利用洛必达法则求解函数极限及其在物理学与经济学中的应用分析

函数在数学和科技发展中具有重要作用,求解函数极限的方法在函数研究中占据重要地位。极限是高等数学的基础,极限问题是研究的核心和难点。现有文献中极限求解方法的研究多集中于理论阐述,缺乏实际应用案例的分析。本文在系统介绍极限求解方法的基础上,结合经济学、物理学等实际应用案例,旨在弥补这一不足。本文探讨了利用洛必达法则求解函数极限的方法,并分析了其在物理学与经济学中的实际应用。首先,概述了洛必达法则的理论基础及其在处理不定式极限问题中的优势。然后,通过具体实例,详细讲解了如何应用洛必达法则解决物理学中的速度和加速度计算问题,以及经济学中的边际成本和边际收益计算问题。通过这些实例,展示了洛必达法则在不同领域的广泛应用和重要性。本文旨在为读者提供一个全面而实用的参考,使其在实际问题中能够有效应用洛必达法则进行函数极限的求解。

第一个重要极限的推广及应用探析

limx→0sinxx=1在高等数学极限知识点中是重要知识点之一,依此主要介绍第一个重要极限的推广,并且简单介绍在极限计算、微分学、常数项级数等中的应用。

薄型隧道专用防火涂料的研制与应用

通过对当前隧道防火涂料所用原料的分析,针对隧道用防火涂料的要求,以水泥和乳胶粉为复合黏结剂,以多种复合材料为隔热填料,在多种助剂的配合下,研制出薄型隧道专用防火涂料。该涂料耐火极限高,粘结强度大,环保无毒,各项技术指标都超过了国家隧道防火涂标准要求,市场前景广阔。

概率论的发展简史及其在生活中的若干应用

介绍了概率论的创立、发展近340年的历史以及数学工作者在概率论发展中所做的贡献,概述了概率论的近期发展方向,并列举了其在生活中的若干应用。

Lagrange中值定理的一个推广

利用区间序列的性质以及极限基础理论研究了微分中值定理中ξ的趋近性质,并证明了Lagrange中值定理中当ab,相互靠近时其中间值ξ→x0的渐进性质.

关于标量衍射理论在二元化透镜设计中的应用极限的探讨

本文分别用菲涅耳衍射公式和瑞利－索末菲衍射公式对不同数值孔径的二元化透镜的衍射问题进行了计算机模拟，给出了计算结果，从中得出了菲涅耳衍射公式和瑞利－索末菲衍射公式各自的应用范围，并对大数值孔径二元化透镜设计中应用严格电磁理论问题作了讨论。

A strength reduction method based on double reduction parameters and its application

In the traditional strength reduction method,the cohesion and the friction angle adopt the same reduction parameter,resulting in equivalent proportional reduction.This method does not consider the different effects of the cohesion and friction angle on the stability of the same slope and is defective to some extent.Regarding this defect,a strength reduction method based on double reduction parameters,which adopts different reduction parameters,is proposed.The core of the double-parameter reduction method is the matching reduction principle of the slope with different angles.This principle is represented by the ratio of the reduction parameter of the cohesion to that of the friction angle,described as η.With the increase in the slopeangle,ηincreases; in particular,when the slope angle is 45°,tηis 1.0.Through the matching reduction principle,different safety margin factors can be calculated for the cohesion and friction angle.In combination with these two safety margin factors,a formula for calculating the overall safety factor of the slope is proposed,reflecting the different contributions of the cohesion and friction angle to the slope stability.Finally,it is shown that the strength reduction method based on double reduction parameters acquires a larger safety factor than the classic limit equilibrium method,but the calculation results are very close to those obtained by the limit equilibrium method.

Application of cusp catastrophe theory to reliability analysis of slopes in open-pit mines

A method of slope reliability analysis was developed by imposing a state equation on the limit equilibrium theory, given the basis of a fixed safety factor technique. Among the many problems of reliability analysis, the most important problem is to find a performance function. We have created a new method of building a limit state equation for planar slip surfaces by applying the mathematical cusp catastrophe theory. This new technique overcomes the defects in the traditional rigid limit equilibrium theory and offers a new way for studying the reliability problem of planar slip surfaces. Consequently, we applied the technique to a case of an open-pit mine and compared our results with that of the traditional approach. From the results we conclude that both methods are essentially consistent, but the reliability index calculated by the traditional model is lower than that from the catastrophic model. The catastrophe model takes into consideration two possible situations of a slope being in the limit equilibrium condition, i.e., it may or may not slip. In the traditional method, however, a slope is definitely considered as slipping when it meets the condition of a limit equilibrium. We conclude that the catastrophe model has more actual and instructive importance compared to the traditional model.

极限思想在数学解题中的应用

本文对极限在初等数学中应用进行了讨论分析,给出一些典型题型,总结了极限在解题时简洁有效的应用。

Application of artificial neural network to predict Vickers microhardness of AA6061 friction stir welded sheets

The application of friction stir welding(FSW) is growing owing to the omission of difficulties in traditional welding processes. In the current investigation, artificial neural network(ANN) technique was employed to predict the microhardness of AA6061 friction stir welded plates. Specimens were welded employing triangular and tapered cylindrical pins. The effects of thread and conical shoulder of each pin profile on the microhardness of welded zone were studied using tow ANNs through the different distances from weld centerline. It is observed that using conical shoulder tools enhances the quality of welded area. Besides, in both pin profiles threaded pins and conical shoulders increase yield strength and ultimate tensile strength. Mean absolute percentage error(MAPE) for train and test data sets did not exceed 5.4% and 7.48%, respectively. Considering the accurate results and acceptable errors in the models' responses, the ANN method can be used to economize material and time.

Degradation process assessment of prestressed concrete continuous bridges in life-cycle

To accurately evaluate the degradation process of prestressed concrete continuous bridges exposed to aggressive environments in life-cycle,a finite element-based approach with respect to the lifetime performance assessment of concrete bridges was proposed.The existing assessment methods were firstly introduced and compared.Some essential mechanics problems involved in the degradation process,such as the deterioration of materials properties,the reduction of sectional areas and the variation of overall structural performance caused by the first two problems,were investigated and solved.A computer program named CBDAS(Concrete Bridge Durability Analysis System) was written to perform the above-metioned approach.Finally,the degradation process of a prestressed concrete continuous bridge under chloride penetration was discussed.The results show that the concrete normal stress for serviceability limit state exceeds the threshold value after 60 a,but the various performance indicators at ultimate limit state are consistently in the allowable level during service life.Therefore,in the case of prestressed concrete bridges,the serviceability limit state is more possible to have durability problems in life-cycle;however,the performance indicators at ultimate limit state can satisfy the requirements.

第一个重要极限的教学

极限是微积分的基础,重要极限是极限学习的重点和难点.重要极限在极限的计算和一些求导公式的证明中起着重要的作用,通过对第一个重要极限的证明和应用的探索,能够更好的理解和应用此重要极限,进而提高教学质量.

Interface stress element method and its application in analysis of anti-sliding stability of gravity dam

The rigid body limit equilibrium method （LEM） and the nonlinear finite element method （NFEM） are often used in the analysis of anti-sliding stability of gravity dam. But LEM cannot reflect the process of progressive instability and mechanical mecha- nism on failure for rock mass while NFEM is difficult to use to solve the displacement discontinuity of weak structural plane. Combining the research with Xiangjiaba Hydropower Station project, the analysis of anti-sliding stability for segment 12# of the dam has been carried out using interface stress element method （ISEM）. The results can reflect the most dangerous location, the scope and distribution of failure zone in weak structural plane, and present the process of progressive failure in dam foun- dation as well as the safety coefficient of possible sliding body. These achievements provide an important technical reference for dam foundation treatment measures. The computational results show that ISEM can naturally describe discontinuous de- formation of rock mass such as dislocation, openness and sliding. Besides, this method is characterized by good adaptability, convenient calculation and high compatibility, thus it is regarded as an effective way to make an analysis of anti-sliding stabil- ity of gravity dam