# Unravelling Three Differential Mean Value Theorems in Calculus

## Authors

• Zhaosheng Fan
• Yiran Fu
• Haoling Xu

## Keywords:

Mean value theorem; Definite integral; Differential calculus.

## Abstract

The theoretical foundation of differential aspect in real analysis is the differential mean value theorem, which connects to both the local and total properties of the copula function. The link between the copula function's local and overall properties is differential calculus. The maximum value, minimum value, extreme value, monotonicity, and other difficulties may all be resolved with the help of these differential mean value theorems. Additionally, they aid in the computation of limits, the demonstration of inequality, and the identification of the curve's inflection point and concave-convex interval. They are also capable of mapping the function and locating the equation's root. To tackle these issues, one can apply a number of significant findings from the differential mean value theorem. This study gives the formulae for solving three distinct mean value theorems. The Cauchy, Lagrange, and Roller theorems are examples of these mean value theorems. Understanding the connections between the three mean value theorems is made easier by these studies, which also provide an explanation of the theory underlying the theorems and provide examples and visuals of their practical application.

## References

Huang Haisong. Research on the Proof and Application of Lagrange's Mean Value Theorem. Journal of Liuzhou Vocational and Technical College, 2018, 18(3): 4-6.

Sun Na. On Proving Methods of Lagrange’s Mean Value Theorem. Study in college mathematics, 2020, 23(5): 2-3.

Xiang Mingyin, Fang Huiping. Using the Mean Value Theorem for Integrals to Calculate Limit. Journal of Huangshan University, 2014, 16(5): 2-3.

Li S. Application of Cauchy’s Mean Value Theorem. Study of College Mathematics. 1999, 2(3): 1-2.

Sayrafiezadeh M. A generalization of Mean Value theorem for integral. The College Mathematics Journal. 2018, 26(3): 223-224.

Guo Shuping, Yuan Daming. The Application of Cauchy’s Mean Value Theorem to the Proof and Construction of the Inequality. Journal of Studies in College Mathematics. 2021, 13(5): 23-29.

Revathy P. Understanding Rolle’s Theorem. The Mathematics Educator 2009, 19(1): 18-26.

Kang Xiaorong. Several Notes on the Rolle Theorem. Study in college mathematics, 2015, 18(5): 1-2.

Wu Ruihua. On Skills of Using Rolle’s Theorem. Study in college mathematics, 2020, 23(5): 1-2.

Zhou Wei. Notes on mean value theorems. Studies in College Mathematics, 2022, 25(5): 14-16.

29-03-2024

Articles

## How to Cite

Fan, Z., Fu, Y., & Xu, H. (2024). Unravelling Three Differential Mean Value Theorems in Calculus. Highlights in Science, Engineering and Technology, 88, 790-795. https://doi.org/10.54097/vap0v228