A Modeling-Based Approach to Calculus

Abstract

In our modern world, we are inundated and grapple with data daily. As mathematicians, we are often more comfortable discussing the behavior of functions presented analytically, in contrast with the data-driven or tabular presentations of functions ubiquitous in our culture. This paper introduces an entry-level course, Mathematical Modeling and Applied Calculus, for students who will most likely take only one calculus-level course in college and who often have a weak algebraic background that suggests they would most likely not be successful in a traditional calculus course. During the first half of the course, students review precalculus ideas while learning techniques for identifying elementary functional models whose graphs correspond to the underlying shapes of data sets, refining their models in response to new data, and using the method of least squares to find “best possible” models of a given type. During the second half of the course, these functional models are analyzed using the tools of calculus. Mathematical Modeling and Applied Calculus focuses on real-life contexts and data sets, uses RStudio to study large data sets, and explores big picture conceptual understandings in order to develop the insights and skills needed to describe, model, and analyze many aspects of our world.

Keywords:

• Applied calculus
• mathematical modeling
• least squares
• curriculum development
• underprepared students
• allied disciplines

Additional information

Notes on contributors

Joel M. Kilty

Joel M. Kilty (B.A. Asbury College, M.A. and Ph.D. University of Kentucky) is Elizabeth Molloy Dowling Associate Professor of Mathematics at Centre College. His research is in the area of differential equations. He has published research articles in several peer-reviewed journals including Mathematische Annalen, Journal of Differential Equations, and the Transactions of the American Mathematical Society. He is an active member of the Mathematical Association of America. He is a dedicated foodie, as well as a dedicated wine and bourbon connoisseur – all of which were consumed during the writing of this article.

Alex M. McAllister

Alex M. McAllister (B.S. Virginia Polytechnic Institute and State University, Ph.D. University of Notre Dame) is H.W. Stodghill Jr. and Adele H. Stodghill Professor of Mathematics at Centre College. He received the MAA’s Kentucky Section Teaching Award in 2015 and Centre’s Kirk Teaching Award in 2009, and his scholarly interests include mathematical logic and foundations, computability theory, and the history of mathematics. He is in the middle of a multi-decade renovation of a 100+ year old house.