解向量问题时的几种典型错误

向量是一个重要的数学概念. 向量不同于数量,它有其自身的一套运算体系,但不少初学者由于对所学知识理解不深,从而导致在解答有关向量问题时,常常出现一些错误. 现分类例析如下,供大家参考.1. 混淆实数 0与零向量0→例 1　有四个式子: (1) 0→·a→ =0→; (2)O·a→ =0; ( 3 ) 0→ -MN=NM; ( 4 )AB+BC+CD+DA=0. 其中正确的个数是A. 3个　B. 1个　C. 2个　D. 4个错解: (1 )、( 2 )、( 3 )、( 4 )中式子全部正确,选D.剖析:考虑 (1)中,0→·a→表示零向量与任意向量a→的数量积,数量积是一个数,而不是向量0→; (2)中, 0·a→表…     

义务教育七年级数学研究性学习的一个案例

本案例依照北师大版义务教育课程标准实验教科书及其教学参考书 (七年级上册 )的指导思想进行设计和教学 ,文中的问题是以该教科书第 1 1 1页复习题C组第 3题为背景进行加工拓展而成的 ,特此说明 .1　学生作业单之一1 1　动手实践 :操作 1　用棋子摆出图 ( 1 )、( 2 )、( 3) ,数一数 :摆图 ( 1 )用个棋子 ,摆图 ( 2 )用个棋子 ,摆图 ( 3)用个棋子 .操作 2　按照上述方式摆出第 ( 4 )个图形和第 ( 5 )个图形 ,数一数 :摆图 ( 4 )用个棋子 ,摆图 ( 5 )用个棋子 .提示 :摆出第 ( 4 )、( 5 )个图形的方式是否唯一呢 ?1 2　自主探索 :问题 1　…     

直觉思维在解题中的运用

直觉思维是一种直接迅速对问题的结果或解决问题的途径作出合理猜测、设想或突然领悟的思维 .数学直觉思维是人脑对数学对象(结构及其关系 )的某种直接的领悟或洞察 ,表现在人们在解决数学问题时 ,不经过逐级分析 ,严谨论证 ,而是直接从整体上把握问题实质 ,迅速敏捷 ,大胆猜想 ,作出判断 .爱因斯坦指出 :“在科学研究中 ,真正可贵的因素是直觉 .”在数学解题中恰当、合理地运用直觉思维 ,可简化思维过程 ,迅速有效地解决问题 .例 1　已知 :ｘ + 1ｙ=1,ｙ + 1ｚ=1,求证 :ｚ + 1ｘ=1.已知两个方程 ,有三个未知数 ,而所求证的等式中只有两个未…     

方差分析法的简易算法

方差分析是一种常用的数理统计方法.但因其计算过程比较复杂。给现场的应用带来一些困难,下面介绍利用电子计算器作单因素方差分析的简易方法. 方差分析的基本作法是将总变差分解为代表因素效应的因素平方和与代表随机偏差的误差平方和.用以上两个平方和的均方构成服从F分布的统计量,来判断因素效应的显著性. 方差分析的一般步骤,在许多书刊上已有详细的介绍。为了与本文介绍的方法进行比较,现简述如下: (1)将因素处于不同状态下所得到的数据,分组列表如表1.设共有K组,各,组样本容量均为n.(3)计算因素平方和SSC (4)计算误差平方和SSE (5…     

2006年高考中的十个易错问题

一、忽视函数单调性的概念致错例1(北京卷)已知f(x)=(3a-1)x 4a,x<1logax,x≥1是(-∞, ∞)上的减函数,那么a的取值范围是()A.(0,1)B.(0,13)C.[71,31)D.[71,1)错解:因为f(x)在(-∞, ∞)上是减函数,所以f(x)在(-∞,1)和(1, ∞)上是减函数,于是3a-1<0且0相似文献   

集合与函数(五题)

1.:eR,集合恤,一2x十2~0}的元素的个数为() (A)l个(B)2个(C)3个(D)0个 2.函数f(:)一2:一月二王的最大值为(^)2(B)l(e)0(D)不存在5.函数,(x)~(x俞)‘,“‘是个()(A)奇函数(B)偶函数(c)亦奇亦偶函数(D)非奇非偶函数4.函数,~3蚤十l的反函数是()(A)夕=2109:(劣十1)(c);=21姐去(‘+1) 5.方程(xZ一4)为()(B)夕~21083(x一1)(D),一告:。g、(·+,)·19(2一二2)=o解的个数(A)4个(B)2个(C)1个(D)无解附:本期“一望而答”答案 1.[望〕集合的元素是一元二次方程. 答:A 2.仁望」函数f(x)是(一oo,l]上的增函数,自变量的最大值1对应着函数的最大值2.…     

非线性最优化条件在企业管理决策中的应用

<正> 在本文中我们讨论非线性的经济管理数学模型(?)这里,n 维向量 x 的第 k 个分量 x_k 表示第 k 个生产活动(共有 n 个基本生产活动)的活动水平;g_i(x)和 h_i(x)表示生产活动中耗费的生产因素,如劳动力、能源、土地、原材料等等.详细一点讲,参量 y_i(i=1,…,p)表示生产运行可能获得的资源数,函数 g_i(x)(i=1,…,p)就表示当生产在活动水平 x 处进行时所需要的资源数;参量 y_i(i=p+1,…,r)是企业运行时需要生产或使用的物料或服务,而函数 h_i(x)(i=p+1,…,r)就定义了活动水平 x 需要生产或使用的物料或服务的量;最后,f(x)表示生产在水平 x处进行时企业的收益.     

2005年高考数学辽宁卷

、选择题 任取10个球,则其中恰有6个红球的概率 为 (1)复数z- 一1十i 1+艺 一1.在复平面内,之所对应 (A) (C) 嵘。·拼。 C}乱 以。·晓。 晓。·碳。 c}8。 晓。·晓。 c{3。 BD 的点在 (A)第一象限(B)第二象限 (C)第三象限(D)第四象限 (2)根限hmf(x)存在是函数f(x)在点x~x0 j,,二。 处连续的 (A)充分而不必要的条件 (B)必要而不充分的条件 (C)充要条件 (D)既不充分也不必要的条件 (3)设袋中有80个红球,20个白球.若从袋中 c}吕。 (4)已知m、n是两条不重合的直线,Q渭、y是三 个两两不重合的平面.给出下列四个命题: ①若m土a,m土夕,则a…     

1988年全国高中数学联赛试题及解答

策一试(1 oj〕,6「}_l:午8:00一9:3()) 一、选择.(木大题共5小题,李诱小题有一个正确答案,选对得7分,选错、不选或多选均得。分) 1.设仃三个函数,第一个是y=甲(x),‘己的反函数就是第二个函数,而第二个函数的图象,歹第二个函数的图象关于直线二+夕二O对称.那么第三个函数是(). (一J)y=一甲(二).(B)y=一叩(一二). (C)少,一甲一‘(二).(D)夕=一甲一‘(一x). 2.已知原兑在椭圆左“x忍+y忍一4kx+Zhy+k恋一1“0的内部,那么参数k的取值范旧是() (月)I介}>1.(刀)1左l笋2. (C)一l相似文献   

关于极值点偏移问题的思考——从一道高考压轴题出发

一、题目展示(2016全国Ⅰ卷理-21) 已知函数f(x)=(x-2)ex+a(x-1)2有两个零点. (1)求a的取值范围; (2)设x1,x2是f(x)的两个零点,证明:x1+x2＜2. 分析:第(1)小题是典型的零点个数问题,利用分离变量的方法可以解决;而第(2)小题属于极值点偏移问题.笔者将重点通过第(2)小题的解决来讨论极值点偏移问题.     

The Steiner ratio of several discrete metric spaces

Steiner's Problem is the “Problem of shortest connectivity”, that means, given a finite set of points in a metric space (X,ρ), search for a network interconnecting these points with minimal length. This shortest network must be a tree and is called a Steiner Minimal Tree (SMT). It may contain vertices different from the points which are to be connected. Such points are called Steiner points. If we do not allow Steiner points, that means, we only connect certain pairs of the given points, we get a tree which is called a Minimum Spanning Tree (MST). Steiner's Problem is very hard as well in combinatorial as in computational sense, but, on the other hand, the determination of an MST is simple. Consequently, we are interested in the greatest lower bound for the ratio between the lengths of these both trees:

which is called the Steiner ratio (of (X,ρ)). We look for estimates and exact values for the Steiner ratio in several discrete metric spaces. Particularly, we determine the Steiner ratio for spaces of words, and we estimate the Steiner ratio for specific graphs.     

股票市盈率水平的决定因素——上海证券交易所的数据验证

本文利用截面分析方法,从多个角度讨论影响股票市盈率水平的因素,并利用上海证券交易所股票数据进行验证。首先,利用上海证券交易所的股票历史数据计算按不同板块 (如行业,每股收益大小等)划分股票类别时,各板块市盈率水平的具体数值,并讨论各不同板块市盈率的分布特征。其次,讨论了市盈率与其直接影响变量(股票价格和每股收益 )之间的相互关系,用数学模型论述两者在影响市盈率变化时所存在的差异。最后,通过对影响市盈率水平的各种因素进行详细的统计特征分析,筛选影响强烈的因素,并建立了描述市盈率与各影响因素之间相关性的多因素模型。     

工件有到达时间及可拒绝下的同类平行机排序问题的近似算法

本文研究工件有到达时间且可拒绝下的同类平行机排序问题。在该问题中, 给定一个待加工工件集, 每个工件在到达之后, 可以被选择安排到$m$台同类平行机器中的某一台机器上进行加工, 也可以被选择拒绝加工, 但需支付一定的拒绝惩罚费用。目标函数是最小化接受工件集的最大完工时间与拒绝工件集的总拒绝费用之和。当$m$为固定常数时, 设计了一个伪多项式时间动态规划精确算法; 当$m$为任意输入时, 设计了一个近似算法, 当接受工件个数大于$(m-1)$时, 该算法近似比为3, 当接受工件个数小于$(m-1)$时, 该算法近似比为$(2+\rho)$, 其中$\rho$为机器加工速度最大值和最小值的比值。最后通过算例演示了算法的运行。     

工件有到达时间及可拒绝下的同类平行机排序问题的近似算法

本文研究工件有到达时间且可拒绝下的同类平行机排序问题。在该问题中, 给定一个待加工工件集, 每个工件在到达之后, 可以被选择安排到$m$台同类平行机器中的某一台机器上进行加工, 也可以被选择拒绝加工, 但需支付一定的拒绝惩罚费用。目标函数是最小化接受工件集的最大完工时间与拒绝工件集的总拒绝费用之和。当$m$为固定常数时, 设计了一个伪多项式时间动态规划精确算法; 当$m$为任意输入时, 设计了一个近似算法, 当接受工件个数大于$(m-1)$时, 该算法近似比为3, 当接受工件个数小于$(m-1)$时, 该算法近似比为$(2+\rho)$, 其中$\rho$为机器加工速度最大值和最小值的比值。最后通过算例演示了算法的运行。     

On the effectiveness of the Harmonic Shelf Algorithm for on-line strip packing

In [J. Csirik, G.J. Woeginger, An on-line algorithm for multidimensional bin packing, Inform. Process. Lett. 63 (1997) 171-175] the authors study the asymptotic worst case ratio between the height of the strip needed to on-line pack a list of boxes by means of the Harmonic Shelf Algorithm and the height of the strip used by an optimal algorithm. In this note we analyze the effectiveness of the former algorithm in terms of the ratio between the unused area inside the strip and the total size of this strip, and we show that the Harmonic Shelf Algorithm is also capable of packing items so that the asymptotic worst case value of this ratio comes arbitrarily close to .     

The Automorphism Conjecture for Small Sets and Series Parallel Sets

The automorphism conjecture for ordered sets states that the automorphism to endomorphism ratio will tend to zero as the size of the ordered set goes to infinity. We show by computer enumeration that up to size 11 the ratio is largest for weakly ordered sets. Subsequently, we derive exact recursive formulas for the number of homomorphisms between two related types of weakly ordered sets and we prove a strong automorphism conjecture for series-parallel ordered sets. We conclude with an example that shows that the automorphism to endomorphism ratio can exceed for arbitrarily large .This work was sponsored by Louisiana Board of Regents RCS grant LEQSF(1999-02)-RD-A-27.     

A note on “A decision model for ranking suppliers in the presence of cardinal and ordinal data, weight restrictions, and nondiscriminatory factors”

Farzipoor Saen (Ann. Oper. Res. 172(1):177–192, 2009) proposed a method based on Data Envelopment Analysis (DEA) for selecting the best supplier in the presence of cardinal and ordinal data, weight restrictions, and non-discriminatory factors. In the present note, we show that the DEA method proposed by Farzipoor Saen (Ann. Oper. Res. 172(1):177–192, 2009) cannot be used for selecting the best supplier in the presence of imprecise data. It is also shown that there are computational errors in calculating the amounts of the preference intensity parameter and the ratio parameter in Farzipoor Saen’s paper.     

A Note on the Approximation of a Minimum-Weight Maximal Independent Set

We consider the polynomial approximation behavior of the problem of finding, in a graph with weighted vertices, a maximal independent set minimizing the sum of the weights. In the spirit of a work of Halldórson dealing with the unweighted case, we extend it and perform approximation hardness results by using a reduction from the minimum coloring problem. In particular, a consequence of our main result is that there does not exist any polynomial time algorithm approximating this problem within a ratio independent of the weights, unless P = NP. We bring also to the fore a very simple ratio guaranteed by every algorithm while no polynomial time algorithm can guarantee the ratio (1 – ). The known hardness results for the unweighted case can be deduced. We finally discuss approximation results for both weighted and unweighted cases: we perform an approximation ratio that is valid for any algorithm for the former and propose an analysis of a greedy algorithm for the latter.     

clever or smart: Strategies for the online target date assignment problem

In this paper, we consider the Online Target Date Assignment Problem (OnlineTDAP) for general downstream problems, where the downstream cost are nonnegative, additive and satisfy the triangle inequality.We analyze algorithm smart, which was introduced by Angelelli et al. [3] and give its exact competitive ratio depending on the number of requests. Since the obtained competitive ratio is at most we answer the question posed in Angelelli et al. [4] if smart has a competitive ratio strictly less than 2.Moreover, we introduce a new algorithm called clever and show that this strategy has a competitive ratio of 3/2. We show that this is asymptotically optimal by proving that no online algorithm can perform better than 3/2−ε.     

Characterizing and bounding the imperfection ratio for some classes of graphs

Perfect graphs constitute a well-studied graph class with a rich structure, which is reflected by many characterizations with respect to different concepts. Perfect graphs are, for instance, precisely those graphs G where the stable set polytope STAB(G) equals the fractional stable set polytope QSTAB(G). The dilation ratio of the two polytopes yields the imperfection ratio of G. It is NP-hard to compute and, for most graph classes, it is even unknown whether it is bounded. For graphs G such that all facets of STAB(G) are rank constraints associated with antiwebs, we characterize the imperfection ratio and bound it by 3/2. Outgoing from this result, we characterize and bound the imperfection ratio for several graph classes, including near-bipartite graphs and their complements, namely quasi-line graphs, by means of induced antiwebs and webs, respectively.