利用微分中值定理证明不等式

给出利用Lagrange中值定理和Cauchy中值定理证明不等式的方法和步骤,同时用一些例子进行说明．     

微积分学中中值定理的统一证明

由一个定理的结论,给出Lagrange中值定理,Cauchy中值定理,积分中值定理和Taylor中值定理的统一证明及一个计算待定型极限的方法.     

积分型中值定理的推广及统一表示

通过减弱连续的条件,推广了一类积分型中值定理,在适当的条件下,用一个式子将Lagrange中值定理、Cauchy微分中值定理、积分型Cauchy中值定理、积分中值定理、积分第一中值定理、Lagrange型积分中值定理、Cauchy型积分中值定理及推广的积分第一中值定理这8个中值定理统一起来.     

关于三个微分中值定理的证明

<正> 在我们所见到的书中,三个微分中值定理的证明顺序依次为Rolle定理,Lagrange定Cauchy定理。本文按与上述完全相反的顺序给出证明,使整个证明显得十分简捷。     

两个微分中值定理证明中辅助函数的多种作法

在数学分析中 ,三个微分中值定理极为重要 .在证明 Lagrange中值定理和 Cauchy中值定理时 ,都少不了作辅助函数 ,各种版本的《数学分析》或《高等数学》书本中 ,都只给出了一种形式的辅助函数 .为了扩展思路 ,给出了多种形式的辅助函数 ,并得出了一般形式 .     

微分中值定理的扩展与证明

1.对拉格朗日(Lagrange)中值定理和柯西(Cauchy)中值定理,从结论的几何意义出发,各列举几种不同几何意义的辅助函数证明定理。2.把拉格朗日中值定理所示的的平面曲线扩展到空间曲线的类似定理及其证明。3.给出拉格朗日中值定理中“ξ”的唯一性和连续性的充分条件,并加以证明。     

从几何的角度看微分中值定理

从平面几何中曲线之间的相切关系不依赖于坐标轴的选取这一基本事实去看数学分析中Rolle中值定理、Lagrange中值定理以及Cauchy中值定理.     

关于微积分中值定理证明方法的注记

<正> 一般的微积分学课本,在证明拉格朗日(Lagrange)中值定理及柯西(Cauchy)中值定理时,都是采用作辅助函数的方法,其所用的辅助函数,就我们所见,有三种不同的类型([1]     

Cauchy微分中值定理的推广的一个简单证明

给出Cauchy微分中值定理的推广的一个简单证明.     

无穷区间上的中值定理

本文将微积分学中的几个中值定理(Rolle定理、Lagrange中值定理、Cauchy中值定理、积分中值定理和推广积分中值定理)全部扩展到无穷区间上去,得到若干个无穷区间上的中值定理,其中值点均在无穷开区间内存在,从而使微积分学的中值定理理论更完善、应用更广泛。     

拉格朗日中值定理的证明方法

分类总结拉格朗日中值定理的各种证明方法,并加以分析讨论,以求深化对微分中值定理的理解．     

对一道数学分析题证明方法的改进

某文献在处理一道关于高阶导数的应用问题时,反复利用Rolle定理来证明高阶导数为零．考虑到这种做法过于繁琐,遂通过对其证明方法的改进,综合使用Lagrange中值定理和Taylor公式,使该问题的解决获得简化．     

A Mean-variance Problem in the Constant Elasticity of Variance (CEV) Mo del

In this paper, we focus on a constant elasticity of variance (CEV) model and want to find its optimal strategies for a mean-variance problem under two con-strained controls: reinsurance/new business an...     

谈泰勒公式的教学

基于函数微分定义,给出了带佩亚诺余项的泰勒公式的教学方案;基于拉格朗日中值定理,给出了带拉格朗日余项的泰勒公式的教学方案,并对两公式在微分学中的应用给出了举例。     

Regularized parametric Kuhn-Tucker theorem in a Hilbert space

For a parametric convex programming problem in a Hilbert space with a strongly convex objective functional, a regularized Kuhn-Tucker theorem in nondifferential form is proved by the dual regularization method. The theorem states (in terms of minimizing sequences) that the solution to the convex programming problem can be approximated by minimizers of its regular Lagrangian (which means that the Lagrange multiplier for the objective functional is unity) with no assumptions made about the regularity of the optimization problem. Points approximating the solution are constructively specified. They are stable with respect to the errors in the initial data, which makes it possible to effectively use the regularized Kuhn-Tucker theorem for solving a broad class of inverse, optimization, and optimal control problems. The relation between this assertion and the differential properties of the value function (S-function) is established. The classical Kuhn-Tucker theorem in nondifferential form is contained in the above theorem as a particular case. A version of the regularized Kuhn-Tucker theorem for convex objective functionals is also considered.     

二阶Hopfield神经网络周期解的存在性

本文讨论了一类带有时滞的二阶Hopfield神经网络周期解的存在性问题。首先利用Brouwer不动点定理证明了平衡点的存在性,通过平衡点和拉格朗日中值定理,将高阶神经网络模型转换为一阶模型,然后利用重合度理论给出了周期解存在的充分条件。     

拉格朗日中值定理的几个应用

本文阐述了拉格朗日中值定理在推导导函数的性质、存在性问题等方面的应用.     

The Singular Function Boundary Integral Method for singular Laplacian problems over circular sections

The Singular Function Boundary Integral Method (SFBIM) for solving two-dimensional elliptic problems with boundary singularities is revisited. In this method the solution is approximated by the leading terms of the asymptotic expansion of the local solution, which are also used to weight the governing partial differential equation. The singular coefficients, i.e., the coefficients of the local asymptotic expansion, are thus primary unknowns. By means of the divergence theorem, the discretized equations are reduced to boundary integrals and integration is needed only far from the singularity. The Dirichlet boundary conditions are then weakly enforced by means of Lagrange multipliers, the discrete values of which are additional unknowns. In the case of two-dimensional Laplacian problems, the SFBIM converges exponentially with respect to the numbers of singular functions and Lagrange multipliers. In the present work the method is applied to Laplacian test problems over circular sectors, the analytical solution of which is known. The convergence of the method is studied for various values of the order p of the polynomial approximation of the Lagrange multipliers (i.e., constant, linear, quadratic, and cubic), and the exact approximation errors are calculated. These are compared to the theoretical results provided in the literature and their agreement is demonstrated.     

On the boundedness of Lagrange multipliers of the Kuhn-Tucker theorem applied to the problem of Tikhonov function minimization

In the proof of the convergence of sequences of approximations derived by the regularized method of linearization, the Kuhn-Tucker theorem with bounded sequences of Lagrange multipliers is applied to sequences of Tikhonov functions. This paper demonstrates that in the case of three existing forms of constrains: (i) functional inequalities strict at some point, (ii) linear functional inequalities, and (iii) a linear operator equality, there exist bounded sequences of Lagrange multipliers of the Kuhn-Thucker theorem applied to the sequences of Tikhonov functions.