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Limit-free derivatives

Published online by Cambridge University Press:  23 January 2015

Christopher J. Sangwin
Christopher J. Sangwin*
Affiliation:
MSOR Network, School of Mathematics, University of Birmingham, Birmingham B15 2TT e-mail:, C.J.Sangwin@bham.ac.uk
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Extract

Algebraic double roots are used by [1] to motivate the following limit-free definition of derivative:

‘A function f(x) has a derivative m at x = a if

for some value c.’

As we shall see later, ‘function’ in this definition will actually be restricted to real polynomials and [1] concludes

‘We have shown how an elementary algebraic principle — double roots — can lead to a complete calculus of polynomials and related functions, without the need for a limit concept.

Type
Articles
Information
The Mathematical Gazette , Volume 95 , Issue 534 , November 2011 , pp. 469 - 482
Copyright
Copyright © The Mathematical Association 2011

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References

1. McAndrew, A., An Rlementary, limit-free calculus for polynomials, Math. Gaz., 94 (March 2010) pp. 67–83.Google Scholar
2. Euler, L., Foundations of differential calculus, Springer (2000). Translated by Blanton, J. from the Latin, Institutiones Calculi Differential, 1755.Google Scholar
3. Needham, T., Visual complex analysis, Oxford oniversity Press (1997).Google Scholar
4. Boyer, C. B., The history of the calculus dnd its conceptual development, Dover (1959).Google Scholar
5. Edwards, C. H., The historical development of the calculus, Springer-Verlag (1979).Google Scholar
6. Robinson, A., Non-standard analysis (revised edition), University Press (1996).Google Scholar
7. Berkeley, O., The Analyst; or a discourse addressed to an Infidel Mathematician, London (1734).Google Scholar
8. Fraser, C. F., Joseph Louis Lagrange's algebraic vision of the calculus, Historia Mathematica 14 (1) (1987) pp. 38–53.Google Scholar
9. Lagrange, J. L., Théorie des Fonctions Analytiques (3rd edn.) Librarie de Bachelier, Paris (1847).Google Scholar
10. Struik, D. J., A source book in mathematics 1200-1800, Harvard College (1969).Google Scholar
11. Hardy, G. H., Weierstrass's non-differentiable function, Transactions of the American Mathematical Society 17 (3) (July 1916) pp. 301–325.Google Scholar
12. Bilodeau, G. G., The origin and early development of non-analytic infinitely differentiate functions, Archive for History of Exact Sciences 27(2) (June 1982) pp. 115–135.Google Scholar
13. Gardiner, A., Understanding infinity: The matnematics of infinite processes, Dover (2003).Google Scholar
14. Shenitzer, A. and Luzin, N., The evolution of function: Part I, The American Mathematical Monthly 105 (1) (1998) pp. 59–67.Google Scholar
15. Luzin, N., The evolution of function: Part II, The American Mathematical Monthly 105 (3) (1998) pp. 263–270.Google Scholar
16. Fowler, J., little o notation, Private correspondence.Google Scholar
17. Lang, S., Undergraduate analysis, Springer (1983).Google Scholar
18. Knuth, D. E., Teach calculus with big O , Notices of the American Mathematical Society 45 (6) (June 1998) pp. 687–688.Google Scholar