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Ukrainian Mathematical Journal, 2014
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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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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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2012
In elementary calculus we learn the mean value theorem: Let f be a real-valued function defined on a closed bounded interval \( \subset \mathbb{R}\) . If f is continuous on and differentiable on (a,b), then there is a point c e (a,b) such that $$f(b) - f(a) =\dot{ f}(c)(b - a).$$
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In elementary calculus we learn the mean value theorem: Let f be a real-valued function defined on a closed bounded interval \( \subset \mathbb{R}\) . If f is continuous on and differentiable on (a,b), then there is a point c e (a,b) such that $$f(b) - f(a) =\dot{ f}(c)(b - a).$$
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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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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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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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Means and the mean value theorem
International Journal of Mathematical Education in Science and Technology, 2009Let I be a real interval. We call a continuous function μ : I × I → ℝ a proper mean if it is symmetric, reflexive, homogeneous, monotonic and internal. Let f : I → ℝ be a differentiable and strictly convex or strictly concave function. If a, b ∈ I with a ≠ b, then there exists a unique number ξ between a and b such that f(b) − f(a) = f ′(ξ)(b − a).
Jorma K. Merikoski +2 more
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Mathematika, 1968
\textit{E. Bombieri} [Mathematika 12, 201--225 (1965; Zbl 0136.33004)] proved the following theorem on the mean value of the remainder term in the prime number theorem for arithmetic progressions. For each positive constant \(A\) there is a positive constant \(B\) such that if \(Q=x^{1/2} \ell^{-s}\) then \[ \sum_{q\le Q} \max_{y\le x} \max_{(a,q)=1 ...
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\textit{E. Bombieri} [Mathematika 12, 201--225 (1965; Zbl 0136.33004)] proved the following theorem on the mean value of the remainder term in the prime number theorem for arithmetic progressions. For each positive constant \(A\) there is a positive constant \(B\) such that if \(Q=x^{1/2} \ell^{-s}\) then \[ \sum_{q\le Q} \max_{y\le x} \max_{(a,q)=1 ...
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