从两道例题看高等数学知识的前后贯通

本文通过探究偏导数几何意义方面的两道例题,将涉及到偏导数,空间曲面,空间曲线,直线,平面,二元函数的标准线性近似等内容融会贯通.     

梯度与等高线的教学体会

根据多元函数的偏导数可以确定等高线凹侧的一个法向量是梯度,该结果进一步揭示了梯度的几何意义,可以作为高等数学教学中方向导数一节的补充     

定义在闭区域上的二元函数求值域问题

若给定的目标函数是线性目标函数或者具有斜率、距离等几何意义,求闭区域上的二元函数的值域或最值,可以考虑利用线性规划知识解决.若给定的二元函数无明显的几何意义,上述方法不再奏效.高等数学中可以采用偏导数知识求解二元函     

圆的面积与周长关系的一点思考

导数是中学教材的重要内容，导数概念是微积分的核心概念之一，它有极其丰富的实际背景和广泛的应用．教材从切线斜率、瞬时速度的角度展开，利用几何直观、归纳类比、抽象概括的形式引入了导数的概念，帮助同学们直观理解导数的背景、思想及其内涵，教材还在“探究与研究”栏目中设置了一些利用几何问题帮助我们进一步理解导数概念的实例和问题．     

有限离散函数的导数和性质

通过引入有限离散函数的导数概念,分别从几何直观和性质两个角度,比较了有限离散函数的导数概念和常规连续函数导数的相似性.结果表明,在局部情况下,有限离散函数导数近似等于连续情形下的导数.在运算性质上,有限离散函数导数的性质非常相似于连续情形时的导数性质.最后的例子给出了有限离散函数导数的一个应用.     

时标上右端函数为两项和的集值微分方程解的平方收敛性

本文讨论了一类时标上右端函数为两项和的集值微分方程解的收敛性问题.首先引入时标上集值函数的Hukuhara导数意义下的Δ偏导数概念,并给出了时标上集值微分系统的比较原理.对于所构造的迭代序列,通过利用广义拟线性化方法和分析技巧,得到了上述问题近似解的平方收敛结果.     

“在”某点处与“过”某点的切线

自从导数内容进入高中教材,导数便成为高考的新热点,应用导数的几何意义求曲线的切线方程是导数的重要应用之一,但同学们在     

浅析北京市2012届上半年期中、期末对导数及其应用的考查

考向一、对导数的概念及导数基本应用的考查 命题规律:以选择题、填空题等客观题目的形式考查导数的基本概念、运算、导数的物理意义、几何意义及利用导数与不等式研究函数的单调性.     

Mathematica数学软件的图形功能在微积分中的应用问题

讨论如何正确使用Math.的图形功能,对微积分中二元函数在一点极限不存在的情形及二元函数在点不连续但偏导数存在的情形给出几何解释。     

二阶导数的几何意义

对一元函数二阶导数的几何意义进行阐释,认为一元函数的二阶导数是描述函数对应曲线的曲率的一个重要指标:二阶导数的绝对值与曲线曲率成正比;在驻点处,二阶导数的绝对值与曲率相等.     

幂向量,复合向量数及其函数理论

本文提出向量为其幂向量和向量幂级数．向量幂级数由一实数和某一向量联合组成的“复合向量数”及其函数有重要涵义．这数也有运算法则．从复合向量数的函数理论分析知其函数有导数和解析函数的必要和充分条件．这些条件构成了“双曲型”方程的特征以及函数的积分性质等．     

Hilbert–Pachpatte type general integral inequalities

In this article we present very general weighted Hilbert–Pachpatte type integral inequalities. These are regarding ordinary derivatives and fractional derivatives of Riemann–Liouiville and Canavati types. Also regarding general derivatives of Widder type and linear differential operators. Our results apply to continuous functions and some to integrable functions.     

Derivative polynomials and closed-form higher derivative formulae

In a recent paper, Adamchik [1] expressed in a closed-form symbolic derivatives of four functions belonging to the class of functions whose derivatives are polynomials in terms of the same functions. In this sequel, simple closed-form higher derivative formulae which involve the Carlitz-Scoville higher order tangent and secant numbers are derived for eight trigonometric and hyperbolic functions.     

Pricing range notes within Wishart affine models

We provide analytic pricing formulas for Fixed and Floating Range Accrual Notes within the multifactor Wishart affine framework which extends significantly the standard affine model. Using estimates for three short rate models, two of which are based on the Wishart process whilst the third one belongs to the standard affine framework, we price these structured products using the FFT methodology. Thanks to the Wishart tractability the hedge ratios are also easily computed. As the models are estimated on the same dataset, our results illustrate how the fit discrepancies (meaning differences in the likelihood functions) between models translate in terms of derivatives pricing errors, and we show that the models can produce different price evolutions for the Range Accrual Notes. The differences can be substantial and underline the importance of model risk both from a static and a dynamic perspective. These results are confirmed by an analysis performed at the hedge ratios level.     

ON THE SHARING VALUES OF ALGEBROID FUNCTIONS AND THEIR DERIVATIVES

In this paper,we define the shared value of an algebroid function and its derivative on its Riemann surface.By considering the relationship between the shared values and the branch points of algebroid functions and their derivatives,we obtain some uniqueness theorems of algebroid functions sharing values with their derivatives,which extend 3 IM shared values theorem of nonconstant meromorphic functions and their derivatives obtained by Mues-Steinmetz and Gundersen.     

导数具四个公共小函数的亚纯函数

该文主要讨论亚纯函数的导数具有四个公共小函数时的唯一性问题，考虑了Nevanlinna 四值定理在亚纯函数的导数具公共小函数时的情形，推广并改进了作者近期的一个结果。      

Close‐to‐convexity of normalized Dini functions

In this paper necessary and sufficient conditions are deduced for the close‐to‐convexity of some special combinations of Bessel functions of the first kind and their derivatives by using a result of Shah and Trimble about transcendental entire functions with univalent derivatives and some newly discovered Mittag–Leffler expansions for Bessel functions of the first kind.     

参数函数和复合函数高阶导数的计算

用函数的链式法则和乘积公式给出了参数函数和复合函数的高阶导数的计算公式．     

Superlinearly convergent approximate Newton methods for LC1 optimization problems

In the literature, the proof of superlinear convergence of approximate Newton or SQP methods for solving nonlinear programming problems requires twice smoothness of the objective and constraint functions. Sometimes, the second-order derivatives of those functions are required to be Lipschitzian. In this paper, we present approximate Newton or SQP methods for solving nonlinear programming problems whose objective and constraint functions have locally Lipschitzian derivatives, and establishQ-superlinear convergence of these methods under the assumption that these derivatives are semismooth. This assumption is weaker than the second-order differentiability. The extended linear-quadratic programming problem in the fully quadratic case is an example of nonlinear programming problems whose objective functions have semismooth but not smooth derivatives.This work is supported by the Australian Research Council.This paper is dedicated to Professor O.L. Mangasarian on the occasion of his 60th birthday.