- [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.