A Survey in Mean Value Theorems

Author

David A. Neuser, Utah State University

Date of Award:

5-1970

Document Type:

Thesis

Degree Name:

Master of Science (MS)

Department:

Mathematics and Statistics

Department name when degree awarded

Mathematics

Committee Chair(s)

Stanley G. Wayment

Committee

Stanley G. Wayment

Abstract

A variety of new mean value theorems are presented along with interesting proofs and generalizations of the standard theorems.

Three proofs are given for the ordinary Mean Value Theorem for derivatives, the third of which is interesting in that it is independent of of Rolle's Theorem. The Second Mean Value Theorem for derivatives is generalized, with the use of determinants, to three functions and also generalized in terms of n-th order derivatives.

Observing that under certain conditions the tangent line to the curve of a differentiable function passes through the initial point, we find a new type of mean value theorem for derivatives. This theorem is extended to two functions and later in the paper an integral analog is given together with integral mean value theorems.

Many new mean value theorems are presented in their respective settings including theorems for the total variation of a function, the arc length of the graph of a function, and for vector-valued functions. A mean value theorem in the complex plane is given in which the difference quotient is equal to a linear combination of the values of the derivative. Using a regular derivative, the ordinary Mean Value Theorem for derivatives is extended into Rⁿ, n > 1.

Checksum

e680e345fbd77521daba3ec0e77bf764

Recommended Citation

Neuser, David A., "A Survey in Mean Value Theorems" (1970). All Graduate Theses and Dissertations, Spring 1920 to Summer 2023. 6842.
https://digitalcommons.usu.edu/etd/6842