Fundamental Theorem Of Calculus Research Articles

The Profound Nature of Derivatives and Integration: A Comprehensive Examination

This article delves into the fundamental concepts of derivatives and integration, two core principles of calculus that govern a wide range of mathematical and real-world applications. Derivatives, which measure the rate of change of a function, and integration, which calculates accumulation and areas under curves, are essential tools in fields such as physics, economics, engineering, and finance. The article offers a detailed exploration of their mathematical definitions, key properties—including linearity, the product and chain rules, and the Fundamental Theorem of Calculus—and the diverse techniques used in integration. The practical applications of these concepts are also examined, with special emphasis on their role in modeling dynamic systems, optimizing processes, and solving real-world problems such as motion analysis, financial modeling, and economic optimization. Through in-depth discussion and examples, this comprehensive article bridges the theoretical underpinnings of derivatives and integration with their vast and critical uses in various scientific and economic disciplines.

Construction of free quasi-idempotent differential Rota-Baxter algebras by Gröbner-Shirshov bases

Differential operators and integral operators are linked together by the first fundamental theorem of calculus. Based on this principle, the notion of a differential Rota-Baxter algebra was proposed by Guo and Keigher. Recently, the subject has attracted more attention since it is associated with many areas in mathematics, such as integro-differential algebras. This paper considers differential Rota-Baxter algebras in the quasi-idempotent operator context. We establish a Gröbner-Shirshov basis for free commutative quasi-idempotent differential algebras (resp. Rota-Baxter algebras, resp. differential Rota-Baxter algebras). This provides a linear basis of a free object in each of the three corresponding categories by the Composition-Diamond lemma. Communicated by P. Kolesnikov

G**β-Continuous and G**β-Irresolute Mappings in Topological Spaces

Topology being somehow very recent in nature but has got tremendous applications over almost all other fields. Theoretical or fundamental topology is a bit dry but the application part is what drives crazy once we get used. Topology has applications in various fields of Science and Technology, like applications to Biology, Robotics, GIS, Engineering, Computer Sciences. Topology though being a part of mathematics but it has influenced the whole world with so strong effects and incredible applications. The concept of continuity is fundamental in large parts of contemporary mathematics. In the nineteenth century, precise definitions of continuity were formulated for functions of a real or complex variable, enabling mathematicians to produce rigorous proofs of fundamental theorems of real and complex analysis, such as the Intermediate Value Theorem, Taylor’s Theorem, the Fundamental Theorem of Calculus, and Cauchy’s Theorem. In the early years of the Twentieth Century, the concept of continuity was generalized so as to be applicable to functions between metric spaces, and subsequently to functions between topological spaces. Topology is an area of mathematics concerned with the properties of space that are preserved under continuous deformations including stretching and bending but not tearing. In 2023, Dr. T. Delcia and M. S, Thillai introduced a new type of closed sets called g**β-closed sets and investigated their basic properties including their relationship with already existing concepts in Topological Spaces. In this paper, we introduce g**β-continuous function, g**β-irresolute function, g**β-open function, g**β-closed function, pre-g**β-open function, and pre-g**β-closed function, and investigate properties and characterizations of these new types of mappings in topological spaces.

A Visualization in GeoGebra of Leibniz’s Argument on the Fundamental Theorem of Calculus

In the literature, it is usually assumed that Leibniz described proof for the Fundamental Theorem of Calculus (FTC) in 1693. However, did he really prove it? If the answer is no from today’s perspective, are there works in which Leibniz introduced arguments that can be understood as formulations and justifications of the FTC? In order to answer this question, we used a historiographic methodology with expert triangulation. From the study of Leibniz’s manuscripts describing the inverse problem of tangents and its relationship with the quadrature problem, we found evidence of a geometrical argument from which the FTC can be inferred. We present this argument using technological resources and modern notation. This result can be used to teach the FTC due to the existence of dynamic and geometrical software, which makes it suitable for the classroom. Moreover, it provides another interpretation of the FTC complementary to the interpretation using Riemann sums.

The Story of Definite Integrals: A Calculus Textbook Narrative Analysis

There is a plethora of textbooks to choose from. While most textbooks contain the same content, teachers need to understand the textbook they have chosen to support their students’ learning, as textbooks can significantly impact student learning and teacher instruction. When searching for a way to understand the various textbooks, my research question became, How do calculus textbooks tell the story of definite integrals? My research is in response to the never-ending search for how teachers can select and use their resources. Also, this study contributes to understanding the relationships between stories and knowledge construction using Dietiker’s (2015) narrative framework to analyze the written calculus curriculum. I wrote the specific stories of introducing the definite integral as told by five calculus textbooks. Analyzing how the stories developed disclosed the similarities and differences among the texts in teaching definite integrals. The analysis also revealed a variety of avenues to introduce and teach definite integrals, including a focus on the area under the curve, Riemann sums, or the Fundamental Theorem of Calculus, and some implications to the context of physics related to these avenues. These insights provide teachers with a deeper understanding of textbook lessons and the variances that can potentially alter student learning.

Compatibility of the method of brackets with classical integration rules

Abstract The method of brackets is a symbolic approach to the computation of integrals over R n {{\mathbb{R}}}^{n} based on a deep result by Ramanujan. Its usefulness to obtain new and difficult integrals has been demonstrated many times in the last few years. This article shows that this method is consistent with most classical rules for the computation of integrals, such as the fundamental theorem of calculus, the Laplace transform, the reduction formula for the integration of functions with spherical symmetry, the Cauchy-Schlömilch transformation, and explicit evaluations for multivariate integrals of product of Bessel functions as obtained by Exton and Srivastava. This work is part of a program dedicated to the derivation of solid theoretical grounds for the use of this attractive integration method.

First-Year Engineering Students’ Interpretations of Differentials and Definite Integrals

In this paper we focus on Norwegian first-year engineering students’ interpretations of differentials and definite integrals. Through interviews with 15 engineering students, we investigated how the students interpreted the different symbols involved in the Fundamental Theorem of Calculus (FTC), as displayed in the textbook used in their calculus course. Through the students’ reflections, we aimed to gain insight into the students’ interpretations of differentials, infinitesimals, and limits, and how these concepts were used to interpret the definite integral. Despite a rather rigorous limit-based approach to definite integrals, provided in teaching and in the students’ textbook, we observed that most students offered conceptual interpretations closer linked with an infinitesimal approach, in terms of integrals being regarded as sums of narrow bars, in line with Leibniz.

Fundamental Theorem of Calculus: Cognitive Demands and Learning Limitations on Mathematical Tasks

Background: Mathematical tasks for university teaching are generally of great cognitive demand, without thinking about the limitations they imply for their students. Objectives: To develop a theoretical analysis of the teaching limitations and cognitive demand of four tasks proposed for teaching calculus, specifically, the Fundamental Theorem of Calculus. Design: Qualitative paradigm, with a descriptive-interpretive approach, according to the nature of the data collected. Setting and Participants: The study is framed in a Colombian University, in the subject “Calculus II” for second-semester engineering students, where a professor designs the tasks to be implemented in this course. Data collection and analysis: The data correspond to the professor's lesson plans the statements of the main mathematical tasks within them. These plans were chosen based on availability and accessibility. A content analysis was conducted, considering as units of analysis the paragraphs or sets of paragraphs of the statement of each school mathematics task. Results: Most of the proposed tasks correspond to high cognitive demand (procedures with connections and mathematical construction) and only one was of low demand (memorisation). Moreover, each of the tasks presents its own cognitive demand and several learning constraints that, some of them, agree with the exposed literature. Conclusions: The work aims to have implications for higher education, since to think of a didactic proposal for a better approach to teaching is necessary to configure lesson plans that mobilise the learning of mathematics in engineering, but from the use of tasks with different cognitive demands, in which will vary from less to more, and lead to meaningful learning for the approach of new tasks.

An Alternative Approach for Defining a Quadratic Function

Defining a quadratic function through the slopes of its secant/tangent lines leads to the fundamental theorem of calculus (FTC) and an alternative way of understanding integration.

On Lr-norm-based derivatives and fuzzy Henstock-Kurzweil integrals with an application

The fundamental theorem of calculus not only plays an important role in the study of differential equation theory, but also in many practical problems solving. In order to enrich its content, in present paper, we extend fuzzy derivatives and fuzzy Henstock-Kurzweil integrals. Firstly, we discuss some properties of Lr-norm-based derivatives for fuzzy Henstock-Kurzweil integrable function (or briefly FLr-derivative). Then, we give some relationships with well-known fuzzy derivatives. After that, we also define a kind of fuzzy Henstock-Kurzweil integral based on FLr-derivative (or briefly Lr-FHK-integral). In addition, we give some important properties and convergence theorems for the Lr-FHK-integral. Meanwhile, we obtain a sufficient and necessary condition for integrability of this integral. As the application, we discuss the fuzzy Fourier series.

On the Basic Theory of Some Generalized and Fractional Derivatives

We continue the development of the basic theory of generalized derivatives as introduced and give some of their applications. In particular, we formulate necessary conditions for extrema, Rolle’s theorem, the mean value theorem, the fundamental theorem of calculus, integration by parts, along with an existence and uniqueness theorem for a generalized Riccati equation, each of which provides simple proofs of the corresponding version for the so-called conformable fractional derivatives considered by many. Finally, we show that for each α>1 there is a fractional derivative and a corresponding function whose fractional derivative fails to exist everywhere on the real line.

A Generalised Continuous Primitive Integral and Some of its Applications

Using the Laplace derivative a Perron type integral, the Laplace integral, is defined. Moreover, it is shown that this integral includes Perron integral and to show that the inclusion is proper, an example of a function is constructed, which is Laplace integrable but not Perron integrable. Properties of integrals such as fundamental theorem of calculus, Hake's theorem, integration by parts, convergence theorems, mean value theorems, the integral remainder form of Taylor's theorem with an estimation of the remainder, are established. It turns out that concerning the Alexiewicz norm, the space of all Laplace integrable functions is incomplete and contains the set of all polynomials densely. Applications are shown to Poisson integral, a system of generalised ordinary differential equations and higher-order generalised ordinary differential equation.

The Dawn of Calculus Occurred in India

It is said that Leibniz and Newton were the inventors of Calculus. Taking into granted the above, if we review it in the light of Indian contributions, we find that Newton and Leibniz cannot be credited as inventors. The very basics of Calculus, i.e., derivatives, integration, Taylor's series, trigonometric series (sine, cosine, tan) and most importantly the fundamental theorem of Calculus were the gifts of Indian Mathematicians. We will be failing in our duty, if the astonishing mathematicians Aryabhatta, Brahmagupta, Bhaskaracharya-II, Madhava and their followers are ignored in this regard. If their contributions be put in a coherent Framework, it will be found that they were the actual fore-father inventors of Calculus. Here, we would especially want to throw glimpses of Madhava of Sangamagrama's contribution.

Toward an optimal theory of integration for functions taking values in quasi-Banach spaces

We present a new approach to define a suitable integral for functions with values in quasi-Banach spaces. The integrals of Bochner and Riemann have deficiencies in the non-locally convex setting. The study of an integral for p-Banach spaces initiated by Vogt is neither totally satisfactory, since there are quasi-Banach spaces which are p-convex for all 0<p<1, so it is not always possible to choose an optimal p to develop the integration. Our method puts the emphasis on the galb of the space, which permits a precise definition of its convexity. The integration works for all spaces of galbs known in the literature. We finish with a fundamental theorem of calculus for our integral.

Using Cauchy theorem to compute complicated real integrals

As the central object in the theory of complex analysis, Holomorphic functions have many elegant mathematical properties. Holomorphic functions are extremely valuable because, on the one hand, they are unexpectedly common, and on the other hand, they may be used to establish extremely powerful theorems. For example, To establish theorems like the prime number theorem, analytic number theorists commonly create holomorphic or meromorphic functions that hold number-theoretic information, such as the Riemann zeta function. Given knowledge about a holomorphic function in a relatively small part of its domain, one may extract information about the function’s behavior in other a priori unrelated sections of its domain, according to Cauchy’s integral formula and the identity theorem (and this is what allows things like contour integration to work). Due to difficulties in obtaining primitives of some real function, the fundamental theorem of Calculus does not work in most cases. In this article, we review the Cauchy theorem and use it as a tool to compute several real integrals.

Generalized discrete operators

We define a class of discrete operators that, in particular, include the delta and nabla fractional operators. Moreover, we prove the fundamental theorem of calculus for these operators.

Derivation of the differential continuity equation in an introductory engineering fluid mechanics course

The differential continuity equation is elegantly derived in advanced fluid mechanics textbooks using the divergence theorem of Gauss, where the surface integral of the mass flux flowing out of a finite control volume is replaced by the volume integral of the divergence of the mass flux within the control volume. To avoid the need for introducing the Gauss divergence theorem in an introductory fluid mechanics course, introductory textbooks in fluid mechanics have opted to use a more simple approach, which depends on the consideration of an infinitesimal control volume and the use of Taylor series expansion. This approach, however, involves a first order truncation of the Taylor series expansion and the use of approximate equality signs which may imply to undergraduate students that the derived continuity equation is an approximate equation. The present study proposes an alternative derivation of the differential continuity equation using a finite control volume and is based on the simple concept of the antiderivative function and the fundamental theorem of calculus. The proposed derivation eliminates the need to formally introduce the Gauss divergence theorem in an introductory engineering fluid mechanics course while avoiding the use of truncated Taylor series expansion and approximate equality signs, hence providing a more simple and sound understanding of the derivation of the differential continuity equation to undergraduate engineering students.

Fractal Calculus and Analysis for Characterizing Geoanomalies Caused by Singular Geological Processes

Integral and differentiation are two mathematical operations in modern calculus and analysis which have been commonly applied in many fields of science. Integration and differentiation are associated and linked as inverse operation by the fundamental theorem of calculus. Both integral and differentiation are defined based on the concept of additive Lebesgue measure although various generations have been developed with different forms and notations. Fractals can be considered as geometry with fractal dimension (e.g., non-integer) which no longer possesses Lebesgue additive property. Accordingly, the ordinary integral and differentiation operations are no longer applicable to the fractal geometry with singularity. This paper introduces a recently developed concept of fractal differentiation and integral operations. These operations are expressed using the similar notations of the ordinary operations except the measures are defined in fractal space or measures with fractal dimension. The calculus operations can be used to describe the new concept of fractal density, the density with fractal dimension or density of matter with fractal dimension. The concept and methods are also applied to interpret the Bouguer anomaly over the mid-ocean ridges. The results show that the Bouguer gravity anomaly depicts singularity over the mid-ocean ridges. The development of new calculus operations can significantly improve the accuracy of geodynamic models.

A Plea for Finite Calculus

There are some wonderful theorems in undergraduate mathematics that everybody should know. We all have learned in school how to apply the fundamental theorem of calculus to solve an integral, say ∫...

Focuses of awareness in the process of learning the fundamental theorem of calculus with digital technologies

This paper brings together three themes: the fundamental theorem of the calculus (FTC), digital learning environments in which the FTC may be taught, and what we term “focuses of awareness.” The latter are derived from Radford’s theory of objectification: they are nodal activities through which students become progressively aware of key mathematical ideas structuring a mathematical concept. The research looked at 13 pairs of 17-year-old students who are not yet familiar with the concept of integration. Students were asked to consider possible connections between multiple-linked representations, including function graphs, accumulation function graphs, and tables of values of the accumulation function. Three rounds of analysis yielded nine focuses in the process of students’ learning the FTC with a digital tool as well as the relationship between them. In addition, the activities performed by the students to become aware of the focuses are described and theoretical and pedagogical implementations are also discussed.