Hostname: page-component-7479d7b7d-q6k6v Total loading time: 0 Render date: 2024-07-12T11:23:30.813Z Has data issue: false hasContentIssue false
  • English
  • Français

Tchebycheff's mean value theorem and some results derivable therefrom

Published online by Cambridge University Press:  11 August 2014

Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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