Fig. 3.25 The folium of Descartes, in Problem 3.24
3.24. The set of points (x, y) that satisfy $x^3 + y^3 - 9xy = 0$ lie on the curve shown in
Fig. 3.25.
(a) Suppose a portion of the curve is the graph of a function y(x). Use the chain rule
to show that $y' = \frac{-x^2 + 9y}{y^2 - 9x}$.
(b) Verify that (2,4) is on the curve. If y(x) is a function having y(2) = 4, find y'(2)
and use the linear approximation of y to estimate y(1.97).
(c) Verify that each line y = mx other than y = -x intersects the curve at one point.
Find the point.
(d) Find the other two points (2,y) on the curve besides (2,4). For functions y(x)
passing through those points, does y'(2) have the same value as you found for
(2,4)?
