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La dérivée est un concept important dans les cours de calcul. Dans certains pays comme au Canada et en France, il est enseigné au niveau collégial tandis qu’en Afrique et en particulier au Cameroun, on l’enseigne au niveau secondaire car la dérivée intervient dans l’étude des variations d’une fonction. Cette thèse dont le titre est : Analyse praxéologique des pratiques des enseignants et de leur utilisation de ressources pour l’enseignement de la dérivée. Une étude de cas dans l’enseignement secondaire général au Cameroun part du constat selon lequel de nombreux élèves et étudiants rencontrent des difficultés avec la notion de dérivée et que ces difficultés s’observent dans leurs résultats scolaires et académiques. Au vu de l’importance de la dérivée pour les différents parcours universitaires en sciences, nous nous intéressons aux pratiques des enseignants de mathématiques et notamment leurs praxéologies didactiques autour de la dérivée mais aussi, nous voulons comprendre les raisons qui justifient les choix faits par ces enseignants. La recherche essaie de répondre à la question de savoir quelles sont les pratiques des enseignants et les ressources utilisées par ces derniers pour l’enseignement de la dérivée? La recherche se propose ainsi d’identifier les praxéologies que les enseignants développent dans leur utilisation des ressources pour préparer leurs leçons et pour enseigner la dérivée. Nous ne nous limitons pas aux praxéologies observables mais nous aimerons comprendre les raisons qui justifient les choix opérés par ces enseignants et les contraintes institutionnelles auxquelles ils font face. Deux cadres théoriques encadrent cette recherche : la Théorie Anthropologique de la Didactique (TAD) qui nous permet d’analyser les types de tâches, les techniques, les technologies et les théories justifiant les technologies employées dans les ressources des enseignants et dans leurs pratiques enseignantes. De même, nous utilisons l’Approche Documentaire du Didactique (ADD) afin d’analyser les ressources institutionnelles et les autres ressources que les enseignants utilisent pour enseigner la dérivée. Pour analyser l’utilisation des ressources des enseignants, nous mettons un accent sur les objectifs et les sous-objectifs visés, les règles d’action, les invariants opératoires et les inférences éventuelles. Sur le plan méthodologique, nous utilisons une étude de cas impliquant trois enseignants qui enseignent la dérivée aux élèves de première. Nous utilisons des entretiens semi-dirigés afin d’analyser les choix que font les enseignants lors de la préparation des cours mais et surtout les raisons qui justifient ces choix; les observations vidéo-filmées permettent de savoir comment les enseignants introduisent la dérivée en un point et sur un intervalle et l’analyse des documents tels que les programmes de mathématiques, les manuels scolaires et d’autres ressources des enseignants employés dans la préparation des cours. Le principal résultat de cette thèse est l’identification du poids des ressources institutionnelles et de l’examen de fin d’année sur les praxéologies mathématiques et sur la manière dont les enseignants utilisent ces ressources. Nous discutons de ce résultat en tenant compte du rapport personnel des enseignants et des praxéologies développées dans les manuels et lors de l’enseignement., Derivatives are one of the most important topics studied in high school (in many countries) and postsecondary mathematics. Many students must learn this topic to pursue their university studies and as a gateway to learn other mathematical topics in various other fields. In Cameroon, secondary teaching lasts from 12 to 18 years old. In this context, what in other countries is considered as pre-university courses, or college, in Cameroon is still called secondary. Derivatives are introduced in the last but one year of this cycle, called première (students are 17), after the topics of function, limit, and continuity. In this thesis, we examined teachers’ practices and their use of resources while teaching derivatives. This study in general secondary education in Cameroon starts from the observation that many pupils and students encounter difficulties with the notion of derivative and that these difficulties are observed in their academic results. The research attempts to identify the tasks, techniques and technologies developed in the resources used by these teachers. It also tries to describe the patterns of use of resources by closely observing the objectives pursued, the rules of action and the operational invariants put in place when using resources. We do not limit ourselves to describing the practices of teachers, but we would like to understand the reasons that justify their choices and the institutional constraints they face. Since we are interested in teachers’ practices and their use of resources to teach derivatives, we use elements of the anthropological theory of the didactic (ATD) to analyze the types of tasks, the techniques, technologies, and theories justifying the technologies used in teachers' resources and teaching practices. By using DA, we understand that teachers, to prepare their teaching, use variety of resources. These resources aiming to teach this content together with the schemes of use of these resources result in a document and these schemes of use of resources is influenced by the teacher’s own personal relationship. The participants in this study were three teachers who teach derivatives to first-class students. We use semi-structured interviews to analyse the choices teachers make when preparing courses, but above all the reasons that justify these choices and video to observe how teachers introduce the derivative. The video shows how teachers introduce the derivative at a point and on an interval. Others analyses helps us understanding how this content is introduced on math programs, textbooks and other teacher resources used in the preparation of courses. The main result of this thesis is the identification of the weight of institutional resources and the end-of-year examination on mathematical praxeology’s and on the way, teachers use these resources. We discuss this result considering the personal relationship of teachers and the praxeology’s developed in textbooks and during teaching.

Викладено теорію границь послідовностей і функцій однієї змінної, диференціальне числення та його застосування. Навчальний посібник «Математичний аналіз: Диференціальне числення функцій однієї змінної» призначено для вивчення тем «Границі послідовностей та функцій, диференціальне числення функцій однієї змінної» студентами 122 «Компьютерні науки». Він містить основні теоретичні положення з названої теми. Разом зі стандартними фактами теорії послідовностей розглядаються методи знаходження границь, що базуються на теоремах Тьоплиця та Штольца, а також теоремі Банаха про стискаючі відображення, що вміщено в Доповненні. Викладено теорію границь функцій, теорію неперервних функцій та диференціальне числення, що ілюструється численними прикладами, в тому числі, підвищеної складності. The textbook "Mathematical Analysis: Differential calculus of functions of one variable" is designed to study the topics "Boundaries of sequences and functions, differential calculus of functions of one variable" by students 122 "Computer Science". It contains the main theoretical provisions on this topic. Along with the standard facts of sequence theory, methods for finding boundaries based on Toplice and Stoltz theorems, as well as Banach's theorem on compressive mappings, which are contained in the Addendum, are considered. The theory of function boundaries, the theory of continuous functions and differential calculus are presented, which is illustrated by numerous examples, including those of increased complexity. Учебное пособие «Математический анализ: дифференциальное исчисление функций одной переменной» предназначено для изучения тем «Границы последовательностей и функций, дифференциальное исчисление функций одной переменной» студентами 122 «Компьютерные науки». Он содержит основные теоретические положения по названной теме. Вместе со стандартными фактами теории последовательностей рассматриваются методы нахождения границ, основанные на теоремах Тёплица и Штольца, а также теореме Банаха о сжимающие отображения, помещены в Приложении. Изложена теория пределов функций, теорию непрерывных функций и дифференциальное исчисление, что иллюстрируется многочисленными примерами, в том числе, повышенной сложности.

RESUMEN. En este trabajo se describe un objeto virtual de aprendizaje (OVA) elaborado bajo una perspectiva interdisciplinar que abarca las materias de Economía y Matemáticas aplicadas a las Ciencias Sociales, correspondientes ambas a la modalidad de Humanidades y Ciencias Sociales del Bachillerato. Dicho OVA está constituido por una secuencia de construcciones elaboradas con el software Geogebra y organizadas en torno al concepto matemático de función derivada y su papel como herramienta básica en la modelización económica (asociado fundamentalmente a lo que suele denominarse análisis marginal). La propuesta se realiza con la intención de ejemplificar el tipo de materiales y actividades que podrían diseñarse y utilizarse de una manera integradora por parte del profesorado de ambas materias. Desde el punto de vista de la docencia de la economía, se considera que este tipo de planteamientos facilitaría la transferencia de los conocimientos de matemáticas al ámbito de la modelización económica, lo que resulta especialmente relevante en relación con la finalidad propedéutica del Bachillerato. Desde el de la docencia de las matemáticas, tanto la investigación relacionada con su didáctica como el propio currículo de la materia para la Educación Secundaria Obligatoria (ESO) y el Bachillerato, reflejan la importancia de incluir la modelización, no sólo por su interés como contenido, sino también por su función vehicular en tanto que facilitadora del aprendizajede los conceptos matemáticos. ABSTRACT.The paper describes a virtual learning object (VLO) built under an interdisciplinary perspective integrating Economíaand Matemáticas aplicadas a las Ciencias Sociales, which are both subjects within the Humanidades y Ciencias Socialesarea in Bachillerato. Such VLO is created out of a sequence of applications designed in the software Geogebra and organized around the mathematical concept of the derivative function and its role as an essential tool in economic modelling (mainly linked to what is usually known asmarginal analysis). The proposal aims to exemplify a set of materials and activities that could be designed and used in an integrated way by teachers of both subjects. From the viewpoint of teaching Economía, this type of approaches may facilitate the transfer of mathematical knowledge to the field of economic modelling, which is especially relevant when taking into consideration the propaedeutic purpose of Bachillerato. Concerning the teaching in Matemáticas, research on its didactics as well as thecurriculum content for the Educación Secundaria Obligatoria (ESO)and Bachilleratohighlight the dual goal of developing students? modelling competency and enhancing their learning of the mathematical concepts involved.

In this article, by using weight functions, the idea of introducing parameters, the reverse extended Hardy–Hilbert integral inequality and the techniques of real analysis, a reverse Hardy–Hilbert-type integral inequality involving one derivative function and the beta function is obtained. The equivalent statements of the best possible constant factor related to several parameters are considered. The equivalent form, the cases of non-homogeneous kernel and some particular inequalities are also presented.

3rd International Conference on New Horizons in Education (INTE) -- JUN 05-07, 2012 -- Prague, CZECH REPUBLIC, WOS: 000313019500082, Effective teaching is one of the most important factors to improve students' achievement. Therefore, many researchers in mathematics education are looking for several ways to improve the quality of learning. The purpose of this paper is to investigate an overview of specific difficulties based on the graph of derived function, another words, how students find the connections between the graph of derived function and the some properties of the original function. The subjects were 102 high school students in grade 11. The study was conducted during the fall semester in two calculus classes. The data of the research has been collected from 5 diagnostic the graph of derived function involved the chance in slope, decreasing, increasing, a local maximum, a local minimum, the point of inflection. The questionnaires were designed to assess how well the students had learned procedures the connections between the graph of derived function and the some properties of original function. Each student was asked explain how you arrived at your answers. While working on the answers of the students, not only the accuracy of the students' answers was examined but also the methods were being analysed even if they were incorrect. The results indicated that the students find it difficult to make connections between the graph of derived function and the original function. Usually, they were to interpret the graph of derived function as the graph of function. The students did not use the mathematical language to describe the graph of derived function Effective teaching in calculus course should be informal, intuitive and conceptually based on graphs and functions in order to improve the quality of learning and develop the understanding of calculus concepts. Selection and/or peer-review under responsibility of The Association of Science, Education and Technology, Sakarya Univ, Istanbul Univ, Suleyman Sah Univ, Natl Pingtung Univ Sci & Technol, Natl Univ Tainan, Inst Politecnico Guarda, Turkish Online Journal Educ Technol, Egitim Teknolojileri Arastirmalari Dergisi, Ohio Univ, Cleveland State Univ, Governors State Univ, Near E Univ, Cyprus Int Univ, Natl Cent Univ, Assoc Sci, Educ & Technol, Online Journal New Horizons Educ

We study statistical inference for the derivative of the nonparametric regression function using local linear model based empirical likelihood. We first derive a normal equation for the derivative through the local linear model and use this equation to construct an empirical likelihood for the derivative. We show that the limiting distribution of the empirical likelihood ratio is a scaled $\chi_1^2$ distribution rather than the usual (unscaled) $\chi_1^2$ distribution. We use this limiting distribution to construct pointwise confidence intervals for the derivative. Such empirical likelihood ratio confidence intervals are easier to obtain than the normal approximation based confidence intervals. A small simulation study also suggests that they are more accurate.