The Mean Value Theorem: Analytical Proof and Computational Approaches

TL;DR

This paper provides formal analytical proofs and computational verification of Rolle's Theorem and the Mean Value Theorem, emphasizing their theoretical significance and practical applications in analysis and modeling.

Contribution

It offers a rigorous analytical proof and computational approaches for the theorems, bridging theory and practice in differential calculus.

Findings

Validated the theorems through analytical proofs

Developed computational methods for theorem verification

Enhanced understanding of geometric interpretations

Abstract

In this paper, we explore two fundamental theorems of differential calculus: Rolle's Theorem and the Mean Value Theorem (MVT). These theorems play a crucial role in the development of theoretical and practical results in mathematics, serving as the basis for various applications in analysis and modeling of real-world phenomena. Initially, we present the formal statements and their respective analytical proofs, highlighting the mathematical rigor necessary for understanding them. Additionally, we discuss the geometric interpretation of both theorems, emphasizing their importance in understanding properties of differentiable functions. The goal of this work is not only to validate these theorems through analytical methods but also to perform their computational verification, providing an integrated view between theory and practice.