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PreprintCC BY-NC-SA 4.0arXiv

OFFER (One-Formula-Facilitates-Every-Relation) for Calculus

  • Samuel S. P. Shen
Open at sourcePDFDOICite

Abstract

The large number of calculus formulas makes learning calculus heavily memory-dependent and can scare away many students. Based on Issac Newton's idea of prediction, this paper shows that calculus students need to master only one critical formula: his proposition that, given boundary data, one can predict the interior and vice versa. We can treat other formulas as consequences of the formula for this proposition. In 1-, 2-, and 3-dimensional spaces, this formula corresponds to the fundamental theorem of calculus, Green's theorem, and the divergence theorem. This approach is summarized as OFFER (one-formula-facilitates-every-relationship) for calculus education. This paper demonstrates the method and advantages of the OFFER pedagogy. The OFFER approach can accelerate the pace of calculus teaching, provide deeper insight into calculus, inspire questions from students, and naturally facilitate the inclusion of real-world data, artificial intelligence (AI), and computer coding in the calculus series. Consequently, OFFER may serve as a way to modernize calculus curricula in the era of AI and the digital economy.

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Cite

BibTeXRIS
BibTeX
@misc{shen2026offer,
  title = {{OFFER (One-Formula-Facilitates-Every-Relation) for Calculus}},
  author = {Samuel S. P. Shen},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.28674},
  eprint = {2610.28674},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.28674},
  note = {CC BY-NC-SA 4.0},
}
RIS
TY  - UNPB
TI  - OFFER (One-Formula-Facilitates-Every-Relation) for Calculus
AU  - Samuel S. P. Shen
PY  - 2026
DA  - 2026/10/19
PB  - arXiv
DO  - 10.48550/arxiv.2610.28674
UR  - https://arxiv.org/abs/2610.28674
AB  - The large number of calculus formulas makes learning calculus heavily memory-dependent and can scare away many students. Based on Issac Newton's idea of prediction, this paper shows that calculus students need to master only one critical formula: his proposition that, given boundary data, one can predict the interior and vice versa. We can treat other formulas as consequences of the formula for this proposition. In 1-, 2-, and 3-dimensional spaces, this formula corresponds to the fundamental theorem of calculus, Green's theorem, and the divergence theorem. This approach is summarized as OFFER (one-formula-facilitates-every-relationship) for calculus education. This paper demonstrates the method and advantages of the OFFER pedagogy. The OFFER approach can accelerate the pace of calculus teaching, provide deeper insight into calculus, inspire questions from students, and naturally facilitate the inclusion of real-world data, artificial intelligence (AI), and computer coding in the calculus series. Consequently, OFFER may serve as a way to modernize calculus curricula in the era of AI and the digital economy.
ER  -