CHRISTINE BREINER: Welcome back to recitation.
In this video, I'd like us to work on the following problem
that has to do with tangent planes and approximations.
So we know that a rectangle-- we'll say a rectangle has
sides x and y, and we know that to find the area of the
rectangle then, we just take x times y, and I would like us
to approximate the area for x equal to 2.1
and y equal to 2.8.
And obviously, with this type of equation, it's not hard to
just compute this, but I'd like us to use the tangent
plane approximation to determine the value, and then
we'll compare it to the actual value, just to give us an idea
of how we can use the tangent plane to approximate things.
And obviously, I'd like to do this near x equal
2 and y equal 3.
So when you're doing your tangent plane approximation,
do the approximation at x equal 2 and y equal 3.
And then when you're done with that, I would like you to
answer this question.
So near x equal 2 and y equal 3, which
has the greater effect?
A change in x or an equal change in y?
So if I change by x in the same value as I change by y,
which will have the greater impact?
So why don't you pause the tape, work on the problems,
and when you're ready to see how I do them, bring it back
up, and I'll come back and show you.
OK, welcome back.
So we're going to work on this problem, and as I mentioned,
the first thing we want to do is do a tangent plane
approximation near x equal 2 and y equal 3.
So let me write down what we're going to need in order
to do that.
First, we'll remind ourselves that in this case, what I'm
going to approximate is the area function for a rectangle,
which is A of x, y is equal to x times y.
So that's the function I'm going to be approximating.
And now the actual approximation, the tangent
plane approximation, has the following form.
So we know A of x, y is approximately--
well, it's going to be the area evaluated at the point
I'm interested in, which we said was 2 comma 3, right,
plus the x-derivative of area evaluated at 2 comma 3 times
the change in x.
And the change in x is where x is now versus where we
started, which is at x equal 2.
And then the same thing in terms of y.
So you do the y-derivative evaluated at 2 comma 3, and
then you do the change in y.
And since we started at y equals 3, the change in y is y
minus 3, OK?
So now we obviously need to fill in three quantities here.
The three things we need to evaluate: area evaluated at 2,
3, its x-derivative evaluated at 2, 3, and its y-derivative
evaluated at 2, 3.
So let me just point out what we have.
Area evaluated at 2, 3, well, that's just 2 times 3, which
we can do, so that's 6.
That's pretty easy.
Now A sub x, so the derivative of the area function with
respect to x.
The derivative of the area function with respect
to x is just y.
y in that case we treat as a constant.
We take this derivative.
We just get y back.
And so, A sub x evaluated at 2, 3 is going to be y, which y
here is 3, so we get A sub x is equal to 3 at 2, 3.
In a similar vein, we can immediately look and see that
A sub y evaluated at 2, 3 is going to be 2.
And the reason for that, of course, is if we look back
here, the derivative of A with respect to y is x, so
evaluating it at 2, 3 gives us 2.
OK.
So now we just have to fill everything in.
So my tangent plane approximation now says I get 6
plus 3 times x minus 2 plus 2 times y minus 3.
And what I wanted was the area at 2.1 comma 2.8.
So if I fill in those values for x and y-- so 2.1 is x and
2.8 is y-- if I fill those in, I get that this is equal to--
well, I'll keep writing
approximately just to be safe--
is 6 plus 3 times--
well, 2.1 minus 2 gives me a 0.1 there and 2 times 2.8
minus 3 gives me a negative 0.2 there, so I get a negative
0.4, I get a positive 0.3, so together this
is a negative 0.1.
6 minus 0.1 gives me 5.9.
So the area based on the approximation
is 5.9 square units.
And, in actuality, if you multiply it out, I think you
get something like 5.88, so the approximation is very
good, right?
We're close to 2, 3 and that is one of the reasons we can
know it's going to be very, it should be very good.
It should be pretty good, OK?
Now, I just have to answer the second part of the question.
So let me remind us what the second part was.
It was near x equal 2 and y equal 3, which
has a greater effect?
A change in x or an equal change in y?
And to do that, it's really easiest to come back and look
at maybe this line.
I'll underline this line right here.
Actually, I'll box it, OK?
This value will represent the change in x.
So we started at 2, and we go somewhere, and that'll
represent the change in x.
This value represents the change in y.
So if we look at which has a greater impact near 2, 3, if
my change in x and my change in y are equal, then obviously
this term has a bigger impact, because there's a 3.
The coefficient here is 3 and the coefficient here is 2.
And so the point is, changes in x will have a greater
effect than changes in y.
And what this corresponds to pictorially is if we make a
slice where we keep y constant and we look at the curve in
the xz-plane, that corresponds to the fact that the
derivative in the xz-plane of the curve there is more
significant-- the curve there has a steeper derivative--
than if I kept the x-value fixed and I looked in the
yz-plane and I looked at the curve there.
So the sort of one-dimensional picture is that the derivative
of a curve in the x-direction, keeping y fixed, is steeper
than the derivative of the curve in the y-direction,
keeping x fixed.
So maybe hopefully that wasn't more confusing than
it should have been.
From here, you can see it right away, but that's sort of
the picture of it as well.
So I think that's where I'll stop.