Investigating Function Extreme Value: Case of Optimization

Abstract

Optimization is a central process in engineering designs. Its core idea is rooted in applying mathematics and calculus techniques to finding a maximum or minimum value of a function, often of several variables, subject to a set of constraints. This study investigated how calculus students formulated and analyzed functions that led them to find dimensions of a rectangle that produced a maximum area enclosed by a string of a fixed length. A group of 23 high school calculus students was immersed in an activity that involved hypothesizing possible outputs, direct measurements, data collecting, model formulating, and optimizing the model values. While typical textbook problems on optimization focus students’ attention on determining unique dimensions that maximize an enclosed area, this activity extended the exploratory part and underpinned not only the behavior of the function of interest but also the behavior of the constraint functions. This phase helped to disclose potential effects of the constraint functions on absolute maximum or minimum. Posttest analysis revealed that STEM activity not only deepened and helped understanding of underlying optimization processes but also challenged students’ mathematical reasoning skills regarded the interpretation of the behavior of the derivative function.