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Current time:0:00Total duration:7:09

Video transcript

so our setup is that we have some kind of two variable function f of XY who has a scalar output and the goal is to approximate it near a specific input point and this is something I've already talked about in the context of a local linearization and I've written out the full local line the full local linearization hard words to say local linearization in its most abstract and general form and it looks like quite the Beast but once you actually break it apart which I'll do in a moment it's it's not actually that bad and the goal of this video is going to be to extend this idea and it literally be just adding terms on to this formula to get a quadratic approximation and what that means is we're starting to allow ourselves to use terms like x squared x times y and y squared and quadratic basically just means anytime you have two variables multiplied together so here you have two x's multiplied together here it's an X multiplied with a Y and here Y squared that kind of thing so let's take a look at this local linearization it seems like a lot but once you actually kind of go through term by term you realize it's a relatively simple function and if you were to plug in numbers for the constant terms it would come out of something relatively simple because this right here where you are evaluating the function at the specific input point that's just going to be some kind of constant that's just going to output some kind of number and similarly when you do that to the the partial derivative this little F sub X means partial derivative at that point you're just getting another number and over here this is also just another number but we've written it in the abstract form so that you can see what you would need to plug in for any function and for any possible input point and the reason for having it like this the reason that it comes out to this form is because of a few important properties that this linearization has let me move this stuff out of the way we'll get back to it in a moment but I just want to emphasize a few properties that this has because it's going to be properties that we want our quadratic approximation to have as well first of all when you actually evaluate this function at the desired point at X naught Y naught what do you get well this constant term isn't influenced by the variable so you'll just get that F evaluated at those points X naught Y naught and now the rest of the terms when we plug in X here this is the only place where you actually see the variable maybe that's worth pointing out right we've got two variables here and there's a lot going on but the only places where you actually see those variables show up where you have to plug in anything is over here and over here when you plug in X not for our initial input this entire term goes to zero right and then similarly when you plug in Y naught over here this entire term is going to go to zero which multiplies out to zero for everything so what you end up with you don't have to add anything else this is just a fact and this is an important fact because it tells you your your approximation for the function at the point about what you are approximating actually equals the value of the function at that point so that's very good but we have a couple other important facts also because this isn't just a constant approximation this is doing a little bit more for us if you were to take the partial derivative of this linearization with respect to X what do you get what do you get when you actually take this partial derivative well if you look up at the original function this constant term is nothing so that just corresponds to a 0 over here this entire thing looks like a constant multiplied by X minus something and if you differentiate this with respect to X what you're going to get is that constant term which is the partial derivative of F evaluated at our our specific point and then the other term has no X's in it it's just a Y which as far as X concerned is a constant so this whole thing would be 0 which means the partial derivative with respect to X is equal to the value of the partial derivative of our original function with respect to X at that point now notice this is not saying that our linearization has the same partial derivative as f everywhere it's just saying that it's partial derivative happens to be a constant and the constant that it is is the value of the partial derivative of F at that specific input point and you can do pretty much the same thing and you'll you'll see that the partial derivative of the linearization with respect to Y is a constant and the constant that it happens to be is the value of the partial derivative of F evaluated at that desired point so these are three facts you know the value of the linearization at the point and the value of its two different partial derivatives and these kind of define the linearization self now what we're going to do for the quadratic approximation is take this entire formula and I'm just literally going to copy it here and then we're going to add to it so that the second partial differential information of our approximation matches that of the original function okay that's kind of a mouthful but it'll become clearer as I actually as I actually work it out and let me just kind of clean it up at least a little bit here so what we're going to do is we're going to extend this and I'm going to change it to name because I don't want it to be a linear function anymore what I want is for this to be a quadratic function so instead I'm going to call it Q of X Y and now we're going to add some terms and what I could do what I could do is add you know a constant times x squared since that's something we're allowed plus some kind of constant times X Y and then another constant times y squared but the problem there is that if I just add these as they are then it might mess things up when I plug in X naught and y naught right it was very important that when you plug in those values that you get the original value of the function and that the partial derivatives of the approximation also match that of the function and that could mess things up because once you start plugging in X naught and Y naught over here that might actually mess with the value so we're basically going to do the same thing we did with the linearization where we put in every time we have an X we kind of attach it we say X minus X not just to make sure that we don't mess things up when we plug in X naught so instead instead of what I had written there what we're going to add as our quadratic approximation is some kind of constant and we'll fill on that constant in a moment times X minus X naught squared and then we're going to add another constant multiplied by X minus X naught times y minus y naught and then that times yet another constant which I'll call C multiplied by Y minus y naught squared all right this is quite a lot going on this is a heck of a function and these are three different constants that we're going to try to fill in to figure out what they should be to most closely approximate the original function f now the important part of making this x minus x naught and y my why not is that when we plug in here when we plug in you know X not for our variable X and when we plug in Y not for our valuable Y all of this stuff is just going to go to 0 and it's going to cancel out and moreover when we take the partial derivative is all of its going to go to 0 as well and we'll see that in a moment I'll just actually show that right now or rather I think I'll call the video done here and then start talking about how we fill in these constants in the next video so I will see you then