数学论文-用高数观点透视近几年的高考数学试题.doc

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用高数观点透视近几年的高考数学试题
(孝感学院数学与统计学院湖北孝感 432000)
摘要随着新课标准的实施,在近几年的高考中出现了一些有着一定的高等数学背景的试题,这主要源于两个主要因素:一是这种题型形式新颖,既能开阔数学视野,有利于高等数学与初等数学的和谐接轨,又能有效地考察学生的思维能力,尤其是创新能力;,命题组成成员中大多是大学教师,.
关键词连续函数;最大(小)值;递推数列;不动点;凹凸性
perspective the mathematics test question in recent years’ college entrance examinationwith the viewpoint in high school mathematics
Wang Zi-peng
(Xiaogan College of Mathematics and Statistics Institute HuBei XiaoGan 432000 )
Abstract: With the implementation of standard courses,College entrance examination in recent years there have been some of the higher mathematics is a certain background questions,This is mainly due to two main factors: First, this form of novel questions, Mathematics can broaden horizons,In favor of higher mathematics and the harmonious integration of Elementary Mathematics,Can effectively study the thinking ability of students, In particular the ability to innovate, Second, test the proposition with the gradual deepening of the reform. Autonomy of the provinces and cities proposition, more and more, posed of members, mostly university teachers. Proposition when they can not be free from the impact of their research background. This article will list a few examples to show that.
Key words: Continuous function; the max(min)value; recursive series; fixed point;concavity and convexity
0 引言
代数推理,递推数列,极限与求导方法的应用,不动点问题,数列极限的一些特性,函数图象的凸性等具有高等数学倾向的问题逐步走进高考,虽然它们对解决问题的逻辑依据不高,但是通过直观化,,意在结合有关研究和分析,尝试着预测今后可能与高数思想相联系的高考试题趋势和方向.
1 2008年高考数学的一个新亮点—猜想题
在近几年的高考数学题中,有不少属于猜想题,它们有的是通过观察猜想结果(不要求证明),,,用高等数学方法解析这些问题,、湖北卷中实例加以分析说明.
例1 (2008年重庆卷第22题) 设各项均为正数的数列{}满足,
.
(1)若,求,并猜想的值(不需要证明);
(2)记,若对恒成立,求的值及
数列{}的通项公式.
参考答案中是用的值,来猜想的值,,我们首先看看另一道2008年广东的高考试题.
例2 (2008年广东卷第21题) 设为实数,是方程
的两个实根,数列{}满足.
(1)证明: ,;
(2)求数列{}的通项公式;
(3)若,求{}的前项和.