Visual Analysis And The Composition Of Functions

Andrew Grossfield

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Conference: 2009 Annual Conference & Exposition
Location: Austin, Texas
Publication Date: June 14, 2009
Start Date: June 14, 2009
End Date: June 17, 2009
ISSN: 2153-5965
Conference Session: Engineering and Mathematics Potpourri
Tagged Division: Mathematics
Page Count: 26
Page Numbers: 14.1354.1 - 14.1354.26
DOI: 10.18260/1-2--4935
Permanent URL: https://peer.asee.org/4935
Download Count: 386

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Andrew Grossfield Vaughn College of Aeronautics

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Throughout his career Dr. Grossfield has combined an interest in engineering design and mathematics. He studied Electrical Engineering at the City College of New York, graduating with a BSEE. During the sixties, he attended the NYU Courant Institute at night, obtaining an M.S. degree in mathematics, while designing circuitry full time during the day for aerospace/avionics companies. He earned his doctorate studying Continuum Mechanics under the direction of L. M. Milne-Thomson, CBE at The University of Arizona. He is a member of ASEE, IEEE, and MAA. Grossfield@IEEE.org is his email address.

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Abstract
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Visual Analysis and the Composition of Functions

Andrew Grossfield Vaughn College of Aeronautics

Abstract

A major problem today concerns educating the next generation of engineers, mathematicians and researchers. Too many of our nation’s students end up neither comprehending nor liking math courses. More intensive drilling of material as currently practiced may be both ineffective and undesirable. In fact, this rigorous drilling may turn more young students away from mathematics and the sciences. Why has this situation developed when mathematics is so interesting?

There are situations and mathematical principles that will enable graphs of functions to be easily produced. This paper will provide and discuss principles that can be applied in graphing a large class of functions. The graphs of polynomials and special basic functions formed by functional composition acting on polynomials will be provided as examples of visual thinking. Engineering students who are encouraged to develop the skills of visual thinking in mathematics may find these skills beneficial in their analytical engineering studies.

A student could find pleasure and confidence in discovering the ability to gain insight into the graphical behavior of a large class of functions. This same student may become more open to studying other aspects of polynomials and other functions. These techniques can provide a quick check of computer-generated graphs or be employed when a computer is unavailable or inconvenient. If we desire to recruit more students into the analytical and other sciences, we need to discover better, easier and more pleasurable ways to present conventional math concepts before attempting to accelerate curricula by moving advanced differential concepts into the lower grades.

Contents 1. Introduction 2. Polynomials 3. Arithmetic operations on functions and their effect on curves Addition, subtraction, multiplication and division 4. Elementary operations on curves and the algebra needed to produce them Translations, stretches, compressions and flips 5. Chains. The composition of functions Properties of chains 6. Functions of polynomials Linear Positive whole number powers

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Grossfield, A. (2009, June), Visual Analysis And The Composition Of Functions Paper presented at 2009 Annual Conference & Exposition, Austin, Texas. 10.18260/1-2--4935