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Multivariable calculus
Course: Multivariable calculus > Unit 3
Lesson 1: Tangent planes and local linearizationWhat is a tangent plane
The "tangent plane" of the graph of a function is, well, a two-dimensional plane that is tangent to this graph. Here you can see what that looks like. Created by Grant Sanderson.
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at2:11, I don't understand how the point on a graph can be described purely by (x,y.) doesn't it need a z coordinate to be properly described as well? likewise, the example of a point on a line in R2 says that that point could be described as (x), but wouldn't a y coordinate be needed to properly show it's location?(8 votes)- It is a bit subtle. Because we assume the function is given, the idea is that "which point we are talking about" is completely determined by choosing (x, y). It's true that if you actually want to locate the final point in R3, then you need to plug in x and y and get f(x,y), which you could think of as the z-coordinate. But since choosing x and y is all the choice you are allowed, and that gives one and only one z, we can say that the point on the surface is picked out entirely once you have chosen (x,y).
Likewise in the one-dimensional case. For example, if f(x) = x², then we can say "find the tangent line at x = 3". Although you actually need to compute the y-value of 9 in order to give the final answer L(x) = 9 + 6(x - 3), still, the "point" was completely determined by "at x = 3", since the function f(x) = x² was given. It was not necessary to say "at the point (3, 9)".(12 votes)
Would that mean people are just tangent objects of the space-time continuum? (When thinking fourth-dimensionally)(1 vote)
Doubt it. The tangent to a 4 dimensional object would be a 3d surface. But, I would think the surface would be highly specific, as the tangent to a 2d graph is a straight line and only a straight line and the tangent to a 3d surface would be a flat plane and only a flat plane. Both the line and plane are infinite in length/size, and people are not "regular" in shape and certainly not infinite in size. Thus, I doubt people are just tangent objects.(7 votes)
Can we say that the plane is the span of the vectors [∂f/∂x,0] (transposed) and [0,∂f/∂y] (transposed)?(1 vote)
I think you are kinda of right but you have to keep in mind that we are in 3d dimensions so I think a more correct answer would be that the plane is the span of
{ [1, 0 ,∂f/∂x] , [0, 1, ∂f/∂y] }
where each vector is transposed.
Remember that the partial derivative with respect to x can be thought of as tangent slope for the curve generated by our graph and a xz plane.
I hope this helps.(2 votes)
Is there a reason why the videos are no longer bookmarked after viewing them? Why was this feature removed?(1 vote)
2:11
Video transcript
- [Voiceover] Hi everyone. Here and in the next few videos
I'm gonna be talking about tangent planes of graphs,
and I'll specify this is tangent planes of graphs
and not of some other thing because in different context
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 wanna find at a
given point what the tangent line to that curve is,
where the tangent line is. You’ll find the equation
for that tangent line and this gives you various
information how to, let's say you wanted to approximate
the function around that point and 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 have some kind of graph of a function, like the one that I have
here, and then 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 gonna be barely 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 move it around and
say that it will just barely be kissing the graph of this
function but at different points. Usually the way that a problem
like this will be framed if you're trying to find such
a tangent plane is first, you think about the specified
input that you want. In the same way that over
in the single variable world what you might do is say,
"What is the input value here?" Maybe you'd name it like x
sub 0, and then you're gonna find the graph of the
function that corresponds to just kissing the graph
at that input point. Over here in the multivariable
world, move things about, you'll choose some 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. Once you decide on what
input point you want, you see where that is on the graph, so we go and say, "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. What you want is the plane that's tangent right at that point. You’ll draw some kind of plane that's tangent right at that point. If we think about what this
inner point corresponds to it's not x sub 0, a single variable input like we have in the single variable world, but instead that red dot
that you're seeing is gonna correspond to some kind of
input here, x sub 0, y sub 0. The ultimate goal over
here in our multivariable circumstance is gonna 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 gonna take in x and y, and we want the graph of that
function to be this plane, and you might specify that this
is depended on the original function that you have and
maybe also specify that it's depended on this input point
in some way, but the basic idea is we're gonna be looking
for a function whose graph is this plane tangent at a given point. In the next couple of videos
I'm gonna 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? It's actually very similar
to the single variable circumstance, and you just
take it one step at a time. See you in next video.