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A Restricted Mean Value Theorem
Journal of the London Mathematical Society, 1969For each prime \(p\) let \(f(p)\) denote the least integer solution \(n\) to the Legendre character conditions \[ \left(\frac{n+a_j}{p}\right) = \varepsilon_j, \quad (j=1,\ldots,k). \] Elliott shows that there exist positive constants \(\alpha\), \(A\) so that \[ \left(\sum_{p\le x} \min(f(p),x^\alpha)\right)/\pi(x) \rightarrow A\quad\text{as }x\to ...
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On a Certain Mean Value Theorem
Moscow University Mathematics Bulletin, 2019zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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The American Mathematical Monthly, 1999
Several theorems go by this name. The present note adds to the assortment an unusual variant (Theorem 1), which involves the shape of the underlying region in an interesting way. We work in Euclidean spaces, although Lemma 2 and the second inequality of Lemma 3 carry over to general Riemannian manifolds.
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Several theorems go by this name. The present note adds to the assortment an unusual variant (Theorem 1), which involves the shape of the underlying region in an interesting way. We work in Euclidean spaces, although Lemma 2 and the second inequality of Lemma 3 carry over to general Riemannian manifolds.
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2002
The derivative of a function f at a point ξ $$f'\left( \xi \right) = \mathop {\lim }\limits_{\Delta x \to 0} {\rm{ }}{{f\left( {\xi + \Delta x} \right) - f\left( \xi \right)} \over {\Delta x}},$$ is the slope of the line tangent to the graph of f at the point P = (ξ ,f (ξ)).
Adi Ben-Israel, Robert Gilbert
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The derivative of a function f at a point ξ $$f'\left( \xi \right) = \mathop {\lim }\limits_{\Delta x \to 0} {\rm{ }}{{f\left( {\xi + \Delta x} \right) - f\left( \xi \right)} \over {\Delta x}},$$ is the slope of the line tangent to the graph of f at the point P = (ξ ,f (ξ)).
Adi Ben-Israel, Robert Gilbert
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The American Mathematical Monthly, 1969
(1969). On Mean Value Theorems. The American Mathematical Monthly: Vol. 76, No. 1, pp. 70-73.
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(1969). On Mean Value Theorems. The American Mathematical Monthly: Vol. 76, No. 1, pp. 70-73.
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1991
Let $$ f\left( \lambda \right) = {{\alpha }_{k}}{{\lambda }^{k}} + \cdots + {{\alpha }_{1}}\lambda $$ be a polynomial of k-th degree with coefficients in J, where \({{\alpha }_{i}} \in M\left( {O({{T}^{{k - i}}})} \right), 1 \leqslant i \leqslant k\) Let $$ s\left( {f\left( \lambda \right)} \right),\xi ,{\text{T}} = s\left( {f,{\text{T ...
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Let $$ f\left( \lambda \right) = {{\alpha }_{k}}{{\lambda }^{k}} + \cdots + {{\alpha }_{1}}\lambda $$ be a polynomial of k-th degree with coefficients in J, where \({{\alpha }_{i}} \in M\left( {O({{T}^{{k - i}}})} \right), 1 \leqslant i \leqslant k\) Let $$ s\left( {f\left( \lambda \right)} \right),\xi ,{\text{T}} = s\left( {f,{\text{T ...
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Mean-Value Theorems in Arithmetic Semigroups
Acta Mathematica Hungarica, 2001zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Lucht, L. G., Reifenrath, K.
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Optimization, 1988
Mean value theorems for nonsmooth functions are presented. Two versions are given, both using the contingent derivative. In. the first one a tangential convexity condition is used. In the second one no convexity assumption is made but the estimate. involves the contingent derivative df (x, b − a) of f at points arbitrarily close to the segment [a, b ...
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Mean value theorems for nonsmooth functions are presented. Two versions are given, both using the contingent derivative. In. the first one a tangential convexity condition is used. In the second one no convexity assumption is made but the estimate. involves the contingent derivative df (x, b − a) of f at points arbitrarily close to the segment [a, b ...
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2014
The main focus of this chapter is the Mean Value Theorem and some of its applications. This is the big theorem in the world of differentiable functions. Many important results in calculus (and well beyond!) follow from the Mean Value Theorem. We also look at an interesting and useful generalization, due to Cauchy.
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The main focus of this chapter is the Mean Value Theorem and some of its applications. This is the big theorem in the world of differentiable functions. Many important results in calculus (and well beyond!) follow from the Mean Value Theorem. We also look at an interesting and useful generalization, due to Cauchy.
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Journal of Mathematical Sciences, 2012
The author proves several extensions of the well-known Lagrange mean value theorem for cases of continuous functions on the real line and in the complex plane. The paper starts with integrating (Denjoy) the equations in the Lagrange mean value theorem and recognizing that the slope of the chord through \((a,f(a))\) and \((b,f(b))\) is equal to the ...
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The author proves several extensions of the well-known Lagrange mean value theorem for cases of continuous functions on the real line and in the complex plane. The paper starts with integrating (Denjoy) the equations in the Lagrange mean value theorem and recognizing that the slope of the chord through \((a,f(a))\) and \((b,f(b))\) is equal to the ...
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