Asymptotic Behavior Of "Intermediate Point" Of Mid-value Theorems With Dini Derivative And The Application Of Dini Derivatives

The differential mid-value theorems are one of the important content of mathematical analysis. The meaning of the differential mid-value theorems is self-evident. It is well known that the mid-value theorems are only confirmed existence of "intermediate point", and does not give the definite position. Even so, it does not affect its wide range of applications. Since 1982, two papers which are published in the American Mathematical Monthly begun to study the asymptotic behavior of "intermediate point" of mid-value theorems when the interval length tends to zero, there have been a lot of researches on asymptotic behavior of "intermediate point" of derivative functions'mid-value theorems, but it is not much to study the asymptotic behavior of "intermediate point" of mid-value theorems with Dini derivatives. At present, only one article does this work is written by Lin Rongfei and Zhang Guocai, so the research on the Dini derivatives and its applications are very necessary. This thesis consists of three parts. Chapter one:The basic definitions and necessary lemmas are posed in this paper. Chapter two:The several L'Hospitals'rules with Dini derivatives(adapt to (?); adapt to (?) adapt to (?)) are established. When the interval length tends to infinite or zero, by using the tool which are established in §2, we study the asymptotic behavior of "intermediate point" of mid-value theorems and Taylor formula with Dini derivatives.Chapter three:By using the Dini derivatives, we study the properties of convex functions and continuous functions.