Off on a Tangent

It was noted in [1] that if a is any positive number other than 1, the tangent line to the curve y = ax at the point (/ In a, e) goes through the origin. The interesting feature here is that the y-coordinate is independent of a. We will show that this result is not as unique as it initially appears; there are many families of curves whose tangent lines at a fixed y-coordinate go through the origin. Shifting attention to the x-coordinate, it turns out that families of curves whose tangent lines at a fixed x-coordinate go through the origin have some interesting properties. In particular, these functions form the kernel of a linear transformation. The tangent line to a differentiable function f at the point (c, f(c)) with c :i 0 goes through the origin if and only if f'(c) = f(c)/c. By shifting, scaling, and tilting a given graph, it is possible to make the modified graph have a tangent line go through the origin at any given value of x or y. These geometric operations on a graph can be performed by modifying the original function by linear factors. This is the main idea used in the next paragraph. Let (a, b) be an open interval that contains the number 1. Suppose that f is differentiable and strictly increasing on (a, b) and that f'(1) < e. (There is nothing special about the choice of the numbers 1 and e; they could be replaced with any positive real numbers.) For each c :f 0, define a function gc on the open interval with endpoints ca and cb by