由高考题引发的对函数极值点教学的一点思考

<正>对于一般函数的极值点,教学中多借助几何直观,用自然语言给出函数极值点的描述性定义:若函数f(x)图象在点P(x1,f(x1))处从左侧到右侧由"上升"变为"下降"(函数由单调递增变为单调递减),我们就称f(x1)为函数f(x)的一个极大值,x1为函数f(x)的一个极大值点;类似的,若函数f(x)图象在点P(x2,f(x2))处从左侧到右侧由"下降"变为"上升"(函数由单调递减变为单调递增),我们就称f(x2)为函数f(x)的一个极小值,x2为函数f(x)的一个极小值点.该定义给出了判断极值点的充要条件,揭示了一般函数极值点的本质特征:极值点附近左侧与右侧函数单调性相反[1].     

打破常规简捷求解

函数f(x)在x = x0 处取得极限的点称之为“极限点”,函数 f(x) 在点 x = x0 处连续的点称之为“连续点”,函数f(x)在x = x0处有导数的点称之为“可导点”,可导函数y = f(x)使f′(x0) = 0 的点 x0 叫做函数f(x)的“驻点”,函数f(x)在x = x0 处取得极值(极大值或极小值) 的点称之为“极值点”,函数f(x)在x = x0 处取得最值(最大值或最小值)的点称之为“最值点”.函数中这五类点很容易混淆,理清它们之间的关系对函数的“极限”和“导数”学习很有帮助.一、函数的“极限点”与“连续点”的关系当自变量x无限地趋近常数x0(但 x不等于x0)时,若…     

能取等号吗?

函数 y=f(x)在 x=x_0处有极值,则它的导数 f′(x)在这点的函数值为零,即 f′(x_0)=0,反过来,函数 y=f(x)的导数在某点的函数值为零时,这点却不一定是函数的极值点.因此,我们必须具体问题具体分析.例1 已知 b>-1,c>0,函数 f(x)=x b 的图象与函数 g(x)=x~2 bx c 的图像相切.(1)求 b 与 c 的关系(用 c 表示 b)(2)设函数 F(x)=f(x)g(x)在(-∞, ∞)内有极值点,求 c 的取值范围.分析:(1)(略);(2)函数 F(x)=f(x)·g(x)在(-∞, ∞)内有极值点,即存在 x_0使F′(x_0)=0,亦即一元二次方程 F′(x)=0有实     

用虚设零点法解函数极值问题

解决有关函数极值问题,一般都是通过求导函数的零点求出极值点来实现,然而,有些时候这一招却不灵啦,请看下例: 例1 已知函数f(x)=ax2-2x+lnx有两个极值点,证明:f(x)的极小值小于-3/2. 分析 第一步:求定义域.函数f(x)=ax2-2x+lnx的定义域为(0,+∞). 第二步:求导.f'(x)=2ax-2+1/x=2ax2-2x+1/x. 第三步:求极值点. 令g(x) =2ax2-2x+1,函数f(x)=ax2-2x+lnx有两个极值点的必要条件是g(x)=2ax2-2x+1=0当x＞0时有两个不等实根.     

例谈函数零点的设而不求问题

<正>函数f(x)的零点,往往是函数f(x)的符号的分界点;对于导函数f′(x)的零点而言,则意味着零点往往可能就是函数f(x)的最值点.但是在实际问题中,函数的零点一般很难具体求出,这就需要我们在求解过程中采用设而不求的策略引入零点,方便解题.本文仅限于讨论需要借助零点存在定理引入零点时设而不求的情况,对于题中明确交代过零点的问题则不属于本文讨论的范围.经笔者总结,现将零点中设而不求的类型归纳如下.一、为确定函数f(x)的符号而引入零     

方程的根、函数的零点的个数问题探讨

高中数学新课程(人教A版)必修一第3.1.1节讲了方程的根、函数的零点问题:方程f(x)=0有实数根函数y=f(x)的图象与x轴有公共点函数y=f(x)有零点,可见函数的零点从不同的角度将数与形,函数与方程有机地联系在一起.从函数的角度来看,零点就是使得函数值为0的实数;从方程的角度来看,零点就是相应方程f(x)=0的实数根;从函数的图象来看,零点就是函数f(x)的图象与x轴交点的横坐标.一些方程不涉及方程     

闭区间上三次函数的最值

函数在闭区间上的最值问题本质上是一个数学规划问题 .高中教材中讨论了二次函数在闭区间上的最值问题 ,现在导数进入了中学教材 ,使得对三次函数最值的讨论成为可能 .本文讨论三次函数 y( x) =x3+ ax2 +bx+ c在闭区间 [α,β]上的最值问题 .记导函数 y′( x) =3x2 + 2 ax+ b的判别式为 Δ.当Δ≤ 0时 ,y( x)没有极值点 ,是单调增函数 ,所以 y( x)在 [α,β]的端点处达到最大、最小值 .当Δ >0时 ,y′( x)有两个零点 ,记为 x1和 x2 ( x1 相似文献   

函数中两个容易混淆的问题

误解1:函数y=f(x)和它的反函数y=f-1(x)的图象的交点在直线y=x上. 教材上例题涉及的函数及我们接触的函数的图象与其反函数的图象的交点大多 直线y=x上,所以不少同学就认为函数若与其反函数不是同一函数,且函数与其反函 的图象有交点,则交点必在直线y=x上,但这种观点是错误的.现举两例,希望同学们 明确这个问题._ 如函数y=7-3x,其图象过(2,1)点,其反函数y= 7-x2 3(x≥0)的图象也过(2,1)点,故函数y=7-3x与其 反函数图象的一个交点为(2,1)点.又由函数与其反函数的 图象关于直线y=x对称,故点(2,1)关于直线y=x的对称 点(1,2)也是函数y=7-3…     

浅谈函数零点的有关问题

<正>一般地,使函数y=f(x)的值为0的实数x称为函数y=f(x)的零点.因此,函数y=f(x)的零点就是方程f(x)=0的实数根.从图象上看,函数y=f(x)的零点就是它的图象与x轴交点的横坐标.一般地,若函数y=f(x)在区间[a,b]上的图象是一条不间断的曲线且f(a)·f(b)<0,则函数y=f(x)在区间(a,b)上有零点.我们经常会遇到函数与方程的有关问题,下面我们看这样几个题目.     

导数问题的高考题型及解法解读

在近几年的高考中,对导数问题的考查力度正在逐年增加,不仅题型在变化,而且设置问题的难度、深度与广度也在不断加大,将导数与其它数学知识的结合已成为高考题的一道靓丽的风景线. 一、对导数定义和求导法则的考查 例1.设函数f(x)=2/x+1nx,则() Ax=1/2为f(x)的极大值点B.x=1/2为f(x)的极小值点 C.x=2为f(x)的极大值点 D.x=2为f(x)的极小值点 解:∵f(x)=2/x+1nx(x＞0),∴f'(x)=-2/x2+1/x,由f'(x)=0解得x=2. 当x∈(0,2)时,f'(x)＜0,f(x)为减函数;x∈(0,+∞)时,f(x)＞0,f(x)为增函数,∴x=2为f(x)的极小值点,所以选D. 点评:本题考查了利用导数确定极值点问题,但首先要利用求导公式对函数顺利求导,才能快速作答.     

线性不等式约束优化问题的强信赖域算法

对一类带有非负边界约束的线性不等式约束优化问题进行了研究，提出了一种新的信赖域算法．该算法在内点法的基础上，把非负边界约束从一般的不等式约束中分离出来，化为信赖域约束的一部分，得到一个简单易解的子问题．在一定的条件下证明了该算法具有强收敛性，并给出了数值结果．     

一类线性不等式约束优化问题的信赖域算法

对一类带有非负边界约束的线性不等式约束优化问题提出了一种新的信赖域算法。此算法以内点法为基础,把非负边界约束从一般的不等式约束中分离出来,化为信赖域约束的一部分,从而得到一个简单易解的子问题。在一定的条件下证明了算法的收敛性,并给出了数值结果。     

Augmenting the Correlated Trait–Correlated Method Model for Multitrait–Multimethod Data

We introduce an approach for ensuring empirical identification of the correlated trait–correlated method (CT–CM) model under a variety of conditions. A set of models are referred to as augmented correlated trait–correlated method (ACT–CM) models because they are based on systematically augmenting the multitrait–multimethod matrix put forth by Campbell and Fiske (1959). We show results from a Monte Carlo simulation study in which data characteristics lead to an empirically underidentified standard CT–CM model, but a well-identified fully augmented correlated trait–correlated method (FACT–CM) model. This improved identification occurs even for a model in which equality constraints are imposed on loadings on each trait factor and loadings on each method factor—a specific case shown to lead to an empirically underidentified CT–CM model.     

General space-efficient sampling algorithm for suboptimal alignment

Suboptimal alignments always reveal additional interesting biological features and have been successfully used to informally estimate the significance of an optimal alignment. Besides, traditional dynamic programming algorithms for sequence comparison require quadratic space, and hence are infeasible for long protein or DNA sequences. In this paper, a space-efficient sampling algorithm for computing suboptimal alignments is described. The algorithm uses a general gap model, where the cost associated with gaps is given by an affine score, and randomly selects an alignment according to the distribution of weights of all potential alignments. If x and y are two sequences with lengths n and m, respectively, then the space requirement of this algorithm is linear to the sum of n and m. Finally, an example illustrates the utility of the algorithm.     

Power duality for multi-antenna OFDM system in broadcast channel with user scheduling

This paper deals with design and analysis of user scheduling and power allocation for multi-antenna OFDM systems with DPC, ZF-DPC, ZF-BF and TDMA transmit strategies. We consider the general multi-user downlink scheduling problem and power minimization with multi-user rate constraints. According to the channel state, it is shown that there is a power optimal policy which selects a subset of users in each scheduling interval. We present user selection algorithms for DPC, ZF-DPC, ZF-BF and TDMA for multi-antenna OFDM system in broadcast channels, and we also present the practical water-filling solution in this paper. By the selected users with the consideration of fairness, we derive the power optimization algorithm with multi-user rate constraints. We also analyze the power duality of uplink-downlink for the transmit strategies of DPC, ZF-DPC and ZF-BF. Simulation results show that the present user-scheduling algorithm and power minimization algorithm can achieve good power performance, and that the scheduling algorithm can guarantee fairness.     

Constrained Maximum Likelihood Estimation for Two-level Mean and Covariance Structure Models

Maximum likelihood is commonly used for estimation of model parameters in analysis of two-level structural equation models. Constraints on model parameters could be encountered in some situations such as equal factor loadings for different factors. Linear constraints are the most common ones and they are relatively easy to handle in maximum likelihood analysis. Nonlinear constraints could be encountered in complicated applications. In this paper we develop an EM-type algorithm for estimating model parameters with both linear and nonlinear constraints. The empirical performance of the algorithm is demonstrated by a Monte Carlo study. Application of the algorithm for linear constraints is illustrated by setting up a two-level mean and covariance structure model for a real two-level data set and running an EQS program.     

基于泄漏约束的DF-GSC语音增强

为了改善广义旁瓣抵消(GSC)语音增强方法的性能,提出了一种带有泄漏约束的判决反馈旁瓣抵消(LCDF-GSC)方法.采用DF-GSC方法以解决GSC对波达方向敏感的问题,在代价函数中引入泄漏因子,以此改善语音失真的问题,而这种问题是由于噪声参考信号中含有语音成份造成的.试验结果表明,尽管经过LCDF-GSC处理后的语音信号信噪比要略低于DF-GSC,IS测度表明这时前者的语音信号失真度要小于后者.MOS分也表明LCDF-GSC方法要优于DF-GSC和单通道Weiner滤波算法.     

Developing rigid constraint for the estimation of pose and structure from a single image

INTRODUCTION Pose and structural parameters analysis of 2Dimages sets find application in many areas such asobject recognition, motion estimation, navigationplanning, and structural analysis of 3D objects(Huang and Netravali, 1994), image communica-tion, and image coding (Mitiche and Aggarwal,1986). Normally, given one set of 3D object pointdata and their corresponding 2D projective imagepoint data, the 3D-2D calibration problem is re-ferred to as the estimation of the camera par…     

Developing rigid constraint for the estimation of pose and structure from a single image

Pose and structure estimation from a single image is a fundamental problem in machine vision and multiple sensor fusion and integration.In this paper we propose using rigid constraints described in different coordinate frames to iteratively estimate structural and camera pose parameters.Using geometric properties of reflected correspondences we put forward a new concept,the reflected pole of a rigid transformation.The reflected pole represents a general analysis of transformations that can be applied to both 2D and 3D transformations.We demonstrate how the concept is applied to calibration by proposing an iterative method to estimate the structural parameters of objects.The method is based on a coarse-to-fine strategy in which initial estimation is obtained through a classical linear algorithm which is then refined by iteration.For a comparative study of performance,we also implemented an extended motion estimation algorithm(from 2D-2D to 3D-2D case)based on epipolar geometry.     

A Bayesian Approach for Multigroup Nonlinear Factor Analysis

The main purpose of this article is to develop a Bayesian approach for a general multigroup nonlinear factor analysis model. Joint Bayesian estimates of the factor scores and the structural parameters subjected to some constraints across different groups are obtained simultaneously. A hybrid algorithm that combines the Metropolis-Hastings algorithm and the Gibbs sampler is implemented to produce these joint Bayesian estimates. It is shown that this algorithm is computationally efficient. The Bayes factor approach is introduced for comparing models under various degrees of invariance across groups. The Schwarz criterion (BIC), a simple and useful approximation of the Bayes factor, is calculated on the basis of simulated observations from the Gibbs sampler. Efficiency and flexibility of the proposed Bayesian procedure are illustrated by some simulation results and a real-life example.