.
5:58
Now at this point you're thinking, (laughing) "Okay, this is ridiculous. I can't talk about F of cat, comma, dog." Well, sure we can. We can say look, because our function is a rule, it says whatever you plug into the first slot, you square that, whatever you plug in the second slot you square and you double and you add the results. So F of cat comma dog would be F of, take your cat, squared it, plus two times the dog, squared.
6:31
And at this point we really can't do anything else because well, we don't know what to do for those operations, but that's what it would be. And if we were talking about symbolically, that would be our answer. So just remember, you know, functions are, they're rules. And so if you have a function of multiple variables, say look, there's a rule where there's multiple inputs and if you change the name of any single input, you just change where it is in the corresponding expression.
6:58
Make sure you count for everything. And so that's the big takeaway from this problem. Okay good. Oh see that's not so bad. Wow, great cats and dogs, life is good.
7:13
Let's try another problem.
7:16
Find and sketch the domain of the following function. So we're not sketching the function that would be a surface up in three dimensions. We're sketching the domain, which means we're looking for something in our inputs. So for us, we look at it and say, hey, it's a function of two variables and the variables are called X and Y. So our answer is going to be some region in the X,Y plane. So that's what we're after. What's the appropriate region in the X,Y plane?
7:50
Okay, so we say, hey, if it's domain, we have some issues, we have divide by zero. We don't want to do that.
8:03
We have square root of a negative, we don't wanna do that. And then we have a log of
8:14
anything less than or equal to zero. We don't wanna do that. These are our problems and we don't see anything else that's a problem. You know, there's no like crazy trig function or anything. So we say, okay look, these are the issues. As long as we avoid those problems, life is good. So let's talk about some of these. 'Cause I think some are easier than others.
8:39
So suppose we start with the numerator. It has a square root in it. We say, okay, look, in order for this to be possible, we need to have two X minus X squared has to be greater than or equal to zero.
8:56
That's the only way this could possibly work. If two X minus X square were was negative, I have a square root of a negative, I'm out, game's over. Okay, so what does that tell us? Well, let's think about it, two X minus X squared. Now you can actually see something here, if you multiply this as X times two minus X. And so if you think of this as a function, and so I'm gonna be a little careful here, so I'm just thinking of this as a function. Here's Y and I want to- whoops, sorry, here's X,
9:32
here's F of X, which is X times two minus X. What would it look like? Well, you can see where the intercepts would be. Where does it hit the axis. It hits it at two places, at zero and at two.
9:53
So it hits there and there.
9:58
It's a parabola. And which way does it open? Up or down? Opens down, negative in front of the X square. So, we see that's what it is. You say, ah. So two X minus X squared is gonna be greater than or equal to zero. That's really telling us that we have to be between zero
10:24
and two. 'Cause that's where we're above the axis. That's where the function is positive. But if we go above two or negative, if we go below zero or negative. So that's a constraint. All right. So that takes care of square roots of negatives. Let's do our log function. Alright, so for our log, what do we need? Well hmm. We need the inside of the log, X minus Y squared
10:59
has to be positive. Or we could say, look X has to be above Y squared, greater than Y squared. Now of course it doesn't, what do you mean by above when we're talking about X? Well, if we think about it, what is that? Well,
11:18
in terms of the coordinate system, if I draw here X and Y, X equals Y squared, I'm gonna do it with a little dotted line. It's a parabola but it's turned sideways. So you know Y equals X squared is this parabola, we're turning it sideways. And so we get this form, this is X equals Y squared.
11:44
Now you just have to think which side are you on? Well, if I want X to be greater than Y squared, it sounds like I should be a larger value. And that's to the right and that's the correct thing. Now if you're not sure, it's very easy to figure out. You just pick a point. Like here's a point. X equals one, Y equals zero. And now you say, is it true or false? Is one greater than zero squared? If it's true, you say good, that's the side we want to be on. So we can say, aha,
12:19
we have to be inside of that parabola turned sideways.
12:24
Alright good good. We should have probably drawn a picture for our previous one as well. Let's do a quick picture for our previous one. So zero less than or equal to X, less than or equal to two. Says what? Well you draw X equals zero and X equals two.
12:45
And then you say everything or X is between zero and two. So that's this strip right up and down.
12:53
Alright, so now we have two things figured out. We know we've dealt with the square root, we've dealt with the long term, so we're good to go. One more thing to figure out, divide by zero. Now let's think about it. When would we divide by zero? That would be when does the log of X minus Y squared equal zero? So we need to have natural log of X minus Y squared is not zero.
13:30
Well okay, so that's what we need. Now, the only time a log of something equals zero says hey, that means I need X minus Y squared is not one.
13:42
All right? Or I say look, X cannot be one plus Y squared. Well what is that? Let's look at that graph. X equals one plus Y squared. Well that's like taking the graph of X equals Y squared, but now we're going to shift it over by one. So that would be just this graph here. But then we say, look, as long as you're not on that parabola, everything else, it's fair game. And so, there we go. That's everything else except that one little curve. Alright, now how do we get our final answer?
14:27
What we have to do is we have to say, look, we've taken all of our problems and we've identified 'em. We can't be on this parabola, we have to be inside of this strip. We have to be to the right of that parabola and we just lay them all on top of each other. And when we do that, we're gonna get one region. And that's the region where all of them are satisfied simultaneously.
14:49
So it's like we're putting everything all on top, stacked together. And so we'll sketch our various curves in that we need to sketch. Our dotted ones. Well the X equals Y squared
15:09
looks something like that.
15:11
The X equals one plus y squared, it looks similar just to shift it over.
15:18
It looks something like that. And that's dotted because we don't want to be there. And we have X equals two. Okay, I am going to draw that one as well. But I'm gonna go back and visit that. And now we start to say, how does it all fit together?
15:35
Well we can include the points on X equals two. So I'm gonna make that solid. Where you have places intersection, like you'll see I have intersections there. If all of the pieces are saying include, then include. If even one piece says, hey don't include, then it's a not included. And the way we mark that, open circle is include, closed circle- sorry, open circle is not include, closed circle means do include. So these are points where we're not going to include.
16:12
And then everything else is shaded inside. So...
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We have that part.
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And we have that part.
16:29
And there we go. That's our region. We could mark some curves if we wanted to. So for example, we could say this is the curve, X equals one plus Y square, over here, this is the curve, X equals Y square and over here is X equals two. But that's it. Every point inside of that shaded part is in our domain. Everything that's not in that shaded part is out of our domain.
16:56
And so there we go. That's it. That's not so bad. Just check your three problem areas. Get rid of all the problems, life is good. Alright, well let's do it again.
17:13
Sketch the domain of the function and now it's very similar. And we say, okay look, let's just go through it. We have logs, we have square roots, we have division. So we say okay, we need to go through our three potential problems. So we'll just say we have a potential divide by zero. We'll have to figure that out.
17:35
We have a square roots of something which is negative and we have a log of something which is less than or equal to zero. Okay, so we just have to go through, piece them together one at a time. And as we do so we'll get a little bit better understanding of what's going on.
18:00
Let's start with the numerator that has a log. Okay, so what do we need? We need the inside part of the log has to be positive.
18:12
So, we need to have Y minus X minus three squared
18:21
is a positive value.
18:25
Another way to say that is that Y has to be greater than X minus three squared.
18:33
Okay, well we can get a good idea of what that is. So X squared, that's our parabola. The X minus three on the inside is a shift and it shifts to the right so that it would come here at three. So we now have this nice parabola coming up and I should have drawn a dotted line because it's strictly greater than. (groans) Retrospect, okay, I'll put a little dash here just to remind myself this is not include
19:05
and now we need to be above it. Okay. So we have to be on top of this. Okay. So that's pretty easy to spot which part we have. Normally if you have a curve, it'll split your plane into two parts. You have to figure out which part do I need. And you do it by testing if you're not sure, but sometimes you can look and say, aha! I see what's going on. No problem. No problem. So that takes care of our log. Our square roots is gonna be a little bit more interesting 'cause we actually have two of them.
19:41
So for each square root we have to think about what happens and it'll probably be something slightly different. That's not too surprising. So let's see what we have.
19:53
The first square root says, look, nine minus Y has to be greater than equal to zero. Okay, well that tells us if we move the Y across, Y has to be less than equal to nine. Oh that was actually pretty straightforward. Here, three minus X, that has to again, greater than equal to zero. Well what does that tell us? So this X has to be less than equal to three? So in terms of our sketch, we say look, here's this line, X equals three, here, there's this line Y equals nine and we wanna do what?
20:36
We wanna be below nine to the right of three, sorry, excuse me, to the left of three, below three. So in terms of our square roots, we want to be in this part right here.
20:52
Alright, so great, beautiful. Okay, well, that takes care of our square roots of negatives. Now are we done?
21:05
Not yet. (chuckles) There's one small, more thing. Our classic divide by zero. We don't wanna do it. We don't wanna do it. Would not be prudent. It would not be prudent. So, how does that happen? Well, if we were to divide by zero, it would have to be the case that these two were somehow canceled.
21:32
So we'd have to have square root of nine minus Y minus square root od three minus X equals zero. Let's just sort of follow what that says. So that would say what? Well, move that across. square root of nine minus Y would have to be the square root of three minus X. Okay, well let's square both sides.
21:58
If we square both sides, we're looking at places where nine minus Y, where it is equal to three minus X.
22:06
Hmm, okay, that's looking promising. Well let's just sort of move things around. So for example, let's move this across and we'll move these other terms around here. So we would get Y would be equal to, and here it'd be X plus six.
22:29
Okay good. So Y equals X plus six. Now that says we have to really, that's where we get a zero downstairs. So this is, we want to avoid.
22:43
Okay. So let's keep that in mind. We don't want to be there. Now where does Y equals X plus six fit. Let's come back to here.
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We also have to be, 'cause of the square roots, you know, inside this part. So Y equals X plus six would be this line there.
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So that's where we need to avoid. Y equals X plus six is ow, we don't want that.
23:13
All right. So, what happens when we now take all of these and put 'em on top of each other? What do we see?
23:24
Well, we're gonna see the following picture. So we'll draw our grid and...
23:36
We need to mark some lines here. Well, we have this parabola, X minus three square. Now notice it intersects here at zero comma nine. Which is right there. Ooh, that's very convenient. Okay, so here is three and here we'll put as nine.
24:00
Then we have this parabola come down and it's that dotted line. Dot, dot, dot, dot, dot ,dot, dot. Okay, because I don't need to worry about what happens above three, I just need to do the left hand side. And now we are gonna have
24:18
another dotted line that goes through six, Y equals X plus six.
24:25
And there's this corner. Now this corner is an open circle. Why? Because Y goes X plus six actually goes right through that corner. That would be zero minus zero, very zero-ey downstairs. Okay? So, these corners up here are also open
24:49
because we're gonna have problems. That's when we'd have a log of zero and, well dotted line, these are all certainly open up, well just mark it as open just to emphasize. But lastly, we say, okay, where do they overlap? And when we do that, we see aha, here's our picture,
25:13
and lo and behold we get our shape.
25:20
So it's kind of a a piece of a parabola. And it's interesting that will occur. Maybe not the most obvious thing, but that's all right. We just work through it patiently and lo and behold, we get out what we need. And so there we go. There we go. Great. Wonderful. All right, well let's move on. So let's try something a little different.
25:53
So here we go. Sketch and label the level curves for the function F of XY equals Y over X squared plus one for the values negative one, negative a half, zero, a half, one. Now what do we mean by level curves? Well we're thinking of these, these are the outputs. So in other words, Z equals these values. So we're looking at the heights and saying, let's slice our function. What does it look like? So, it's actually not so bad. We say, look, we're just gonna set our function
26:29
equal to each one of these values and simplify and see what we get. So if we start Y over X squared plus one equals minus one,
26:41
well that says Y equals minus X squared minus one.
26:46
Y over X squared plus one equals minus a half, that's Y equals minus a half X squared minus a half. Y over X squared plus one equals zero. Well that's Y equals zero. See we're clearing our denominators, multiplying across. And you probably can say, wow, I spot a pattern here, I think I can see where we're going. Good. That's what we want. And so we say, all right, these are our level curves.
27:22
What are they? Well, they're curves in the plain. So those are the actual functions. They're all parabolas of some sort. Okay great. Well, but now what? Well now we have to actually draw what the problems are. So we'll just mark, here we go, X and Y.
27:45
And what do we have? Well let's start with the easy one. Y equals X square plus one, it's our standard parabola shifted up by one. So we'll draw that and let's say here's one. So there we go.
28:03
And a half X square plus half, it's taking our standard parabola and it's squished it down by a factor of a half. But then we've also moved it up by a half. So it's gonna be a little bit broader. So it's gonna look something like that. Y equals zero. That's pretty easy to sketch. Y equals minus a half X square minus a half. Well that's very similar to this. It's just a negative of it.
28:35
And minus X squared minus one. Again, very similar to what we started with before. Just a negative.
28:43
Now we're not done yet. We did our sketch. But we also should label. Now when we talk about labeling, we just say for each curve there's a value. In other words, what's these numbers here? So starting with negative one, that's this curve, negative a half zero, one half,
29:08
and one. And that's our answer.
29:14
That's not so bad. Actually, level curves tend not to be too bad. We just say, look, set our function equal to a constant and then draw. It'll either be something which we could draw or maybe it's kind of hopeless. But you know, even if it's hopeless, we can still at least talk about the theory. So we'll have sometimes when we talk about that in future sessions.
29:36
Alright, well let's do another one like that. That was pretty quick, right? So we can do at least two like that.
29:44
Sketch and label the level curves for the function Z is four Y over X square plus Y squared plus one. Now you notice we don't say F of XY equals, we say Z equals well, sort of what we're implicitly saying, this is our function of X and Y. Alright, well it should be a little bit more interesting, I suspect. Let's figure it out. So, but the process is the same. You see, these values here, these are our Z values, or if we wanna sound fancy, "Zed", and we say, okay, so what do we do?
30:23
Well, we simply set our function equal to each one of them and simplify. So four Y over X squared plus Y squared plus one equals two.
30:37
Well we can move it across. So I'm gonna move this across and I'm gonna move the two across, in other words divide by two. This is the same as saying two Y equals X squared plus Y squared plus one.
30:52
Alright, well now what? Well that's really the same as X squared plus Y squared minus two Y plus one equals zero, right?
31:08
Yeah, it is. And what is that? Well that's X squared plus Y minus one squared equals zero. Well that's gonna be an interesting level curve. We'll get to that one in a minute. Okay, now we'll repeat similar process as we go along. How about for one? Well four Y over X squared plus Y squared plus one
31:35
equals one.
31:38
That would give us what? Well, it says four Y equals X squared plus Y squared
31:46
plus one, 'cause you multiply across. Or you can say, hey, that's really X squared plus Y squared minus four Y. And now what would we need? Because we can kind of see, if you see an X squared Y squared together, it's a circle. So you'd wanna say, well let's complete the square. And if you were to make this a perfect square, Y squared minus four Y, you would need to add plus four. Y plus four, take the middle term divide by two and square. You might say, okay, but wait,
32:23
there's really only a plus one. So that says you've got three too many. Well, move and put three more on the other side, you can check, see if you subtract three, you have a negative plus one. So what is this? Well, it's X squared plus Y minus two squared equals three.
32:46
Well, okay, that'll be something. All right, keep going. And the next ones aren't so bad. Four Y over X squared plus Y squared plus one equals zero. Well that's easy. Multiply across, that goes away. Four Y equals zero is just another fancy way of saying Y equals zero.
33:08
Okay, four Y over X squared plus Y squared plus one equals negative one. Well you can see there's not much that's gonna change other than the sign. And so it's gonna feel very similar to what we did above.
33:28
So instead of being minus four Y, it becomes plus four Y, but you still do the same four equals three or X squared plus Y plus two squared equals three.
33:44
And our last one, four Y over X squared plus Y squared plus one equals minus two, is the same as minus two Y
33:55
equals X squared plus Y squared plus one, which becomes the same as X squared plus Y squared plus two Y plus one equals zero or X squared plus Y plus one squared equals zero.
34:13
Alright, so we did the following. We took our function and set each one equal to the outputs that we were interested in. So the outputs were specified two, one, zero, minus one, minus two.
34:29
Now what we're going to do is say, okay, we've rewritten them in a way that we can actually plot them. So our next step is to draw them and then we have to do the label. So we've sort of done the prep work so that we can now finish the sketch and the label. So if we do this, what's gonna happen? Well, we'll get the following. We draw our grid. So, let's start with maybe the one in the middle. That's probably the easiest. Y equals zero, that's super easy. That's the X-axis. So we say, okay good.
35:06
Here we are, Y equals zero. And now let's move up, let's do one.
35:15
X squared plus Y minus two squared equals three. Well that says our center is at zero, two, so it's the circle, radius is root three. Okay. So it's going to be something, ah, square to three, it's less than two. So it's gonna be something that looks like this.
35:40
All right.
35:43
If you come down here, it's a similar situation, except instead of being at zero, two, the center, now the center's at zero, minus two. It's a essentially the mirror image.
35:58
All right. Now the interesting one, what does X squared plus Y minus one squared equals zero look like? Does it look like anything? Okay, that's actually not a trivial question. Well what can we say about all the points where X squared plus Y minus one squared equals zero?
36:19
Because they're both squared. Both of these terms have to be greater than or equal to zero. When we add them up, we have to end up with something greater than or equal to zero. Yeah, that's true. Okay, so what does that lead us to? Well the only way we can get zero is if both of these were zero to begin with, which means we need X to be zero and y equal to one. In other words, it's a single point.
36:49
Just one point. So our level curve matches at just a single value. Wow. And similar down here.
37:02
Okay, now you might be saying, well what is that really telling us? What's actually happening is it turns out that this function, we haven't proven yet, but eventually we'll be able to, the highest it can ever get to is two
37:19
and the lowest it can ever get to is negative two. So the function is bounded between negative two and positive two.
37:27
And so we're really seeing sort of a peak here. And if you think about what's going on, so this is our peak here at two, and then here's at one, here's at zero, minus one and minus two. You could imagine what some other curves would look like. So for example, if you did a half, it'd be a circle that you'd expect would look like that. And as you got closer to zero, the circles would get bigger and bigger. And of course if you were between one and two, the circles would start to sort of
37:58
zoom in on that point there. So there we go. That's our level curves. Good good. Well let's do another problem.
38:13
Okay.
38:16
Describe the domain of the following function. It's kind of like the ones we just did except... (evil laugh) It's a function of three variables. So instead of having just two variables, we'll have three. So we don't necessarily have to sketch it 'cause it's gonna be a three dimensional space, but maybe we can get a vague sketch going.
38:42
So we have a couple of things. We see a square root, a log, and a division. So we say, well let's talk about our log. Here, we say we need to have Z be positive
38:59
because whenever we plug into our log has to be positive.
39:03
and we need log of Z, not equal to zero. Well that's the same as saying Z can't be one because when Z is one then you'd have log of one downstairs and that's a division by zero.
39:20
Okay, so that takes care of our log and our division.
39:25
Let's turn to our square root. So for our square root we say, hey look we need the inside,
39:33
we need to make sure that that's positive. So that says one minus X squared minus Y squared minus Z squared has to be greater than or equal to zero or X squared plus Y squared plus Z squared is less than or equal to one.
39:52
And we say, wait a second, we recognize that. X square plus y squared plus C squared equals one is a unit sphere. The less than equal says we're on the inside. So we're inside the unit sphere. And now we're starting to see, okay, all right. So we can say, look, we'll put it all together. We say we want all the points
40:17
with X squared plus Y squared plus Z squared
40:24
less than or equal to one. And we say, well look, we need Z to be positive and Z not equal to one. Well if you think about the sphere, the highest point it could ever be in the unit sphere is one. And so we can say look, and zero, less than Z, less than one. So that's a description in terms of inequality. If we wanted to sketch a picture, it's a little bit hard to sketch in three dimensions, but we can vaguely talk about it.
40:55
Alright, we could do that. We would say, well we have the bottom part,
41:02
which is our Z equals zero. So the bottom here
41:07
and we'll just do a little dotted. Do do do do do do do. Okay. Don't include the bottom.
41:18
That's our little side note here. But other than that we do have the sphere coming up.
41:30
All right, so, we'll just make it look somewhat sphere-like as we come along. So the other thing is we don't include this one point at the very tippy top.
41:47
So we don't include the 0.001. Now you might say, well Steve, wait, when you plug in 001, yeah sure you get a zero downstairs, but you also get a zero upstairs, right? Zero over zero. That's okay. Right? No. Come on. We would never do that. Once you had divided by zero, forget it. All bets are off. Okay, so it's the top half of a sphere. Just (clicks tongue) knock off that top point, (clicks tongue) knock off that bottom surface and you're good to go. That's our domain.
42:24
Okay, well one more, one more problem.
42:31
All right. Find and sketch some level surfaces for this function, Z minus X squared minus Y squared. So again, it's similar before but instead of level curves, it's level surfaces. And so it doesn't say which ones we have to do and it doesn't say how many. So it's kind of a very vague question,
42:58
but isn't that great. It means whatever we do, we gotta be right. We have to do at least two 'cause it says some, so that means more than one. Let's do three. So let's look at our surfaces corresponding to negative one, zero and positive one.
43:17
Alright, so what would happen? So for negative one, we'd have negative one would be equal to Z minus X squared minus Y squared or what does that become? That says Z equals X squared plus Y squared minus one- whoop Z. For zero, now we get zero equals Z minus X squared minus Y squared or Z equals X squared plus Y squared. And for one, one equals Z minus X squared minus Y squared
43:57
or Z equals X squared plus Y squared plus one. We say well look, it's really Z equals, so that's good. And Z equals X squared plus Y squared, that's one we actually know about. 'Cause we've talked about when we talked about quadric surfaces. It's our bowl. So it's a bowl. And now we just have to think about, well what are these doing here? The minus one plus one. Well they are shifts. So what we have is the following. So if here's our grid, so Z equals X square plus y squared is a bowl.
44:38
So this would correspond to zero. And now Z equals X squared plus Y squared plus one and that's a bowl that's shifted up.
44:51
But just move it straight up by one.
44:55
All right, good. And one more, the minus one. Well that's the shift down.
45:03
You see the reason it's the shift down there, we go, is we're just affecting the Z coordinate by moving everything down by one.
45:13
So what are we forming? Well what we're forming is kind of like a big, infinite stack of, if you ever use like Tupperware, where you have nested Tupperware and you have 'em all nicely stacked one inside of another and we get that shape. So there's a sketch of some level surfaces. Now generally speaking, drawing surfaces is not very easy, so we're not gonna do it very much. But we are gonna talk about the idea of level surfaces. So it's the concept that's important.
45:48
So when we talk about level surfaces, it's a function equals some constant. And from there we're often good enough to make progress in what we're doing. Alright, well that's it for now. Thank you for coming for the practice. See you again in another session. Bye.
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