Differentiation of parametric and implicit functions

problem
Squareness

Age

16 to 18

Challenge level

The family of graphs of x^n + y^n =1 (for even n) includes the circle. Why do the graphs look more and more square as n increases?
problem
Implicitly

Age

16 to 18

Challenge level

Can you find the maximum value of the curve defined by this expression?
problem
Folium of Descartes

Age

16 to 18

Challenge level

Investigate the family of graphs given by the equation x^3+y^3=3axy for different values of the constant a.