The following graph is the member of the family of folia in Problem 51 corresponding to c 1 Discuss How can the DE in Problem 51 help in finding points on the graph of x 3 y 3 3xy where the tangent line is vertical How does knowing where a tangent line is vertical help in determining an interval I ...

A tangent line will be vertical where y is undefined or in this case where x ...

The following graph is the member of the family of folia in Problem 51 corresponding to c

Question:

The following graph is the member of the family of folia in Problem 51 corresponding to c = 1. Discuss: How can the DE in Problem 51 help in finding points on the graph of x³+ y³= 3xy where the tangent line is vertical? How does knowing where a tangent line is vertical help in determining an interval I of definition of a solution Ï† of the DE? Carry out your ideas, and compare with your estimates of the intervals in Problem 50.

У4 х 1

Data from problem 51

The graphs of members of the one-parameter family x³ + y³ = 3cxy are called folia of Descartes. Verify that this family is an implicit solution of the first-orde differential equation

У(у — 2х3) dy x(2y³ – x³)