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Current time:0:00Total duration:10:40

Video transcript

hello everyone so I'm talking about how to find the tangent plane to a graph and I think the first step of that is to just figure out how we control planes in three dimensions in the first place so what I have pictured here is a red dot representing a point in three dimensions and the coordinates of that point easily enough are 1 2 3 so the x coordinate is 1 the y coordinate is 2 and the Z coordinate is 3 and then I have a plane that passes through it and the goal of the video is going to be to find a function a function that I'll call L that takes in a 2-dimensional input what x and y and this function L should have this plane as its graph now the first thing to notice is that there's lots of different planes that could be passing through this point right at the moment it's one that's got a certain kind of angle you could think of it going up in one direction but you could give this a lot of different directions and get a lot of different planes that all pass through that one point so we're going to need to find some way of distinguishing the specific one that we're looking at which is this one right here from other possible planes that can pass through it and as we work through you'll see you'll see how this is done in terms of partial derivatives but as we are getting our head around what the formula for this graph can be let's just start observing properties that it has the first property is that the graph actually passes through this point 1 2 3 and what that means in terms of you know functions over here is that if you evaluate it at the point 1 2 the input pair were both where X is 1 and Y is 2 then it should equal 3 it should equal 3 because that's telling you that when you go to x equals 1 and y equals 2 and then you say what's the height of the graph above that point it should be the the z coordinate of the desired point so this right here is kind of fact number 1 that we can take into consideration and beyond that let's start thinking about what makes planes what makes these kind of flat graphs different from the sort of curvy graphs that you might be used to in other multi variable functions the main idea is that if you intersect it with another plane so here I'm going to intersect it with y equals 2 this kind of constant plane and so I'll go ahead and write that down that plane that you're looking at is y equals 2 and you can think of this as representing you know what is movement in the x-direction look like as we move along the x-direction this kind of has a slope it the two planes intersect along a line and that's one of the crucial features of a plane is that if you intersect it with another plane you just get a straight line meaning the slope is constant as you move in the x-direction but it's also constant that same slope if you move in the y-direction if I had chosen a different plane you know if instead I chosen y equals 1 which looks like this then you get a line with the same slope and no matter what constant value of y you choose it's always intersecting that plane with a line that has the same slope and now if you look back to the videos on partial derivatives and in particular on how you interpret the partial derivative of a function with respect to its graph what this is telling you is that when we take the partial derivative of L with respect to X because constant Y means you're moving in the X direction this should just be some kind of constant some kind of constant a I don't kind of emphasize that is a constant value here and the same the same goes in the other direction right let's say instead of intersecting it with constant values of Y you say well what if you intersected it with a constant value of x like x equals 1 well in that case what you should get because you're intersecting it with a plane is another straight line so these two planes are intersecting along a straight line which means as you move in the Y direction this slope won't change but also as you move in the X direction if you imagine slicing it with a bunch of different planes all representing different constant values of X you would be getting a line with that same slope and that's telling you another powerful fact that the partial derivative of L with respect to Y you know if you're moving in the Y Direction that's equal to some other constant that I'm going to call B and now keep in mind these are very these are very powerful statements because the partial derivative of L with respect to X is a function this is a function of x and y and that might actually be worth emphasizing here that this partial derivative of X with respect to Y is something that you evaluate at you know a point in two dimensional space and we're saying that that's equal to some kind of constant value now that's a pretty powerful thing right because it's telling you it's giving you control over the function at all possible input points you know for movement in a specified direction and the same goes over here this is this is telling you that a function is constantly equal to you know some value B and we're not sure what this value B is but just geometrically we can kind of estimate what these things should be so if I take back the plane representing a constant Y value and we say what's this slope you know you're moving in the X direction we've got a constant Y value what is the slope at which this plane intersects our graph I would estimate this as about a slope of two you know you kind of go over one and it goes up two so what that would tell you is that at least in the in the specific graph that we're looking at this is at least approximately equal to two and then similarly if we look at constant constant x value and we say that represents movement in the Y direction what is the slope there this this looks to me like about 1 as a slope you kind of move over one unit you go up one unit so I'd say down here that the constant value of the partial derivative with respect to Y is about equal to one so we have three different facts here the value of the function at the point 1 2 the value of its partial derivative with respect to X everywhere and the partial derivative with respect to Y everywhere and this information is actually going to be enough to tell us what the function as a whole should equal now specifically this idea that the partial derivative with respect to X is constant tells us that the function function L of X Y is going to equal 2 times X plus something that doesn't have any X's in it something that as far as X concerned is a constant because the only thing whose derivative with respect to X is the constant 2 is 2 X plus something that's constant as far as X is concerned and then similarly over here if the partial with respect to Y is the constant 1 then that tells you that the whole function looks like you know this is a click a constant as far as Y is concerned so once we once we bring in Y it's going to be 1 times y plus something that's constant as far as Y is concerned you know this part is already constant as far as Y is concerned so anything that I add beyond here has to be constant as far as both x and y is concerned so that part has to actually literally be a constant so I'm just going to put in I'm just going to put in C for that to represent constant so this is you can see this is a very restrictive property on our function because the only place X can show up is this this linear term and the only place Y can show up is as this linear term and when I use the word linear you can pretty much interpret it as saying the the term X shows up without an exponent or without anything fancy happening to it it's just x times a constant that's pretty much what I mean by linear it's got more precise formulations in other context but as far as we're concerned here you can just think of it as meaning variable times a constant so the question is what should this see be and you can imagine that we can based on this property based on the value at the point 1 2 we can uniquely determine C and you can plug in x equals 1 y equals 2 know that this has to equal 3 and solve for C which we could do but I'm going to actually do something a little bit more convenient I'm going to kind of shift around where the constants show up and I'm going to say that the whole function should equal 2 times and then I'm going to put a constant in with X I'm going to say X minus 1 and then I'm going to do the same thing with Y I'm going to say plus 1 here's the partial derivative with respect to Y y minus and then I'm going to say 2 and the reason I'm doing this notice this doesn't take change the partial derivative information it's just if we take the partial derivative with respect to X this will still be 2 and when we take it with respect to Y this will still be 1 but the reason I'm putting these in here is because we're going to evaluate it at the point 1 2 so I want to make it look as easy as possible to evaluate at the point 1 2 and then from here I'm just going to add another constant so instead of saying C because this is going to be slightly different from C I'll call it K but the idea is I'm just moving around constants if you imagined distributing the multiplication here and having you know 2 times that negative 1 and 1 times that negative 2 you're just changing what the value of the constant stuck on the end here is now the important part the reason that I'm writing it this way which is only slightly different is because then when I evaluate this at L of 1 2 this whole first part cancels out because plugging in x equals 1 means this whole part goes to 0 same with the second part because when I plug in y equals to this part goes to 0 so K this other constant that I'm tagging on the end is going to completely specify what what happens when I evaluate this at the point one two and of course I want it to be the case that when I evaluated at 1/2 I get three I want it to be the case when I evaluate it one two I get three so that tells me that this constant K here should just equal three so notice the way the way that I've written the function here is actually quite powerful we have a lot of control this term too was telling us the slope with respect to X so when you move purely in the X direction and that was kind of illustrated here purely in the X direction that's telling us the slope with respect to X and then this term 1 here was telling us the slope with respect to Y so when we moved purely in the Y direction that's telling us the slope there and we could just turn those knobs if we change the two and we change the one that's what's going to allow us to basically change what the slopes of the plane are I'm going to say slopes plural because it's with respect to the X and the y direction and that will give us control over various different planes that pass through you know if I'm looking at the one of Jesus I don't let's say the one right here then the movement in the Y Direction is very shallow so that would be turning this knob lower and instead of one it might be point zero one and if I were looking at movement in the X direction you know this looks actually negative so this would tell you that it can be some kind of negative number so you can kind of dial these knobs and that changes the different planes to pass through that same point and then plugging in this one two and three tells us what point we're specifying but we're basically saying when you input x equals one and you input y equals two the whole thing should equal three so this form right here is powerful enough that I want you to remember it for the next video I want you to remember the idea of writing things down in this way where you specify the point it's passing through with its x coordinate y coordinate and Z coordinate placed where they are and then you tweak the slopes using these coefficients out front so with that I will see you next video