“以学生为本”是课堂有效教学的核心

1从"复合函数求导"的认知冲突设计谈起1.1教学的要求根据"浙江省数学学科教学指导意见",对于复合函数的求导公式是不要求证明的(也包括基本初等函数的导数公式、导数运算法则),但要求     

例谈构造法解导数与不等式问题

<正>导数的思想最初是由法国数学家费马提出的,在中学数学中,导数的思想为研究函数的图像和性质起到了重要的作用,是高考数学的重要考点之一,本文从构造法的角度来谈一谈解导数与不等式的问题.一、基本求导法则与公式要想通过题目中的导数构造辅助函数,就必须对基本求导法则与公式非常熟悉,现在把基本求导法则与基本初等函数的导数公式列     

反函数导数教学中的两点建议

提出了反函数导数教学中的两点建议:其一是用复合函数的极限运算法则合理解释求导公式的推导中求极限时用到的换元法;其二是在计算反函数的高阶导数时正确分析低阶导数的复合结构.同时为加深学生对求导公式的理解,教学中可引入具体例题验证公式.     

导数公式推导的启发式教学方法

针对一般微积分教材中函数的积、商求导法则、反三角函数的求导公式和参数式函数的二阶求导公式,提供几种简明易懂的启发式教学方法,对导数的运算有很好的参考价值.     

借助函数图形加深对定理的理解

利用函数图形解释复合函数的极限运算法则及其证明思路、由参数方程所确定的函数的导数以及复合函数求导法则、柯西中值定理的证明思路。     

浅谈新课程下中学导数的学习

近几年的高考中,对导数的考查主要包括三个层次:1．考查导数的概念、求导的公式和求导的法则;2．导数的简单应用包括求函数的极值,求函数的单调区间,证明函数的单调性等; 3．综合考查,包括解决应用问题,将导数内容和     

幂指函数的求导法则

<正> 形如y=u(x)~(v(x))(u(x)>0)的函数称为幂指函数。幂指函数属于初等函数的一部分,它是微分学中经常迂到的函数类型之一。求幂指函数的导数通常有两种方法,一是利用“对数求导法”,也即先对函数表达式取对数,然后再求导数;二是利用对数恒等式将幂指函数化为以e为底的复合指数函数形式,再求其导数,本质上都是用复合函数求导法则。先看下面的例子:     

方向导数的计算公式的注记

方向导数本质上也是函数的一种变化率.利用向量的Schmidt正交化方法进行坐标变换,将方向导数转换为对新变量的偏导数,再结合多元复合函数的求导法则,给出方向导数计算公式的一种新的证明.     

从《导数与微分》的教学谈技能技巧和能力的培养

人们做事总想“快”又“巧”,事半功倍。数学形式千变万化,方法繁多。在解题时如何灵活地运用知识,使解题既快又巧,这不仅有利于加深对基础知识的理解,更重要的是学到灵活解题的思想与方法。因此,数学教学中,“巧”字不容忽视。下面结合《导数与微分》的数学,谈谈自己的一些体会。一、要“巧”,首先概念要清,要清晰地把握住数学规律的本质。导数和微分是微积分中的基本概念,求初等函数的导数是该章的重点,是学习微积分必备的基本技能。要求导,就必须利用基本初等函数的求导公式及法则,而每个公式及法则都是直接或间接根据导数     

有理函数的高阶导数公式

导数是微积分学的重要研究对象,寻求函数的高阶导数公式是困难的,只有极少数类型的函数可以得到高阶导数公式.本文给出了有理函数的高阶导数的公式,我们从分解真分式入手,由于任意一个真分式都可以分解成最高分式之和,再利用导数的运算法则及公式,把复数部分转换为三角式进行化简,进而推算出有理函数的高阶导数的公式.最后本文结合实例对有理函数的高阶求导做出了讨论.     

利用二阶方向导数证明极值的充分条件

介绍二元函数二阶方向导数的概念与计算方法.利用线性代数中的二次型知识,对二元函数在驻点处是否取得极值的充分性定理给出有几何意义的证明.     

A minimal set-valued strong derivative for vector-valued Lipschitz functions

A set-valued derivative for a function at a point is a set of linear transformations whichapproximates the function near the point. This is stated precisely, and it is shown that, in general, there is not a unique minimal set-valued derivative for functions in the family of closed convex sets of linear transformations. For Lipschitz functions, a construction is given for a specific set-valued derivative, which reduces to the usual derivative when the function is strongly differentiable, and which is shown to be the unique minimal set-valued derivative within a certain subfamily of the family of closed convex sets of linear transformations. It is shown that this constructed set may be larger than Clarke's and Pourciau's set-valued derivatives, but that no irregularity is introduced.The author would like to thank Professor H. Halkin for numerous discussions of the material contained here.     

模糊有界变差函数全变差的积分表示与距离导数

定义和讨论了模糊数值函数的距离导数,给出了模糊有界变差函数全变差的积分表示.发现模糊绝对连续函数是几乎处处距离可导的,距离导数的积分等于其原函数的总变差,从而给出了模糊有界变差函数全变差的积分表示.     

Numerical scheme with convergence for a generalized time‐fractional Telegraph‐type equation

In this work, we design and analyze a numerical scheme for solving the generalized time‐fractional Telegraph‐type equation (GTFTTE) which is defined using the generalized time fractional derivative (GTFD) proposed recently by Agrawal. The GTFD involves the scale and the weight functions, and reduces to the traditional Caputo derivative for a particular choice of the weight and the scale functions. The scale and the weight functions play an important role in describing the behavior of real‐life physical systems and thus we study the solution behavior of the GTFTTE by varying the weight and the scale functions in the GTFD. We investigate the solution profile of the GTFTTE under some of these choices. We also provide the stability and the convergence analysis of the proposed numerical scheme for the GTFTTE. We consider two test examples to perform numerical simulations.     

分数阶光滑函数线性和二次插值公式余项估计

本文在局部分数阶导数定义的基础上给出了高阶局部分数阶导数定义,并据此得到了一般形式的分数阶Taylor公式.用该公式给出了分数阶光滑函数线性和二次插值公式余项的表达式,并进一步导出了分段线性插值的收敛阶估计.针对分数阶导数临界阶计算困难的问题,本文利用线性插值余项设计了一种外推算法,能够比较准确地求出函数在某点的局部分数阶导数的临界阶.最后通过编写算法的Mathematica程序,验证了理论分析的正确性,并用实例说明了算法的有效性.     

Mathematical features of the Calculus

The fundamental theorems of the calculus describe the relationships between derivatives and integrals of functions. The value of any function at a particular location is the definite derivative of its integral and the definite integral of its derivative. Thus, any value is the magnitude of the slope of the tangent of its integral at that position, and any two subtracted values are the area under its derivative. The slope formula of secant lines actually is the mean value theorem for the derivative function in addition to representing the well-known Fermat definition of the derivative. The sine and other functions are discussed.     

Determination of jumps for functions via derivative Gabor series

Recently, Shi Xianliang and Hu Lan published the method of concentration factors for determination of jumps of functions via MCM conjugate wavelets. Usually, it is difficult to calculate the Hilbert transform of general window functions. The aim of this paper is to discuss determination of jumps for functions based on derivative Gabor series. The results will simplify the calculation of jump values.     

特殊类型函数高阶导数与原函数的统一公式

本文通过研究几种特殊类型函数的高阶导数与原函数的求法 ,获得了由该类函数自身及其一阶导数的特征 ,即可快速写出该类函数的 n阶导数 y( n) 与原函数 y( - 1 ) 的统一公式 y( n) ( n=-1 ,1 ,2 ,3 ,… ) .该公式可给实际运算带来许多简化与方便 .     

有理函数的积分

应用导数运算,给出有关有理函数积分的一种计算方法.     

Calculating the derivative of piecewise functions

Exercises involving the calculation of the derivative of piecewise defined functions are common in calculus, with the aim of consolidating beginners’ knowledge of applying the definition of the derivative. In such exercises, the piecewise function is commonly made up of two smooth pieces joined together at one point. A strategy which avoids using the definition of the derivative is to find the derivative function of each smooth piece and check whether these functions agree at the chosen point. Showing that this strategy works together with investigating discontinuities of the derivative is usually beyond a calculus course. However, we shall show that elementary arguments can be used to clarify the calculation and behaviour of the derivative for piecewise functions.