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我们利用解析几何中的定比分点公式证明一组代数不等式。我们这样作的目的,是把数和形结合在一起,使证明既严密又直观。已知 a,b 是正实数,求证: 相似文献
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牛保林 《中学生数理化(高中版)》2004,(10):15-16
我们知道,若设点P分有向线段→P1P2所成的比为λ,则有(Ⅰ)λ>0时,P内分→P1P2;(Ⅱ)λ<0(λ≠-1)时,P外分→P1P2;(Ⅲ)λ=0时P与P1重合;(Ⅳ)P与P2重合时,λ不存在. 相似文献
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定比分点公式是解析几何中重要的公式之一。公式本身及证明过程都具有丰富的内涵和重要的数学思想。本文着重谈用定比分点公式所包含的数学思想来证明一类不等式题。 相似文献
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我们知道,若P_1(x_1,y_1),P_2(x_2,y_2),P(x,y),且P分P_1P_2的比为λ(λ=-1),见y=y_1 λy_2/1 λ或λ=y-y_1/y_2-y。由公式易得: 1°.λ>0(?)y介于y_1、y_2之间。 相似文献
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1.巧求值域例1求函数y=(1 cosx)/(3-2cosx)的值域.分析观察上式可联想到定比分点公式x=(x_1 x_2λ) /(1 λ)得y=(1/3 (-(1/2))(-(2/3)cosx))/(1 (-(2/3)cosx)),即P(y,0)分起点为P_1(1/3,0),终点为P_2(-(1/2),0)的有向线段(?)的比为 相似文献
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在解决代数问题时,如果能将其中的常数与变数巧妙地转化为数轴或坐标平面内的定点与动点,则可以通过这些点在“运动”中的相对位置和区域来了解和判断各种数量间的关系。而定比分点公式正是促成这种转化的不可多得的媒介,本文拟就以下几个方面谈谈它在代数解题中的应用。 相似文献
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于万俊 《数学学习与研究(教研版)》2008,(9)
定比分点公式是平面解析几何中的重要公式,在解析几何中的应用非常广泛.在平面直角坐标系中分点的坐标是以二维变量(x,y)形式出现的,在数轴上定比及定比分点公式显得更简洁和新颖,分点的坐标是以一维变量x的形式出现的.所以在高中数学的其他章节内容中,若能灵活运用定比及定比分点公式求解, 相似文献
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公式 若正三棱锥的侧棱长为l,侧面顶角为θ,则高h =33l 1 2cosθ ( 0 <θ<2π3)。证 如图 ,已知在正三棱锥P -ABC中 ,PO⊥平面ABC ,用向量法证明如下 :∵PO =PA AO =PB BO=PC CO ,∴ 3PO =(PA PB PC) (AO BO CO)。又因点O是正△ABC的中心 ,易证AO BO CO =O ,∴PO =(PA PB PC) / 3。∴ |PO|2 =( |PA|2 |PB|2 |PC| 2 |PA|·|PB|cosθ 2 |PA||PC|cosθ 2 |PB||PC|cosθ) / 9=(l2 l2 l2 2l2 cosθ 2l2 cosθ 2l2 … 相似文献
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郭书力 《呼伦贝尔学院学报》2001,9(2):72-73
本文主要介绍了利用积分证明不等式 ,其核心内容是导数与积分的应用 ,目的是使学生充分掌握导数及积分的性质 ,锻炼思维 ,扩大视野。 相似文献
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给出了Binet-Cauchy公式的三个应用:用此公式证明恒等式;用此公式不等式;用此公式计算行列式。 相似文献
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指出了同济大学第五版《高等数学》教材的配套参考书上([1]、[2]、[3]、[4]、[5]),关于计算曲面积分一题的解法错误所在,分析了错误的原因,给出了正确解法。告诫学生使用高斯公式计算曲面积分时一定不能忽视条件,否则可能导致错误。 相似文献
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Louis van Schalkwijk Theo Bergen Arnoud van Rooij 《Educational Studies in Mathematics》2000,43(3):293-311
The Mathematics Department of the University of Nijmegen in collaboration with the Graduate School of Education has developed
a math course in the field of fractals and dynamic processes for volunteer students in the second phase of secondary education
in the Netherlands. The students, of approximately 16 years of age, show a special interest in, and an aptitude for mathematics
and informatics. One of the main goals of the course was to highlight the deductive aspect of mathematics, an aspect that
is neglected in the ordinary math curriculum of secondary education in the Netherlands. That goal was pursued by giving the
students ample opportunity to conduct investigations on their own and in a way that they would be responsible for judging
the correctness of their arguments in making mathematical deductions. In that way proving is imbedded in a larger structure
and becomes a tool for the students to convince each other. During the courses we searched for the right way for teachers
to coach these investigations, that is: to find a balance between mere concentration on guiding the process of the students'
investigations and active intervention in the learning process of proving. In this article we illustrate with two examples
– the first from the '95–'96 course and the second from the '96–'97 course – in what way we adjusted our coaching. Our results
are explorative, but our approach appears promising and we are convinced that investigations as learning environment for proving
would also be a valuable part for the regular secondary math education in the Netherlands.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
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