Using Tangent Lines to Define Means

We found these results to be quite intriguing and wondered if other means could be generated in this way by using different functions. As it turns out, this idea can be applied to a wide class of functions and the locations of the corresponding points c exhibit a surprising degree of simplicity and elegance. We will provide the details in this paper under the assumption that the reader has only a limited knowledge of the vast field of means. At the end of the paper, we will relate the means defined here to other types of means that have been considered in the literature. Given two distinct real numbers a and b, a mean M(a, b) is a number that lies between a and b. A mean is symmetric if M(a, b) = M(b, a) for all a and b. In this paper, we will only consider symmetric means defined for positive numbers. The most familiar example of a mean is the arithmetic mean (or average) of two numbers, but there are many more examples. For instance, the geometric mean of two positive numbers a and b is /a-b. This number is the side length of a square whose area is equal to that of a rectangle with lengths a and b. The value between a and b, often called c, whose existence is asserted by the Mean Value Theorem from differential calculus, also provides a way to define a mean of two numbers. As with the point of intersection of the tangent lines, a mean defined in this way depends on the function f. We will mention a connection between these two types of means.