One-sided differentiability: a challenge for computer algebra systems

Enrique Ferres-López; Eugenio Roanes-Lozano; Angélica Martínez-Zarzuelo; Fernando Sánchez; Enrique Ferres-López; Eugenio Roanes-Lozano; Angélica Martínez-Zarzuelo; Fernando Sánchez

doi:10.3934/era.2023090

Electronic Research Archive

2023, Volume 31, Issue 3: 1737-1768. doi: 10.3934/era.2023090

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Research article Special Issues

One-sided differentiability: a challenge for computer algebra systems

1.
Área de Ingeniería, Centro Universitario de Tecnología y Arte Digital, Madrid, Spain
2.
Universidad Complutense de Madrid, Madrid, Spain
3.
Instituto de Matemática Interdisciplinar (IMI), Universidad Complutense de Madrid, Madrid, Spain
4.
Departamento de Didáctica de Ciencias Experimentales, Sociales y Matemáticas, Facultad de Educación-Centro de Formación del Profesorado, Universidad Complutense de Madrid, Madrid, Spain
5.
Departamento de Matemáticas, Facultad de Ciencias, Universidad de Extremadura, Badajoz, Spain

Received: 22 October 2022 Revised: 21 January 2023 Accepted: 28 January 2023 Published: 06 February 2023

Computer Algebra Systems (CASs) are extremely powerful and widely used digital tools. Focusing on differentiation, CASs include a command that computes the derivative of functions in one variable (and also the partial derivative of functions in several variables). We will focus in this article on real-valued functions of one real variable. Since CASs usually compute the derivative of real-valued functions as a whole, the value of the computed derivative at points where the left derivative and the right derivative are different (that we will call conflicting points) should be something like "undefined", although this isn't always the case: the output could strongly differ depending on the chosen CAS. We have analysed and compared in this article how some well-known CASs behave when addressing differentiation at the conflicting points of five different functions chosen by the authors. Finally, the ability for calculating one-sided limits of CASs allows to directly compute the result in these cumbersome cases using the formal definition of one-sided derivative, which we have also analysed and compared for the selected CASs. Regarding teaching, this is an important issue, as it is a topic of Secondary Education and nowadays the use of CASs as an auxiliary digital tool for teaching mathematics is very common.
- computer algebra systems,
- one-sided differentiability,
- real-valued functions of one real variable,
- limits,
- software comparison
Citation: Enrique Ferres-López, Eugenio Roanes-Lozano, Angélica Martínez-Zarzuelo, Fernando Sánchez. One-sided differentiability: a challenge for computer algebra systems[J]. Electronic Research Archive, 2023, 31(3): 1737-1768. doi: 10.3934/era.2023090

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Abstract

Computer Algebra Systems (CASs) are extremely powerful and widely used digital tools. Focusing on differentiation, CASs include a command that computes the derivative of functions in one variable (and also the partial derivative of functions in several variables). We will focus in this article on real-valued functions of one real variable. Since CASs usually compute the derivative of real-valued functions as a whole, the value of the computed derivative at points where the left derivative and the right derivative are different (that we will call conflicting points) should be something like "undefined", although this isn't always the case: the output could strongly differ depending on the chosen CAS. We have analysed and compared in this article how some well-known CASs behave when addressing differentiation at the conflicting points of five different functions chosen by the authors. Finally, the ability for calculating one-sided limits of CASs allows to directly compute the result in these cumbersome cases using the formal definition of one-sided derivative, which we have also analysed and compared for the selected CASs. Regarding teaching, this is an important issue, as it is a topic of Secondary Education and nowadays the use of CASs as an auxiliary digital tool for teaching mathematics is very common.

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