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Abstract for Jim Reynolds' Talk

Title:  Equal Areas, Apple Orchards, and Fun with The Fundamental Theorem

Abstract:  One part of the Fundamental Theorem of Calculus states conditions about the differentiability of functions defined as the integral (with respect to t) of f(t) from t = a to t = x.  Because many students' first impression of this result is that it is abstract and useless, problems that utilize it in unexpected ways have always been favorites of mine.This presentation deals with two such problems.

One problem starts with functions g(x) = kxⁿ and h(x) = xⁿ  with n>0 and k>1 then seeks to determine a function y = f(x) (passing through the origin) such that for any Quadrant I point P on g(x) the area bounded by g, h, and the vertical line through P is always equal to the area bounded by f, g and the horizontal line through P.  The other problem looks at a typical yield function Y(t), in tons per year, for an apple orchard and considers when the orchard should be replanted.