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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.

The article reports that the theoretical particle physicists with vexing calculus problems called Feynman integrals which are central to every calculation of how subatomic particles interact, and theorists in China have found a general numerical method to solve the integrals differential equations.

We give two extensions of the classical formula for sums of powers on arithmetic progressions. This is achieved by using an identity involving binomial mixtures, which can be viewed as a generalization of the binomial transform. [ABSTRACT FROM AUTHOR]

This article deals with some existence and Ulam-Hyers-Rassias stability results for a class of functional differential equations involving the Hilfer-Hadamard fractional derivative. An application is made of a Schauder fixed point theorem for the existence of solutions. Next we prove that our problem is generalized Ulam-Hyers-Rassias stable. [ABSTRACT FROM AUTHOR]

We use a trigonometric identity and integral calculus to evaluate several infinite series; in particular, we deduce the corresponding partial fraction decomposition of,,and csch x. [ABSTRACT FROM PUBLISHER]

The article discusses solutions to the inverse curvature problem in relation to the subject of calculus. Topics covered include computing for curvatures through differential equations and Chebyshev conditions, the origin of curvatures in describing shapes which was linked to the study "The Method of Fluxions and Infinite Series, with its Applications to the Geometry of Curved-Lines," and the significance of the inverse curvature problem in differential geometry.

We use a trigonometric identity and elementary calculus to obtain simple proofs of the following identities: and. [ABSTRACT FROM AUTHOR]

This study investigates interactions between calculus learning and problem-solving in the context of two first-semester undergraduate calculus courses in the USA. We assessed students’ problem-solving abilities in a common US calculus course design that included traditional lecture and assessment with problem-solving-oriented labs. We investigate this blended instruction as a local representative of the US calculus reform movements that helped foster it. These reform movements tended to emphasize problem-solving as well as multiple mathematical registers and quantitative modelling. Our statistical analysis reveals the influence of the blended traditional/reform calculus instruction on students’ ability to solve calculus-related, non-routine problems through repeated measures over the semester. The calculus instruction in this study significantly improved students’ performance on non-routine problems, though performance improved more regarding strategies and accuracy than it did for drawing conclusions and providing justifications. We identified problem-solving behaviours that characterized top performance or attrition in the course. Top-performing students displayed greater algebraic proficiency, calculus skills, and more general heuristics than their peers, but overused algebraic techniques even when they proved cumbersome or inappropriate. Students who subsequently withdrew from calculus often lacked algebraic fluency and understanding of the graphical register. The majority of participants, when given a choice, relied upon less sophisticated trial-and-error approaches in the numerical register and rarely used the graphical register, contrary to the goals of US calculus reform. We provide explanations for these patterns in students’ problem-solving performance in view of both their preparation for university calculus and the courses’ assessment structure, which preferentially rewarded algebraic reasoning. While instruction improved students’ problem-solving performance, we observe that current instruction requires ongoing refinement to help students develop multi-register fluency and the ability to model quantitatively, as is called for in current US standards for mathematical instruction. [ABSTRACT FROM PUBLISHER]

We use freshman calculus to prove thatfor θ ∈ (0, π) andn= 0, 1, 2, … ; in particular, we obtain a simple unified method for evaluating the following infinite series: [ABSTRACT FROM PUBLISHER]

This paper presents a new twist on a familiar paradox, linking seemingly disparate ideas under one roof. Hilbert's Grand Hotel, a paradox which addresses infinite set comparisons is adapted and extended to incorporate ideas from calculus – namely infinite series. We present and resolve several variations, and invite the reader to explore his or her own variations. [ABSTRACT FROM PUBLISHER]

We give a detailed derivation of a generalization of the second derivative test of single-variable calculus which can classify critical points as local minima or local maxima (or neither), whenever the traditional second derivative test fails, by considering the values of higher-order derivatives evaluated at the critical points. The enhanced test is local, in the sense that it is only necessary to evaluate all relevant derivatives at the critical point itself, and it is reasonably robust. We illustrate an application of the generalized test on a trigonometric function where the second derivative test fails to classify some of the critical points. [ABSTRACT FROM PUBLISHER]

The article discusses the study conducted to examine the asymptotic behavior of solutions for a general closed system of tanks that were connected by pipes using mathematical calculus. Topics discussed include finding the eigenvalues and eigenvectors for the analytical assessment using differential equation, Gershgorin circle theorem and recurrence relations, and the heat equation and circulant and Laplacian matrices.

The article presents information on a report which suggests that nonroutine problems in algebra may alter impressions of mathematics for students and allows a keener sense of understanding and more productive lines of thinking. It suggests mathematics teachers to involve nonroutine calculus task in the present curricula that will help in developing students' approach of dealing with a mathematical problem.

The fragment of the polymodal provability logic GLP in the language with connectives ┬, Λ, and 〈 n〉 for all n ∈ ω is considered. For this fragment, a deductive system is constructed, a Kripke semantics is proposed, and a polynomial bound for the complexity of a decision procedure is obtained. [ABSTRACT FROM AUTHOR]

The article presents a numerical solution on the singularly perturbed Steklov-type problem with asymptotically degenerate spectrum in the case where the unperturbed problem has nonsingular matrix. An overview of the history of the Steklov problem is provided which goes back to the nineteenth century, noting the consideration of mathematical authors of its behavior spectrum's various aspects. The problem consists a rapidly alternating boundary conditions, the Dirichlet and Steklov. It offers a description of the domains of the problem as well as highlights the boundary value problem and the convergence of solutions.

The article presents a mechanical engineering problem on the perturbation of an eigenvalue in the continuous spectrum of a waveguide with an asymmetric obstacle. The statement of the eigenvalue problem is provided, which has an obstacle closed set with a Lipschitz boundary. It looks at the solution of the limit problem, restricted to the upperwaveguide since the given problem is symmetric about the horizontal axis. Also illustrated is the augmented scattering matrix, the asymptotics of the augmented scattering matrix, and the search for an eigenvalue. A discussion for the conditions of the described theorem is included.

Abstract: This Letter is devoted to the integrability problem of planar nonlinear differential equations. We introduce a new method to detect local analytic integrability or to construct a singular series expansion of the first integral around a singular point for planar vector fields. The method allows to find new variables (essential variables) where the integrability problem is more feasible. The new method can be used in different context and is an alternative to all the methods developed up to now for any particular case. [ABSTRACT FROM AUTHOR]

This paper reports on the results of an observational parallel study conducted simultaneously at 2 universities - one each in New Zealand and Germany. It deals with university engineering students’ difficulties in the formulation step of solving a typical application problem from a first-year calculus course. Two groups of students (54 in New Zealand and 50 in Germany) completed a questionnaire about their difficulties in solving the problem which was set as part of a mid-semester test. The research endeavoured to find reasons most of the students could not use their knowledge to construct a simple function in a familiar context. It was neither lack of mathematics knowledge nor an issue with the context. The students’ difficulties are analyzed and presented along with their suggestions on how to improve their skills in solving application problems. [ABSTRACT FROM AUTHOR]

We present two Markov models of the spreading of rumors. Specifically, we determine the limiting distribution as the population becomes large for the time to spreading of the rumor to the full population. For the first model, all particles are identical, but one initially knows the rumor. The limiting distribution is the convolution of two double exponential distributions. For the second model, there are two kinds of individuals: spreaders, who can spread the rumor, and ordinary individuals, who can only learn the rumor. Here, the limiting distribution is simply a double exponential distribution. [ABSTRACT FROM AUTHOR]

In this paper, we continue the research on limit theorems for the Lerch zeta-function L( λ ; 1 2 + it) when the pair of parameters ( λ, α) are near to (0; 0), (0; 1 2 ), (0; 1), ( 1 2 ; 0) and (1; 0). [ABSTRACT FROM AUTHOR]

An overview is given of three conceptual lessons that can be incorporated into any first-semester calculus class. These lessons were developed to help promote calculus students' ability to think conceptually, in particular with regard to the role that infinity plays in the subject. A theoretical basis for the value of these lessons is provided, and both indirect and direct evidence, that demonstrates that these lessons were successful in beginning to achieve their goal, is described. Ideas for how to incorporate these lessons into the standard calculus curriculum are presented as well as thoughts on how to further improve the lessons' ability to promote students' conceptual understanding. [ABSTRACT FROM AUTHOR]

The article presents the solution for solving boundary value problems for fourth-order equation using fractional derivatives. It cites that the Caputo fractional derivative operator is expressed as a Riemann-Liouville fractional integral. It refers the theorem f(x,t) ∈ C1 [ 0,p], f(0,t) = f(p,t) = 0, ∂ f(x,t)/ ∂ x ∈ Lipϒ [0,p].

This article offers information on the distribution of relatively prime numbers. It also aims to prove that if one picks an ordered pair of integers at random, then the probability that these integers are relatively prime or approximately 60.8 percent. It also notes that this article will be useful for further reading in a probability or number theory course. The article also discusses some preliminary results starting with the recollection of the famous results of Leonard Euler using the formula, known as the Basel Problem.

The article discusses the author's formulation of the calculus course for honors college students. The author had used the historical approach inorder to create awareness and interest to students. It was used as a motivation to allow students to appreciate the subject more and pursue calculus in farther levels. The scrutinous study of the history of calculus generated a debate over the real father of calculus whether its Isaac Newton or Gottfried Leibniz. The author can not determine the success rate of the subject due to lack of evidential data but he is confident that it had created an active participation on his honors students.

Two original problems in geometry are presented with solutions utilizing to differential calculus: (a) rectangle inscribed in a sector; (b) point on the ray of the angle. The possibility of applying mathematics in general and differential calculus in particular for solution of practical problems is discussed. [ABSTRACT FROM AUTHOR]

We investigate one solvable subclass of quantified formulas in pure predicate calculus and obtain a necessary and sufficient condition for the satisfiability of formulas from this subclass. [ABSTRACT FROM AUTHOR]

The article presents a study aiming to prove the theorem that residuated algebra is isomorphically embeddable into a powerset residuated algebra. It considers the generalized Lambek calculus (GLC) which is related to arbitrary residuated algebras using a special labeled deductive system. It argues that the theorem in stronger than of J. M. Dunn due to the use of more special powerset frame than the concrete resituated algebra and use different methods in achieving the completeness of GLC.

The article discusses a calculus problem of finding optimal paths with Elvis, a Welsh corgi. The problem asks what route should Elvis take in order to take the tennis ball at Lake Michigan as fast as possible. Particular focus is also given to the existence of optimal choice in solving optimization problems.

Some classical examples of problem solving, where an extreme value condition is required, are here considered and/or revisited. The search for non-calculus solutions appears pedagogically useful and intriguing as shown through a rich literature. A teacher, who teaches both maths and physics, (as happens in Italian High schools) can find in these kinds of problems a mind stimulating exercise compared with the standard solution obtained by the differential calculus. A good link between the geometric and analytical explanations is so established. [ABSTRACT FROM AUTHOR]

The article presents some examples of problem solving methods from precalculus and calculus I courses. In an algebraic equation involving two square roots, students are taught to solve it by isolating one of the radicals and squaring each side of the resulting equation. It illustrates various methods of solving equations, including standard method, algebraic method, brute force method and elimination method. It also describes the look before you leap method.

We provide an easy argument, using the second fundamental theorem of calculus, for computing the area of the unit circle [ABSTRACT FROM AUTHOR]