Abstract
In this chapter, which is independent of all subsequent chapters, we allow ourselves a brief diversion. We have met and used Rolle’s Theorem (Theorem 5.1), its extension the Mean Value Theorem (Theorem 5.2), and its extension Cauchy’s Mean Value Theorem (Theorem 5.11). Here we consider other Mean Value – type theorems. Each of these, as with their namesake, has an appealing geometric interpretation.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
One cannot fix one’s eyes on the commonest natural production without finding food for a rambling fancy.
—Mansfield Park, by Jane Austen
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Boas, R.P., Jr.: A Primer of Real Functions. Mathematical Association of America, Washington, DC (1981)
Flanders, H.: In: Larson, L.L. (ed.) Review of Problem-Solving Through Problems. Springer, New York (1983). Am. Math. Mon. 92, 676–678 (1985)
Flett, T.M.: A mean value theorem. Math. Gaz. 42, 38–39 (1958)
Gillman, L.: Order relations and a proof of L’Hôpital’s rule. Coll. Math. J. 28, 288–292 (1997)
Martinez, S.M., Martinez de la Rosa, F.: A generalization of the mean value theorem. Wolfram Demonstrations Project. (http://demonstrations.wolfram.com)
Mercer, P.R.: On a mean value theorem. Coll. Math. J. 33, 46–48 (2002)
Meyers, R.E: Some elementary results related to the mean value theorem. Coll. Math. J. 8, 51–53 (1977)
Nadler, S.B., Jr.: A proof of Darboux’s theorem. Am. Math. Mon. 117, 174–175 (2010)
Neuser, D.A., Wayment, S.G.: A note on the intermediate value property. Am. Math. Mon. 81, 995–997 (1974)
Olsen, L.: A new proof of Darboux’s theorem. Am. Math. Mon. 111, 713–715 (2004)
Pompeiu, D.: Sur une proposition analogue au theoreme des accroissements finis. Mathematica (Cluj.) 22, 143–146 (1946)
Reich, S.: On mean value theorems. Am. Math. Mon. 76, 70–73 (1969)
Sahoo, P.K., Riedel, T.: Mean value theorems and functional equations. World Scientific, Singapore/River Edge (1998)
Talman, L.A.: Simpson’s rule is exact for cubics. Am. Math. Mon. 113, 144–155 (2006)
Tong, J.: The mean value theorems of Lagrange and Cauchy. Int. J. Math. Educ. Sci. Tech. 30, 456–458 (1999)
Tong, J.: The mean value theorem of Lagrange generalized to involve two functions. Math. Gaz. 84, 515–516 (2000)
Tong, J.: The mean-value theorem generalised to involve two parameters. Math. Gaz. 88, 538–540 (2004)
Trahan, D.H.: A new type of mean value theorem. Math. Mag. 39, 264–268 (1966)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer Science+Business Media New York
About this chapter
Cite this chapter
Mercer, P.R. (2014). Other Mean Value Theorems. In: More Calculus of a Single Variable. Undergraduate Texts in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-1926-0_7
Download citation
DOI: https://doi.org/10.1007/978-1-4939-1926-0_7
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4939-1925-3
Online ISBN: 978-1-4939-1926-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)