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凸函數(shù)、琴生不等式及其在中學(xué)數(shù)學(xué)中的應(yīng)用

發(fā)布時間:2018-10-22 12:43
【摘要】:凸函數(shù)不僅是一類非常重要的函數(shù),而且是聚集諸多優(yōu)良性質(zhì)的典型函數(shù),對于函數(shù)凸性的研究,在數(shù)學(xué)的許多領(lǐng)域和分支中都有非常重要的應(yīng)用,特別是有關(guān)不等式的推導(dǎo)問題、求最值和取值范圍問題以及三角函數(shù)問題這三個方面,凸函數(shù)都有著十分重要的作用。除此之外,凸函數(shù)還涉及了許多數(shù)學(xué)概念、性質(zhì)、理論、命題的證明和應(yīng)用,而且凸函數(shù)在高考和數(shù)學(xué)競賽中也是有著極其重要的理論價值和應(yīng)用價值。本文首先簡單介紹了凸函數(shù)的定義、幾何意義、等價定義、性質(zhì)、判定以及琴生不等式的證明、變形、推廣;其次,闡述了凸函數(shù)、琴生不等式在中學(xué)教學(xué)中所體現(xiàn)的教育價值和功能;最后,對凸函數(shù)與琴生不等式在中學(xué)數(shù)學(xué)中的應(yīng)用進(jìn)行系統(tǒng)、全面、明確的劃分。例如:對于相似的題型給出統(tǒng)一的解題方法,即簡化了證明過程,又開拓了學(xué)生的解題思路;對于有些題型,還給出了相應(yīng)的推廣,如由“求解圓的內(nèi)接、外切多邊形的面積的最值問題”推廣到“求解橢圓的內(nèi)接、外切多邊形面積的最值問題”。經(jīng)過系統(tǒng)、全面、明確的整理,有助于學(xué)生對這部分知識所涉及的相關(guān)題型有更深入的了解,學(xué)會利用凸函數(shù)的性質(zhì)和琴生不等式更加便捷簡單的解決相關(guān)問題,同時為學(xué)生提供一些新的解題思路和技巧。
[Abstract]:Convex function is not only a kind of very important function, but also a typical function that gathers many excellent properties. For the research of function convexity, it has very important applications in many fields and branches of mathematics. In particular, convex functions play a very important role in the derivation of inequalities, the problem of finding the maximum value and the range of values, and the trigonometric function problem. In addition, convex functions also involve many mathematical concepts, properties, theories, propositions and applications, and convex functions also have extremely important theoretical and applied value in college entrance examination and mathematics competitions. In this paper, the definition of convex function, geometric meaning, equivalent definition, property, judgment and proof, deformation and generalization of Qin Sheng inequality are introduced briefly. Finally, the application of convex function and Qinsheng inequality in middle school mathematics is systematically, comprehensively and clearly divided. For example, a unified method for solving similar problem types is given, which simplifies the proof process and opens up the students' thinking of solving problems. For some problem types, it also gives corresponding generalizations, such as "solving the inner connection of the circle," The problem of the minimum value of the area of a tangent polygon is extended to "solving the problem of the interior connection of an ellipse and the area of an outer polygon". Through systematic, comprehensive and definite arrangement, it is helpful for students to have a deeper understanding of the related question types involved in this part of knowledge, and to learn to use the properties of convex functions and Qin Sheng's inequality to solve the related problems more conveniently and simply. At the same time for students to provide some new ideas and skills to solve problems.
【學(xué)位授予單位】:西北大學(xué)
【學(xué)位級別】:碩士
【學(xué)位授予年份】:2016
【分類號】:G633.6

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