Starting to write things in a more vectorized way and the convenience here is that, that if you're dealing with a function with three input variables And what that is, it's a vector containing the two numbers x nought, y nought. To just kind of indicate that it's a constant of some kind. Which we are approximating, you would call, see I'll make it a nice bold-faced x nought. X is this input vector, and then similarly, the specified, ah, specified input about Um, but I'll try to emphasize it, just making it bold. To emphasize bold-faced x equals this vector, and where's, that's confusing cause x is already one of the input The name that we give this, it's very common to just call it x, and maybe a bold-faced x and that would be easier to do typing than it is writing, so I'll just kind of try These as its components, and we just wanna capture that all, and I wanna give that a name. There's some vector, some vector that has Input as being a pair of points, what I wanna say is that So the input, ah, rather than talk about the So first of all, let's think about how we would start describing everything going on here with vectors. Now you might see this inĪ more complicated form, or what's at first a moreĬomplicated form using vectors. More technical meaning of the word linear, this is all it really It's not squared, there's no square root, it's not in an exponent It's just being multiplied by a constant, and similarly this y term it's just being multiplied by a constant. Here, this variable term, doesn't have anythingįancy going on with it. But loosely what it means, and the reason people call it linear, is that this x term Of linear algebra, and admittedly, this is not actually a linear function in the technical sense. Very precise formulation, especially in the context So what do I mean by this word linear? The word linear has a Cause hopefully if you'reĪpproximating it near a point, then at that point, it's actually equal. That your linearization actually equals the function itself at the local point. This way, you can just think about adding whatever the function itselfĮvaluates to at that point. Reason we kind of paired up these terms and organized This terms goes to zero, and this is the whole In x nought and y nought, this terms goes to zero, cause x nought minus x nought is zero. And then to this entire thing because you wanna make sure that when you evaluate this functionĪt the input point itself. You take the partialĭerivative with respect to y, you evaluate it at the input point, the point about which you are linearizing, and then you multiply And then we add to that, basically doing the same thing with y. So the only variable right here, everything is a constant, but the only variable part is that x. You evaluate it at the point about which you're approximating and then you multiply thatīy x minus that constant. And you evaluate that at x of o or x nought, y nought. The way you do this local linearization is first you find the partial derivative of f with respect to x, which I'll write with More abstractly this time, rather than a specific example. The last couple videos, I'll write a little bit So just to remind us of where we were, and what we got to in That have more than just, just two input variables It'll be both more compact, and hopefully easier to remember, and also it's more general. Video is gonna be to show how we write this local linearization here in vector form, because Very complicated function with something that's much easier, something that has constant Way to approximate a function, which is potentially a The whole reason for doing this, is that this is a really good In an abstract 3D spaced to some kind of graph. We don't really care about, you know, tangent planes You're approximating the function with something simpler, with something that's actually linear, and I'll tell you what Point x nought, y nought, and the idea of a linearization, a linearization, means And what this basically means, the word local means you're looking at a specific input point. Name Local Linearization, Local linearization, this is And you want the graph of that function to be a plane tangent to the graph. Point ends up on the graph, and you wanna find a new function, a new function which we were calling L, and maybe you say L sub f, which also is a function of x and y. In the last couple videos, I showed how you can take a function, ah, just a function with two inputs, and find the tangent plane to its graph, and the way that you think about this, you first find a point, some kind of input point, which is, you know I'll just write abstractly as x nought and y nought.
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