在教学过程中,我们发现拉格朗日中值定理是学生学习微积分的巨大障碍,这是因为拉格朗日中值定理是微分中值定理的核心内容,是研究函数与导数之间联系的理论工具,在微积分学中起着至关重要的作用,应用十分广泛。本文重点研究拉格朗日中...在教学过程中,我们发现拉格朗日中值定理是学生学习微积分的巨大障碍,这是因为拉格朗日中值定理是微分中值定理的核心内容,是研究函数与导数之间联系的理论工具,在微积分学中起着至关重要的作用,应用十分广泛。本文重点研究拉格朗日中值定理在证明导数极限定理、求函数极限问题、证明不等式以及证明函数单调性方面的应用,以及拉格朗日中值定理的两个推广。希望本文可以对学生学习微积分有所帮助。During the teaching process, we found that the Lagrange Mean Value Theorem is a significant obstacle for students learning calculus. The Lagrange Mean Value Theorem is the core content of the Mean Value Theorem in differential calculus. It is a theoretical tool for studying the relationship between functions and their derivatives and plays a crucial role in calculus, with a wide range of applications. This paper focuses on the application of the Lagrange Mean Value Theorem in proving the derivative limit theorem, solving limit problems of functions, proving inequalities, and proving the monotonicity of functions, as well as two extensions of the Lagrange Mean Value Theorem. It is hoped that this article can be of assistance to students in their study of calculus.展开更多
应变-旋转(Strain-Rotation,S-R)和分解定理为分析几何非线性问题提供了合理可靠的理论基础,但用有限元求解时会遇到大变形发生后的网格畸变问题。近年提出的虚单元法(Virtual element method,VEM)适用于一般的多边形网格,因此,该文尝...应变-旋转(Strain-Rotation,S-R)和分解定理为分析几何非线性问题提供了合理可靠的理论基础,但用有限元求解时会遇到大变形发生后的网格畸变问题。近年提出的虚单元法(Virtual element method,VEM)适用于一般的多边形网格,因此,该文尝试使用一阶虚单元求解基于S-R和分解定理的二维几何非线性问题,以克服网格畸变的影响。基于重新定义的多项式位移空间基函数,推演获得一阶虚单元分析线弹性力学问题时允许位移空间向多项式位移空间的投影表达式;按照虚单元法双线性格式的计算规则,分析处理基于更新拖带坐标法和势能率原理的增量变分方程;进而建立离散系统方程及其矩阵表达形式,并编制MATLAB求解程序;采用常规多边形网格和畸变网格,应用该文算法分析均布荷载下的悬臂梁和均匀内压下的厚壁圆筒变形。结果与已有文献和ANSYS软件的对比表明:该文算法在两种网格中均可有效执行且具备足够数值精度。总体该文算法为基于S-R和分解定理的二维几何非线性问题求解提供了一种鲁棒方法。展开更多
文摘在教学过程中,我们发现拉格朗日中值定理是学生学习微积分的巨大障碍,这是因为拉格朗日中值定理是微分中值定理的核心内容,是研究函数与导数之间联系的理论工具,在微积分学中起着至关重要的作用,应用十分广泛。本文重点研究拉格朗日中值定理在证明导数极限定理、求函数极限问题、证明不等式以及证明函数单调性方面的应用,以及拉格朗日中值定理的两个推广。希望本文可以对学生学习微积分有所帮助。During the teaching process, we found that the Lagrange Mean Value Theorem is a significant obstacle for students learning calculus. The Lagrange Mean Value Theorem is the core content of the Mean Value Theorem in differential calculus. It is a theoretical tool for studying the relationship between functions and their derivatives and plays a crucial role in calculus, with a wide range of applications. This paper focuses on the application of the Lagrange Mean Value Theorem in proving the derivative limit theorem, solving limit problems of functions, proving inequalities, and proving the monotonicity of functions, as well as two extensions of the Lagrange Mean Value Theorem. It is hoped that this article can be of assistance to students in their study of calculus.
基金Supported by the National Natural Science Foundation of China(12171335,12301603)the Science Development Project of Sichuan University(2020SCUNL201)the Scientific Foundation of Nanjing University of Posts and Telecommunications(NY221026)。
基金Supported by the National Natural Science Foundation of China(12171335,12301603)the Science Development Project of Sichuan University(2020SCUNL201)the Scientific Foundation of Nanjing University of Posts and Telecommunications(NY221026)。
文摘应变-旋转(Strain-Rotation,S-R)和分解定理为分析几何非线性问题提供了合理可靠的理论基础,但用有限元求解时会遇到大变形发生后的网格畸变问题。近年提出的虚单元法(Virtual element method,VEM)适用于一般的多边形网格,因此,该文尝试使用一阶虚单元求解基于S-R和分解定理的二维几何非线性问题,以克服网格畸变的影响。基于重新定义的多项式位移空间基函数,推演获得一阶虚单元分析线弹性力学问题时允许位移空间向多项式位移空间的投影表达式;按照虚单元法双线性格式的计算规则,分析处理基于更新拖带坐标法和势能率原理的增量变分方程;进而建立离散系统方程及其矩阵表达形式,并编制MATLAB求解程序;采用常规多边形网格和畸变网格,应用该文算法分析均布荷载下的悬臂梁和均匀内压下的厚壁圆筒变形。结果与已有文献和ANSYS软件的对比表明:该文算法在两种网格中均可有效执行且具备足够数值精度。总体该文算法为基于S-R和分解定理的二维几何非线性问题求解提供了一种鲁棒方法。
