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02:07

Fangjun Z.

If $f(x)=x+\sqrt{2-x}$ and $g(u)=u+\sqrt{2-u},$ is it true that $f=g ?$

00:56

Felicia S.

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Lowie T.

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All right, So we're going to define one of the coolest things in this course, and it's gonna be essential for the rest of this topic Is gonna be a crucial tool is going to allow us to do a lot. And that's the Grady int vector. And so let me get a little bit of motivation of where this is even coming from. So let's suppose that we have a function of two variables X and Y, and so are domain of our function is going to be some subset of the X y plane. So it is important to realize that when you see you know this X Y coordinate system, I'm not actually going to be plotting a function on here. This is actually going to be something in the domain. Now what I can do is to find, uh, level curves, control level curves on my coordinate axes in the domain, if my function f So let's say I have some kind of level curve like this going to draw something strange. So this is like maybe F is equal to zero. So these air all the values in the domain of my function that give me an output of zero. That's a level curve. And so we can keep going. Maybe we have something else like this. I don't know. Maybe this is f equals one. And then maybe we have some more level curves One right here and one right here. And so maybe thats f equals two, but that's the idea. So you kind of get this sense that that this is lower elevation out here. We're kind of going in toe higher elevation over here. All right, So if I pick a point somewhere in the domain of my function, let's say right here now what I can dio is Aiken define the partial derivatives, so f x and F sub y. Now what f sub x x f sub y. Actually, you're telling me geometrically is the rate of change of the function. If I cut the function along the plane parallel to the X axis and a parallel to the plane, a plane parallel to the Y axis like that. So these air actually giving me the rates of change in these two directions Now, there's nothing really special about the X direction in the Y direction. I could think about taking the rate of change of the function. If I cut along this direction or this direction, you know, wherever I wanted to find, sort of. How is my elevation of my function changing in that direction? Well, I can do that, and the way I do that is I'm going to use this Grady in Vector. But the Grady in Vector is going to do something very cool for us. So let me just tell you what the Grady in Vector is. So the Grady in vector of a function of two variables and we'll have a similar definition for functions of more than two variables. But we'll start just with the function of two variables. Is denote ID by this crazy looking upside down triangle f. And now this Just a side note. This triangle thing is called Nab Wow, Okay. And it's a Greek word that describe sort of an ancient Hebrew harp, and it sort of looks like ah harp if you've seen ah harp. But what it means for us is its degrade e int. So it's a vector. It has an X component in the y component. The X component, but simply is the partial derivative with respect to X in the Y component is the partial derivative with respect to why and now notice that this is going to be a function of X and y here. And it may be that there's certain points where the Grady in function doesn't exist. If, say, the partial derivative with respect to X doesn't exist or the partial derivative with respect, why doesn't exist? But when it does exist, I can evaluate. So if this is some point a B well, then I can talk about the Grady int of the function evaluated at the point, maybe. And so what is this? I mean what I mean. Here it is. Here's the definition. So I just put the X component are the partial derivative with respect to X and the X component partial derivative with respect to y and the Y component. But what is this vector At some point? Well, this vector is going to point in a specific direction. It points in the direction off. Steepest ascent. Okay, so what do I mean by that? Let's say I'm at this point here. So where does the Grady in vector points? So notice that this is on a level curve corresponding to F equaling zero. And so the uphill direction is this way. This is like the top of a hill up here. So the Grady Int Vector is going to point basically straight uphill in the direction of steepest ascent. So if I'm climbing a mountain and I say all right, what's the most direct path to the summit? Well, I just want to aim straight uphill in the direction that's actually the steepest. And so if I pick another point, so here's Grady in of F right there. So if I pick another point right here, the Grady in Vector is going to point straight uphill. Or if I go say here, Green in Vector is gonna point straight uphill or over here, the greedy in Vector will point straight uphill. And there's another important property. So it points in the direction of steepest ascent. But also, if I take some point a B and I look at the Grady in Vector, it's going to be orthogonal. Just remember, that means perpendicular to the level curve, and we're gonna use this fact later when we define a tangent plane. So what I mean, is if we look here at the Grady in at this point, well, it's going to be perpendicular to the level curve like that. So in other words, if I take a direction, it's pointing in the direction of my level curve. Well, the greedy in is gonna be pointing in direction that's perpendicular to that direction. So these air just two of the most important properties of the gravy in Okay, so geometrically it points in the direction of steepest ascent, but maybe a little bit more. Technically, it's orthogonal to the level curve, and that's going to be extremely important. So let's just make a couple notes about the great again. So first thing we define the Grady int for a function of two variables. But for a function of, say, three variables, we can also to find the Grady in Vector, and you can probably guess what it's gonna be. It's gonna be the vector. It's now going to have three components. The X component is gonna be the partial derivative with respect to X. The Y component will be the partial with respect to why, and the Z component will be the partial with respect to Z, and we don't have quite as nice of, uh, kind of a geometric interpretation has the direction of steepest ascent. But the Grady in vector points in general. Yeah, in the direction and of greatest change. And now notice. I'm not using steepest ascent anymore because we don't have quite the geometric way of visualizing greatest changes. Deepest ascent. But the Grady it will point in the direction of greatest change. That's the first thing. And then, secondly, three Grady int will be orthogonal. Well, we don't have little curves and three dimensions. We have local services, but the Grady in Vector in three dimensions will be orthogonal to the level surface, which is really cool. Okay, great. Okay, so let's talk about a really cool application of the Grady in Vector. So we said the great Grady in vector points in the direction of steepest ascent or greatest change. But let's say we have f a function of two or more variables and you a vector. Then we define what's called the directional derivative. It's of F in the direction of you as follows. So of course we're going to assume that F is differential at the point where we're defining this, because what we want to do is we want to take the grading of death. So the notation is diese bju of the function f. And now what this is is we take the Grady in a bath ingredient of F and project it into the U direction. What we really do is we just take the scaler projection of the Grady int in the direction of you. So it's a dot product of the Grady int with Vector you divided by the length of you. And this really is just the component of the Grady int that lies in the direction of you. And so there's another way to think about the direction directional derivative. That's very helpful. So if the Grady int points in the direction of greatest change, then the directional derivative is really nothing more than the rate of change in the direction of you. So we can think about it this way. If I take a step in the U direction, then the directional derivative tells me by approximately how much my function is going to change in that direction. All right, so we're going to continue talking about functions of just two variables, and it's important to remember that all of these ideas will be will extend to functions of three variables or functions of more variables. But the mantra is, if you can understand it in two dimensions, then you'll be golden. You'll be able to sort of just adapt. The idea is to higher dimensions. So suppose we have a function of two variables, and we have talked about several things involving the derivative. We talked about the partial derivative with respect to X, and now we can think about this is the directional derivative in the extraction. We've talked about the partial with respect to why this is the directional derivative in the Y direction. We've talked about the Grady int of death. This is the directional derivative. Uh, that's actually in the direction of greatest change. Okay, so if you have your own a point, you want to know which way do I go? Toe have the function increase the most? That's what the Grady is going to get, which is gonna be very helpful for lots of different things. Uh, but actually, for now, the Grady int of a function and sort of a unique way is going to give us a way to actually just nail down what it means for a function to be differential at a point. Okay, so in one variable, let me just remind you, Ah, function is differential. It's, um ffx is differential. At some point X equals a ISS. Well, this limit exists basically. So the limit his ex a purchase a f of X minus f of a over X minus a exists. Okay, so we've already seen. That's a lot more complicated, even just in two variables. Because remember to check if this lemon exists, we kind of want to check from the left and from the right, eso limit will exist that the left hand limit equals the right hand limit. But in more than one variables saying two variables, there is actually infinitely many directions that we have to check and make sure theme The this sort of rate of change exists. Here's three of them. We have the partial with respect to X, the partial with respect to why we have the Grady int, I's gonna be one of those directions, and so we could just go through and check all those infinitely many directions. But that's really inefficient. And what's really happening is if you think about all these infinitely many directions at a point, we're really filling out a not a tangent line, like in one variable but a tangent plane. So in two variables, the idea is going to be that a function is different chewable at now. Here again, we're in two variables. So at some point in the plane, if F has a tangent plane at this point A. B. So, in other words, to say, another way to say that this function as a tangent plane A B is to say it has a linear approximation. And this is really connecting back to the one variable case because for the derivative to exist in the one variable case, we also needed toe have a tangent line, and we had the interpretation that the derivative was the slope of that tangent line. Okay, now we're in two variables. We have a lot more direction. Well, what is a linear approximation of a function of two variables? Well, it's not a tangent line is a tangent plane, so if we have a tangent plane approximation, then this function is said to be differential. Okay? And I realized that I haven't really given you away toe, you know, find detained a plane or anything like that. But, you know, I just want to make this connection that different ability. It is really related to this idea of a linear approximation at a point. And in two variables, that's going to be a plane. And again, this idea will generalize to multiple dimensions. More. Sorry, more variables other than to. But if you can understand it into variables, then you're on on your way to understanding the idea of difference ability. Okay, so I'm gonna do my best here to sort of give you a visualization of the tangent plane. And I hope this really isn't coming out of nowhere. I hope it's very natural when you see this. So we have our X y Z coordinate system, and so I'm gonna draw our function here. So our function of two variables is going to be some kind of surface like this. Okay, hopefully you can visualize that it's gonna have It's gonna be curved right in general. I mean, it could be a plane. It could be flat But here's kind of our function plotted here, and it's gonna be plotted kind of above the X Y plane. So here's our domain down here, and here's the surface that's being formed or being graft by this function here. And so if I take a point, say right here now, we already talked about that. In any given direction, there could be tangent lines, but sort of the collection of all those lines will live in a plane like that, and that plane is gonna be tangent to the surface. And if you can't visualize this, take a basketball and a piece of cardboard and put the piece of cardboard on the basketball and have the cardboard just touched the basketball at one point, and that will be a representation you have. The basketball is the surface, and then that piece of cardboard will be the tangent plane. Okay, so hopefully you're kind of tracking with this idea of tantra plane. So let's talk about how we would actually find a tangent plane and again if we can find a tangent plane, that means their function is differential there or it has a linear approximation there, and a lot of times. You will want to use that linear approximation, uh, to sort of estimate function values just like you did in one variable. Okay, so here's what we dio We know that Z can be written as function of X y if we have a function of two variables. So, in other words, we can plot r z coordinate, uh, to be just the scaler output of our function of two variables. Here's what I'm gonna do. I'm going to define a new function of three variables. Okay? And it's going to be f of X Y minus Z, okay. And so this looks a little weird, But why did I do this? Notice that our our surface or the graph of F is the level surface. Mhm capital F equals zero. Okay, so now maybe you see the connection. I'm sort of giving a generalization of our function. I'm saying our function is just one of the level surfaces of this function of three variables. Okay. And why am I doing this? Well, I'm going to skip some of the technical details, but assuming that are derivative, our function F has continuous partial derivatives. Uh, I think that's all we need. Well, what is that going to allow us to dio? It's gonna allow us to form the tangent plane at any point. So at the point a B if I take the Grady int of capital F and evaluate it at a be and then I need to evaluate it a z coordinate, I'm just gonna evaluate it at the Z coordinate That's going to be on this level surface or, in other words, just f evaluated at a B. So the Z coordinate of the function So this Grady int is orthogonal. So we already saw this thio the surface to the level surface f equals zero. But to the surface, Z equals f of X y, hence orthogonal to the tangent plane. At this point a B. But this is amazing, right? Because how do we describe plane? Well, describe to describe the plane. We just need a point on the plane, but this is our point on the plane right here. We know the plane is going to touch the surface at this point. You know, just think about the basketball example at that one point on our surface, our plane is gonna touch And then this Grady in Vector is going to be our vector. That's worth agonal to the Plains That's going to allow us to write down the equation of the plane. But then that's easy. All we have to do is just say, Well, let's call this vector in, Okay, This is just some vector. I could just plug in the numbers, so I just take my vector in and I take the dot product with X minus. Hey, why might be and Z minus f a baby. Mhm. And I set that equal to zero. And this is my equation of the plane tangent to my surface. The graph of my function at this point a b. Okay, so now that you've listened to me, give a little bit of background on the change of plane and differential ability, let me sort of give you the final answer. So if I want the equation of the tangent plane so the tangent plane of a function at appoint a B is just Well, if I what I had in the last slide that dot product, if I expand it out, this is what I get. I get at the Max evaluated a B So the partial that with respect to X evaluated at that point Times X minus a plus partial with respect to y evaluated at a B times Why might this be and then plus f a baby? So this is almost identical to the linear approximation in one variable recall that it's that prime of a This is the one variable case times X minus a What's f of a So you see, all we've done is added in the change in the Y direction. So we have the change in the extraction, the change in the Y direction and as long as thes partial derivatives air continuous, this tangent plane will be tangent to the surface on We'll waken say that the function is differential there Same thing we could do it one variable if we have more than two variables Well, imagine what we're going to dio We're just gonna add on born more terms we're gonna have a f c evaluated at the point times of Z minus whatever See point that is Is this idea eyes called, um so this tangent finding the tangent plane or in one variable of the tangent line This is also called finding well, linear approximation. Fox Mission Thio F at the point. Maybe this is very useful in practice because, you know, you probably remember this from the one variable case. Maybe you have a function. You don't really know much about the function. But if you can find some sort of linear approximation, then it's really easy to approximate nearby values of the function. So this is extremely important and just basic function approximation theory, this linear approximation. That's what the change of plan really is. And here is sort of the cookie cutter definition that, you know, if you just wanna plug and chug into, that's the danger plane.

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