, The folium of Descartes is an algebraic
curve carved out by a certain equation. By
which equation? This equation, x cubed
plus y cubed minus 3axy equals 0. It's the
points on the plane that satisfy this
equation.
So, what's a folium? Well, folium is just
a Latin word for leaf, you know, the sorts
of things that grow on trees. So, where's
the leaf? Well, here's the leaf. I've
plotted the points on the plane that
satisfy x cubed plus y cubed minus 9xy
equals 0. And this is the curve that I
get, and you can see it looks kind of like
a leaf. This is not the graph of a
function, it's really a relation. x cubed
plus y cubed minus 9xy is a polynomial in
two variables, in both x and y, in both. I
can't solve for x in terms of y. Look,
this graph fails the vertical line test.
For a given value of x, there's
potentially multiple values of y which
will satisfy this equation. So, what's the
point of all these? Well, once upon a
time, Descartes challenged Fermat to find
the tangent line to this folium. And
Descartes couldn't do it but Fermat could.
And now, so can you. And you can do it
with implicit differentiation. So, let's
use implicit differentiation on this,
thinking of y secretly as a function of x.
So, the derivative of x cubed is 3x
squared. The derivative of y cubed, well,
that's 3y squared times dy dx, that's
really the Chain rule in action, minus,
now it's got to differentiate this. It
will be 9 times the derivative x, which is
1y minus 9x times the derivative of y,
which is dy dx, and that's equal to 0.
Alright, now I can rearrange this, the
things with the dy dx, and the things
without the dy dx, and you gather it
together. So, 3x squared minus 9y plus,
and the things with the dy dx term, 3y
squared minus 9x dy dx equals 0. Now, I'm
going to subtract this from both sides.
So, I'll have 3y squared minus 9x times dy
dx equals minus 3x squared plus 9y. And
I'm going to divide both sides by this, so
I'll have dy dx equals minus 3x squared
plus 9y over 3y squared minus 9x. And note
that we're calculating dy dx but the
answer involves both x and y. And you can
see, it's really working. I can pick a
point on this curve like a point 4, 2
satisfies this equation. Then, I can ask
what's the slope of the tangent line to
the curve through the point 4, 2? When I
go back to our calculation of the
derivative and if I plug in 4 for x and 2
for y, I get that the derivative is 4/5.
And indeed, I mean, this graph is somewhat
stretched, but, you know, yeah, I mean
that doesn't look terribly unreasonable
for the slope of this line. Problems like
this one, which once stumped the smartest
people on earth can now be answered by
you, by me, by lots and lots of people.
Calculus is part of a human tradition of
making not just impossible things
possible, but things that were once really
hard much easier.
Well, in any case, there's plenty more
questions that you can just ask about
different kinds of curves besides this
folium of Descartes. You can write down
some polynomial with x's and y's, like y
squared minus x cubed minus 3x squared
equals = 0 and then you can ask about the
points, the x comma y's that satisfy this
equation.
And if you want to know the slope of the
tangent line, use implicit
differentiation. The trick is just to use
the Chain rule and to treat y as a
function of x.