What is a tangent plane

Hey everyone, so here and in the next few videos, I'm going to be talking about tangent planes. Tangent planes of graphs. I'll specify that this is tangent planes of graphs and not of some other thing because in different contexts of multivariable 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.

In the single-variable world, a common problem that people like to ask in calculus is, you 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. 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.

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. Instead of having a tangent line—because the line is a very one-dimensional thing and here it's a very two-dimensional surface—instead, you'll have some kind of tangent plane. 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.

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. 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 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 o. Then you're going to find the graph of the function that corresponds to kind of just kissing the graph at that input point. Over here in the multivariable world, you can kind of move things about. You'll choose some kind of input point, like this little red dot, and that could be at various different spots. It doesn't have to be where I put it; you could imagine putting it somewhere else.

But once you decide on what input point you want, you see where that is on the graph. 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. If we think about what this input point corresponds to, it's not x sub o, a single variable input like we have in the single-variable world. 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 multivariable circumstance is going to be to find some kind of new function—so 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. You might specify that this is dependent on the original function that you have, and maybe you 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. In the next couple of 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!