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Tchebycheff's mean value theorem and some results derivable therefrom

Published online by Cambridge University Press:  11 August 2014

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Extract

f(x), f′(x),…, f(n+1)(x) are one-valued, finite and continuous (v≤x≤w).

Define

Then*

u (x) is one-valued, finite and continuous (v ≤ x ≤ w).

by writting

Type
Research Article
Information
Journal of the Staple Inn Actuarial Society , Volume 7 , Issue 2 , October 1947 , pp. 70 - 80
Copyright
Copyright © Institute of Actuaries Students' Society 1947

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References

page 70 note * By successive partial integration of the equation

page 70 note † A sufficient Gegenbeispiel for the hypothesis that v≤xp≤w, p=1, 2, …, n, is provided by: u (x) = 1/(1·01–x), n = 2, v = 0, w = 1. The solution is m 1 = ·793, m 2 = ·693 giving x 1 = 1·046, x 2 = ·541. In any such case it is necessary to extend the bounded interval for which f (x) is one-valued, finite and continuous to include all values of xp .

page 71 note * Because

page 75 note * Some of this information was kindly supplied by John Brown and Co. of Clydebank.

page 75 note † . See Biometrika, Vol. IV (1905/1906), Part III, p. 379 Google Scholar; J.I.A. Vol. XL (1906), p. 117 Google Scholar; Attwood's, Theoretical Naval Architecture, p. 12 Google Scholar; A.M. p. 307; M.A.S., Part II, p. 196.

page 76 note *

page 76 note †

page 80 note * Accurate values were found from the formulae