scholarly journals Calculus for the intermediate point associated with a mean value theorem of the integral calculus

2020 ◽  
Vol 28 (1) ◽  
pp. 59-66
Author(s):  
Emilia-Loredana Pop ◽  
Dorel Duca ◽  
Augusta Raţiu
Keyword(s):  
Continuous Functions ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Integral Calculus ◽  
Intermediate Point

AbstractIf f, g: [a, b] → 𝕉 are two continuous functions, then there exists a point c ∈ (a, b) such that\int_a^c {f\left(x \right)} dx + \left({c - a} \right)g\left(c \right) = \int_c^b {g\left(x \right)} dx + \left({b - c} \right)f\left(c \right).In this paper, we study the approaching of the point c towards a, when b approaches a.

scholarly journals Properties of the intermediate point from a mean value theorem of the integral calculus - II

2019 ◽  
Vol 27 (1) ◽  
pp. 29-36
Author(s):  
Emilia-Loredana Pop ◽  
Dorel Duca ◽  
Augusta Raţiu
Keyword(s):  
Continuous Functions ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Point Function ◽  
Integral Calculus ◽  
Intermediate Point

Abstract In this paper we consider two continuous functions f, g : [a, b] → ℝ and we study for these ones, under which circumstances the intermediate point function is four order di erentiable at the point x = a and we calculate its derivative.

The Multidirectional Mean Value Theorem in Banach Spaces

1997 ◽  
Vol 40 (1) ◽  
pp. 88-102 ◽  
Author(s):  
M. L. Radulescu ◽  
F. H. Clarke
Keyword(s):  
Banach Spaces ◽  
Hilbert Spaces ◽  
Continuous Functions ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Locally Lipschitz Functions ◽  
Bump Function ◽  
Lipschitz Continuous ◽  
Lower Semi ◽  
Locally Lipschitz

AbstractRecently, F. H. Clarke and Y. Ledyaev established a multidirectional mean value theorem applicable to lower semi-continuous functions on Hilbert spaces, a result which turns out to be useful in many applications. We develop a variant of the result applicable to locally Lipschitz functions on certain Banach spaces, namely those that admit a C1-Lipschitz continuous bump function.

scholarly journals Sensitivity of the “intermediate point” in the mean value theorem: an approach via the Legendre-Fenchel transformation

2021 ◽  
Vol 71 ◽  
pp. 114-120
Author(s):  
Jean-Baptiste Hiriart-Urruty
Keyword(s):  
Asymptotic Behavior ◽  
Convex Functions ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Intermediate Point ◽  
Mild Conditions ◽  
The Mean

We study the sensitivity, essentially the differentiability, of the so-called “intermediate point” c in the classical mean value theorem $ \frac{f(a)-f(b)}{b-a}={f}^{\prime}(c)$we provide the expression of its gradient ∇c(d,d), thus giving the asymptotic behavior of c(a, b) when both a and b tend to the same point d. Under appropriate mild conditions on f, this result is “universal” in the sense that it does not depend on the point d or the function f. The key tool to get at this result turns out to be the Legendre-Fenchel transformation for convex functions.

scholarly journals An Integral Mean Value Theorem concerning Two Continuous Functions and Its Stability

2015 ◽  
Vol 2015 ◽  
pp. 1-4 ◽  
Author(s):  
Monea Mihai
Keyword(s):  
Continuous Functions ◽  
Mean Value ◽  
Mean Value Theorem ◽  
The Mean ◽  
The Stability

The aim of this paper is to investigate an integral mean value theorem proposed by one of the references of this paper. Unfortunately, the proof contains a gap. First, we present a counterexample which shows that this theorem fails in this form. Then, we present two improved versions of this theorem. The stability of the mean point arising from the second result concludes this paper.

scholarly journals A note on the first mean value theorem in integral calculus

1969 ◽  
Vol 094 (2) ◽  
pp. 199-201
Author(s):  
David Preiss ◽  
Jaromír Uher
Keyword(s):  
Mean Value ◽  
Mean Value Theorem ◽  
Integral Calculus
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