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Limits, Singularities and other concerns in the Elementary Functions of Calculus

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2021 Fall ASEE Middle Atlantic Section Meeting


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Publication Date

November 12, 2021

Start Date

November 12, 2021

End Date

November 13, 2021

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Andrew Grossfield Ph. D Vaughn College of Aeronautics and Technology

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ANDREW GROSSFIELD Throughout his career Dr. Grossfield has combined an interest in engineering design and mathematics. He earned his BEE at CCNY, an MS at the Courant Institute and a doctorate from the University of Arizona. He was licensed in NYS as a Professional Engineer and belongs to the MAA, the ASEE and the IEEE.

Seeing the differences between the mathematics memorized in schools and the math understood and needed by engineers has led him to a career presenting alternative mathematical insights and concepts. is his email address.

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Conventional calculus textbooks needlessly introduce the concepts of singularities and limits in a strange way that confuses many students who could be practical engineers and technicians. This expository paper is intended to provide a view of the singularities or problem points of calculus functions that a beginning student can grasp.

The elementary functions of calculus, which are the functions most used by most engineers, are single-valued, smooth and have relatively few extreme points. The singularities occur when problems arise in the computing the functional values. Perhaps for some of the horizontal values, a division by zero may occur. Perhaps over some interval of horizontal values, even roots of negative values may require computation. Computation at these problem points can result in the graph of the function having infinite jumps, finite jumps or point gaps. Another kind of discontinuity is an excessive number of extreme points, that is a pile-up of maxima and minima, as occurs at the origin of the function y = sin(1/x).

This paper provides an intuitive, visual interpretation of the concepts of continuity, discontinuity, limits, right and left-hand limits. The paper finishes with the advance of mathematics over the past two centuries and how math pedagogy changed to go beyond the grasp of the ordinary engineer.

Conventional math pedagogy obfuscates the limit concepts treated forcing students to memorize what they do not understand. An initially clear presentation of the simple cases, lets in fresh air, enabling the student to envision intuitively, the arc of the study.

Grossfield, A. (2021, November), Limits, Singularities and other concerns in the Elementary Functions of Calculus Paper presented at 2021 Fall ASEE Middle Atlantic Section Meeting, Virtually Hosted by the section.

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