Ideas Of Calculus Research Articles

Focusing on foundational Calculus ideas to understand the derivative concept via problem-solving tasks that involve the use of a Dynamic Geometry System

The aim of this paper is to review recently calculus curriculum reforms and research studies that document what types of understanding students develop in their precalculus courses. We argue that it is important to characterize what difficulties students experience to solve tasks that include the use of foundational calculus concepts and to look for possible ways for students to develop a way of reasoning to work on problems that involve variational phenomena. Thus, we identified tasks in which calculus students exhibit limited understanding of essential concepts to approach and solve those tasks. The purpose is to illustrate and discuss how the systematic use of a Dynamic Geometry System (DGS) could provide a set of affordances for students to develop ways of thinking to grasp calculus foundational ideas and to study the derivative concept. Here, we relied on Thurston’s seminal work that emphasizes the relevance for learners to identify, connect, and coordinate different dimensions and meanings (intuitive, symbolic, algorithmic, geometric, physical, and formal) to construct, understand, and apply the concept of derivative.

Promoting Productive Understandings of Rate of Change in Calculus Courses

A productive understanding of rate of change concept is essential for constructing a robust understanding of derivatives. There is substantial evidence in the research that students enter and leave their Calculus courses with naive understandings of rate of change. Implementing a short unit on “what is rate of change” can address these issues and better support students in understanding the fundamental ideas of Calculus. This paper provides a short teaching intervention that instructors can utilize to help students build a productive understanding of rate of change that will aid them in understanding derivatives as instantaneous rates of change.

Evaluation of Limits Involving Trigonometric Functions by L’ Hospital Rule

Limit is the basic idea of calculus and it is the theoretical basis of a series of important concepts such as derivatives, continuity, definite integral, etc., In addition, it is a tool for solving many problems, thus correctly mastering the calculation methods and skills of the limit is of great significance to learn more advanced mathematics well. Through induction and conclusion, this article mainly introduces the application of several methods of solving the limit in light of L'Hopital's rule, as well as gives the analysis of example problems for each method. For this purpose, it is expected that readers can get some inspiration for solving problems from them, so that they can solve problems more clearly. Furthermore, a brief introduction is made to the main content of the L'Hospital's rule in advanced mathematics and an analysis is given of the misunderstandings in the teaching and application of the L'Hospital's rule, and a detailed description of many specific application of the L'Hospital's rule in the analysis of limit problems.

The additional proof and application of Cauchy-Riemann Equation

As far back as the seventeenth century, Newton and Leibniz introduced and refined the idea of calculus, and the concepts of derivative and differentiation gradually entered the mathematical world as powerful tools. However, after the appearance of the theory of complex functions in the 18th century, the conventional methods used to analyze the properties of complex functions were somewhat limited. Thus, mathematicians Cauchy and Riemann came up with the idea of the “C-R equation” and provided the necessary and sufficient conditions in derivable complex functions based on the research done by d'Alembert. This equation laid the foundation for the analytic function. This paper introduces the idea of the limit of two variables function and shows both the concept of and graphical meaning partial derivative, paving way for the concept of complex function and that of C-R equation. In addition, the paper specifically proves the C-R equation by a much more rigorous method with the routes from the linear equation and applies this important equation to determine the number of and the locations of the differentiable points in a complex plane. It is also one of the factors to determine the analyticity of complex functions. At the end of the paper, an example will be provided to illustrate the application of the C-R equation.

UAV-Aided Dual-User Wireless Power Transfer: 3D Trajectory Design and Energy Optimization.

Unmanned aerial vehicles (UAVs) have been widely considered to enhance the communication coverage, as well as the wireless power transfer (WPT) of energy-constrained communication networks to prolong their lifetime. However, the trajectory design of a UAV in such a system remains a key problem, especially considering the three-dimensional (3D) feature of the UAV. To address this issue, a UAV-assisted dual-user WPT system was investigated in this paper, where a UAV-mounted energy transmitter (ET) flies in the air to broadcast wireless energy to charge the energy receivers (ERs) on the ground. By optimizing the UAV's 3D trajectory toward a balanced tradeoff between energy consumption and WPT performance, the energy harvested by all ERs during a given mission period was maximized. The above goal was achieved through the following detailed designs. On the one hand, on the basis of previous research results, there is a one-to-one correspondence between the UAV's abscissa and height, so only the relationship between the height and time was focused on in this work to obtain the UAV's optimal 3D trajectory. On the other hand, the idea of calculus was employed to calculate the total harvested energy, leading to the proposed high-efficiency trajectory design. Finally, the simulation results demonstrated that this contribution is capable of enhancing the energy supply by carefully designing the 3D trajectory of the UAV, compared to its conventional counterpart. In general, the above-mentioned contribution could be a promising way for UAV-aided WPT in the future Internet of Things (IoT) and wireless sensor networks (WSNs).

Analysis of the Method and Principle of Measuring the Average Power of Equipment Based on DC Power Analyzer

To solve the problem that it is difficult to accurately measure the average power of equipment with large instantaneous power change, this paper analyzes the principle of equipment power change and the mechanism of measurement error and proposes a reliable average power measurement method based on the idea of calculus. Through test verification, compared with the common three-measurement average methods, the uncertainty of the results calculated by the method proposed in this paper is reduced, and the results are accurate and reliable, which can reflect the real power of the equipment and meet the requirements of the test and test support.

Development of Extreme Thoughts in Calculus and Its Application in Mathematical Analysis

Extreme thought is the basic idea of calculus. A series of important concepts in mathematical analysis, such as the continuity of functions, derivatives and definite integrals, are all defined with the help of extreme thought. The generation and improvement of extreme thought is under the need of society and it has created a new impetus for the development of mathematics, and has become the basis and starting point of modern mathematical thought and methods. In order to deepen the public’s understanding of extreme thought, this article has studied the development and application of extreme thought. First of all, the concept and emergence of extreme thought are briefly introduced, and then the development process of extreme thought is analyzed from three stages: the germination period, the development period, and the perfect period, and finally the application of extreme thought in mathematical analysis is explained from three aspects.

Integrated Optimization on Energy Saving and Quality of Service of Urban Rail Transit System

Optimizing to increase the utilization ratio of regenerative braking energy reduces energy consumption, and can be done without increasing the deviation of train running time in one circle. The latter entails that the train timetable is upheld, which guarantees that the demand for passenger transport services is met and the quality of services in the urban rail transit system is maintained. This study proposes a multi-objective optimization model for urban railways with timetable optimization to minimize the total energy consumption of trains while maximizing the quality of service. To this end, we apply the principles and ideas of calculus to reduce the power of the velocity in the train energy consumption model. This greatly simplifies the complexity of the optimization model. Then, considering the conflicting requirements of decision-makers, weight factors are added to the objective functions to reflect decision-makers’ preferences for energy-saving and the quality of service. We adopt the nondominated sorting genetic algorithm-II (NSGA-II) to solve the proposed model. A practical case study of the Yizhuang urban railway line in Beijing is conducted to verify the effectiveness of the proposed model and evaluate the advantages of the optimal energy saving timetable (OEST) in comparison to the optimal quality of service timetable (OQOST). The results showed that the OEST reduced total energy consumption by 8.72% but increased the deviation of trains running time in one circle by 728 s. The total energy consumption was reduced by 6.09%, but there was no increase in the deviation of train running time in one circle with the OQOST.

A Definite Explanation of the Concept of Limit in Teaching

The limit idea is the basic idea of calculus. Almost all concepts in mathematical analysis are inseparable from the limit, but the limit is a concept that is difficult to understand accurately. By assuming that there is a certain point closest to the limit value, this paper provides a reasonable explanation for the infinitesimal paradox and a new answer to the question why the limit value is accurate in teaching. At the same time, this method is applied to the derivative and used to understand a common practical problem in mathematics. The analysis shows that this method is effective for the accurate understanding of limit problems.

The pedagogical power of context: extending the Epidemiology of Eyam

This paper describes the further development of the Epidemiology of Eyam project at Winchester College since the publication of our previous paper (French et al 2019 Phys. Educ. 54 045008). A much wider cadre of students have been involved with this recent phase of the study, and we have also benefitted from a sabbatical collaboration with Oxford University Mathematical Institute. It is hoped that the research presented here can be used as a case study for an Extended Project Qualification (EPQ) or equivalent in a range of schools around the UK and worldwide. We feel that a project such as this, with a strong humanities and public health rooted context, could incentivise students of mathematics and sciences to participate in inter-disciplinary teams over an educationally significant period, and offer an opportunity to develop vital independent research skills that are required at University, but are often difficult to experience in a School context. The main educational goal of our previous paper was to provide an suitable context to incentivise the introduction of Calculus ideas. In this paper we assume a slightly higher level of mathematical skill, and aim to present a more comprehensive analysis of the epidemiological model that we will refer to as the 'Eyam Equations.' We describe a 'semi-analytic' solution, and make the connection to the approximation of Kermack and McKendrick (Kermack and McKendrick 1927 Proc. R. Soc. Lond. A 115 700–21) which held sway for much of the early twentieth century. We revisit the Eyam 1660s plague data of William Mompesson, and also apply the model to Ebola data collected in Liberia during the 2014–16 epidemic in West Africa. Motivated by uncertainty in the size of the 'at-risk' Susceptible population, we re-parameterize the model in terms of alternative inputs which enable a curve-fitting mechanism to be conducted more efficiently, with a much more tightly bounded range of possible parameter values. In addition to a spreadsheet model, we have created a software application in the MATLAB environment which has dynamic tools that could potentially enable a sensible curve fit to be calculated very rapidly in response to new data. From a time series describing the Infective population of the 2014–16 Liberia Ebola epidemic, we can predict Infective I(t), Susceptible and Dead populations, calculate the associated total population size , calculate the Kendall Susceptible threshold , and hence the Basic Reproduction Number . For the Liberia Ebola data, these are: . Note this suggests that out of an ‘at risk’ population of , about 75% may ultimately have been infected by the Ebola virus. In the WHO Ebola Response Team report (WHO Ebola Response Roadmap Situation Report), for the 2014–16 Liberia outbreak.

Motivating Calculus Through a Single Question

The Corvette Problem is a nonroutine investigation, one that provides an authentic context to explore many interrelated calculus ideas. In our years of sharing this problem with students and colleagues, we have found additional topics from the calculus curriculum to have rich interpretations in this environment. The Corvette Problem contextualizes much of the material in first-semester undergraduate calculus by way of unifying theorems and definitions into a powerful mosaic of ideas.

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On the use of history of mathematics: an introduction to Galileo's study of free fall motion

ABSTRACTIn this paper, we report on an experimental activity for discussing the concepts of speed, instantaneous speed and acceleration, generally introduced in first year university courses of calculus or physics. Rather than developing the ideas of calculus and using them to explain these basic concepts for the study of motion, we led 82 first year university students through Galileo's experiments designed to investigate the motion of falling bodies, and his geometrical explanation of his results, via simple dynamic geometric applets designed with GeoGebra. Our goal was to enhance the students’ development of mathematical thinking. Through a scholarship of teaching and learning study design, we captured data from students before, during and after the activity. Findings suggest that the historical development presented to the students helped to show the growth and evolution of the ideas and made visible authentic ways of thinking mathematically. Importantly, the activity prompted students to question and rethink what they knew about speed and acceleration, and also to appreciate the novel concepts of instantaneous speed and acceleration at which Galileo arrived.

The skin effect: a fresh look

Based on network analysis, a simple method has been suggested to explain the skin effect to undergraduate students. The mathematics required is an elementary idea of calculus giving a surprising insight into how nature behaves.

A “Close” Look at Linear Approximations

Approximation is a concept that underpins many big ideas in calculus. Students approximate the slope of a tangent line using secant-line slopes and eventually evaluate limits of those approximations. Linear approximation has uses in differential calculus. To help my students understand linear approximation, I adapted an activity from the Instructor's Guide for Stewart's Calculus (Shaw 2015). Students construct multiple representations of four functions and analyze each function's behavior near the origin using Desmos® (Desmos Graphing Calculator 2016) or similar technology, for better visualization.

Comparing Calculus Communication across Static and Dynamic Environments Using a Multimodal Approach

In this article, a thinking-as-communicating approach is used to analyse calculus students’ thinking in two environments. The first is a ‘static’ environment in the sense of static visual representations, such as those found in textbook diagrams, while the second is a dynamic environment as exploited by the use of dynamic geometry environments (DGEs). The purpose of the article is to compare calculus students’ communication as it is facilitated by each of these two environments, and to explore the role of paper- and digital-mediated representations for positioning certain ways of thinking about calculus. The analysis provides evidence that the participants employed different modes of communication – utterances, gestures and touchscreen-dragging – and they communicated about fundamental calculus ideas differently when prompted by different types of representations. The study presents implications for teaching dynamic aspects of functions and calculus, and argues for a multimodal view of communication to capture the use of gestures and dragging for communicating dynamic and temporal mathematical relationships.

CÁLCULO NO ENSINO MÉDIO: UMA ABORDAGEM POSSÍVEL E NECESSÁRIA COM AUXÍLIO DO SOFTWARE GEOGEBRA

http://dx.doi.org/10.5902/2179460X14523This article is the result of the dissertation of the Professional Masters in Mathematics in National Network, held in March 2013 at the Federal University of Santa Maria (UFSM). The work aimed to verify the possibility of inclusion in high school, the intuitive ideas of the Differential and Integral Calculus: limits of a function, the average rate of change, instantaneous variation and calculation of areas under the graph of positive functions, limited the x-axis and vertical lines, or even between positive functions in a particular area of the same range. To facilitate the understanding of these ideas, activities were developed using the Software GeoGebra as support learning tool. The activities were applied to an experimental group of students from the first year of High School, combined with the study of quadratic functions. It was found that you can open the horizons within the teaching and learning of mathematics in high school, to the intuitive ideas of calculus. In this work, the objectives of this research will be presented, some proposed activities and the results achieved, and propose developments to deepen and apply the issues discussed here.

They'll Need It for Calculus

A calculus teacher with a middle school background challenges the assumptions that algebraic manipulation skills are the most important foundations for calculus. He suggests the use of a smaller set of big middle school ideas that will be important to later success with the ideas of calculus.

Generalising calculus ideas from two dimensions to three: how multivariable calculus students think about domain and range

We analysed multivariable calculus students' meanings for domain and range and their generalisation of that meaning as they reasoned about the domain and range of multivariable functions. We found that students' thinking about domain and range fell into three broad categories: input/output, independence/dependence, and/or as attached to specific variables. We used Ellis' actor-oriented generalisations framework to characterise how students generalised their meanings for domain and range from single-variable to multivariable functions. This framework focuses on the process of generalisation – what students see as similar between ideas in multiple contexts. We found that students generalised their meanings for domain and range by relating objects, extending their meanings, using general principles and rules, and using/modifying previous ideas. Our findings suggest that the domain and range of multivariable functions is a topic instructors should explicitly address.

Where does the calculus go? An investigation of how calculus ideas are used in later coursework

Teachers of post-calculus courses frequently lament their students’ proficiency in calculus. While research outlines the topics mathematicians consider to be the core concepts and skills of calculus, there is no consensus on what calculus might be needed for students in different disciplines nor how we might want our students to understand these ideas. We argue that teachers may be lamenting the fact that students were not prepared to know calculus in a way that it is compatible with how it must be used post-calculus. Using differential equations as an exemplary post-calculus course, we explore the consequences of an analysis of foundational calculus topics in terms of how such topics are used. We present evidence and characterize elements of an epistemological mismatch between calculus and post-calculus.