The folium of Descartes is a curve defined by equation x3 + y 3 = 3xy:
The curve was first proposed by Descartes in 1638 in a very controversial manner: Fermat had
claimed to have recently discovered a method for finding slopes of tangent lines and Descartes
challenged Fermat to find the slope of the tangent line to this curve at an arbitrary point. We will
complete Descartes’ challenge.
A. Find the equation of the tangent line to x3 + y 3 = 3xy at the point ( 32 , 23 ).
Think first. What is your strategy? Talk/think about it and write down most important steps.
Implement your strategy.
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Aftermath. Sketch the tangent line on the graph and explain how it compares to your answer.
Write out in words and full sentences how you would explain to someone the process/steps by
which you solved this problem.
B. Find the point(s) in first quadrant at which tangent line to the folium is horizontal.
Think first. What is your strategy? Talk/think about it and write down most important steps.
Implement your strategy.
Aftermath. Sketch the tangent line on the graph and explain how it compares to your answer.
C. Go through your solution of part 1 and circle places where you’ve had the most difficulty or
where you think another student might make a mistake.
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