Electronic Proceedings of the Seventh Annual International Conference on Technology in Collegiate MathematicsOrlando, Florida, November 17-20, 1994Paper C030Families of Linear Functions and Their Envelopes |
Steve LighDepartment of Mathematics Southeastern Louisiana University Hammond, LA 70402 USA Phone: (504) 549-2175 LIGH@SELU.EDU list of all papers by this author | Randall G. WillsDepartment of Mathematics Southeastern Louisiana University Hammond, LA 70402 USA Phone: (504) 549-2660 rwills@selu.edu list of all papers by this author |
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Consider the family of lines y=a*x+f(a), generated by a differentiable function y=f(x). We define the envelope of this set of lines to be a function y=g(x) such that every line in the family is tangent to the graph of y=g(x) at some point and through each point on the graph of y=g(x), there passes a line in the family that is tangent to y=g(x). We use the notation f(x)->g(x) to denote that the envelope of the family of lines y=a*x+f(a) is the function y=g(x). We present two main results:
Theorem 1 will describe the effect that translations and dilations of f(x) have on the envelope g(x), and Theorem 2 will give a closed form expression for the envelope g(x) for a certain class of functions f(x).
Theorem 1: Suppose f(x)->g(x), and let F(x)=A*f(B*x+C)+D*x+E, and G(x)=A*g(x/(A*B)+D/(A*B))-(C/B)*x+(-(C*D)/B+E) where A, B, C, D, E are real numbers and A,B<>0. Then F(x)->G(x).
Theorem 2: Suppose f(x) is a differentiable function such that f'(x) is injective. f(x)->g(x) if and only if g(x)=f((f')^(-1)(-x))+x*(f')^(-1)(-x). Note: (f')^(-1)(x) is the inverse of f'(x).