Teaching Calculus with Infinitesimals and Differentials

Keywords: Calculus; differentials; infinitesimals; nonstandard analysis; definite integral

Abstract: Several new approaches to calculus in the U.S. have been studied recently that are grounded in infinitesimals or differentials rather than limits. These approaches seek to restore to differential notation the direct referential power it had during the first century after calculus was developed. In these approaches, a differential equation like dy = 2 xdx is a relationship between increments of x and y , making dy / dx an actual quotient rather than code language for the limit of (1/h)[ f ( x + h ) - f ( x )] as h approaches 0. A definite integral of 2 x dx is a sum of pieces of the form 2 x dx , not the limit of a sequence of Riemann sums. In this article I motivate and describe some key elements of differentials-based calculus courses, and I summarize research indicating that students in such courses develop robust quantitative meanings for notations in single- and multi-variable calculus. (Click here for article)

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# Rob Ely

11.06.2021

Teaching Calculus with Infinitesimals and Differentials

Keywords: Calculus; differentials; infinitesimals; nonstandard analysis; definite integral

Abstract: Several new approaches to calculus in the U.S. have been studied recently that are grounded in infinitesimals or differentials rather than limits. These approaches seek to restore to differential notation the direct referential power it had during the first century after calculus was developed. In these approaches, a differential equation like dy = 2 xdx is a relationship between increments of x and y , making dy / dx an actual quotient rather than code language for the limit of (1/h)[ f ( x + h ) - f ( x )] as h approaches 0. A definite integral of 2 x dx is a sum of pieces of the form 2 x dx , not the limit of a sequence of Riemann sums. In this article I motivate and describe some key elements of differentials-based calculus courses, and I summarize research indicating that students in such courses develop robust quantitative meanings for notations in single- and multi-variable calculus. (Click here for article)

## 00 MatRIC Resources

 1 Pat Thompson & Guershon Harel 8:10 2 David Bressoud 4:09 3 Tin Lam Toh 6:09 4 Kristin Frank & Pat Thompson 4:46 5 Imeme Ghedamsi & Thomas Lecorre 6:22 6 Michael Tallman, Zackery Reed, Michael Oehrtman, & Marilyn P. Carlson 7:17 7 Rob Ely 6:34 8 Ricardo Scucuglia R. da Silva, Lara Martins Barbosa, Marcelo C. Borba, & Andre Luis Andrejew Ferreira 4:39 9 Fernando Hitt & Sarah Dufour 5:17 10 Gilbert Greefrath, Reinhard Oldenburg, Hans-Stefan Siller, Volker Ulm, & Hans-Georg Weigand 6:10 11 Hyunkyoung Yoon, Younggon Bae, Woong Lim, & Oh Nam Kwon 6:28 12 Tommy Dreyfus, Anatoli Kouropatov, & Gila Ron 5:51 13 Rafael Martinez-Planell & Maria Trigueros 6:17 14 Guershon Harel 7:01