在地震资料反演中,反褶积是一种重要的压缩地震子波、提高薄层纵向分辨率的地震数据处理方法。由于地层为层状结构,反射系数可视作稀疏的脉冲序列,因此地震反褶积可以描述为稀疏求解问题。然而,反褶积问题通常是病态的,需要引入正则化...在地震资料反演中,反褶积是一种重要的压缩地震子波、提高薄层纵向分辨率的地震数据处理方法。由于地层为层状结构,反射系数可视作稀疏的脉冲序列,因此地震反褶积可以描述为稀疏求解问题。然而,反褶积问题通常是病态的,需要引入正则化约束以获得稳定和准确的解。本研究介绍了几种不同的正则化方法,包括L1正则化、L2正则化、Cauchy正则化以及结合L1和L2正则化的方法,给出了它们的数学模型,并着重比较了Cauchy正则化与结合L1和L2正则化的方法。通过简单的一维模型和复杂的Marmousi2 (二维)模型的实验,我们评估了这些正则化方法在稀疏脉冲反褶积中的性能表现。结果表明,结合L1和L2正则化的联合方法在噪声抑制和分辨率提升方面表现优异,能够更准确地恢复地下结构的真实反射特性。本文的研究为选择适当的正则化策略以优化地震数据的反褶积处理提供了理论支持和实用指导。In seismic data inversion, deconvolution is an important seismic data processing method that compresses seismic wavelets and improves the vertical resolution of thin layers. Due to the layered structure of the strata, the reflection coefficient can be regarded as a sparse pulse sequence, so seismic deconvolution can be described as a sparse solution problem. However, deconvolution problems are often pathological and require the introduction of regularization constraints to obtain stable and accurate solutions. This study introduces several different regularization methods, including L1 regularization, L2 regularization, Cauchy regularization, and methods combining L1 and L2 regularization. Their mathematical models are given, and the comparison between Cauchy regularization and methods combining L1 and L2 regularization is emphasized. We evaluated the performance of these regularization methods in sparse pulse deconvolution through experiments using a simple one-dimensional model and a complex Marmousi2 (two-dimensional) model. The results show that the joint method combining L1 and L2 regularization performs well in noise suppression and resolution improvement, and can more accurately restore the true reflection characteristics of underground structures. This study provides theoretical support and practical guidance for selecting appropriate regularization strategies to optimize the deconvolution processing of seismic data.展开更多
针对结构化的非凸非光滑优化问题,提出了一种改进的惯性近端交替方向乘子法(Modified Inertial Proximal Alternating Direction Method of Multipliers, MID-PADMM)。该问题在多个领域,包括机器学习、信号处理和经济学中具有重要应用...针对结构化的非凸非光滑优化问题,提出了一种改进的惯性近端交替方向乘子法(Modified Inertial Proximal Alternating Direction Method of Multipliers, MID-PADMM)。该问题在多个领域,包括机器学习、信号处理和经济学中具有重要应用。现有算法在处理这类问题时,往往面临收敛速度慢或无法保证收敛的挑战。为了克服这些限制,引入了一种双重松弛项,以增强算法的鲁棒性和灵活性。理论分析表明,MID-PADMM算法在适当的条件下能够实现全局收敛,并且具有O(1/k)的迭代复杂度,其中k代表迭代次数。数值实验结果表明,与现有的状态最优算法相比,MID-PADMM在多个实例中展现出更快的收敛速度和更高的求解质量。展开更多
地震勘探技术在地震学研究的有关领域中起着重要作用,地震数据处理正确与否,直接影响了解释的准确性与精确度。为了验证动校正速度对最终叠加剖面的影响,对实际某地区的地震资料处理分析,最终得到了最佳动校正速度的叠加图像。实验结果...地震勘探技术在地震学研究的有关领域中起着重要作用,地震数据处理正确与否,直接影响了解释的准确性与精确度。为了验证动校正速度对最终叠加剖面的影响,对实际某地区的地震资料处理分析,最终得到了最佳动校正速度的叠加图像。实验结果表明,选择正确的动校正速度改善了叠加剖面图的信噪比。同时,选择最佳的动校正速度是得到信噪比高的图像的必要选择,这对现实中处理野外采集的地震数据处理有一定的借鉴作用。Seismic exploration techniques play a crucial role in seismic research, directly affecting the accuracy and precision of interpretation based on the correctness of seismic data processing. In order to verify the impact of different dynamic correction speeds on the final superimposed profile, seismic data processing analysis was conducted in a specific region. The optimal dynamic correction speed was finally obtained for the superposition image. The experimental results show that selecting the correct dynamic correction speed can improve the signal-to-noise ratio of the superimposed profile image. At the same time, choosing the optimal dynamic correction speed is a necessary choice to obtain high signal-to-noise ratio images, which has certain reference significance for the seismic data processing of field acquisition in reality.展开更多
随着信息量的激增,深度学习已成为深化知识信息处理和提高学习效率的关键途径。本文基于深度学习理论,探讨了教育学领域中深度学习的内涵,并针对自适应过滤预测法设计了深度学习的实施框架。将这一教学模式应用于自适应过滤预测法的实...随着信息量的激增,深度学习已成为深化知识信息处理和提高学习效率的关键途径。本文基于深度学习理论,探讨了教育学领域中深度学习的内涵,并针对自适应过滤预测法设计了深度学习的实施框架。将这一教学模式应用于自适应过滤预测法的实践教学中,旨在通过深度学习技术提升学生的学习成效和满意度,进而为学生实践技能的提高、创新能力的培养以及终身学习能力的构建打下坚实的基础。With the surge of information, deep learning has become a key way to deepen knowledge and information processing and improve learning efficiency. Based on the theory of deep learning, this paper discusses the connotation of deep learning in the field of pedagogy, and designs the implementation framework of deep learning for the adaptive filter prediction method. By applying this teaching mode to the practical teaching of adaptive filtering and prediction method, the purpose is to improve students’ learning effectiveness and satisfaction through deep learning technology, and then lay a solid foundation for the improvement of students’ practical skills, the cultivation of innovation ability and the construction of lifelong learning ability.展开更多
自动控制原理是自动化及相关专业的学科基础课程,具有典型的多学科交叉特点,其先修课程包括高等数学、计算机、电力电子、机械等。其中,数学问题是该课程的核心问题。通过对数学问题的探究,帮助学生清楚地认识工程背景,激发学生的爱国...自动控制原理是自动化及相关专业的学科基础课程,具有典型的多学科交叉特点,其先修课程包括高等数学、计算机、电力电子、机械等。其中,数学问题是该课程的核心问题。通过对数学问题的探究,帮助学生清楚地认识工程背景,激发学生的爱国和求知热情,进而运用正确的学习方法较好地掌握自动控制的核心技术,对培养具有创新能力的工程型复合人才具有良好的指导意义。Automatic control theory is a basic subject of Automation and related majors, which has a typical character of multidisciplinary cross. Its pre-requisite courses include mathematics, computer, power electronics, mechanics, etc. Math is the central issue of this subject. Through the investigation of math problems, the students can better understand the engineering background, so that their patriotism and curiosity will be inspired. In this sense, the students can use the right learning method to delicately grasp the key knowledge of automatic control, thereby facilitating cultivation of engineering composite talents with innovation ability.展开更多
抽象思维能力是数学素养的一个重要方面,它不仅是解决数学问题的核心能力,也是学生进行科学研究与创新的基础。《高等代数》作为高中《代数》的进阶,具有高度抽象性。因此,在《高等代数》课程中如何培养抽象思维能力,使学生实现高中到...抽象思维能力是数学素养的一个重要方面,它不仅是解决数学问题的核心能力,也是学生进行科学研究与创新的基础。《高等代数》作为高中《代数》的进阶,具有高度抽象性。因此,在《高等代数》课程中如何培养抽象思维能力,使学生实现高中到大学学习的顺利过渡是值得研究的课题。本文分析了《高等代数》中抽象性思维四个方面的体现,并从抽象概念的理解、逻辑推理的训练、推广与类比、可视化、解决问题、以及探索与创新等方面给出了抽象性思维培养的途径和方法。Abstract thinking ability is an important aspect of mathematical literacy. It is not only a core competence for solving mathematical problems but also the foundation for students’ scientific research and innovation. As a progression from high school Algebra, Advanced Algebra is highly abstract. Therefore, how to cultivate abstract thinking skills in the Advanced Algebra course to facilitate a smooth transition for students from high school to university learning is a worthy topic of research. This paper analyzes several aspects of abstract thinking in Advanced Algebra and provides approaches and methods for cultivating abstract thinking from the perspectives of understanding abstract concepts, training logical reasoning, generalization and analogy, visualization, problem-solving, and exploration and innovation.展开更多
非线性Ablowitz-Kaup-Newell-Segur方程是一类应用广泛的非线性偏微分方程。(2 + 1)维空时分数阶Ablowitz-Kaup-Newell-Segur方程常用于描述孤立波在光纤中传播的物理过程,本文利用复行波变换和扩展的Tanh-函数展开法,获得了(2 + 1)维...非线性Ablowitz-Kaup-Newell-Segur方程是一类应用广泛的非线性偏微分方程。(2 + 1)维空时分数阶Ablowitz-Kaup-Newell-Segur方程常用于描述孤立波在光纤中传播的物理过程,本文利用复行波变换和扩展的Tanh-函数展开法,获得了(2 + 1)维空时分数阶Ablowitz-Kaup-Newell-Segur方程的系列新的精确行波解。The Ablowitz-Kaup-Newell-Segur (AKNS) equations, a class of nonlinear partial differential equations, find their utility in a wide array of applications. The space-time fractional (2 + 1)-dimensional AKNS equation, in particular, is capable of describing the physical process of solitary wave propagation in optical fibers. A new class of exact traveling wave solutions of (2 + 1)-dimensional generalized fractional AKNS equation are obtained by employing complex traveling wave transformation and extended Tanh expansion method.展开更多
多元函数的梯度是微积分中的一个重要概念,在分析学中占有举足轻重的地位,它允许我们在多维空间中对函数进行深入的理解和操作。梯度不仅揭示了函数在特定点的局部行为,还为优化问题提供了方向性指导。在数学、物理学、工程学以及其他...多元函数的梯度是微积分中的一个重要概念,在分析学中占有举足轻重的地位,它允许我们在多维空间中对函数进行深入的理解和操作。梯度不仅揭示了函数在特定点的局部行为,还为优化问题提供了方向性指导。在数学、物理学、工程学以及其他科学领域,梯度的概念和应用都极为广泛。The gradient of multivariate functions is an important concept in calculus and occupies a pivotal position in the field of analysis. It allows us to deeply understand and manipulate functions within multidimensional spaces. The gradient not only reveals the local behavior of a function at specific points but also provides directional guidance for optimization problems. The concept and application of the gradient are extremely broad in mathematics, physics, engineering, and other scientific fields.展开更多
机器学习在股票领域的应用日益成为研究和实践的热点。通过分析相关上市公司的股票价格、财务数据、市场情绪和宏观经济因素等多维数据,机器学习算法能够建立预测模型,帮助投资者做出更明智的投资决策。对反映股价信息的个股、大盘、财...机器学习在股票领域的应用日益成为研究和实践的热点。通过分析相关上市公司的股票价格、财务数据、市场情绪和宏观经济因素等多维数据,机器学习算法能够建立预测模型,帮助投资者做出更明智的投资决策。对反映股价信息的个股、大盘、财务数据3类构建了共15项指标,然后采用机器学习中的K-Means聚类算法对我国A股收益数据进行了聚类分析,分析识别出对提高股票投资获胜概率的关键性指标以及相应合适的取值范围。在得到相关结论后,先使用板块以及指数数据对结论进行了检验,得到0.8及以上的盈利概率;然后采用2015~2022历年来的历史交易数据进行了二次验证,策略累计回报率显著优于沪深300基准指数。Machine learning’s application in the stock market field is increasingly becoming a focal point in both research and practice. By analyzing multidimensional data such as stock prices, financial data, market sentiments, and macroeconomic factors related to listed companies, machine learning algorithms can establish predictive models to assist investors in making wiser investment decisions. This article constructs 15 indicators across three categories—individual stocks, market indices, and financial data—to reflect stock price information. Then, using the K-Means clustering algorithm in machine learning, it conducts cluster analysis on the returns data of A-shares in China, identifying crucial indicators and their appropriate value ranges that enhance the probability of successful stock investments. After obtaining these conclusions, the study validates them initially using sector and index data, achieving a profitability probability of 0.8 or higher. Subsequently, historical trading data from 2015 to 2022 is used for a secondary validation, showing significantly better cumulative returns compared to the benchmark Shanghai and Shenzhen 300 Index.展开更多
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
文摘在地震资料反演中,反褶积是一种重要的压缩地震子波、提高薄层纵向分辨率的地震数据处理方法。由于地层为层状结构,反射系数可视作稀疏的脉冲序列,因此地震反褶积可以描述为稀疏求解问题。然而,反褶积问题通常是病态的,需要引入正则化约束以获得稳定和准确的解。本研究介绍了几种不同的正则化方法,包括L1正则化、L2正则化、Cauchy正则化以及结合L1和L2正则化的方法,给出了它们的数学模型,并着重比较了Cauchy正则化与结合L1和L2正则化的方法。通过简单的一维模型和复杂的Marmousi2 (二维)模型的实验,我们评估了这些正则化方法在稀疏脉冲反褶积中的性能表现。结果表明,结合L1和L2正则化的联合方法在噪声抑制和分辨率提升方面表现优异,能够更准确地恢复地下结构的真实反射特性。本文的研究为选择适当的正则化策略以优化地震数据的反褶积处理提供了理论支持和实用指导。In seismic data inversion, deconvolution is an important seismic data processing method that compresses seismic wavelets and improves the vertical resolution of thin layers. Due to the layered structure of the strata, the reflection coefficient can be regarded as a sparse pulse sequence, so seismic deconvolution can be described as a sparse solution problem. However, deconvolution problems are often pathological and require the introduction of regularization constraints to obtain stable and accurate solutions. This study introduces several different regularization methods, including L1 regularization, L2 regularization, Cauchy regularization, and methods combining L1 and L2 regularization. Their mathematical models are given, and the comparison between Cauchy regularization and methods combining L1 and L2 regularization is emphasized. We evaluated the performance of these regularization methods in sparse pulse deconvolution through experiments using a simple one-dimensional model and a complex Marmousi2 (two-dimensional) model. The results show that the joint method combining L1 and L2 regularization performs well in noise suppression and resolution improvement, and can more accurately restore the true reflection characteristics of underground structures. This study provides theoretical support and practical guidance for selecting appropriate regularization strategies to optimize the deconvolution processing of seismic data.
文摘针对结构化的非凸非光滑优化问题,提出了一种改进的惯性近端交替方向乘子法(Modified Inertial Proximal Alternating Direction Method of Multipliers, MID-PADMM)。该问题在多个领域,包括机器学习、信号处理和经济学中具有重要应用。现有算法在处理这类问题时,往往面临收敛速度慢或无法保证收敛的挑战。为了克服这些限制,引入了一种双重松弛项,以增强算法的鲁棒性和灵活性。理论分析表明,MID-PADMM算法在适当的条件下能够实现全局收敛,并且具有O(1/k)的迭代复杂度,其中k代表迭代次数。数值实验结果表明,与现有的状态最优算法相比,MID-PADMM在多个实例中展现出更快的收敛速度和更高的求解质量。
文摘地震勘探技术在地震学研究的有关领域中起着重要作用,地震数据处理正确与否,直接影响了解释的准确性与精确度。为了验证动校正速度对最终叠加剖面的影响,对实际某地区的地震资料处理分析,最终得到了最佳动校正速度的叠加图像。实验结果表明,选择正确的动校正速度改善了叠加剖面图的信噪比。同时,选择最佳的动校正速度是得到信噪比高的图像的必要选择,这对现实中处理野外采集的地震数据处理有一定的借鉴作用。Seismic exploration techniques play a crucial role in seismic research, directly affecting the accuracy and precision of interpretation based on the correctness of seismic data processing. In order to verify the impact of different dynamic correction speeds on the final superimposed profile, seismic data processing analysis was conducted in a specific region. The optimal dynamic correction speed was finally obtained for the superposition image. The experimental results show that selecting the correct dynamic correction speed can improve the signal-to-noise ratio of the superimposed profile image. At the same time, choosing the optimal dynamic correction speed is a necessary choice to obtain high signal-to-noise ratio images, which has certain reference significance for the seismic data processing of field acquisition in reality.
文摘随着信息量的激增,深度学习已成为深化知识信息处理和提高学习效率的关键途径。本文基于深度学习理论,探讨了教育学领域中深度学习的内涵,并针对自适应过滤预测法设计了深度学习的实施框架。将这一教学模式应用于自适应过滤预测法的实践教学中,旨在通过深度学习技术提升学生的学习成效和满意度,进而为学生实践技能的提高、创新能力的培养以及终身学习能力的构建打下坚实的基础。With the surge of information, deep learning has become a key way to deepen knowledge and information processing and improve learning efficiency. Based on the theory of deep learning, this paper discusses the connotation of deep learning in the field of pedagogy, and designs the implementation framework of deep learning for the adaptive filter prediction method. By applying this teaching mode to the practical teaching of adaptive filtering and prediction method, the purpose is to improve students’ learning effectiveness and satisfaction through deep learning technology, and then lay a solid foundation for the improvement of students’ practical skills, the cultivation of innovation ability and the construction of lifelong learning ability.
文摘自动控制原理是自动化及相关专业的学科基础课程,具有典型的多学科交叉特点,其先修课程包括高等数学、计算机、电力电子、机械等。其中,数学问题是该课程的核心问题。通过对数学问题的探究,帮助学生清楚地认识工程背景,激发学生的爱国和求知热情,进而运用正确的学习方法较好地掌握自动控制的核心技术,对培养具有创新能力的工程型复合人才具有良好的指导意义。Automatic control theory is a basic subject of Automation and related majors, which has a typical character of multidisciplinary cross. Its pre-requisite courses include mathematics, computer, power electronics, mechanics, etc. Math is the central issue of this subject. Through the investigation of math problems, the students can better understand the engineering background, so that their patriotism and curiosity will be inspired. In this sense, the students can use the right learning method to delicately grasp the key knowledge of automatic control, thereby facilitating cultivation of engineering composite talents with innovation ability.
文摘抽象思维能力是数学素养的一个重要方面,它不仅是解决数学问题的核心能力,也是学生进行科学研究与创新的基础。《高等代数》作为高中《代数》的进阶,具有高度抽象性。因此,在《高等代数》课程中如何培养抽象思维能力,使学生实现高中到大学学习的顺利过渡是值得研究的课题。本文分析了《高等代数》中抽象性思维四个方面的体现,并从抽象概念的理解、逻辑推理的训练、推广与类比、可视化、解决问题、以及探索与创新等方面给出了抽象性思维培养的途径和方法。Abstract thinking ability is an important aspect of mathematical literacy. It is not only a core competence for solving mathematical problems but also the foundation for students’ scientific research and innovation. As a progression from high school Algebra, Advanced Algebra is highly abstract. Therefore, how to cultivate abstract thinking skills in the Advanced Algebra course to facilitate a smooth transition for students from high school to university learning is a worthy topic of research. This paper analyzes several aspects of abstract thinking in Advanced Algebra and provides approaches and methods for cultivating abstract thinking from the perspectives of understanding abstract concepts, training logical reasoning, generalization and analogy, visualization, problem-solving, and exploration and innovation.
文摘非线性Ablowitz-Kaup-Newell-Segur方程是一类应用广泛的非线性偏微分方程。(2 + 1)维空时分数阶Ablowitz-Kaup-Newell-Segur方程常用于描述孤立波在光纤中传播的物理过程,本文利用复行波变换和扩展的Tanh-函数展开法,获得了(2 + 1)维空时分数阶Ablowitz-Kaup-Newell-Segur方程的系列新的精确行波解。The Ablowitz-Kaup-Newell-Segur (AKNS) equations, a class of nonlinear partial differential equations, find their utility in a wide array of applications. The space-time fractional (2 + 1)-dimensional AKNS equation, in particular, is capable of describing the physical process of solitary wave propagation in optical fibers. A new class of exact traveling wave solutions of (2 + 1)-dimensional generalized fractional AKNS equation are obtained by employing complex traveling wave transformation and extended Tanh expansion method.
文摘多元函数的梯度是微积分中的一个重要概念,在分析学中占有举足轻重的地位,它允许我们在多维空间中对函数进行深入的理解和操作。梯度不仅揭示了函数在特定点的局部行为,还为优化问题提供了方向性指导。在数学、物理学、工程学以及其他科学领域,梯度的概念和应用都极为广泛。The gradient of multivariate functions is an important concept in calculus and occupies a pivotal position in the field of analysis. It allows us to deeply understand and manipulate functions within multidimensional spaces. The gradient not only reveals the local behavior of a function at specific points but also provides directional guidance for optimization problems. The concept and application of the gradient are extremely broad in mathematics, physics, engineering, and other scientific fields.
文摘机器学习在股票领域的应用日益成为研究和实践的热点。通过分析相关上市公司的股票价格、财务数据、市场情绪和宏观经济因素等多维数据,机器学习算法能够建立预测模型,帮助投资者做出更明智的投资决策。对反映股价信息的个股、大盘、财务数据3类构建了共15项指标,然后采用机器学习中的K-Means聚类算法对我国A股收益数据进行了聚类分析,分析识别出对提高股票投资获胜概率的关键性指标以及相应合适的取值范围。在得到相关结论后,先使用板块以及指数数据对结论进行了检验,得到0.8及以上的盈利概率;然后采用2015~2022历年来的历史交易数据进行了二次验证,策略累计回报率显著优于沪深300基准指数。Machine learning’s application in the stock market field is increasingly becoming a focal point in both research and practice. By analyzing multidimensional data such as stock prices, financial data, market sentiments, and macroeconomic factors related to listed companies, machine learning algorithms can establish predictive models to assist investors in making wiser investment decisions. This article constructs 15 indicators across three categories—individual stocks, market indices, and financial data—to reflect stock price information. Then, using the K-Means clustering algorithm in machine learning, it conducts cluster analysis on the returns data of A-shares in China, identifying crucial indicators and their appropriate value ranges that enhance the probability of successful stock investments. After obtaining these conclusions, the study validates them initially using sector and index data, achieving a profitability probability of 0.8 or higher. Subsequently, historical trading data from 2015 to 2022 is used for a secondary validation, showing significantly better cumulative returns compared to the benchmark Shanghai and Shenzhen 300 Index.
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.
