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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.

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]

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]

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]

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.

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