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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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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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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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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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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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The Converse of Pólya’s Mean Value Theorem
SIAM Journal on Mathematical Analysis, 1987let L be a linear differential operator of the form \(Lu=u^{(n)}+a_ 1(t)u^{(n-1)}+...+a_ n(t)u,\) which is disconjugate on an interval I, where \(a_ i(t)\), \(i=1,2,...,n\) are continuous real functions on an interval \(J\supset I\). Pólya proved that if v is any n times differentiable function on I which has \(n+1\) zeros, then \(Lv(p)=0\) for some ...
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2017
This chapter is dedicated entirely to the Mean Value Theorem and its complex history. The opening section offers modern statements of the Mean Value Theorem and some of its variants, proofs of these results, their interrelations, and some applications. The second section provides to some extent the prehistory of the Mean Value Theorem, from Apollonius ...
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This chapter is dedicated entirely to the Mean Value Theorem and its complex history. The opening section offers modern statements of the Mean Value Theorem and some of its variants, proofs of these results, their interrelations, and some applications. The second section provides to some extent the prehistory of the Mean Value Theorem, from Apollonius ...
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Funny Forms of the Mean Value Theorem
The American Mathematical Monthly, 2015(2015). Funny Forms of the Mean Value Theorem. The American Mathematical Monthly: Vol. 122, No. 8, pp. 780-780.
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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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1986
Giver a curve, y = f(x), we shall use the derivative to give us information about the curve. For instance, we shall find the maximum and minimum of the graph, and regions where the curve is increasing or decreasing. We shall use the mean value theorem, which is basic in the theory of derivatives.
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Giver a curve, y = f(x), we shall use the derivative to give us information about the curve. For instance, we shall find the maximum and minimum of the graph, and regions where the curve is increasing or decreasing. We shall use the mean value theorem, which is basic in the theory of derivatives.
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