Student: Molly Siddall
Faculty Mentor: Dr. Greg Boudreaux, Mathematics Department
Research Title:  USING FLANKING CIRCLES TO INVESTIGATE TANGENCY

USING FLANKING CIRCLES TO INVESTIGATE TANGENCY

 Molly A. Siddall (Greg Boudreaux) Department of Mathematics, University of North Carolina at Asheville, One University Heights, Asheville, North Carolina 28804

In this research, an alternate approach to computing derivatives is used to prove several basic results from the theory of differentiation, such as the rules for differentiating simple monomials, differentiating vertical stretches of functions, and differentiating vertical shifts of functions. The alternate approach utilizes flanking circles: two circles that share a single point with a curve and lies on either side of the curve. This method may be considered a generalization of Euclid's definition of a tangent to a circle – a line that touches the circle at a single point, but does not pierce it. Not surprisingly, this method relies on some basic results Euclid proved about tangents to circles in his Elements, particularly results on mutually tangent circles. It has been shown that if flanking circles exist for a particular function at given point on the function, then the function is differentiable at that point. It is also known that the converse of this statement is not true. Necessary and sufficient conditions which guarantee the existence of flanking circles are being investigated as well. A benefit of using the flanking circle approach is that it provides a practical way to "define" tangents to arbitrary curves without the notion of a limit, which is simple to implement, even at inflection points. Furthermore, flanking circles make it possible to demonstrate in an elementary fashion the uniqueness of a tangent to a function at a point. Until now, it was fairly impossible to prove this fundamental result in the typical calculus classroom.