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The fundamental theorem of calculus: visually?

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Abstract

The paper wants to show how it is possible to develop based on an adequate basic idea (so-called “Grundvorstellung”) of the derivative a visual understanding of the (first) Fundamental theorem of Calculus.

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References

  • Bender, P. (1990). Zwei “Zugänge” zum Integral-Begriff? Mathematica Didactica, 13(34), 102–127.

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  • Breuker, U. (1991). Was heißt den hier anschaulich? Mathematisch-naturwissenschaftlicher Unterricht, 44, 274–284.

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  • Kaiser-Meßmer, G. (1986). Anwendungen im Mathematikunterricht (Vol. 2). Bad Salzdetfurth: Franzbecker.

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  • Kirsch, A. (1976). Eine “intellektuell ehrliche” Einführung des Integralbegriffs in Grundkursen. Didaktik der Mathematik, 4, 87–105.

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Authors and Affiliations

  1. Department of Mathematics, University of Kassel, Kassel, Germany

    Arnold Kirsch

Authors
  1. Arnold Kirsch
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Correspondence to Arnold Kirsch.

Additional information

A. Kirsch: Deceased, October 14, 2013.

Translated version of the paper “Der Hauptsatz—anschaulich?”, which appeared 1996 in the journal mathematik lehren, issue 78, pp. 55–59.

Appendix

Appendix

1.1 Worksheet 1

  1. 1.

    Draw the graph of the “area collection function” F 0 for the given function f, as in the example above. Calculate the area under f geometrically, first at individual points, then generalize for x. Also state the term for F 0(x)!

  1. 2.

    Graph the “area collection” function F 1 (starting at point a = 1) for all four functions f. Compare the general shape of F 1 to that of F 0.

  2. 3.

    The function F 1(x) also has a value for x = 0 (in general: for x < 1). What does it mean?

1.2 Worksheet 2

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Kirsch, A. The fundamental theorem of calculus: visually?. ZDM Mathematics Education 46, 691–695 (2014). https://doi.org/10.1007/s11858-014-0608-9

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  • DOI: https://doi.org/10.1007/s11858-014-0608-9

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