Problem Of Calculus Research Articles

How teaching experience and physics and mathematics content knowledge impact professional noticing skills of STEM graduate students

AbstractThe framework of professional noticing describes three components (attending, interpreting, and deciding) that allow teachers to better understand the thinking of their students. Via this method, teachers attend to their classroom by observing relevant cues from students, interpret these cues based on their knowledge of student development, and decide how best to proceed in their lesson. This study utilized an open‐response survey to collect data regarding the professional noticing skills, physics and mathematics content knowledge, and teaching experience of Science, Technology, Engineering, and Mathematics graduate students. Participants were given a physics and calculus problem to solve to assess their level of content knowledge and then watched a video‐based scenario of a teacher and student discussing the same problems. After, participants were prompted to answer questions corresponding with the attending, interpreting, and deciding components of professional noticing. We found significant results that suggest teaching experience alone is not enough to employ professional noticing skills when attending physics scenarios, and that possessing content knowledge has a positive impact on professional noticing ability, both overall and within the components.

On family of the Caputo-type fractional numerical scheme for solving polynomial equations

Fractional calculus can be used to fully describe numerous real-world situations in a wide range of scientific disciplines, including natural science, social science, electrical, chemical, and mechanical engineering, economics, statistics, weather forecasting, and particularly biomedical engineering. Different types of derivatives can be useful when solving various fractional calculus problems. In this study, we suggested a single step modified one parameter family of the Caputo-type fractional iterative method. Convergence analysis shows that the proposed family of methods' order of convergence is . To determine the error equation of the proposed technique, the computer algebra system CAS-Maple is employed. To illustrate the accuracy, validity, and usefulness of the proposed technique, we consider a few real-world applications from the fields of civil and chemical engineering. In terms of residual error, computational time, computational order of convergence, efficiency, and absolute error, the test examples' acquired numerical results demonstrate that the newly proposed algorithm performs better than the other classical fractional iterative scheme already existing in the literature. Using the computer program Mathematia 9.0, we compare the draw basins of attraction of the suggested fractional numerical algorithm to those of the currently used fractional iterative methods for the graphical analysis. The graphical results show how quickly the newly developed fractional method converges, confirming its supremacy to other techniques.

A novel local Hermite radial basis function‐based differential quadrature method for solving two‐dimensional variable‐order time fractional advection–diffusion equation with Neumann boundary conditions

AbstractA novel Hermite radial basis function‐based differential quadrature (H‐RBF‐DQ) method is presented in this paper based on 2D variable order time fractional advection–diffusion equations with Neumann boundary conditions. The proposed method is designed to treat accurately for derivative boundary conditions, which considerably improve the approximation results and extend the range of applicability for the method of RBF‐DQ. The advantage of the present method is that the Hermite interpolation coefficients are only dependent of the point distribution yielding a substantially better imposition of boundary conditions, even for time evolution. The proposed algorithm is thoroughly validated and is demonstrated to handle the fractional calculus problems with both Dirichlet and Neumann boundaries very well.

Do Dogs Know Calculus With Early Transcendentals?

Summary A standard applications problem in differential calculus involves minimizing the travel time needed for a dog on a beach to reach a toy floating in the water, taking into account the different velocities the dog can achieve when running versus swimming. The solution usually presented to such problems are formulated in terms of a Cartesian coordinate. Here, a solution that is based in trigonometry is presented that, while quite natural, appears to be relatively unknown. The solution to the optimization problem in the trigonometric setting has a clear interpretation which is lacking in the Cartesian solution.

On the influence of the cental trend on the nature of the density distribution of maximum entropy in machine learning

The principle of maximum entropy (ME) has a number of advantages that allow it to be used in machine learning. The density distribution of maximum entropy (WEO) requires solving the problem of calculus of variations on the a priori distribution, where the central tendency can be used as a parameter. In Lebesgue space, the central tendency is described by the generalized Gelder average. The paper shows the evolution of the density of the ME distribution depending on the given norm of the average. The minimum Kulback — Leibler divergence between the WEO and the a prior density is achieved at the harmonic mean, which is effective in reducing the dimensionality of the training sample. At the same time, this leads to a deterioration in the function of loss in the conditions of machine learning by precedents.

AiMA - An AI-Based Mobile System to Assist College Students with Math-Related Issues

Introduction: As technology continues to progress, the importance and influence of mathematics in people's lives are growing. The emergence of mobile systems has brought about several benefits, such as portability, convenience, and accessibility, especially in the field of education. It is crucial to develop a mobile system that can cater not only to high school students but also to those enrolled in colleges and universities, offering solutions for basic calculations and highly specialized algorithms. To address this need, this research presents AiMA, an AI-based mobile system that assists college students with math-related issues. Methods: In this paper, we discuss the limitations of current math-solving mobile applications and propose a solution that caters to specialized algorithms for university students. AiMA employs React Native on the front-end side to render the user interface and Flask-server on the back-end side to handle numerical data. Additionally, an AI model is implemented to handle input images stored in a cloud-based media management platform for detection. Results: The system's functionality evaluation indicates that AiMA provides advantages over existing solutions by enabling users to solve complex calculus problems and receive tailored solutions for specific areas of mathematics. Conclusion: This makes AiMA a highly valuable tool for college and university students who require specialized assistance with their math-related studies.

Exploration of Prospective Mathematics Teachers’ Mathematical Connections When Solving the Integral Calculus Problems Based on Prior Knowledge

<p style="text-align:justify">Mathematical connection ability is very important to be mastered by prospective mathematics teacher students as competency to teach in secondary schools. However, the facts show that there are still many students who have weak mathematical connection abilities. This qualitative descriptive study aimed to explore how the process, and product of the mathematical connection made by prospective mathematics teacher students when solving the integral calculus problems based on their prior knowledge. The research subjects were 58 students who were prospective high school mathematics teachers at the University of Jember, Indonesia. Data were collected using documentation, questionnaire, test, and interview methods. After the test results of all subjects were analyzed, six students were interviewed. To find the match between the results of the written test and the results of the interview, a triangulation method was carried out. Data analysis used descriptive qualitative analysis with steps of data categorization, data presentation, interpretation, and making conclusions. The results show that the research subjects have connected and used mathematical ideas in the form of procedures, facts, concepts/principles, and representations in solving integral calculus problems. Students with high prior knowledge abilities can make better mathematical connections than students with moderate and low prior abilities. From these results, it is recommended that lecturers need to improve students' prior knowledge and train the students more intensely to solve integral calculus problems so all students can develop their mathematical connection abilities into very strong categories.</p>

A generalized analytical approach for highly accurate solutions of fractional differential equations

A generalized homotopy-based approach is developed to give highly accurate solutions of fractional differential equations. By introducing a scaling transformation, the computational domain of the nonlinear Riccati differential equations with fractional order changes from [0,+∞) to [0,1]. Analytical approximation of arbitrary accuracy is achieved, whose convergence is proved theoretically. The effectiveness and accuracy of our solution is strictly checked via error analysis. The proposed method is expected to be as a new and reliable analytical approach to give highly accurate solutions of strongly nonlinear problems in fractional calculus.

Students' Mathematical Understanding Ability Through the M-Apos Approach on Derivative Materials

This study aimed to improve students' mathematical understanding skills. The understanding ability is students' ability to understand the concept of the material, not just memorize the material. When students understand the concepts of differential calculus, students can quickly solve problems in differential calculus. The students' understanding gained in the differential calculus material will be the basis for understanding the integral and advanced calculus material. However, the current conditions where the Coronavirus that has hit the world, including Indonesia, have resulted in students having to study from home with a learning process that is carried out online. Students cannot access books in the library and limited credit to access teaching materials from the internet because the learning materials owned by students are limited. It makes it difficult for students to learn, so students' mathematical understanding abilities are low. The M-APOS learning approach was used in this study to answer this problem. The study results found that learning with this approach has succeeded in increasing students' mathematical understanding abilities on derivative material. This research is an experimental study with a pretest-posttest control group research design.

Necessary conditions for a minimum in classical calculus of variations in the presence of various types of degenerations

In the paper, a new approach is offered for studying an extremal in the problems of classical calculus of variations in the presence of various types of degenerations. This approach: firstly, is based on using Weierstrass type variations in two forms: in the form of variations on the right with respect to the given point, and in the form of variations on the left with respect to the same point; secondly, allows to get new and stronger results. The research is conducted under the assumption that along the considered extremal the Weierstrass and Legendre conditions as well as other necessary conditions for minimum degenerate, i.e. they are fulfilled as equalities at points or on some intervals. Two types of new necessary conditions are obtained: of equality type and of inequality type conditions for a strong and also a weak local minimum. Specific examples and a counterexample show that some of the necessary conditions for minimum obtained in this article are strengthening and refining of the corresponding known results in this direction.

Simulations and Solutions of Calculus Problem: An Effort to Improve Students' Innovative and Quality Computing Skills

One of the impacts of online learning that is not optimal is learning loss. Therefore, the aims of this study, we conduct computing practicum with Maple and Matlab to improve students' ability to develop learning media based on Network Virtual Laboratory (NVL). The implications of this practicum can improve students' innovative attitudes in developing various learning media NVL-based, with still through the validation process and field tests so that quality learning media is produced . The subject of this study consisted of 61 students who took the integrated mathematics computing course calculus . This is a development research (R&D), where each student develops learning media then validated and field trials. We assess each media resulting from the development using questionnaires and field observations. The results showed that as many as 85% of students have been able to innovate well in developing learning media NVL-based including function limit materials, derivatives, and integrals. The results of the assessment of the expert team showed that the ability of students in developing learning media reached an average score of 3.97 which is categorized as "innovative", while the results of product trials for early-level students obtained an average of 4.07 blessings "very qualified". Furthermore, the practicum module and all media results of student development have been uploaded to the University's Learning Management System Moodle-based for use in the learning process in the future. We hope that the learning media that has been developed can be applied to calculus learning in the future.

A Poincaré type inequality with three constraints

In this paper, we consider a problem in calculus of variations motivated by a quantitative isoperimetric inequality in the plane. More precisely, the aim of this article is the computation of the minimum of the variational problem $$\begin{aligned}\inf _{u\in {\mathcal {W}}} \frac{\displaystyle \int _{-\pi }^{\pi }[(u')^2-u^2]d\theta }{\displaystyle \left[ \int _{-\pi }^{\pi } |u| d\theta \right] ^2} \end{aligned}$$ where a function $$u\in {\mathcal {W}}$$ is a $$H^1(-\pi ,\pi )$$ periodic function, with zero average on $$(-\pi ,\pi )$$ and orthogonal to sine and cosine.

How to correctly answer ‘Is the optimal critical point a local minimizer?’ in Calculus courses

This short note discusses how the optimality conditions for minimizing a multivariate function subject to equality constraints have been covered in some undergraduate Calculus courses. In particular, we will focus on the most common optimization problems in Calculus of several variables: the 2 and 3-dimensional cases. So, along with sufficient conditions for a critical point to be a local minimizer, we also present and discuss counterexamples for some statements that can be found in the literature of undergraduate Calculus related to Lagrange Multipliers, such as ‘between the critical points, the ones which have the smallest image (under the function) are minimizers’ or ‘a single critical point (which is a local minimizer) is a global minimizer’.

CUT-OFF DISTRIBUTIONS AND PROBABILISTIC INTERPRETATIONS OF THE GENERAL FRACTIONAL DERIVATIVES WITH MEMORY EFFECT

The physical/probabilistic meanings of fractional derivatives or integrals are very basic and important problems in fractional calculus. In this paper, we investigate the probabilistic interpretations of the general fractional derivatives with memory effect and their connections with cut-off distributions. Some examples like the cut-off Rayleigh distribution and the Atangana–Gómez fractional derivative are especially discussed. This paper shows that the kernels of the general fractional derivatives with memory effect may be not the same as the density functions of the physical processes described. For example, the kernel of Atangana–Gómez fractional derivative is a Gaussian function but its corresponding probabilistic distribution is a cut-off Rayleigh distribution.

The Reciprocal Relationship Between Conceptual and Procedural Knowledge—A Case Study of Two Calculus Problems

The literature describes different stances concerning the focus on how mathematics should be taught, with some preferring a conceptual knowledge approach and others a procedural knowledge approach. The current study investigated the relationship between students’ conceptual and procedural knowledge in a calculus context. To better understand the relationship between students’ conceptual and procedural knowledge, we conducted a content analysis of the solution approaches of three subject specialists, as well as student responses to two mathematical problems. The students (n =192) were enrolled for a first-year mathematics module which forms part of an extended engineering degree in South Africa. The solutions to the two problems were analysed based on the number and nature (conceptual or procedural) of experts’ and students’ steps to solve each problem. Each step in the solution was categorised based on the approaches used to solve the problem. The study found that solutions are not unique and could follow more than one approach. More importantly, the study found that the relationship between conceptual and procedural knowledge is complex and integrated as solutions require both procedural and conceptual knowledge. The findings reveal that calculus problems cannot be uniquely described as mainly conceptual or procedural. Both procedural and conceptual thinking is required to solve calculus problems and is often iterative. Student techniques to solve the calculus problems included algebraic, graphical and unexpected approaches. The analyses of student solutions suggest that lecturers and teachers should compare and discuss multiple solution strategies with their students to enhance mathematical proficiency and understanding.

Optimizing the Use of Matlab GUI Attributes in the Creation of Calculus Learning Media: An Effort to Measure Students' Innovative Attitudes

Calculus learning requires a high level of visualization in instilling concepts optimally to students. Therefore, the purpose of this study is to look at students' innovative attitudes in developing matlab GUI-based learning media in solving calculus problems and see the influence of the ability to use matlab GUI attributes on the level of innovation of the students in developing learning media. In the early stages, the students developed media according to the given topic. In the second stage, the product was assessed in terms of its validity. The students' innovative attitudes were assessed by a team of experts using questionnaires with 5-point Likert scale. From the 16 media successfully developed, validation results were obtained with an average score of 3.83 which means "valid". Meanwhile, from the assessment of innovative attitudes, the students’ attitudes are in the category of "innovative" with an average score of 3.61. In addition, the results of the regression tests revealed that the innovative attitudes are influenced by the ability to use the matlab GUI attributes by 84.2%. The rest is influenced by other factors with similarities: . Lastly, the results of the assessment showed that the media belonged to the category of "very effective" (score 81.2%).

An eigenvalue problem in fractional h-discrete calculus

In this paper, we prove existence of solutions for an eigenvalue problem in the fractional h-discrete calculus. Dirichlet type boundary conditions are considered. Several properties for Green’s function of the associated problem are proven. A fixed point theorem in Cone theory is a main tool to obtain sufficient conditions on upper and lower bounds for eigenvalues of the boundary value problem so that the existence of positive solutions are guaranteed.

Assessing the impact of visualization media on engagement in an active learning environment

The growing adoption of active learning pedagogical methods has shown significant positive effects in higher education as they promote students' engagement and meeting of learning outcomes. Similarly, problem-based learning and media-based learning allow instructors to model educational experiences that prepare students with knowledge and skills necessary for the real world. In this study, the authors wanted to investigate the impact of a group problem-based learning activity using visualization media on students' learning motivation in a mathematics course. In this activity, students were grouped and asked to solve volumes of revolution calculus problems while being either in a traditional classroom or in a classroom equipped with a large-scale visualization display system. The authors assessed the students' self-efficacy, effort regulation, perceived usefulness, task attraction, and user experience after participating in the activity. The authors found a significant difference in students' self-efficacy and task attraction based on the results' analysis. The authors did not find a significant difference in students' effort regulation and perceived usefulness. The students did report a significantly positive experience while using the large-scale visualization display system. The analysis illustrated a medium to large effect size of the results' statistical significance. This study supports future instructors and researchers in designing and developing impactful pedagogical active learning experiences.

On positivity and monotonicity analysis for discrete fractional operators with discrete Mittag–Leffler kernel

We consider conditions under which the positivity of a fractional difference implies either positivity, monotonicity, or convexity, and we consider both the non‐sequential and sequential cases. The particular difference we study is the discrete nabla ABC‐type difference, and it possesses a Mittag–Leffler kernel; this leads to some different results when compared to other types of kernels. We conclude by applying our results to certain classes of initial value problems in discrete fractional calculus.

Enhancing flipped classroom with peer teaching to promote students’ conceptual understanding and self-efficacy in calculus courses

Calculus differential is the basis of mathematics before entering the level of further mathematics, which has a high analysis level. Students have difficulty understanding concepts and have not been able to analyze/predict/realize (self-efficacy) that impact challenges in working on calculus problems. On the other hand, technology has been developed in education. It is offered as a solution to the issues, namely peer teaching flipped classrooms. This learning model is a technology-based approach to student centre learning. This research compared conventional flipped classrooms and peer teaching flipped classrooms. This research investigated the differences in students' conceptual understanding and self-efficacy in both classes in terms of majors’ background and prior achievement level. This research used a nonequivalent-group pretest and posttest design using flipped classroom method. The type of data in the study was quantitative, namely test and questionnaire. The ANCOVA results showed that the peer teaching flipped method was better than the conventional flipped classroom method in terms of students' conceptual understanding and self-efficacy. Moreover, self-efficacy had a positive correlation with conceptual understanding in both methods. Furthermore, the student's prior achievement level influenced students' conceptual understanding and self-efficacy, while students' major's background influenced students' self-efficacy only