- [Instructor] Consider the closed curve in the xy-plane
given by this expression here.
Find the coordinates of the two points on the curve
where the line tangent to the curve is vertical.
So pause this video, and see if you could have a go at it.
So I don't know what the exact shape
of this closed curve is.
But if I were to draw some type of a closed curve,
maybe it looks something like this.
This isn't the one that's right over here.
This one also has two points
where my tangent line is vertical.
One point would be right over there.
Another point would be right over there.
Now, how do we figure this out?
Well, what we could do is use implicit differentiation
to find the derivative of y with respect to x,
and think about the x and y values
that would give us a situation where that derivative
is nonzero in the numerator and zero in the denominator.
So let's do that.
So let's just, let me rewrite everything.
I have x squared plus two x
plus y to the fourth plus four y
is equal to five.
I want to take the derivative with respect to x
of both sides of this equation.
I'm trying to find an expression
for the derivative of y with respect to x.
And so what am I going to get?
This is going to be equal to two x plus two plus,
derivative of this with respect to y is four y to the third
and then times the derivative of y with respect to x,
that's just straight out of the chain rule,
plus derivative of this with respect to y is four,
times the derivative of y with respect to x, once again,
straight out of the chain rule, is equal to, whoops,
I want to take the derivative with respect to x here,
is equal to
zero.
And now we just have to solve for dy/dx.
So there's a couple things we could do.
We could take the two x plus two
and subtract it from both sides,
and we could also factor out a four dy/dx
out of this stuff right over here.
So let's do that.
Let's subtract the two x plus two from both sides
and factor out the four times dy/dx.
So we will get four times dy/dx
times
y to the third
plus one
is equal to,
I subtracted this from both sides, so I'll get negative two.
And actually, let me factor out a negative two,
so negative two times x plus one.
And now I just have to divide both sides
by four times all of this business.
And I am going to get the derivative of y
with respect to x is equal to negative two times
x plus one
divided by,
divided by
four times y to the third
plus one.
And actually, this can be rewritten
as being equal to negative of x plus one
divided by two times y to the third plus one.
I just divided the numerator and the denominator by two.
Now, why is this useful?
Well, we could think about what y values, because y is
the only variable we have in the denominator here,
would make the denominator equal zero,
and then find the corresponding x values for those y values
by going to our original equation.
Well, this is going to be zero,
zero when
y is equal to negative one.
So when y is equal to negative one,
let's figure out what x is.
And to do that, we just have to substitute y
equals negative one back in our original equation
and then solve for x.
So let's do that.
Let me clear this out since I need that real estate.
So if we go back and we substitute
y equals negative one up here,
we're going to get x squared plus two x plus one,
plus one
minus four is equal to five.
This is going to be negative three.
Subtract five from both sides, you get x squared
plus two x minus eight is equal to zero.
This is just simple factoring.
So it's going to be x plus four times x minus two.
What two numbers, when I take their product,
I get negative eight?
Four, negative two.
When I add four, negative two, I get a positive two there,
is equal to zero.
So x is equal to negative four,
or x is equal to positive two
when y is equal to negative one.
So to answer their question,
find the coordinates of the two points on the curve
where the line tangent to the curve is vertical,
well, the answer here would be,
get a little bit of a drum roll,
it would be the points
x equals negative four
and y is equal to negative one and
x equals two and y is equal to negative one.
And we're done.