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Folium of Descartes

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

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Witch of Agnesi

Sketch the members of the family of graphs given by y = a^3/(x^2+a^2) for a=1, 2 and 3.

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Quick Route

What is the quickest route across a ploughed field when your speed around the edge is greater?

Least of All

Age 16 to 18 Challenge Level:

This could just be a straightforward calculus exercise except that you are asked first to consider the geometric situation and make conjectures about the result before you do any calculus.

To appreciate and fully understand the mathematics you are learning it is good practice to make conjectures if you can and then review your first thoughts in light of further work on the problem.

It is important to ask yourself how many answers you might expect to find (one only or perhaps more than one?) and to ask yourself how you will know if you have found ALL the answers.