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- [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.