Means and the mean value theorem
Authors: Merikoski, Jorma1; Halmetoja, Markku2; Tossavainen, Timo3
Source: International Journal of Mathematical Education in Science and Technology, Volume 40, Number 6, January 2009 , pp. 729-740(12)
Publisher: Taylor and Francis Ltd
- Free Content
- New Content
- Subscribed Content
- Free Trial Content
Abstract:
Let I be a real interval. We call a continuous function μ : I × I → a proper mean if it is symmetric, reflexive, homogeneous, monotonic and internal. Let f : I → be a differentiable and strictly convex or strictly concave function. If a, b ∈ I with a ≠ b, then there exists a unique number ξ between a and b such that f(b) - f(a) = f '(ξ)(b - a). We study under what conditions ξ is a proper mean of a and b, and what kind of means are obtained by applying certain f 's. We also study the converse problem: Given a proper mean μ(a, b), does there exist f such that f(b) - f(a) = f '(μ(a, b))(b - a) for all a, b ∈ I with a ≠ b?Keywords: mean; mean value theorem; teaching of mathematics; undergraduate mathematics
Document Type: Research article
DOI: 10.1080/00207390902825328
Affiliations: 1: Department of Mathematics and Statistics, University of Tampere, Finland 2: Mantta Senior High School, FI-35801 Mantta, Finland 3: Department of Teacher Education, University of Joensuu, FI-57101 Savonlinna, Finland
- Free Content
- New Content
- Subscribed Content
- Free Trial Content
Click here for Page Help