How to use the Fundamental Theorem of Calculus?

Now we can put the two part of the Fundamental Theorem of Calculus together named the Fundamental Theorem of Calculus:Suppose f（x）is continuous on[a,b].1.If F（x）=∫₀^xf（t）dt,then F’（x）=f（x）.2.∫_α^bf（x）dx=F（b）-F（a）,where     

What does Part 2 of the Fundamental Theorem of Calculus mean?

We defined the indefinite integral as an anti-derivative,and defined the definite integral as the limit of Riemann sums.Both of them are very different and seem to be little in common.Part 1of the Fundamental Theorem of Calculus shows how indefinite integration and definite integration are related.In the other words,it shows how anti-derivative and the area are related.Today,we’ll learn Part 2 of the Fundamental Theorem of Calculus.     

Monads and distributive laws for Rota–Baxter and differential algebras

In recent years, algebraic studies of the differential calculus and integral calculus in the forms of differential algebra and Rota–Baxter algebra have been merged together to reflect the close relationship between the two calculi through the First Fundamental Theorem of Calculus. In this paper we study this relationship from a categorical point of view in the context of distributive laws which can be tracked back to the distributive law of multiplication over addition. The monad giving Rota–Baxter algebras and the comonad giving differential algebras are constructed. Then we obtain monads and comonads giving the composite structures of differential and Rota–Baxter algebras. As a consequence, a mixed distributive law of the monad giving Rota–Baxter algebras over the comonad giving differential algebras is established.     

Four calculus qualms

In the teaching of calculus, some misunderstandings can arise, which can be alleviated by slightly deeper investigation, to be shared with students as occasions demand. In this note, four such explorations are described: (1) the Fundamental Theorem of Calculus is not stated and then proven, but calculated. (2) The integral calculus could conceivably be developed before the diff erential calculus. (3) The Taylor series about any point for any ‘good’ function is not stated and then proven, but calculated from scratch. (4) The graphing of the functions b?x and x?b (for any given b > 0) on the same set of axes is done more accurately, and more interestingly.     

Areas and the Fundamental Theorem of Calculus

We present a concise, yet self-contained module for teaching the notion of area and the Fundamental Theorem of Calculus for different groups of students. This module contains two different levels of rigour, depending on the class it used for. It also incorporates a technological component.     

How to Use the First FTC

<正>At the last time,we’ve learned the First Part of the Fundamental Theorem of Calculus.It shows how indefinite integration and definite integration are related.In the other words,it shows how anti-derivative and the area are related.Today,you’ll deeply understand the First FTC by using it.     

关于变上限函数求导的教学实例

从变上限函数的基本概念以及求导公式出发,通过几组教学实例,阐述对简单变上限函数、复合变：上（下）限函数、以及需要作变量替换才能求导的变上限函数求导的方法．     

Filling an area discretely and continuously

The literature dealing with student understanding of integration in general and the Fundamental Theorem of Calculus in particular suggests that although students can integrate properly, they understand little about the process that leads to the definite integral. The definite integral is naturally connected to the antiderivative, the area under the curve and the limit of Riemann sums; these three conceptualizations of the definite integral are useful in different contexts and provide students with what it takes to interpret the definite integrals. Research shows that students rarely invoke the multiplicatively-based summation conception of the definite integral although it is essential for evaluating line integrals, surface integrals and volumes. This paper describes a teaching module that promotes understanding as well as activating all three conceptualizations of the definite integral through motivating the accumulation area function and the results in the Fundamental Theorems of Calculus.     

Improved pedagogy for linear differential equations by reconsidering how we measure the size of solutions

For over 50 years, the learning of teaching of a priori bounds on solutions to linear differential equations has involved a Euclidean approach to measuring the size of a solution. While the Euclidean approach to a priori bounds on solutions is somewhat manageable in the learning and teaching of the proofs involving second-order, linear problems with constant co-efficients, we believe it is not pedagogically optimal. Moreover, the Euclidean method becomes pedagogically unwieldy in the proofs involving higher-order cases. The purpose of this work is to propose a simpler pedagogical approach to establish a priori bounds on solutions by considering a different way of measuring the size of a solution to linear problems, which we refer to as the Uber size. The Uber form enables a simplification of pedagogy from the literature and the ideas are accessible to learners who have an understanding of the Fundamental Theorem of Calculus and the exponential function, both usually seen in a first course in calculus. We believe that this work will be of mathematical and pedagogical interest to those who are learning and teaching in the area of differential equations or in any of the numerous disciplines where linear differential equations are used.     

Intra-mathematical connections made by high school students in performing Calculus tasks

In this article, we report the results of research that explores the intra-mathematical connections that high school students make when they solve Calculus tasks, in particular those involving the derivative and the integral. We consider mathematical connections as a cognitive process through which a person relates or associates two or more ideas, concepts, definitions, theorems, procedures, representations and meanings among themselves, with other disciplines or with real life. Task-based interviews were used to collect data and thematic analysis was used to analyze them. Through the analysis of the productions of the 25 participants, we identified 223 intra-mathematical connections. The data allowed us to establish a mathematical connections system which contributes to the understanding of higher concepts, in our case, the Fundamental Theorem of Calculus. We found mathematical connections of the types: different representations, procedural, features, reversibility and meaning as a connection.     

初基演算

<正> 命题演算的构成,通常有三步骤的说法,即从 Johanson 的“极小演算”到 Heyting的构造论命题演算再到二值演算.此外,Lewis 从模态或严格蕴涵出发,也分别了许多步骤,以达到二值演算为其极限;特别值得注意的是最后三个步骤,即从 S4 到 S5 到二值演算.这两个三步骤就某意义说乃是通常命题演算的构成中最本质的步骤.综合这两个三步骤,会带来许多便利,而本文所提出的也就是作为二者共同基础的初基演算.     

Exploring students' mathematical performance,metacognitive experiences and skills in relation to fundamental theorem of calculus

Several studies have explored students’ understanding of the relationships between definite integrals and areas under curve(s). So far, however, there has been less attention to students’ understanding of the Fundamental Theorem of Calculus (FTC). In addition, students’ metacognitive experiences and skills whilst solving FTC questions have not previously been explored. This paper explored students’ mathematical performance, metacognitive experiences and metacognitive skills in relation to FTC questions by interviewing nine university and eight Year 13 students. The findings show that several students had difficulty solving questions related to the FTC and that students’ metacognitive experiences and skills could be further developed.     

Noether's Theorem on Surfaces

In this article we prove a version of Noether's Theorem (of Calculus of Variations) which is valid for a general regular (compact) surface. As a special feature, the Lie group of transformations is allowed to act on the Cartesian product of the surface and the functional space. Additionally, we apply the Theorem to a problem in Classical Differential Geometry of surfaces. The given application is actually an example showing how Noether's Theorem can be used to construct invariant properties of the solutions to variational problems defined on surfaces, or equivalently, of the solutions to the associated Euler-Lagrange equations resulting from them.     

Fundamental Theorem of Calculus

A simple but rigorous proof of the Fundamental Theorem of Calculus is given in geometric calculus, after the basis for this theory in geometric algebra has been laid out. Various classical examples of this theorem, such as the Green’s and Stokes’ theorem are discussed, as well as the theory of monogenic functions which generalizes analytic functions of a complex variable to higher dimensions.     

统计方法在研究微积分与后续课程相关性中的应用与实证分析

着眼于本科生后期专业课程的学习效果,探讨微积分课程学习效果的有效性.以北京理工大学某经管类专业全部学生大一大二两学年的学习成绩为依据,分析微积分课程学习与后续理科课程学习的相关性,提出评估微积分课程学习效果的量化指标θ值的概念,并基于回归分析和相关性分析给出其算法.在实例分析中,通过研究,发现学生微积分课程学习效果的好坏会影响他们对后续相关课程的学习,同时也发现学生将微积分知识运用到间接相关科目的能力比运用到直接相关的科目的能力薄弱.     

Learning the integral concept by constructing knowledge about accumulation

We propose an approach to the integral concept for advanced high school students and provide evidence for the potential of this approach to support students in acquiring an in-depth proceptual view of the integral. The approach is based on the mathematical idea of accumulation. A ten-lesson unit has been implemented with four pairs of students. The students’ learning processes were micro-analysed using the methodological–theoretical framework of Abstraction in Context. In this paper, we focus on the lessons in which the notions of approximation and accumulation are introduced. The work of one student pair is analysed in detail, and the work of the other pairs is summarized. Our results show that most of the students reached a proceptual understanding of the integral that prepared them for the next step in the curriculum, namely the Fundamental Theorem of Calculus.     

解读“微积分算术”

2011年在科学网博客上出现的微积分算术,把抽象而高深的微积分看作函数的算术,只用几步高中代数,就能避开极限而又不失严格讲解微积分.首先,用等式讲解多项式的微积分;然后用不等式讲解显式初等函数的微积分.但是,某些读者可能会存在疑问:真的能让微积分的门槛降低,而又不失严格?它到底具备什么样的特点?对学生群体的定位如何?需要进一步的解读,这就是本文的目的.     

Fundamental Theorems of Weak Doi–Hopf Modules and Semisimple Weak Smash Product Hopf Algebras

Abstract

This paper, mainly gives a Fundamental Theorem of weak Doi–Hopf modules, which is not only generalizes the Fundamental Theorem of weak Hopf modules but also generalizes the Fundamental Theorem of relative Hopf modules. Moreover, it gives a sufficient and necessary condition for weak smash product algebras to be weak bialgebras, and a sufficient condition for weak smash product algebras to be semisimple weak Hopf algebras.     

Improving student learning in calculus through applications

Nationally only 40% of the incoming freshmen Science, Technology, Engineering and Mathematics (STEM) majors are successful in earning a STEM degree. The University of Central Florida (UCF) EXCEL programme is a National Science Foundation funded STEM Talent Expansion Programme whose goal is to increase the number of UCF STEM graduates. One of the key requirements for STEM majors is a strong foundation in Calculus. To improve student learning in calculus, the EXCEL programme developed two special courses at the freshman level called Applications of Calculus I (Apps I) and Applications of Calculus II (Apps II). Apps I and II are one-credit classes that are co-requisites for Calculus I and II. These classes are teams taught by science and engineering professors whose goal is to demonstrate to students where the calculus topics they are learning appear in upper level science and engineering classes as well as how faculty use calculus in their STEM research programmes. This article outlines the process used in producing the educational materials for the Apps I and II courses, and it also discusses the assessment results pertaining to this specific EXCEL activity. Pre- and post-tests conducted with experimental and control groups indicate significant improvement in student learning in Calculus II as a direct result of the application courses.