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Current time:0:00Total duration:4:42

in the last couple videos I talked about the local linearization of a function and in terms of graphs there's a nice interpretation here where if you imagine the graph of a function and you want to approximate it near a specific point so you picture that point somewhere on the graph and it doesn't have to be there you know it could choose to be kind of anywhere else along the graph but if you have some sort of point and you want to approximate the function near there you can have another function whose graph is just a flat plane and specifically a plane which is tangent to your original graph at that point and that's kind of visually how you think about the local linearization and what I'm going to start doing here in this next video and in the ones following is talking about quadratic approximations so quadratic approximations and these these basically take this to the next level and first I'll show what they look like graphically and then I'll show you what that actually means in formulas but graphically instead of having a plane that's flat you have a few more parameters to deal with and you can give yourself some kind of surface that hugs the graph a little bit more closely it's still going to be simpler in terms of formulas it can still be notably simpler than the original function but this actually hugs it closely and as we move around the point that its approximating near the way that it hugs it can look pretty different and if you want to think graphically what a quadratic approximation is you can basically say if you slice this surface this kind of ghostly white surface in any direction it'll look like a parabola of some kind and notice like given that we're dealing in multiple dimensions that can make things look pretty complicated like this right here you know if you slice it kind of in this direct moving things about if you look at it from this angle it kind of looks like a concave up parabola but if you were looking at it from another direction it kind of looks concave down and all in all you get a surface that actually has quite a bit of information carried within it and you can see that by hugging the graph very closely this approximation is going to be well it's going to be even closer because near the point where you're approximating you can go out you can take a couple steps away and the approximation is still going to be very close to what the graph is and it's only when you step really far away from the original point that the approximation starts to deviate away from the graph itself so this is going to be something that although it takes more information to the scribed in a local linearization it gives us a much closer approximation so a linear function which you know one that just draws a plane like this in terms of actual functions what this means so I'll kind of write linear this is going to be some kind of function of x and y and what it looks like is some kind of constant which I'll say a plus another constant times the variable X plus another constant times the variable Y this is sort of the basic form of linear functions and technically this isn't linear if one was going to be really pedantic they would say that that's actually a fine because I'm strictly speaking linear functions shouldn't have this constant term it should be purely X's and Y's but in the context of approximations people usually usually would call this the linear term so a quadratic term what this is going to look like ah drat ik we are allowed to have all the same terms is that linear one so you can have you know a constant you can have these two linear terms B X and C Y and then you're allowed to have anything that has two variables multiplied into it so maybe I'll have D times x squared and then you can also have something times X Y this is considered a quadratic term which is a little bit weird at first because usually we think of quadratics is associated with that exponent two but really it's just saying anytime you have two variables multiplied in and then we can add some other constants say F times y squared where you know now we're multiplying two y's into it so all of these guys these are what you would call your quadratic terms things that you know either x squared Y squared or x times y anything that has two variables in it so you can see this gives us a lot more control because previously as we tweaked the constants a B and C you're able to get yourself you know that gives you control over all sorts of planes in space and if you choose the most optimal one you'll get one that's tangent to your curve at the specific point and kind of it'll depend on where that point is you'll get different planes but they're all tangent so what we're going to do in the next couple videos is talk about how you tweak all of these six different constants so that you can get functions that really closely hug the curve right and as you if they're all going to depend on the original point because as you move that point around what it takes to hug the curve is going to be different it's going to have to do with partial differ information about your original function the function whose graph this is and it's going to look pretty similar to the local linearization case just you know added complexity so we have to add a few more steps in there and I'll see you next video talking about that