What is a tangent plane

everyone so here and in the next few videos I'm going to be talking about tangent planes tangent planes of graphs and I'll specify that this is tangent planes of graphs and not of some other thing because in different contexts of multi variable calculus you might be taking a tangent plane of say a parametric surface or something like that but here I'm just focused on graphs I'm so in the single variable world you a common problem that people like to ask in calculus is you'd have some sort of curve and you want to find at a given point what the tangent line to that curve is what the tangent line is and you'll find the equation for that tangent line and this gives you various information kind of how to let's say you wanted to approximate the function around that point it turns out to be a nice simple approximation and in the multivariable world it's actually pretty similar in terms of geometric intuition it's almost identical you'll have some kind of graph of a function like the one that I have here and instead of having a tangent line because the line is very one-dimensional thing and here it's a very two-dimensional surface instead you'll have some kind of tangent plane so this is something where it's just going to barely be kissing the graph in the same way that the tangent line just barely kisses the function graph in the one dimensional circumstance and it could be at various different points rather than just being at that point you could kind of move it around and say that okay it'll just barely be kissing the graph of this function but at different points and usually the way that a problem like this would be framed if you're trying to find such a tangent plane is first you think about the the specified input that you want so in the same way that over in the single variable world what you might do is say okay what is the input value here maybe you'd name it like X sub zero and then you're going to find the graph of the function that corresponds to kind of just kissing the graph of that input point over here in the multi variable world kind of move things about you'll choose some kind of input point like this little red dot and that could be a various different spots it you know doesn't have to be where I put it you could imagine putting it somewhere else but once you decide on where what input point you want umm you see where that is on the graph so you kind of go and say oh that input point corresponds to such and such a height so in this case it actually looks like the graph is about zero at that point so the output of the function would be zero and what you want is a plane that's tangent right at that point so you'll draw some kind of plane that's tangent right at that point and if we think about what this what this input point corresponds to it's not X sub Oh a single variable input like we have in the single variable world but instead that red dot that you're seeing is going to correspond to some kind of input pair X sub o and Y sub o so the ultimate goal over here in our multi variable circumstance is going to be to find some kind of new function so I'll I'll write it down here some kind of new function that I'll call L for linear that's going to take in x and y and we want the graph of that function to be this plane and you might specify that it's this is dependent on the original function that you have and maybe also specify that it's dependent on this input point in some way but the basic idea is we're going to be looking for a function whose graph is this plane tangent at a given point and in the next couple videos I'm going to talk through how you actually compute that it might seem a little intimidating at first because how do you control a plane in three dimensions like this but it's actually very similar to the single variable circumstance and you just kind of take it one step at a time see you next video