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An elementary, limit-free calculus for polynomials

Published online by Cambridge University Press:  23 January 2015

Alasdair McAndrew
Alasdair McAndrew*
Affiliation:
Victoria University, PO Box 14428, Melbourne, Victoria 8001, Australia, e-mail:Alasdair.McAndrew@vu.edu.au
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Extract

This work grew out of a simple problem: how could I introduce the basic concepts of calculus to students who were mathematically under-prepared? It has been pointed out by many authors (see [1] and its references) that limits are a major stumbling block for many students beginning their study of calculus. Part of the difficulty with limits is the conceptual confusion between process and value. Some limits can be evaluated by a simple process of substitution, for example,

while others cannot, for example,

Type
Articles
Information
The Mathematical Gazette , Volume 94 , Issue 529 , March 2010 , pp. 67 - 83
Copyright
Copyright © The Mathematical Association 2010

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References

1. Goos, Merrilyn, Stillman, Gloria, and Vale, Colleen, Teaching Secondary School Mathematics, Research and practice for the 21 st century, Allen & Unwin (2007).Google Scholar
2. Keisler, H. Jerome, Elementary calculus: an approach using infinitesimals (2000). Available at: http.//www.rnath.wisc.edu/~keisler/calc.html, accessed 6 February 2008.Google Scholar
3. Tall, David, Intuitive infinitesimals in the calculus. Fourth International Congress on Mathematics Education, 1980. Available at http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1980c-intuitive-infls%.pdf, accessed 6 February 2008.Google Scholar
4. Machover, Moshe, The place of nonstandard analysis in mathematics and mathematics teaching, The British Journal for the Philosophy of Science, 44 (1993) pp. 205–212.Google Scholar
5. Sullivan, Kathleen, The teaching of elementary calculus using the nonstandard analysis approach, Amer. Math. Monthly, 83(5) (May 1976) pp. 370–375.Google Scholar
6. Sedinger, Harry, Derivatives without limits, Two Year College Mathematics Journal, 11(1) (January 1980) pp. 55–56.Google Scholar
7. Marsden, Jerrold and Weinstein, Alan, Calculus unlimited (1981). Available at http://caltechbook.library.caltech.edu/197/, accessed 7 February 2008.Google Scholar
8. Traylor, D. Reginald and Roman, Julia S., Finding derivatives without the notion of limits. In Seventh Annual International Conference on Technology in Collegiate Mathematics (1994). Available at http://archives.math.utk.edu/ICTCM/VOL07/C010/paper.pdf Google Scholar
9. Dovermann, Karl Heinz. Applied Calculus (1999). Available at http://www.math.hawaii.edu/~heiner/calculus.pdf, accessed on 5 February 2008.Google Scholar
10. Invernizzi, Sergio and Rinaldi, Maurizio, A limit-free approach to derivatives: Report on a classroom project. In 2nd International Conference on the teaching of mathematics (2002).Google Scholar
11. Brand, Neal. Derivatives without limits. Available at http://www.math.unt.edu/~brand/projects/calc1/algder.pdf Google Scholar
12. French, Doug, Derivatives without limits, Math. Gaz., 86 (July 2002) pp.279–281.CrossRefGoogle Scholar
13. Suzuki, Jeff, The lost calculus (1637-1670): Tangency and optimization without limits. Maths. Mag. 78(5) (December 2005) pp. 339–353.Google Scholar
14. Hendel, Russell Jay, Six proofs, The College Mathematics Journal, 21(4) (September 1990) pp. 312–313.Google Scholar