- [Voiceover] Hey guys, so in the last video
I was talking about how you can define a function
whose graph is a plane,
and moreover a plane that passes through a specified point
and whose orientation you can somehow specify.
And we ended up seeing how specifying that orientation
comes down to certain partial derivative information.
And first let me just kind of repeat what the conclusion was
but I'll put it in more abstract terms
since I did it in a very specific example last time.
So basically if you want some kind of function
which gives you a plane that
passes through a certain point.
Well first let's see what that point is right.
Let's say the point was x nought, y nought, and z nought.
So these are constant values
and this is my way of abstractly describing
a single point in space using
x nought to represent a constant x value,
y nought to represent constant y value, that kind of thing.
So what it is, is it's gonna be some sort of other constant
a multiplied by x minus x nought.
So this white x here is the variable
and then x nought is just a constant.
Now let me go ahead and make that parentheses there.
Then we add to that b multiplied by
and then b is just some other constant
just like a is some other constant
multiplied by y minus y nought
and then all of that you add z nought.
And now if you were just presented this as it is
it's kind of a lot right,
there's five different constants going on.
But really what this is saying is
you want something where the partial derivative
with respect to x is just some constant
and you want to be able to specify what that constant is.
And similarly, the partial derivative
with respect to y is another constant.
And you just want to ensure that this
passes through this point x nought, y nought, z nought.
And if you imagine plugging in x the variable equals x nought
the constant this part goes to zero.
Similarly plugging in y nought the constant
makes this part go to zero.
So this is a way of specifying that when you
evaluate the function x nought y nought equals z nought,
and that's what makes sure that the graph
passes through that point.
So with that said let's start thinking about
how you can find the tangent plane to a graph.
And first of all let's think about what that point is,
how you specify such a point.
Instead of specifying any three numbers in space,
because you have to make sure the point
is somewhere on the graph,
you instead only specify two.
You're basically gonna say what's the x coordinate
and in this case let's say the x coordinate was one,
and then the y coordinate which
looks about like negative two.
To make it easier I'm just gonna say it is negative two.
Then the z coordinate is specified
because this is a graph.
Z coordinate is forced to be whatever the output
of the function is at one negative two.
So this is gonna be whatever the output of our function is
at one negative two.
And f here, f is going to be whatever
function gives us this graph.
So maybe I should write down the actual function
that I'm using for this graph.
In this case f which is a function of x and y
is equal to three minus one third
of x squared minus y squared.
So this is the function that we're using
and you evaluate it at that point
and this will give you your point in three dimensional space
that our linear function,
that our tangent plane has to pass through.
So we can start writing out our linear function.
We can say okay so our linear function
has a function of x and y.
It's gotta make sure it goes through
that one and that negative two,
so this is gonna be some constant a
that we'll fill in in a moment,
multiplied by x minus that one,
plus and then b also a constant we'll specify in a moment,
times y minus that negative two,
so it's minus a negative two
and then the thing that we add to it
is f of one negative two.
Now let's just go ahead and evaluate that.
Let's say we plug in one to negative two.
So if we go up here and we plug in three minus one third of
if x equals one, one third of one squared,
so that's one third one squared,
and then y is negative two.
So that would be minus negative two squared.
So that's three minus a third minus four
so the whole thing is equal to
three minus four is negative one minus another third
is negative four thirds.
So that's what we add to this entire thing.
We add negative four thirds
or maybe I should just kind of
make clear the separation here.
So this is our function but we don't know what a and b are.
Those are things that we need to plug in.
Now the whole idea of the tangent plane
is that the partial derivative with respect to x
should match that of the original function.
So if we go over to the graph here
and start thinking about partial derivative information.
If we want the partial derivative with respect to x
then you imagine moving purely in the x direction here.
This intersects the graph along some kind of curve
and what the partial derivative with respect to x
at this point tells you, is the slope of the tangent line,
in that direction of that point.
So that's what the partial derivative with respect to x
is telling you and what you want when you look
at the tangent plane is that the tangent plane
also has that same slope.
If I lined things up here,
you'd want it also to have that same slope.
So you can specify over here and say a.
You want a to be equal to the partial derivative
of the function with respect to x
evaluated at this one negative two.
Evaluated at that special point, one negative two.
And similarly b for pretty much the same reasons
and I'll draw it out here so let's erase this line.
So instead of intersecting it with that slice
let's see what movement in the y direction looks like.
So in this case it looks like a very steep slope right
because in this case the tangent line in that direction
is a pretty steep slope and now when we bring in
the tangent plane it should intersect with that
constant x value plane along that same slope.
Made it kind of messy there
but you can see the line formed
by intersecting these two planes
should be that desired tangent,
and what that corresponds to in formulas
is that this b which represents the partial derivative of l,
l is the tangent plane function,
that should be the same as if we took the partial derivative
of f with respect to y at that point,
at this point one negative two.
And this is stuff that we can compute
and that we can figure out.
So let's start plugging that in.
First let me just copy this function
because we're gonna need it.
Now let's go on down here.
I'm just gonna, let's paste it down here
in the bottom because that's what we'll need.
So let's compute the partial derivative of f
with respect to x.
So we look down here, the only place where x shows up
is in this negative one third of x squared context
so the partial derivative of f with respect to x
is gonna be just the derivative of this little guy
which is negative, we bring down the two,
negative two thirds of x.
So when we go ahead and plug in x equals one
to see what it looks like when we evaluate at this point
that's just gonna be equal to negative two thirds.
So that tells us that a is gonna
have to be negative two thirds.
Now for similar reasons,
let's go ahead and compute the
partial derivative with respect to y.
We look down here, the only place that y shows up
in the entire expression is this negative y squared.
So the partial derivative of f with respect to y
is equal to negative two y, negative two y.
And now when we plug in y equals negative two
what we get is negative two multiplied coincidentally
by negative two, didn't have to be the case
that those were the same, and that whole thing equals four.
So the partial derivative of f with respect to y
evaluated at this point one negative two
is equal to four.
So if we were to plug this information back up
into our formula we would replace
a with negative two thirds.
It would say negative two thirds.
And we would replace b with four, replace b with four.
And that would give us the full formula,
the full formula for the tangent plane.
And this can be kind of a lot to look at at first
because we have to specify the input point one negative two.
And then we had to figure out where
the function evaluates at that point.
And then we had to figure out both of the
partial derivatives with respect to x and with respect to y.
But all in all, there's not actually
a lot to remember from how you go about computing this.
Looking at the graph actually makes
things seem a lot more reasonable
because each of those terms has an actual meaning.
If we look at the one and negative two,
that's just telling us the input,
the kind of x and y coordindates of the input
and of course we have to evaluate that
because that tells us the z coordinate
that will put us on the graph corresponding to that point
and then to get a tangent plane
you just need to specify the two bits of
partial differential information
and that will tell you how this graph
needs to be oriented,
and once you start thinking of things in that way,
geometrically, even though there's a lot going on here,
there's five different numbers you have to put in,
each one of them feels like of course you need that number
otherwise you couldn't specify a tangent plane.
There's kind of a lot of information required
to put it on the appropriate spot.
So with that I will see you next video.