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JOEL LEWIS: Hi.

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Welcome back to recitation.

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In lecture, you've been learning
about using gradients

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to compute tangent planes
to surfaces.

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So I have an example of a
practice problem here for you.

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So what I'd like you to do in
part a is to use gradients to

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find the tangent plane to the
surface z equals x cubed plus

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3xy squared at the
point 1, 2, 13.

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And in part b, I'd like you to
do something similar, which is

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to use gradients to find the
tangent line to the curve x

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cubed plus 2xy plus y squared
equals 9 at the point 1, 2.

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So why don't you pause
the video, have a

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couple goes at those.

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Come back and we can work
on them together.

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So hopefully, you had some good
luck working on these

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problems. Let's just take
a look at them.

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So for part a, you're given a
function in the sort of usual

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form that we use to graph it,
which is you're given z equals

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a function of x and y.

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But in order to apply this
gradient method, what we

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really want is we want to look
at this surface as if it were

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a level surface of some function
of three variables.

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So in order to do that, what
we want to do always is to

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bring the x, y and z all
together on the same side with

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just a zero or a constant
on the other side.

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So let me do that.

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So I'm going to rewrite the
defining equation of this

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surface as 0 equals x cubed plus
3xy squared minus z, and

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I'm going to define this
right-hand side to be a new

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function w of x, y, z.

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All right?

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So if I call this thing w, then
our surface in question

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is just a level surface of w.

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It's the level surface
w equals 0.

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And so we know in that situation
that the gradient of

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w is perpendicular to
its level surfaces.

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It's orthogonal to its
level surfaces.

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So the normal to our surface is
exactly the gradient of w.

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All right?

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So gradient of w is the normal
to our surface, and a normal

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is what we use to write down
the equation for a tangent

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line-- oh, tangent
plane, excuse me.

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So, OK, so let's compute
the gradient of w.

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Well, that's not hard to do.

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We just take the partial
derivatives with

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respect to x, y and z.

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So the partial derivative of
w with respect to x is 3x

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squared plus 3y squared.

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The partial derivative with
respect to y is 6xy, and the

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partial derivative with respect
to z is minus 1.

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So one thing to notice is that
when you do this method, when

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you have the function given at
z, when you have the surface

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given in the form z as a
function of x and y, you're

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going to bring the z over, and
you always have a minus 1

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there when you set the
problem up this way.

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Because you'll have a minus z,
and then you'll just take the

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partial with respect to z, and
the other terms will only

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involve x and y, so they'll
be killed by the partial

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derivative.

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So in any case, this is our
gradient, so we want the

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normal vector.

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We were asked for the
tangent plane at a

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particular point, I believe.

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Yes, at the point 1, 2, 13.

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So we need to compute the
gradient at that particular

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point and that will be
our normal vector.

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So the gradient at
this point is--

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well, we just plug in, so the
gradient at 1, 2, 13.

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So x is 1.

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So this is 3 times 1 plus 3
times 4, so that's going to be

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15, and 6xy is 12, and minus
1 is just minus 1.

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So this is the gradient vector
at our point 1, 2, 13.

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So now we have a point, the
point 1, 2, 13, and we have

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the normal vector 15, 12, minus
1, so that gives us the

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equation for the tangent
plane right off.

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So the equation for the tangent
plane, I just dot the

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normal vector with the vector
connecting our point to the

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point x, y, z, so that gives us
15 times-- well, our point

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is 1, 2, 3-- so it's 15 times
x minus 1 plus 12 times y

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minus 2 minus 1 times
z minus 13 equals 0.

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So in point-normal
form, this is the

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equation for that plane.

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Great.

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And if you wanted, you could
rewrite this a whole bunch of

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different ways, but I'll
just leave it there.

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So let's do part b.

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I guess I'll just start
it right below here.

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So for part b, we have a curve
x cubed plus 2xy plus y

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squared equals 9.

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So this is a curve that is
defined by this implicit

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relationship between x and y.

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All right?

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And so what I want
to do is I can do

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exactly the same process.

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We're going to do exactly
the same thing.

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We're going to find the
normal-point form for the

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tangent line, and so we're going
to do that by defining a

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function f of x, y.

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In this case, it's a function of
just two variables, because

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we're only working with a
curve in two dimensions.

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Before, we had a surface in
three dimensions so we had a

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function of three variables.

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So f of x, y, and so then our
curve is exactly a level curve

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of the graph of f, right?

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It's the level curve
f equals 9.

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So in order to find the tangent
line, I can do exactly

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the same thing.

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I can find the gradient.

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The gradient is normal to the
tangent line and then I can

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use normal-point form.

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So the gradient of f is-- again,
f is just a polynomial

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function so its gradient
is easy to compute.

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It's 3x squared plus 2y
comma 2x plus 2y.

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And so we're interested in
this tangent line at a

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particular point so we're
interested at the point 1, 2.

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So the gradient of f at 1, 2,
well, I just plug in again, so

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I get 3 plus 4.

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That's 7.

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And 2 plus 4 is 6.

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And so again, the same analysis
as we used in the

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tangent plane case works in
the tangent line case.

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Let's come over here.

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So xy is on the tangent line
if and only if we have that

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the gradient dot, so that's
the gradient 7, 6 dot the

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vector x minus 1, y minus 2--
this is the vector connecting

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the point x, y to
our point 1, 2--

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is equal to 0, if and only
if those two things are

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orthogonal.

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So this is i.e., 7 times x minus
1 plus 6 times y minus 2

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is equal to 0.

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So this is the point-normal
form for the

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equation of that line.

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And again, you could, you know,
expand out and rewrite

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this in whichever forms you
happen to like to see your

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equations of lines.

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So there you go.

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Using the gradient,
we can compute

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tangent planes to surfaces.

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Similarly, we can use the
same idea to compute

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tangent lines to curves.

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The point is that the gradient
vector of a function is

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orthogonal to the level curves
of that function.

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And so we use that to get the
normal vectors to our curves

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or our surfaces, and with the
normal vector, we can then

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easily compute the tangent plane
or the tangent line.

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So I'll stop there.

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