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Journal articles on the topic 'The Mean Value Theorem of Differentiation'

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Published: 7 June 2025

Last updated: 16 July 2025

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1

Raikhola, Sher Singh. "An explorative study on "Differentiation, Rolle's theorem, and the mean value theorem: In Vedic and modern methods." International Journal of Statistics and Applied Mathematics 9, no. 5 (2024): 137–44. http://dx.doi.org/10.22271/maths.2024.v9.i5b.1830.

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2

Deng, Yashuang, and Yuhui Shi. "An Error Compensation Method for Improving the Properties of a Digital Henon Map Based on the Generalized Mean Value Theorem of Differentiation." Entropy 23, no. 12 (2021): 1628. http://dx.doi.org/10.3390/e23121628.

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Continuous chaos may collapse in the digital world. This study proposes a method of error compensation for a two-dimensional digital system based on the generalized mean value theorem of differentiation that can restore the fundamental performance of chaotic systems. Different from other methods, the compensation sequence of our method comes from the chaotic system itself and can be applied to higher-dimensional digital chaotic systems. The experimental results show that the improved system is highly consistent with the real chaotic system, and it has excellent chaotic characteristics such as high complexity, randomness, and ergodicity.

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3

BURGIN, MARK. "FUZZY OPTIMIZATION OF REAL FUNCTIONS." International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 12, no. 04 (2004): 471–97. http://dx.doi.org/10.1142/s021848850400293x.

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The main goal of this paper is to develop such means of analysis that allows us to reflect and model vagueness and uncertainty of our knowledge, which result from imprecision of measurement and inaccuracy of computation. To achieve this goal, we use here neoclassical analysis to problems of optimization. Neoclassical analysis extends the scope and results of the classical mathematical analysis by applying fuzzy concepts to conventional mathematical objects, such as functions, sequences, and derivatives. Basing on the theory of fuzzy limits, we construct a fuzzy extension for the classical theory of differentiation in the context of computational mathematics. It is done in the second part of this paper, going after introduction. Two kinds of fuzzy derivatives of real functions are considered: weak and strong ones. In addition, we introduce and study extended fuzzy derivatives, which may take infinite values. In the third part of this paper, fuzzy derivatives are applied to a study of maxima and minima of real functions. Different conditions for maxima and minima of real functions are obtained. Some of them are the same or at least similar to the conditions for the differentiable functions, while others differ in many aspects from those for the standard differentiable functions. Many classical results are obtained as direct corollaries of propositions for fuzzy derivatives, which are proved in this paper. Such results as the Fuzzy Intermediate Value theorem, Fuzzy Fermat's theorem, Fuzzy Rolle's theorem, and Fuzzy Mean Value theorem are proved. These results provide better theoretical base for computational methods of optimization.

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4

Yu, Xiao, Yan Hua, and Yanrong Lu. "Observer-based robust preview tracking control for a class of continuous-time Lipschitz nonlinear systems." AIMS Mathematics 9, no. 10 (2024): 26741–64. http://dx.doi.org/10.3934/math.20241301.

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<p>In this paper, a novel observer-based robust preview tracking controller design method is proposed for a class of continuous-time Lipschitz nonlinear systems with external disturbances and unknown states. First, a state observer is designed to reconstruct unknown system states. Second, using differentiation, the state lifting technique, the differential mean value theorem, and several ingenious mathematical manipulations, an augmented error system (AES) containing the previewable information of a reference signal is constructed, thereby transforming the tracking control problem into a robust $ H_{\infty} $ control problem. Based on linear parameter-varying (LPV) system theory, a sufficient condition for asymptotic stability of a closed-loop system with a robust $ H_{\infty} $ performance level is established in terms of the linear matrix inequality (LMI). Furthermore, a tracking controller, which includes observer-based feedback control, integral control, and preview feedforward compensation, is established for the original system. In particular, the tracking controller design is simplified by computing the observer and tracking controller gains simultaneously via only a one-step LMI algorithm. Finally, numerical simulation results demonstrate that the proposed controller leads to superior improvement in the output tracking performance compared with the existing methods.</p>

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5

Yang, Man, Qiang Zhang, Ke Xu, and Ming Chen. "Adaptive Differentiator-Based Predefined-Time Control for Nonlinear Systems Subject to Pure-Feedback Form and Unknown Disturbance." Complexity 2021 (July 27, 2021): 1–12. http://dx.doi.org/10.1155/2021/7029058.

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In this article, by utilizing the predefined-time stability theory, the predefined-time output tracking control problem for perturbed uncertain nonlinear systems with pure-feedback structure is addressed. The nonaffine structure of the original system is simplified as an affine form via the property of the mean value theorem. Furthermore, the design difficulty from the uncertain nonlinear function is overcome by the excellent approximation performance of RBF neural networks (NNs). An adaptive predefined-time controller is designed by introducing the finite-time differentiator which is used to decrease the computational complexity problem appeared in the traditional backstepping control. It is proved that the proposed control method guarantees all signals in the closed-loop system remain bound and the tracking error converges to zero within the predefined time. Based on the controller designed in this paper, the expected results can be obtained in predefined time, which can be illustrated by the simulation results.

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6

Trokhimchuk, Yu Yu. "Mean-Value Theorem." Ukrainian Mathematical Journal 65, no. 9 (2014): 1418–25. http://dx.doi.org/10.1007/s11253-014-0869-z.

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7

Tokieda, Tadashi F. "A Mean Value Theorem." American Mathematical Monthly 106, no. 7 (1999): 673. http://dx.doi.org/10.2307/2589498.

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8

Tokieda, Tadashi F. "A Mean Value Theorem." American Mathematical Monthly 106, no. 7 (1999): 673–74. http://dx.doi.org/10.1080/00029890.1999.12005102.

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9

PALES, ZSOLT. "A general mean value theorem." Publicationes Mathematicae Debrecen 89, no. 1-2 (2016): 161–72. http://dx.doi.org/10.5486/pmd.2016.7443.

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10

Tong, Jingcheng. "On Flett's mean value theorem." International Journal of Mathematical Education in Science and Technology 35, no. 6 (2004): 936–41. http://dx.doi.org/10.1080/00207390412331271339.

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11

Mercer, Peter R. "On A Mean Value Theorem." College Mathematics Journal 33, no. 1 (2002): 46. http://dx.doi.org/10.2307/1558980.

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12

Penot, J. P. "On the mean value theorem." Optimization 19, no. 2 (1988): 147–56. http://dx.doi.org/10.1080/02331938808843330.

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13

Wooley, Trevor D. "On Vinogradov's mean value theorem." Mathematika 39, no. 2 (1992): 379–99. http://dx.doi.org/10.1112/s0025579300015102.

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14

de Camargo, André Pierro. "The geometric Mean Value Theorem." International Journal of Mathematical Education in Science and Technology 49, no. 4 (2017): 613–15. http://dx.doi.org/10.1080/0020739x.2017.1394503.

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15

Trokhimchuk, Yurii Yu. "To the mean-value theorem." Journal of Mathematical Sciences 188, no. 2 (2012): 128–45. http://dx.doi.org/10.1007/s10958-012-1112-9.

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16

Sand, Mark. "MEAN VALUE THEOREM NO MORE!!" PRIMUS 5, no. 4 (1995): 339–42. http://dx.doi.org/10.1080/10511979508965798.

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17

de Reyna, Juan Arias. "A generalized mean-value theorem." Monatshefte für Mathematik 106, no. 2 (1988): 95–97. http://dx.doi.org/10.1007/bf01298830.

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18

Hutník, Ondrej, and Jana Molnárová. "On Flett’s mean value theorem." Aequationes mathematicae 89, no. 4 (2014): 1133–65. http://dx.doi.org/10.1007/s00010-014-0311-5.

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19

Kim, Sung Soo, and John Holbrook. "A Very Mean Value Theorem." Mathematical Intelligencer 25, no. 1 (2003): 42–47. http://dx.doi.org/10.1007/bf02985637.

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20

Zhou, Yingcheng. "Mean Value Theorem and Its Uses." Highlights in Science, Engineering and Technology 72 (December 15, 2023): 926–30. http://dx.doi.org/10.54097/48pkqt70.

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The foundation of all mathematical analysis is calculus. The mean value theorem of differential equations and the mean value theorem of integral equations are crucial concepts in calculus. They establish the framework for the entire calculus. This essay discusses three different types of mean value theorems. They have some degree of generalization and proof. Thus, the mean value theorems of Lagrange, Rolle, and Cauchy are all correctly demonstrated in this study. The foundation for these three theorems further establishes the calculus fundamental theorem. They can expand the differential mean value theorem and demonstrate the Newton-Leibniz formula. The application of mean value theorems is discussed in this paper. This research is really important. Because mathematics uses the most fundamental and significant tool used in human investigation. Like the progression from the mean value theorem of the differential to the mean value theorem of the integral, from the known to the unknown.

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21

Molnárová, Jana. "On Generalized Flett's Mean Value Theorem." International Journal of Mathematics and Mathematical Sciences 2012 (2012): 1–7. http://dx.doi.org/10.1155/2012/574634.

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We present a new proof of generalized Flett's mean value theorem due to Pawlikowska (from 1999) using only the original Flett's mean value theorem. Also, a Trahan-type condition is established in general case.

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22

Koliha, J. J. "Mean, Meaner, and the Meanest Mean Value Theorem." American Mathematical Monthly 116, no. 4 (2009): 356–61. http://dx.doi.org/10.1080/00029890.2009.11920948.

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23

Bailey, D. F., and G. J. Fix. "A generalization of the mean mean value theorem." Applied Mathematics Letters 1, no. 4 (1988): 327–30. http://dx.doi.org/10.1016/0893-9659(88)90143-7.

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24

Koliha, J. J. "Mean, Meaner, and the Meanest Mean Value Theorem." American Mathematical Monthly 116, no. 4 (2009): 356–61. http://dx.doi.org/10.4169/193009709x470227.

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25

Bhandari, Thaneshor. "A Study on Flett’s Mean Value Theorem." Kaladarpan कलादर्पण 3, no. 1 (2023): 138–41. http://dx.doi.org/10.3126/kaladarpan.v3i1.55264.

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The aim of this study is to evaluate the parallels between the results of Flett's Mean Value Theorem, Lagrange's Mean Value Theorem, and Rolle's Theorem, as well as their geometrical significance. It also covers Thomas M. Flett's 1958 Mean Value Theorem of Differential and Integral Calculus and its various extensions. We intend to present a thorough analysis of various (existing as well as new) adequate conditions for this theorem's validity because it is a topic of interest in many areas of mathematics, including functional equations.

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26

Chen, Zhibing, and Shusen Ding. "A Higher Mean Value Theorem: 10935." American Mathematical Monthly 110, no. 6 (2003): 544. http://dx.doi.org/10.2307/3647923.

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27

Chubarikov, V. N. "On a Certain Mean Value Theorem." Moscow University Mathematics Bulletin 74, no. 1 (2019): 35–37. http://dx.doi.org/10.3103/s0027132219010078.

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28

Merikoski, Jorma K., Markku Halmetoja, and Timo Tossavainen. "Means and the mean value theorem." International Journal of Mathematical Education in Science and Technology 40, no. 6 (2009): 729–40. http://dx.doi.org/10.1080/00207390902825328.

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29

Barbut, Erol, and Gene Denzel. "Meandering around the mean value theorem." International Journal of Mathematical Education in Science and Technology 19, no. 1 (1988): 139–43. http://dx.doi.org/10.1080/0020739880190118.

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30

Wooley, Trevor D. "Corrigendum: On Vinogradov's mean value theorem." Mathematika 40, no. 1 (1993): 152. http://dx.doi.org/10.1112/s0025579300013796.

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31

Wooley, Trevor D. "On Vinogradov's mean value theorem. II." Michigan Mathematical Journal 40, no. 1 (1993): 175–80. http://dx.doi.org/10.1307/mmj/1029004681.

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32

GUO, PENG, CHANGPIN LI, and GUANRONG CHEN. "ON THE FRACTIONAL MEAN-VALUE THEOREM." International Journal of Bifurcation and Chaos 22, no. 05 (2012): 1250104. http://dx.doi.org/10.1142/s0218127412501040.

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In this paper, we derive a fractional mean-value theorem both in the sense of Riemann–Liouville and in the sense of Caputo. This new formulation is more general than the generalized Taylor's formula of Kolwankar and the fractional mean-value theorem in the sense of Riemann–Liouville developed by Trujillo.

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33

Penot, J. P. "Mean-Value Theorem with Small Subdifferentials." Journal of Optimization Theory and Applications 94, no. 1 (1997): 209–21. http://dx.doi.org/10.1023/a:1022672005994.

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34

Nikonorov, Yu G. "On the integral mean value theorem." Siberian Mathematical Journal 34, no. 6 (1993): 1135–37. http://dx.doi.org/10.1007/bf00973476.

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35

He, Yiyang, and Lanxin Liu. "Cauchy Mean Value Theorem in Hyperbolic Plane." Highlights in Science, Engineering and Technology 107 (August 15, 2024): 383–89. http://dx.doi.org/10.54097/n7rb7j50.

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A Hyperbolic number is a generalization of real number. It is a commutative ring with zero factor generated by two real numbers. It is of great value in practical application. Differential mean value theorems play an important role in real analysis, one of which is Cauchy mean value theorem. By studying the correlative properties of hyperbolic numbers, this paper extends the Cauchy mean value theorem in real analysis to the hyperbolic plane, obtains the hyperbolic mean value theorem, and gives a strict proof. Hyperbolic Cauchy mean value theorem lays a further theoretical foundation for the development of hyperbolic analysis, injects new impetus to hyperbolic analysis, and improves the practical application value of hyperbolic analysis.

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36

Lu, Jian, and Qi Wan. "Lagrange's Mean Value Theorem and Taylor's Theorem and Their Applications." Journal of Education and Culture Studies 8, no. 3 (2024): p42. http://dx.doi.org/10.22158/jecs.v8n3p42.

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Lagrange's mean value theorem and Taylor's theorem are two important and widely used formulas in calculus courses. In this paper, we introduce the method for proving Lagrange's mean value theorem and Taylor's theorem using Rolle's theorem, and the application of these two theorems in estimating the value of integrals, determining the concavity and convexity of functions, and solving the limits of functions.

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37

de Camargo, André Pierro. "A New Proof of the Equivalence of the Cauchy Mean Value Theorem and the Mean Value Theorem." American Mathematical Monthly 127, no. 5 (2020): 460. http://dx.doi.org/10.1080/00029890.2020.1722552.

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38

Zuhairoh, Faihatuz, and Kasmila Ulansari. "The Second Mean Value Theorem for Integrals." Jurnal Matematika, Statistika dan Komputasi 21, no. 3 (2025): 608–26. https://doi.org/10.20956/j.v21i3.43133.

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This article discusses the Second Mean Value Theorem for integrals by presenting a comprehensive mathematical proof using a deductive-mathematical approach that involves the Extreme Value Theorem and the Comparison Theorem. Given a continuous function and an integrable function that does not change sign on the interval , it is proven that there exists at least one point such that: \[ \int_a^b f(x)g(x)\,dx = f(c) \int_a^b g(x)\,dx \] The article also provides various examples of the theorem’s application, including numerical computations using the Newton-Raphson method to determine the value of in certain cases. In addition, case studies are presented that link the theorem to modeling in probability, economics, and engineering, thereby demonstrating its relevance in data analysis and dynamic systems. The results of this study not only enrich the theoretical foundation of integral analysis but also offer practical contributions to problem solving in various disciplines.

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39

Wu, Shuchen. "Proof and Application of the Mean Value Theorem." Highlights in Science, Engineering and Technology 72 (December 15, 2023): 565–71. http://dx.doi.org/10.54097/nw2nd028.

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In calculus, mean value theorem (MVT) connects a function's derivative and its rate of change over a certain interval. This paper delves into the mathematical intricacies of the MVT and its multifaceted applications. Through rigorous proofs and illustrative examples, this study establishes the MVT's fundamental role in calculus and its relevance in understanding the behavior of functions. The paper extends its exploration to encompass related theorems, including extreme value theorem, which connects function’s continuity and extrema, Intermediate Value Theorem, which states that the function value within an interval of a continuous function must be between the maximum and minimum values, local extreme value theorem, Rolle’s theorem, a specific situation of the theorem, and the integral MVT, an application in integral aspect of MVT, further enriching the comprehension of these pivotal concepts. These theorems provide powerful tools for understanding the properties of continuous functions, identifying critical points, and establishing relationships between function values and their derivatives. This paper highlights the significance of proving these theorems and solving mathematical problems as applications. Through a systematic exploration of the mathematical foundations, this paper contributes to a deeper comprehension of the core principles underlying calculus and their applied theorems in different contexts.

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40

Mercer, Peter R. "101.07 Cauchy's mean value theorem meets the logarithmic mean." Mathematical Gazette 101, no. 550 (2017): 108–15. http://dx.doi.org/10.1017/mag.2017.15.

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41

Fejzic, H., and D. Rinne. "More on a Mean Value Theorem Converse." American Mathematical Monthly 106, no. 5 (1999): 454. http://dx.doi.org/10.2307/2589151.

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42

Matkowski, Janusz. "Mean-value theorem for vector-valued functions." Mathematica Bohemica 137, no. 4 (2012): 415–23. http://dx.doi.org/10.21136/mb.2012.142997.

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43

Lozada-Cruz, German. "Some variants of Lagrange's mean value theorem." Selecciones Matemáticas 7, no. 1 (2020): 144–50. http://dx.doi.org/10.17268/sel.mat.2020.01.13.

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44

Davitt, R. M., R. C. Powers, T. Riedel, and P. K. Sahoo. "Flett's Mean Value Theorem for Holomorphic Functions." Mathematics Magazine 72, no. 4 (1999): 304. http://dx.doi.org/10.2307/2691225.

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45

Schaumberger, Norman. "More Applications of the Mean Value Theorem." College Mathematics Journal 16, no. 5 (1985): 397. http://dx.doi.org/10.2307/2687000.

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46

Shishkina, E. L. "Mean-Value Theorem for B-Harmonic Functions." Lobachevskii Journal of Mathematics 43, no. 6 (2022): 1401–7. http://dx.doi.org/10.1134/s1995080222090232.

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47

Wooley, Trevor. "Vinogradov's mean value theorem via efficient congruencing." Annals of Mathematics 175, no. 3 (2012): 1575–627. http://dx.doi.org/10.4007/annals.2012.175.3.12.

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48

Kupka, Ivan. "Topological generalization of Cauchy's mean value theorem." Annales Academiae Scientiarum Fennicae Mathematica 41 (February 2016): 315–20. http://dx.doi.org/10.5186/aasfm.2016.4120.

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49

Tong, Jingcheng. "Cauchy's Mean Value Theorem Involving n Functions." College Mathematics Journal 35, no. 1 (2004): 50. http://dx.doi.org/10.2307/4146885.

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50

Ovan, A. Saputra, and N. Tasni. "Integral mean value theorem for discontinuous function." Journal of Physics: Conference Series 1918, no. 4 (2021): 042033. http://dx.doi.org/10.1088/1742-6596/1918/4/042033.

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