HM/MA 004, Assignment 6, due Friday, 18 October 2002, 10:00 AM

(Note on collaboration: If you do the homework in pairs or groups, please list the name(s) of your collaborator(s) on what you each turn in.)

1. The mathematical vocabulary of curves. (8 points) Define the following terms as they are used in the "analytic geometry" of the seventeenth-century mathematicians whose works we have been reading. Be concise but clear about meaning (sample definition: "normal---the line perpendicular to the tangent line at the point where the tangent touches the curve").
   • axis
   • ordinate
   • quadrature
   • geometric progression

2. Understanding descriptions of mathematical properties of curves. (12 points)
   • i. Section 11.B1(a) of the Reader (pp. 351--352) describes what is called Debeaune's "inverse tangent" problem. In what way does Debeaune's curve depend on a property of its tangent? (Give just a qualitative verbal description---no named points or line segments or construction instructions.)
   • ii. Give a similar qualitative verbal description of the relationship between Barrow's curves in section 11.E3 (pp. 378--379). Would you call this an "inverse area" problem? Why or why not?

3. Identifying properties of tangents to curves. (10 points) In class, we deduced that the slope of the tangent line to Fermat's parabola (section 11.C1, pp. 358--359) at a given point was twice the length of the ordinate at that point. (Note that I'm not talking about Fermat's result a=2d on the location of the tangent line.) Consider now the tangent line to the ellipse, whose location you found in Problem 2 of Assignment 5. What is its slope in terms of the ordinate at the point where it touches the ellipse?