罗尔中值定理的一些新证法_英文_

　R eceived d ate :2006207217

第24卷第4期

大　学　数　学Vol.24,№.42008年8月COLL EGE MA T H EMA TICS Aug.2008

So me New Ways to Prove Rolle ’s Theorem

YA O J i n g 2s un

(Dept.of Math.,Anhui Normal University ,Wuhu 241000,China )

Abstract :We give three new methods proving Rolle ’s Theorem.The second simple way is only dependent on the well 2known Heine 2Borel Covering Theorem.This implies that Rolle ’s Theorem is the direct consequence of completeness of real numbers.

K ey w ords :Rolle ’s theorem ;completeness of real numbers ;f ull cover ;Heine Borel covering theorem ;

δ2fine tagged partition

C LC Number :O171 Document Code :C Article I

D :167221454(2008)0420131203

The st udy on Rolle ’s Theorem as well as ot her mean value t heorems of differentials is a very att ractive issue and it was also involved in calculus reform in U SA.Many scholars have done a great deal of work during t he past decade [1-3].We know t hat if Rolle ’s Theorem is proved ,it can be used to p rove Lagrange Mean Value Theorem and Cauchy Mean Value Theorem so long as a corresponding auxiliary f unction is const ructed.Therefore ,it is better to say Rolle ’s Theorem is t he essence and basis of t he next two t heorems t han to say t he conclusions of t he next two t heorems seem to have wider applicability t han t hat of Rolle ’s Theorem.To make t hings simpler ,people lay emp hasis on discussing t he ways to p rove Rolle ’s Theorem.The articles of professor Xu Ji 2hong [4]and t he aut hor [5]respectively give a new way to p rove Rolle ’s Theorem.In t he paper ,we shall give some met hods p roving Rolle ’s Theorem by some forms of completeness of real numbers.

Def inition 1　A collection C of clo sed subintervals of [a ,b]is a f ull cover of [a ,b]if to each x ∈[a ,b]t here corresponds a number δ(x )>0such t hat every closed subinterval of [a ,b ]t hat contains x and has lengt h less t hat δ(x )belongs to C [6].

Lemm a 1　If C is a f ull cover of [a ,b],t hen C contains a partition of [a ,b],i.e.,t here exist a =x 0,x 1,…,x n =b such t hat x k -1

Def inition 2　A partition of an interval [a ,b]is a finite collection of non 2overlapping closed intervals whose union is [a ,b].A tagged partition of [a ,b]is a partition of [a ,b]wit h one point ,referred to as t he tag ,cho sen f rom each interval comprising t he partition.A tagged partition of [a ,b]will be denoted by {(c i ,[x i -1,x i ]):1≤i ≤n},where

a =x 0

and c i ∈[x i -1,x i ]is t he tag of t he interval [x i -1,x i ]for each index i .Now let δbe a positive f unction defined on [a ,b].A δ2fine tagged partition of [a ,b]is a tagged partition {(c i ,[x i -1,x i ]):1≤i ≤n}of

[a ,b]t hat

satisfies