微积分在证明和式不等式中的应用

利用微积分的思想和方法,简单、快捷地解决了一类和式不等式的证明。     

James J. Kaput (1942–2005) imagineer and futurologist of mathematics education

Jim Kaput lived a full life in mathematics education and we have many reasons to be grateful to him, not only for his vision of the use of technology in mathematics, but also for his fundamental humanity. This paper considers the origins of his ‘big ideas’ as he lived through the most amazing innovations in technology that have changed our lives more in a generation than in many centuries before. His vision continues as is exemplified by the collected papers in this tribute to his life and work.

William Wales and the 1769 transit of Venus: puzzle solving and the determination of the astronomical unit

According to Thomas Kuhn, a significant part of “normal science” is the fact gathering, empirical work which is intended to illustrate an existing paradigm. Some of this effort focuses on the determination of physical constants such as the astronomical unit (AU). For Kuhn, normal science is also what prepares students for membership in a particular scientific community and is embodied in some form in our science textbooks. However, neither Kuhn nor the textbook says much about the individuals who practice normal science, especially those who had been relegated to the “hack” duties of long and arduous measurement and calculation. In this paper, to provide a context for students of astronomy, I will outline the story of the determination of the AU and in particular the contribution of William Wales, an obscure British astronomer. Wales, toiling in the shadow of Halley (of Halley’s comet fame), Mason and Dixon (of Mason and Dixon line fame) and the infamous Captain Cook endured a brutal winter in northern Canada for a brief glimpse of the 1769 transit of Venus. In the end, Wales supplied one small piece of the puzzle in the determination of the AU and he exemplified the human spirit and persistence of a Kuhnian “puzzle solver”.

微积分在经济学最优化问题中的应用

通过文献和案例研究,揭示经济领域中的最优化现象,并把经济学现象分析归纳到数学领域中,利用微积分中的理论,讨论“完全竞争、卖方垄断、买方垄断”三种情况下的利润最大化问题,给出严格的数学逻辑推理。     

《竹书纪年》真伪考

《竹书纪年》为汲冢《竹书》中的一种,为晋武帝时汲郡人不准盗掘魏襄王之墓时所发现。故又统称《汲冢书》。因此书系科斗文竹简,有“数十车”之多,后世又无传存;加之其出土年代和出土之王墓说法不一,故有人怀疑它的真实性,而斥之为“汉后人伪作”。《竹书纪年》所载王年及其历史事件均与相关历史典籍和出土文物及青铜铭器所载完全一致,其天象记载亦与历史的实际情形相吻合,足见《竹书纪年》真实不伪。     

甘岑后继式演算系统与其自然演绎系统的比较

《逻辑演绎研究》(“Untersuchungen über das Logische Schliessen”)是甘岑提交哥廷根大学的就职论文,文中甘岑提出了两种逻辑演算系统,即“自然演绎”和“后继式演算”系统。从自然演绎开始甘岑尝试构造一个接近数学实际推理的逻辑演算,而后针对自然演绎系统的不足进行了转化和改进,构造了后继式演算系统,并证明了该逻辑演算系统的合理性。甘岑的逻辑演绎思想在逻辑史上具有十分重要的地位,对现代逻辑和证明论的发展具有深远的影响。     

《微积分》如何计算圆周率π的近似值

圆周率π可以说是在数学中最为常见的一个无理数,在计算圆周长、圆面积、球体积以及很多相关图形(比如扇形、椭圆和椭球等)计算中起到了关键作用。它最早被定义为圆周长与直径的比值。而如何计算圆周率π也引起了古今中外众多数学家们的关注。利用"割圆术",我国古代著名数学家祖冲之得到了两个圆周率的近似值,分别为约率22/7和密率335/113。其中密率335/113足足比欧洲早了1000年。然而,由于"割圆术"方法的局限性,改进已有结果的难度变得越来越大。在本文中,我们主要介绍在微积分中利用无穷级数计算圆周率π的一些公式。利用计算机编程,人们甚至可以将圆周率计算到小数点后10万亿位。     

基于过程考核的微积分教学质量提高

针对微积分(Ⅱ)课程教学质量提高的要求,提出了课程过程考核的基本思路和具体措施。以两个教学对比班作为比较研究对象,在不同过程考核的情况下,比较研究了两个对比班学生的期末卷面成绩。从比较结果看出,加强了过程考核的班级成绩明显优于未加强过程考核的班级。加强微积分(Ⅱ)过程考核有助于教学质量的提高,为微积分教学改革提供依据。     

数学分析教学中常见问题的解读

针对数学分析教学的经典概念作较本质的剖析,对典型问题从几何或物理的角度进行较形象的处理,并用哲学的思想对重要概念或定理作较深入的思考,借以提高学生的学习兴趣、理解能力与创新能力.     

《天官书》志疑

《天官书》记载天文星象,内容冷僻,文辞古奥;三家注征引各家之书,或不标出处,其标注者,今亦多已失传,且内容芜杂,难以一一稽考,是以校勘颇为不易.此文以景祐本、耿本、黄本、彭本、柯本、凌本、殿本诸本与金陵书局本对校,稽核《汉书》、《晋书》、《隋书》、《旧唐书》、《新唐书》、《宋史》及相关文献,参考前人研究成果,结合文字、音韵、训诂、词汇、文例、史实等,对《天官书》中50余处文字作了考辨.