Mathematical Culture and Thought

Mathematical Culture and Thought

‎Characterizations of ‎Q‎uadratic Polynomials

Document Type : Translation

Authors
A. M. Kohestani 1
A. K. Asboei 2
1 Graduated in pure mathematics from University of Mazandaran, Iran
2 ‎Department ‎of‎ Mathematics Education‎, Farhangian University‎, ‎Iran
10.30504/mct.2022.1304.1911
Abstract
This note was motivated by a student’s misconception that the average velocity over a time interval
of a particle in rectilinear motion is the arithmetic mean of its velocities at the ends of the interval. We present
an algebraic proof that this property holds only if the particle has constant acceleration, thereby recovering a
well-known result. A characterization is also provided of differentiable functions on (−∞,∞) whose restrictions
to the intervals (-∞ 0]‎‎ and ‎‎[0,∞)‎‎ are quadratic polynomials. In addition, a calculus-free proof is presented to
show that a certain feature of the mean value theorem holds only for quadratic polynomials.
Keywords
quadratic polynomials
average speed
differentiability
instantaneous speed

Subjects

Holland, F., ‎Characterizations of quadratic polynomials, ‎‎Math. Mag., 89 (5) ‎(2016)‎‎, ‎352-357‎.

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Mathematical Culture and Thought
Volume 42, Issue 2 - Serial Number 73
December 2023
Pages 175-183

  • Receive Date 18 July 2022
  • Revise Date 11 August 2022
  • Accept Date 21 August 2022
  • Publish Date 22 December 2023