Afin de mieux cerner les enjeux de la transition entre le secondaire et le postsecondaire, nous proposons un examen du passage de la notion de fonction à celle de dérivée. À la lumière de plusieurs travaux mettant en évidence des difficultés inhérentes à ce passage, et nous basant sur les recherches de Carlson et ses collègues (Carlson, 2002; Carlson, Jacobs, Coe, Larsen et Hsu, 2002; Carlson, Larsen et Jacobs, 2001; Oehrtman, Carlson et Thompson, 2008) sur le raisonnement covariationnel, nous présentons une analyse de la dynamique du développement de ce raisonnement chez des petits groupes d’élèves de la fin du secondaire et du début du collégial dans quatre situations-problèmes différentes. L’analyse des raisonnements de ces groupes d’élèves nous a permis, d’une part, de raffiner la grille proposée par Carlson en mettant en évidence, non seulement des unités de processus de modélisation (ou unités de raisonnement) mises en action par ces élèves lors des activités proposées, mais aussi leurs rôles au sein de la dynamique du raisonnement. D’autre part, cette analyse révèle l’influence de certaines caractéristiques des situations sur les interactions non linéaires entre ces unités. / To better understand the transitional challenges between high-school and post-secondary education, we propose a study of the passage from the notion of function to the notion of derivative. Based on numerous studies on the difficulties related to this passage and, more specifically, on the work of Carlson and colleague’s (Carlson, 2002; Carlson, Jacobs, Coe, Larsen & Hsu, 2002; Carlson, Larsen & Jacobs, 2001; Oehrtman, Carlson & Thompson, 2008) on covariational reasoning, we present an analysis of the dynamics of the development of covariational reasoning. By submitting four different problem-situations to small groups of students ending secondary school and beginning college (15-18 years old), we were able to examine that development. On one hand, the analysis of the reasoning of those students allowed us to refine the grid proposed by Carlson, bringing out, not only the reasoning units used by those students during the proposed activities, but also their role in the dynamic of the reasoning. On the other hand, this analysis reveals the influence of certain characteristics of the situations on the non-linear interactions between those units.

This thesis comprises three original papers about learning rate of change in school mathematics. The overall aim of the thesis is to contribute with knowledge in this particular area of research. Within this aim, each of the original papers contribute with its own perspective. The theoretical framework used in the thesis is variation theory of learning (Marton & Booth, 1997; Marton, 2015), by which learning is seen as experiencing a phenomenon in a new way. This theoretical point of departure is reflected in the research question of the thesis: What is critical to discern to use and express rate as a measure of the relation of changes in two quantities? The empirical study was conducted as a learning study (Pang & Marton, 2003). A learning study is an iterative, interventional research arrangement in which teachers and researchers collaboratively explore a specific ability, the object of learning, worthwhile for the students to learn. The object of learning related to this thesis, ‘to express the quantitative rate of change of a linear relation’, was explored in a series of three research lessons at a secondary school. Data consists of students’ responses to written pre- and post-tests, and lesson videos. Some data have been analysed during the on-going empirical study and some data have been analysed after it was concluded. Principles from variation theory have been used as tools for analysis throughout the study. Main results of Paper I demonstrate how two critical aspects are identified and revised through the process of learning study. In Paper II the results indicate that qualitatively different questions in a task may affect students’ ways to relate changes in two quantities. The results of Paper III suggest how different perspectives of slope may promote homogeneity as an aspect of rate. Results also comprise six critical aspects of the object of learning, four of which was identified by revisiting the results of Paper II. In summary, the critical aspects also specify the meaning of a covariational perspective of rate. Results are discussed in relation to previous educational research about rate of change, covariation of quantities and students’ conceptions of rate and slope. Further research directions are suggested. / Den här sammanläggningsuppsatsen omfattar tre artiklar om förändringstakt i matematik. Uppsatsens övergripande syfte är att bidra med kunskap till detta specifika forskningsfält. Inom ramen för detta syfte bidrar var och en av de tre artiklarna med sitt eget perspektiv. Det teoretiska ramverket i uppsatsen är variationsteori (Marton & Booth, 1997; Marton, 2015). Ur detta teoretiska perspektiv ses lärande som ett erfarande av ett fenomen på ett nytt sätt. Denna teoretiska utgångspunkt har färgat uppsatsens övergripande forskningsfråga: Vad är kritiskt att urskilja för att använda och uttrycka förändringstakt somett mått på relationen mellan förändringar i två storheter? Den empiriska studien genomfördes som en learning study (Pang et al., 2003). Learning study är en iterativ, intervenerande forskningsansats där lärare och forskare i samarbete utforskar en specifik förmåga, lärandeobjektet, något som är värdefullt för elever att lära sig. Lärandeobjektet som behandlas i denna uppsats, ’att uttrycka förändringstakten i en linjär relation kvantitativt’, utforskades i en serie av tre lektioner på en högstadieskola. Data består av elevers svar till skriftliga uppgifter i för- och eftertest, samt videoinspelade lektioner. Data har analyserats både under den pågående empiriska studien, och efter att den avslutades. Principer från variationsteori har använts som analysverktyg genom studiens gång. Huvudresultaten av Artikel I visar hur två kritiska aspekter identifieras, revideras och förfinas genom learning study som process. I Artikel II tyder resultaten på att kvalitativt olika frågor i en uppgift påverkade elevernas sätt att relatera förändringar i två storheter. Resultaten i Artikel III visar hur olika perspektiv på lutningen hos en graf kan föra fram homogenitet som en aspekt av förändringstakt. Uppsatsens resultat omfattar framför allt sex aspekter som är kritiska att urskilja för att utveckla förmågan att använda och uttrycka förändringstakt som ett mått på relationen mellan förändringar i två storheter. Fyra av dem identifierades då resultaten av Artikel II tolkades utifrån ett variationsteoretiskt perspektiv. Som helhet specificerar också de kritiska aspekterna innebörden av ett samvarierande perspektiv på förändringstakt. Resultaten diskuteras i förhållande till tidigare forskning om lärande om förändringstakt, samvariation mellan storheter och elevers uppfattningar av lutning. Några vidare forskningsriktningar pekas ut.

[pt] Os sistemas elétricos de grande porte estiveram sempre fortemente associados à utilização de máquinas síncronas para a geração de energia e atendimento às cargas, tendo como outros benefícios, a possibilidade de prover o controle dinâmico do perfil de tensão da rede e a regulação da frequência elétrica. Em relação ao controle da frequência, as máquinas síncronas agregam resposta inercial, ou seja, convertem em energia elétrica parte da energia cinética armazenada em seus rotores, e vice-versa, sempre que caracterizado um desequilíbrio entre carga e geração. Este comportamento determina as taxas iniciais de variação da frequência elétrica (RoCoF – Rate of Change of Frequency), as quais podem ser definidas principalmente por dois fatores: a magnitude do desbalanço de potência ativa e a inércia global do sistema, sendo esse último dependente do número de geradores sincronizados e da inércia de cada gerador. O crescimento cada vez maior na matriz de energia elétrica de usinas eólicas e solares fotovoltaicas, que usualmente conectam-se à rede por meio de conversores eletrônicos e, portanto, não agregam resposta inercial ao sistema, trará impactos na regulação da frequência e nos valores de RoCoF. Nesse cenário, o principal objetivo deste trabalho é apresentar uma análise quantitativa da evolução das taxas de variação da frequência no SIN durante perdas de grandes blocos de geração. Os valores de RoCoF são analisados tanto do ponto de vista global, através da frequência no centro de inércia do sistema, quanto do ponto de vista regional, a partir de barras que representam as regiões do SIN. / [en] Large electrical systems have always been strongly associated with the use of synchronous machines for power generation and load service, with other benefits, possibility of providing dynamic control of the system voltage profile and frequency regulation. Regarding frequency control, synchronous machines aggregate inertial response, i.e., convert part of the kinetic energy stored in their rotors into electrical energy (and vice-versa), whenever there is an imbalance between load and generation. This behavior determines the initial Rate of Change of Frequency (RoCoF), determined mainly by two factors: a magnitude of the active power unbalance and the global inertia of the system, that is dependent on the number of synchronized generators and the inertia of each generator. The increase of wind and solar photovoltaic plants in electrical energy matrix, which are usually connected to the system through inverters and do not aggregate inertial response to the system, will impact on frequency regulation and RoCoF values. In this scenario, the main objective of this thesis is to present a quantitative analysis of the evolution of RoCoF in the Brazilian Electrical System during outages of large generation blocks. RoCoF values are analyzed from a global point of view, through the frequency at the center of inertia of the system, and from a local point of view, through buses that represent the Brazilian Electrical Subsystems.

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.

This thesis addresses the absence of explicit pricing formulae and the complexity of proposed models (incomplete markets framework) in the area of mortality risk management requiring the application of advanced techniques from the realm of Financial Mathematics and Actuarial Science. In fact, this is a multi-essay dissertation contributing in the direction of designing and pricing mortality-linked derivatives and offering the state of art solutions to manage longevity risk. The first essay investigates the valuation of Catastrophic Mortality Bonds and, in particular, the case of the Swiss Re Mortality Bond 2003 as a primary example of this class of assets. This bond was the first Catastrophic Mortality Bond to be launched in the market and encapsulates the behaviour of a well-defined mortality index to generate payoffs for bondholders. Pricing this type of bond is a challenging task and no closed form solution exists in the literature. In my approach, we adapt the payoff of such a bond in terms of the payoff of an Asian put option and present a new methodology to derive model-independent bounds for catastrophic mortality bonds by exploiting the theory of comonotonicity. While managing catastrophic mortality risk is an upheaval task for insurers and re-insurers, the insurance industry is facing an even bigger challenge - the challenge of coping up with increased life expectancy. The recent years have witnessed unprecedented changes in mortality rate. As a result academicians and practitioners have started treating mortality in a stochastic manner. Moreover, the assumption of independence between mortality and interest rate has now been replaced by the observation that there is indeed a correlation between the two rates. Therefore, my second essay studies valuation of Guaranteed Annuity Options (GAOs) under the most generalized modeling framework where both interest rate and mortality risk are stochastic and correlated. Pricing these types of options in the correlated environment is an arduous task and a closed form solution is non-existent. In my approach, I employ the use of doubly stochastic stopping times to incorporate the randomness about the time of death and employ a suitable change of measure to facilitate the valuation of survival benefit, there by adapting the payoff of the GAO in terms of the payoff of a basket call option. I then derive general price bounds for GAOs by employing the theory of comonotonicity and the Rogers-Shi (Rogers and Shi, 1995) approach. Moreover, I suggest some `model-robust' tight bounds based on the moment generating function (m.g.f.) and characteristic function (c.f.) under the affine set up. The strength of these bounds is their computational speed which makes them indispensable for annuity providers who rely heavily on Monte Carlo simulations to calculate the fair market value of Guaranteed Annuity Options. In fact, sans Monte Carlo, the academic literature does not offer any solution for the pricing of the GAOs. I illustrate the performance of the bounds for a variety of affine processes governing the evolution of mortality and the interest rate by comparing them with the benchmark Monte Carlo estimates. Through my work, I have been able to express the payoffs of two well known modern mortality products in terms of payoffs of financial derivatives, there by filling the gaps in the literature and offering state of art techniques for pricing of these sophisticated instruments.

Made available in DSpace on 2016-04-27T16:57:45Z (GMT). No. of bitstreams: 1 EDM - Claudio dallanese.pdf: 1511562 bytes, checksum: e7b4467bf7d57ecbf0c536f800f0c19b (MD5) Previous issue date: 2006-10-27 / The aim of this investigation was to identify and to analyze arguments and metaphors used by a group of students of a master s degree course in Mathematical Education for the rate of change, to understand how they learn this topic. The option of working with those subjects relapsed in the fact that they are all teachers of Mathematics and they have already seen Calculus in their graduation. To those subjects, tasks were offered in a learning scenery where the dialogue was privileged among teacher, students and technology. The vision adopted regarding the technology was of a prothesis, in the sense that it makes it possible for the student to do things different from the way that he would do without it. With the intention of working with different texts, sometimes we offered tasks that the students interacted with the computer, sometimes we offered a task which the prosthesis was a small canal made from a PVC tube, tennis ball, ping-pong ball, chronometer and tape measure. The classes that the students worked in those tasks were filmed using a VHS camera. Notes of some speeches and interventions of the students and the teacher written on a notebook helped to enrich the collection of data. The analysis was based on Embodied Cognition Theory and on the Model of the Argumentative Strategy. We conclude that the process of understanding medium rate of change and instantaneous rate of change is not only the case of just a passage from one to another analytical formula or from a graph to a formula. There is a difference among the cognitive mechanisms to understand the graph and the analytic formula, which contributes to the students' difficulty with that topic. It is not just the formal definition that is responsible for that difficulty. We observed that with the aid of the computer science technology, it was possible to create an environment where the fictive motion, intrinsic of the language, became a factive movement. That is, when secants straight lines coincided with a tangent straight line for successive approaches, and when the tangent straight line to the curve in a point could move, at the same time the values of the slope of those straight lines could be seen in the screen / Esta investigação teve por objetivo identificar e analisar argumentos e metáforas utilizadas por um grupo de alunos de um curso de pós-graduação em Educação Matemática para taxa de variação, para entender como é que eles aprendem esse tópico. A opção de trabalhar com esses sujeitos recaiu no fato de serem todos professores de Matemática do ensino fundamental e/ou médio e já terem visto Cálculo em sua graduação. A esses sujeitos foram oferecidas tarefas num cenário de aprendizagem onde se privilegiou o diálogo entre professor, alunos e tecnologia. A visão adotada com relação à tecnologia foi a de prótese, no sentido de que ela possibilita ao aluno fazer coisas diferentes do modo que faria sem ela. Com o intuito de trabalhar com textos distintos, ora oferecemos tarefas em que os alunos interagiram com o computador, ora oferecemos uma tarefa em que a prótese era uma canaleta feita de cano de PVC, bola de tênis, bola de pingue-pongue, cronômetro e trena. As aulas em que os alunos trabalharam nessas tarefas foram filmadas utilizando uma filmadora VHS. Apontamentos por escrito em um diário de classe de algumas falas e intervenções dos alunos e da professora ajudaram a enriquecer a coleta de dados. A análise baseou-se na Teoria da Cognição Corporificada e no Modelo da Estratégia Argumentativa. Concluímos que o processo de compreender taxa média e taxa instantânea de variação não é o caso apenas de uma passagem de uma fórmula analítica a outra ou de um gráfico para uma fórmula. Existe uma diferença entre os mecanismos cognitivos para compreender o gráfico e a fórmula analítica, diferença esta que contribui com a dificuldade dos alunos com esse tópico. Não é apenas a definição formal que é responsável por essa dificuldade. Observamos que com o auxílio da tecnologia informática foi possível criar um ambiente onde o movimento fictivo, intrínseco da linguagem, se transformou em um movimento factivo. Isto é, quando retas secantes coincidiam com uma reta tangente por sucessivas aproximações e quando a reta tangente à curva num ponto podia se mover, ao mesmo tempo os valores do coeficiente angular dessas retas podiam ser vistos na tela

Este trabalho traz uma proposta de atividades, a serem desenvolvidas em sala de aula, com o objetivo de introduzir o conceito de derivadas para os alunos da primeira série do Ensino Médio. Antes das atividades, estão presentes algumas breves pesquisas. O histórico da presença de tópicos do Cálculo Diferencial e Integral no Ensino Médio no Brasil, assim como a análise de alguns livros didáticos, serve para mostrar como o assunto já foi e está sendo tratado no país. Também são exibidos aspectos sobre o Ensino Médio na Alemanha e nos Estados Unidos, países onde o cálculo está presente na Escola Secundária, embora de formas bastante diferentes. Um capítulo sobre a preparação adequada para as aulas também foi incluído, uma vez que a simples inserção da derivada poderia causar problemas de tempo para o cumprimento do cronograma e não trazer os resultados esperados. São necessários algum grau de adequação dos conteúdos ministrados e de cooperação com professores de Física. As atividades visando o ensino dos conceitos iniciais de derivada são motivadas por um problema físico de movimento. O foco é dado na intuição e na visualização de gráficos, para que haja uma melhor compreensão dos conceitos envolvidos. A utilização de um software de geometria dinâmica é requerida em boa parte do tempo, como importante ferramenta de apoio pedagógico / This paper presents a proposal of activities to be developed in the classroom, with the goal of introducing the concept of derivative for students in the first grade of secondary school. Before the activities, some brief researches are presented. The historical presence of the topics of Differential and Integral Calculus in brazilian High Schools, as well as the analysis of some textbooks, serves to show how it has been and is being treated in the country. Aspects of the High School are also shown in Germany and the United States, countries where the calculus is present in High School, though in quite different ways. A chapter about the proper preparation for these classes was also included, since the simple insertion of the derivative could cause problems for meeting the schedule and could not bring the expected results. Some degree of adequacy of the contents and cooperation with Physics teachers are needed. The activities aiming at teaching the initial concepts of derivatives are motivated by a physical problem of motion. The focus is given on intuition and visualization of graphs, so there is a better understanding of the concepts involved. The use of a dynamic geometry software is required for much of the time, as an important tool for pedagogical support

Avec le paradigme du réseau intelligent, les ingénieurs de protection ont maintenant à leur disposition une large gamme de nouvelles technologies de communication. Parmi elles, la norme CEI 61850-9-2 a introduit le concept de bus de procédé qui permet l'envoi de valeurs échantillonnées horodatées à un temps absolu depuis les transformateurs de mesure du terrain jusqu’aux relais de protection numériques. Ces derniers peuvent intégrer la fonction d'unité de mesure de phaseur qui peut être utilisé pour échanger des synchrophaseurs entre les fonctions de protection et pour une nouvelle protection anti-îlotage. Les relais de fréquence et de dérivée de fréquence sont, de nos jours, les méthodes anti-îlotage les plus couramment employées, mais leurs performances ne sont pas satisfaisantes. Dans ce contexte, une nouvelle génération de techniques de traitement du signal pour les relais de protection ayant des échantillons horodatées comme signal d'entrée et intégrant la mesure de synchrophaseurs est nécessaire. Cette thèse étudie d'abord l'impact des valeurs échantillonnées sur le traitement du signal. Trois solutions sont ensuite proposées pour calculer les phaseurs, les fréquences et les dérivées de fréquence dans diverses conditions statiques et dynamiques, puis testées par simulation. Enfin, un algorithme de mesure de synchrophaseurs incorporé dans le traitement de signal initial est proposé. Cet algorithme a été testé selon la dernière version de la norme d'unité de mesure de phaseur et les résultats obtenus sont conformes aux exigences de mesure. / With the Smart Grid paradigm, protection engineers now have available to them a large range of new communication technologies. Among them, IEC Standard 61850-9-2 has introduced the process bus concept which permits sending of absolute time-stamped digitized analogue values from the instrument transformers of the field to numerical relays. The latter can incorporate the phasor measurement unit function which can be used for exchanging synchrophasors between protection functions and for new anti-islanding protection. Frequency and rate-of-change-of-frequency relays are, nowadays, the most commonly employed anti-islanding methods but their performance is not satisfactory. In this context, a new generation of signal processing techniques for protection relays having time-stamped digitized analogue values as input signal with synchrophasors measurement capability is required. This thesis first studies the impact of sampled measured values on the signal processing. Three solutions are then proposed to compute phasor, frequency and rate-of-change-of-frequency estimates under various static and dynamic conditions, and tested via simulation. Finally, a synchronized phasor measurement algorithm incorporated into the initial signal processing is proposed. This algorithm has been tested following the latest version of the phasor measurement unit standard and the results obtained comply with the measurement requirements.

Este trabalho traz uma proposta de atividades, a serem desenvolvidas em sala de aula, com o objetivo de introduzir o conceito de derivadas para os alunos da primeira série do Ensino Médio. Antes das atividades, estão presentes algumas breves pesquisas. O histórico da presença de tópicos do Cálculo Diferencial e Integral no Ensino Médio no Brasil, assim como a análise de alguns livros didáticos, serve para mostrar como o assunto já foi e está sendo tratado no país. Também são exibidos aspectos sobre o Ensino Médio na Alemanha e nos Estados Unidos, países onde o cálculo está presente na Escola Secundária, embora de formas bastante diferentes. Um capítulo sobre a preparação adequada para as aulas também foi incluído, uma vez que a simples inserção da derivada poderia causar problemas de tempo para o cumprimento do cronograma e não trazer os resultados esperados. São necessários algum grau de adequação dos conteúdos ministrados e de cooperação com professores de Física. As atividades visando o ensino dos conceitos iniciais de derivada são motivadas por um problema físico de movimento. O foco é dado na intuição e na visualização de gráficos, para que haja uma melhor compreensão dos conceitos envolvidos. A utilização de um software de geometria dinâmica é requerida em boa parte do tempo, como importante ferramenta de apoio pedagógico / This paper presents a proposal of activities to be developed in the classroom, with the goal of introducing the concept of derivative for students in the first grade of secondary school. Before the activities, some brief researches are presented. The historical presence of the topics of Differential and Integral Calculus in brazilian High Schools, as well as the analysis of some textbooks, serves to show how it has been and is being treated in the country. Aspects of the High School are also shown in Germany and the United States, countries where the calculus is present in High School, though in quite different ways. A chapter about the proper preparation for these classes was also included, since the simple insertion of the derivative could cause problems for meeting the schedule and could not bring the expected results. Some degree of adequacy of the contents and cooperation with Physics teachers are needed. The activities aiming at teaching the initial concepts of derivatives are motivated by a physical problem of motion. The focus is given on intuition and visualization of graphs, so there is a better understanding of the concepts involved. The use of a dynamic geometry software is required for much of the time, as an important tool for pedagogical support

La presente investigación de carácter cuantitativo tiene el propósito de demostrar que el uso de la Metodología de Valor Ganado (Earned Value Management - EVM) para controlar los costos en una obra influye en la optimización de la rentabilidad de un proyecto inmobiliario; en este caso en la ejecución de la especialidad de estructuras de la obra: Proyecto Multifamiliar Los Fresnos. Para ello, se determinó el desempeño de la obra mediante los índices propuestos por el EVM, para conocer el estado del proyecto; se calculó el rendimiento y las variaciones de costo y tiempo de la ejecución de obra para estimar el rendimiento al cierre de la obra; y por último, se tomó acciones en obra para optimizar la gestión de costos. La investigación se aborda de acuerdo al tipo de estudio correlacional - transversal, ya que se midió el desempeño de costo y tiempo de la ejecución de la obra usando los índices de Rendimiento y Variación que ofrece la teoría del EVM tabulados en formatos de tablas y gráficos (validado por las ecuaciones propuestas en la Guía del PMBOK ® 5ta versión del PMI®) determinando el estado económico, la fidelidad del cronograma en el que se encontraba la obra para determinados periodos y la proyección de cómo terminaría la ejecución de la especialidad de estructuras, luego se tomaron acciones en función de los resultados obtenidos. Los resultados finales para el último periodo de aplicación del EVM en la especialidad de estructuras fue de un ahorro de hasta S/. 60 mil nuevos soles con un ligero retraso del 3% con respecto al cronograma planificado. Concluyéndose que la aplicación de la Metodología del Valor Ganado resultó efectiva para controlar las desviaciones que presentó la obra Los Fresnos, medir el progreso de la obra y actuar de forma correctiva en esta investigación. This quantitative research aims to demonstrate that the use of the Earned Value Management (EVM) to control costs in a job influences the optimization of the performance of a real estate project; in this case the execution of the specialty structures of the job: Multifamily Project Los Fresnos. To do this, the performance of the job was determined by the rates proposed by the EVM, to find out the status of the project; calculations were taken of the performance and changes in cost and time of the execution of work to estimate the performance at the end of the job; and finally, action was taken on the job to optimize cost management. The research is discussed according to the type of correlational study - transversely, as the performance of cost and time of execution of the job was measured using indexes of performance and variation offered by the theory of EVM tabulated formats charts and graphs (validated by the equations proposed in the PMBOK ® Guide 5th Version PMI®) determining the economic status, the fidelity of schedule in which the work was at for certain periods and projecting how to end the execution of the specialty structures, then actions were taken depending on the results obtained. The final results for the latest period of EVM implementation specializing in structures were savings of up 60,000 nuevos soles with a slight delay of 3% compared to the planned schedule. Concluding that the application of Earned Value Management was effective in controlling the deviations that the Los Fresnos job presented, measuring the progress of the job, and acting in the correct manner for this investigation.