Concepts Of Calculus Research Articles

Fractional Order EOQ Model with Linear Trend of Time-Dependent Demand

In this paper we introduce the classical EOQ model with a linear trend of time-dependent demand having no shortages using the concept of fractional calculus. The application of fractional calculus has been already used in classical EOQ model where the demand is assumed to be constant. In this present article fractional differential calculus can be used to describe EOQ model with time-dependent linear trend of demand to develop more generalized EOQ model. Here, we want to discuss more deeply its role as a tool for describing the traditional classical EOQ model with time dependent demand.

Differential Calculus Teaching through Virtual Learning Objects in the Field of Management Sciences

This paper examines the use of Virtual Learning Objects (VLO) in the teaching of differential calculus in the area of Management Sciences as a teaching strategy to improve the assimilation of the theoretical knowledge acquired in the classroom courses. This is accomplished by the construction of virtual pedagogical practices concerning to optimization problems through GeoGebra. It is free software that generates VLOs with an adequate contrast between the theoretical mathematical knowledge and its practical application in Management Sciences. Thus, different dynamic, algorithmic and heuristic processes are encouraged, being useful for logic and constructivist student training. The proposed methodology is based on the construction of VLOs (Virtual Learning Objects). It allows choosing variables in the GeoGebra software where students can interact with the formulation of mathematical problems applied to their curricula. So, they strengthen their knowledge and have a greater autonomy in the analysis of optimization problems in financial and management field. The results show how these teaching strategies facilitate the assimilation of the concepts of differential calculus in Management Sciences students. This allows a greater development of student skills in front of the understanding of the geometric behaviour of a curve.

Calculus & GeoGebra, an Interesting Partnership

In engineering careers, the study of the Differential and Integral Calculus begins in the first course of Calculus. In order to understand the definitions and theorems involving such contents, students need to handle symbolic and graphical representations. With GeoGebra, it is possible to create different interactive applications that can be used as teaching tools to illustrate math classes. In this paper, some animations of the fundamental concepts of Calculus are shown, emphasizing the analysis of their geometric interpretations. These tools were designed for the Calculus I course at the Facultad Regional San Nicolás, Universidad Tecnológica Nacional, Argentina.

New concepts of fractional quantum calculus and applications to impulsive fractional q-difference equations

In this paper we define new concepts of fractional quantum calculus by defining a new q-shifting operator. After giving the basic properties we define the q-derivative and q-integral. New definitions of Riemann-Liouville fractional q-integral and q-difference on an interval $[a,b]$ are given and their basic properties are discussed. As applications of the new concepts, we prove existence and uniqueness results for first and second order initial value problems for impulsive fractional q-difference equations.

A q-analogue of the multiplicative calculus: Q-multiplicative calculus

In this paper, we propose q-analog of some basic concepts of multiplicative calculus and we called it as q-multiplicative calculus. We successfully introduced q-multiplicative calculus and some basic theorems about derivatives, integrals and infinite products are proved within this calculus.

STARLIKENESS OF q-DIFFERENTIAL OPERATOR INVOLVING QUANTUM CALCULUS

In the present paper, we investigate starlikeness condi- tions for $ q$ − differential operator by using the concept of quantum calculus in the unit disk.

Use of GeoGebra in primary math education: a theoretical approach

GeoGebra as cost-free, multi-platform mathematics software, dynamically connecting concepts in geometry, algebra and calculus is applied in different school’s levels, and primary school is no exception. The aim of the paper is to present a theoretical approach of application of GeoGebra in Lithuanian’s primary math education. It analyzes features of GeoGebra, its possible integration into primary school education according to Educational Plan and Programme. Also some examples how this software can be effectively incorporated into teaching and learning are presented. With a view to evaluate GeoGebra’s suitability to primary education primary school teachers were interviewed and results of that exploratorystudy are presented. Also the study seeks to identify the conditions to successfully support teachers in using GeoGebra and learning objects prepared by it in their teaching process including organisational and technical support required.

Information disclosure on social networking sites: An intrinsic–extrinsic motivation perspective

This study applies an intrinsic–extrinsic perspective to examine information disclosure intentions by online social network users. We argue that voluntary disclosure is tied to the intrinsic value that users attribute to their social networking activities. We propose opposing hypotheses regarding the effect of extrinsic motivations on disclosure intentions. On one side reward motivation may promote disclosure intentions. Alternatively, rewards may raise suspicion and concern among users, potentially deterring disclosure intentions. We build a hypothetical model that draws from the theory of planned behavior, the concept of privacy calculus and the incentive theory. We come across an unexpected contradiction in our results; we find a positive correlation between reward-level and disclosure intentions on one hand, and an insignificant effect of extrinsic motivations on disclosure intentions on the other hand. We deduce that information ambiguity and reward-amount ambiguity are likely to be important influencers of disclosure intentions.

GeoGebra와 미분적분학 개념의 시각화

최근 테크놀로지가 발달함에 따라 대학 수학교육에서 시각화를 통한 추상적인 수학적 개념의 직관적 이해는 그 비중이 점점 커지고 있다. 본 연구에서는 대학 미분적분학을 학습하는 학생들의 이해를 도울 수 있고 웹만 연결되어 있으면 무료로 사용할 수 있는 그리고 필요한 내용을 직접 개발할 수 있는 도구 중 그 장점이 잘 알려져 있는 지오지브라(GeoGebra)를 활용한 무료 시각화 자료들을 개발하여 소개한다. 특히 미분적분학 교재의 순서에 맞추어 개발된 시뮬레이션 도구를 활용하여 개념의 이해와 함께 새로운 지식을 생산할 수 있는 교육환경을 제공하고자 하였다. 마지막으로 미적분학 개념들의 시각화를 마친 이번 시도가 대학수학의 다른 강좌와 중등 수학교육으로 확대될 수 있는 가능성에 대하여 논의하였다. Recently, with the development of technology, intuitive understanding of abstract mathematical concepts through visualizations is growing in popularity within college mathematics. In this study, we introduce free visualization tools developed for better understanding of topics which students learn in Calculus. We visualize important concepts of Calculus as much as we can according to the order of most Calculus textbooks. In this process, we utilized a well-known, free mathematical software called GeoGebra. Finally, we discuss our experience with visualizations in Calculus using GeoGebra in our class and discuss how it can be effectively adopted to other university math classes and high school math education.

Reading the World with Calculus

It is now increasingly recognized that mathematics is not a neutral value-free subject. Rather, mathematics can challenge students’ taken-for-granted realities and promote action. This article describes two issues, namely deforestation and income inequality. These were specifically chosen because they can be related to a range of calculus concepts including rates of change, optimization, Riemann sums, the Fundamental Theorem of Calculus, and partial derivatives. The aim is to give ideas about how an issue may be examined from different mathematical perspectives. The impact of discussing relevant issues in the calculus classroom is described through student reflections and course evaluation.

Russell's Principles of Mathematics and the Revolution in Marburg Neo-Kantianism

Bertrand Russell's Principles of Mathematics offered a sustained criticism of Hermann Cohen's foundations of neo-Kantianism, evoking a radical rethinking of the basis of Marburg thought by Cohen's student, Ernst Cassirer. Russell's criticisms were directed at the understanding of the concepts of the differential calculus, which grounded the objectivity of scientific thought in Cohen's effort to resurrect the Kantian transcendental project. On the basis of Russell's criticisms, Cassirer abandoned Cohen's foundations and instead drew on the general relational form, borrowed from Russell's work in mathematical logic, to serve as the a priori source of objectivity. However, Cassirer's reliance on Russell's thought was limited, for he rejected Russell's Logicism, the principal philosophical thesis of his Principles. Instead, Cassirer specified that the serial relation, as defined by the Formalist mathematicians, was the source of objectivity of the advanced mathematical sciences. The result was a structuralist...

Calculus from a Statistics Perspective

SummaryWe present a statistics perspective on calculus, which can help students understand calculus concepts and analyze a function defined by data or sampling values from a given function, rather than an explicit mathematical formula. Our approach uses the arithmetic and graphic means for the notions of integral and derivative, respectively. An intuitive approach to the law of large numbers replaces the notion of limit for this introductory level.

On conformable fractional calculus

Recently, the authors Khalil et al. (2014) introduced a new simple well-behaved definition of the fractional derivative called conformable fractional derivative. In this article we proceed on to develop the definitions there and set the basic concepts in this new simple interesting fractional calculus. The fractional versions of chain rule, exponential functions, Gronwall’s inequality, integration by parts, Taylor power series expansions, Laplace transforms and linear differential systems are proposed and discussed.

An Analytic Exact Form of the Unit Step Function

In this paper, the author obtains an analytic exact form of the unit step function, which is also known as Heaviside function and constitutes a fundamental concept of the Operational Calculus. Particularly, this function is equivalently expressed in a closed form as the summation of two inverse trigonometric functions. The novelty of this work is that the exact representation which is proposed here is not performed in terms of non – elementary special functions, e.g. Dirac delta function or Error function and also is neither the limit of a function, nor the limit of a sequence of functions with point wise or uniform convergence. Therefore it may be much more appropriate in the computational procedures which are inserted into Operational Calculus techniques.

Application of Fractional Calculus Theory to Robust Controller Design for Wind Turbine Generators

This paper presents a robust controller design method for wind turbine generators using the concepts of fractional calculus. It also compares features of fractional order control systems with those of classic integer order controllers. The proposed method uses isodamping feature, which desensitizes the phase-frequency variations about a gain crossover frequency. This increases the robustness of a fractional order control system against uncertainties. In conventional integer order control systems, realization of isodamping feature requires a controller with very high-order transfer function whereas a fractional order system can readily realize this feature in a compact form. The proposed method is applied to a study system consisting of a permanent magnet wind turbine generator. The test system investigates the tracking performance of the control system considering the backlash and aging phenomenon within the dynamic model of the wind turbine generator. The study results based on a time-domain simulation show the superior capabilities of fractional order controller compared with classic controllers in the presence of model uncertainties. It has been shown that the concept of fractional calculus can be used as a promising robust control approach for the future high performance feedback control systems with application to wind energy systems.

On the stability of iteration methods for special solution of time-fractional generalized nonlinear ZK-BBM equation

An investigation of a possible description of non-linear wave was performed inside the scope of fractional order derivative. A clear motivation underpinning the extension of this study within the concept of fractional calculus was presented. The special solution of the extended equation was derived using an efficient iteration method called homotopy decomposition method. A theorem underpinning the uniqueness of the special solution was proposed and proved in detail. An investigation for stability of the iteration method for solving the extended equation was presented in detail. Numerical simulations were carried out using the proposed algorithm for different values of alpha and n.

Reducing risk of closed loop control of blood glucose in artificial pancreas using fractional calculus.

Healthcare costs in the US are among the highest in the world. Chronic diseases such as diabetes significantly contribute to these extensive costs. Despite technological advances to improve sensing and actuation devices, we still lack a coherent theory that facilitates the design and optimization of efficient and robust medical cyber-physical systems for managing chronic diseases. In this paper, we propose a mathematical model for capturing the complex dynamics of blood glucose time series (e.g., time dependent and fractal behavior) observed in real world measurements via fractional calculus concepts. Building upon our time dependent fractal model, we propose a novel model predictive controller for an artificial pancreas that regulates insulin injection. We verify the accuracy of our controller by comparing it to conventional non-fractal models using real world measurements and show how the nonlinear optimal controller based on fractal calculus concepts is superior to non-fractal controllers in terms of average risk index and prediction accuracy.

Exploring Support Strategies for High School Mathematics Teachers from Underachieving Schools

Algebra, trigonometry and calculus concepts are dealt with in grades 10 to 12 of South African high school topics, as prescribed by the Curriculum and Assessment Policy Statement. This paper reports on a study exploring practicing teachers' subject matter knowledge of certain topics they were expected to teach. The theoretical perspectives for the design and analysis of this study were based on Shulman's conceptual framework. This study was a qualitative case study (n = 28). Practising teachers from the Pinetown district in KwaZulu-Natal participated in this mathematics' teacher intervention. Focused tasks on algebra, trigonometry and calculus were designed and administered to those teachers. The researchers analysed the data from those tests and found that certain areas of their pedagogical content knowledge required attention.

Fractional Alexander polynomials for image denoising

Image denoising is an important task in image processing. The interest in using a fractional mask window operator based on fractional calculus has grown for image denoising. This paper mainly introduces the concept of fractional calculus and proposes a new mathematical method in using fractional Alexander polynomials for image denoising. The structures of n×n fractional mask windows on eight directions of this algorithm are constructed. Finally, we measure the denoising performance by employing experiments based on visual perception and by using peak signal-to-noise ratios. The experiments illustrate that the improvements achieved are compatible with other standard smoothing filters.

A discrete approach to the concept of derivative

The concept of derivative is a basic concept of calculus. It is closely related to the concept of function, the idea of rate of change, and the limit concept. In recent decades, teaching the concept of derivative in mathematics classrooms has changed: a quite formal approach—closely linked to the teaching of calculus at university and based on the sequence concept—has been transformed to or substituted by a new one. This means working with rates of change, an intuitive access to the concepts of limit and derivative. It includes working with real functions right from the beginning, a great emphasis on graphs, and the use of digital technologies. The meaning of sequences has decreased to a point where they are sometimes no longer even taught in the calculus course. In recent years this concept has been criticized for not developing adequate perceptions of the basic concepts of calculus and not sufficiently preparing the students for scientific courses at university. In this paper we present an alternative discrete step-by-step approach to the basic concepts of calculus by working with sequences and difference sequences, functions defined on \( \mathbb{Z}\) and discrete domains of \( \mathbb{Q}\), and by subsequently developing the concept of rate of change in a discrete learning environment. The paper is based on general theoretical considerations and empirical investigations by the author and is meant as a contribution to classroom design-research or “design science” (Wittmann, Educ Stud Math 29(4):355–374, 1995).