A Survey in Mean Value Theorems

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\u27s Theorem. The Second Mean Value Theorem for derivatives is generalized, with the use of determinants, to three functions and also generalized in terms of nth 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\u3e1