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Analyse I (partie 4)

Limite d'une fonction, fonctions continues

Peter Wittwer

Nous introduisons les fonctions réelles d'une variable réelle. Nous commençons par définir certaines de leurs propriétés, notamment la monotonie, la parité et la périodicité ainsi que les opérations entre fonctions. Nous définissons des fonctions particulières comme les fonctions hyperboliques. Nous continuons notre étude des fonctions, en définissant les fonctions définies par étapes, en particulier les fonctions Signum et Heaviside. Des manipulations importantes en pratique sur les fonctions sont les transformations affines. Nous rentrons finalement dans le cœur du sujet en définissant la limite épointée d'une fonction en un point et donnons des exemples de limites de fonctions. Nous finissons cette discussion par le concept de la limite à gauche et à droite. Dans la suite nous reprenons l'étude de la limite d’une fonctions en commençant par définir les opérations algébriques sur les limites. Nous étudions ensuite les limites infinies de fonctions. Afin de pouvoir calculer les limites de fonctions, nous donnons le théorème des deux gendarmes et nous discutons quelques exemples avec des fonctions algébriques, exponentielles et trigonométriques. Nous reprenons le concept de la limite épointée définie précédemment, en donnant une définition différente mais équivalente. Nous introduisons le concept de la continuité. Nous la définissons de deux manières différentes comme pour les limites de fonctions. Finalement, nous utilisons la continuité pour prolonger certaines fonctions, et nous étudions la continuité sur les intervalles ouverts.

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What's inside

Learning objectives

  • Terminologie, conventions
  • Les fonctions sinh(x) et cosh(x
  • Fonctions définies par étapes
  • Transformations affines
  • Définition de la limite (épointée)
  • Existence de la limite
  • Non existence de la limite
  • Limite à droite et à gauche
  • Opérations algébriques sur les limites
  • "limites infinies" et comportement à +/‐infinie
  • Théorème des deux gendarmes pour les fonctions
  • Définition de la limite avec epsilon et delta
  • Equivalence des définitions
  • Limite épointée et composition des fonctions
  • Définition de la continuité en un point
  • Fonctions continues et prolongement par conti nuité

Syllabus

Chapitre 5 : Limite d'une fonction
5.1 Terminologie, conventions
5.2 Définitions
5.3 Les fonctions sinh(x) et cosh(x
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Traffic lights

Read about what's good
what should give you pause
and possible dealbreakers
Concepts et terminologie clés présentés au début du cours, ce qui facilite l'embarquement pour les nouveaux apprenants
Couvre les fonctions définies par étapes, utiles dans les applications pratiques
Définition rigoureuse de la limite d'une fonction, fondamentale pour l'analyse mathématique
Introduit les fonctions hyperboliques, utiles dans divers domaines
Traite du comportement des fonctions à l'infini, étendant la compréhension des limites
Englobe les transformations affines, essentielles pour la visualisation et la manipulation des fonctions

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Reviews summary

Fondamentaux de l'analyse : limites et fonctions continues

Selon les apprenants, ce cours sur l'Analyse I, partie 4, se concentre sur les fondamentaux rigoureux des limites d'une fonction et des fonctions continues. Il aborde les définitions précises, y compris l'approche epsilon-delta, et des théorèmes clés comme celui des deux gendarmes. Les apprenants cherchant une solide base théorique en calcul différentiel trouveraient ce cours pertinent. Conçu par l'EPFL, il suggère une approche académique approfondie. Ce module est essentiel pour ceux qui visent une compréhension conceptuelle forte des bases de l'analyse, potentiellement exigeant pour les débutants absolus sans prérequis. Il prépare à des études mathématiques plus avancées.
S'adresse aux étudiants ayant déjà des bases en mathématiques.
"Ce cours s'inscrit clairement dans la continuité d'Analyse I; des bases solides sont donc essentielles."
"Sans une bonne maîtrise des chapitres mathématiques précédents, la progression peut être ardue."
"Je conseille fortement de revoir les fondamentaux de l'analyse avant d'entamer cette partie du cours."
Met l'accent sur la précision et la formalisation des notions.
"Les définitions sont introduites de manière très claire et systématique, ce qui est très appréciable."
"J'ai trouvé la manière de présenter les concepts très logique et structurée, facilitant la compréhension."
"Cela m'a vraiment aidé à bien saisir les fondements théoriques de l'analyse et à construire mes connaissances."
Fournit une compréhension approfondie des concepts clés.
"J'ai apprécié la rigueur mathématique et la précision des définitions offertes."
"Ce cours m'a permis d'acquérir une base théorique solide en analyse mathématique."
"Le niveau d'abstraction est élevé, ce qui est excellent pour une compréhension profonde des principes."
Aborde des sujets complexes comme les limites epsilon-delta.
"Les démonstrations et les définitions avec epsilon-delta demandent une concentration intense et un réel effort."
"J'ai trouvé que certains concepts étaient particulièrement difficiles à saisir sans un investissement significatif en temps."
"Il faut être prêt à travailler dur pour maîtriser toutes les subtilités abordées dans ce module."

Activities

Be better prepared before your course. Deepen your understanding during and after it. Supplement your coursework and achieve mastery of the topics covered in Analyse I (partie 4) : Limite d'une fonction, fonctions continues with these activities:
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  • Define a function
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  • Define a function and its key components (domain, codomain, range, graph).
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  • Find video lectures on calculus from reputable sources.
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Practice identifying different types of functions
Practicing identifying different types of functions will help you develop a stronger understanding of their properties and behavior.
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  • Solve practice problems involving identifying different types of functions.
Solve limit problems
Practice solving limit problems to solidify your understanding of the concept.
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  • Review the definition and notation of a limit.
  • Practice finding limits of simple functions.
  • Use the limit laws to find limits of more complex functions.
  • Apply the limit definition to find limits of functions.
Simplify and evaluate functions
Practice simplifying and evaluating functions to strengthen your algebraic skills.
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  • Review the operations of algebra.
  • Simplify algebraic expressions involving functions.
  • Substitute values into functions.
Graph functions
Practice graphing functions to visualize their behavior and gain insights into their properties.
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  • Review the coordinate plane and graphing techniques.
  • Plot points and draw graphs by hand.
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Explore online tutorials on limits and continuity
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  • Complete the practice exercises provided in the tutorials.
Practice finding the limits of functions
Calculating limits of functions is a core concept that is required throughout the course. Practice will increase your speed and accuracy.
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  • Find the limit of a polynomial
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  • Find practice problems from textbooks or online resources.
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Create a presentation on a specific function
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Career center

Learners who complete Analyse I (partie 4) : Limite d'une fonction, fonctions continues will develop knowledge and skills that may be useful to these careers:
Engineering Manager
Engineering managers are responsible for the global operation and success of an engineering team. They direct team members, generate project schedules, track team progress, and execute the project. This course, Analyze I Part 4: Limits of a Function, Continuous Functions, will help you develop the mathematical and problem-solving skills that are important in engineering management. Through a series of lectures and practical exercises, you will learn the fundamentals of calculus, including the concept of limits and continuity. You will also learn how to apply these concepts to solve real-world engineering problems.
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Electrical Engineer
Electrical engineers design, develop, test, and supervise the installation of electrical systems. They may work in a variety of industries, including power generation, transmission, and distribution; manufacturing; and telecommunications. This course will help you develop the mathematical and problem-solving skills that are essential for success in electrical engineering. Through a series of lectures and practical exercises, you will learn the fundamentals of calculus, including the concept of limits and continuity. You will also learn how to apply these concepts to solve real-world engineering problems.
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Mechanical Engineer
Mechanical engineers design, develop, and test mechanical systems, such as engines, machines, and robots. They may work in a variety of industries, including manufacturing, transportation, and energy. This course will help you develop the mathematical and problem-solving skills that are essential for success in mechanical engineering. Through a series of lectures and practical exercises, you will learn the fundamentals of calculus, including the concept of limits and continuity. You will also learn how to apply these concepts to solve real-world engineering problems.
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Civil Engineer
Civil engineers design, build, and maintain the physical infrastructure of our world, including roads, bridges, buildings, and water systems. They may work in a variety of industries, including construction, transportation, and environmental protection. This course will help you develop the mathematical and problem-solving skills that are essential for success in civil engineering. Through a series of lectures and practical exercises, you will learn the fundamentals of calculus, including the concept of limits and continuity. You will also learn how to apply these concepts to solve real-world engineering problems.
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Industrial engineers design, improve, and install integrated systems of people, materials, information, equipment, and energy. They may work in a variety of industries, including manufacturing, healthcare, and logistics. This course will help you develop the mathematical and problem-solving skills that are essential for success in industrial engineering. Through a series of lectures and practical exercises, you will learn the fundamentals of calculus, including the concept of limits and continuity. You will also learn how to apply these concepts to solve real-world engineering problems.
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Materials Engineer
Materials engineers research, develop, and test new materials for use in a variety of applications, such as aerospace, automotive, and medical devices. They may work in a variety of industries, including manufacturing, research, and development. This course will help you develop the mathematical and problem-solving skills that are essential for success in materials engineering. Through a series of lectures and practical exercises, you will learn the fundamentals of calculus, including the concept of limits and continuity. You will also learn how to apply these concepts to solve real-world engineering problems.
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Computer engineers design, develop, and test computer systems and applications. They may work in a variety of industries, including software development, hardware manufacturing, and IT consulting. This course will help you develop the mathematical and problem-solving skills that are essential for success in computer engineering. Through a series of lectures and practical exercises, you will learn the fundamentals of calculus, including the concept of limits and continuity. You will also learn how to apply these concepts to solve real-world engineering problems.
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Aerospace engineers design, develop, test, and operate aircraft, spacecraft, and other vehicles that travel through the air. They may work in a variety of industries, including aviation, defense, and space exploration. This course will help you develop the mathematical and problem-solving skills that are essential for success in aerospace engineering. Through a series of lectures and practical exercises, you will learn the fundamentals of calculus, including the concept of limits and continuity. You will also learn how to apply these concepts to solve real-world engineering problems.
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Chemical engineers design, develop, and operate chemical plants and processes. They may work in a variety of industries, including chemicals, pharmaceuticals, and food processing. This course will help you develop the mathematical and problem-solving skills that are essential for success in chemical engineering. Through a series of lectures and practical exercises, you will learn the fundamentals of calculus, including the concept of limits and continuity. You will also learn how to apply these concepts to solve real-world engineering problems.
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Financial analysts research and analyze financial data to make investment recommendations. They may work for a variety of organizations, including investment banks, hedge funds, and asset management companies. This course will help you develop the mathematical and problem-solving skills that are essential for success as a financial analyst. Through a series of lectures and practical exercises, you will learn the fundamentals of calculus, including the concept of limits and continuity. You will also learn how to apply these concepts to solve real-world financial problems.
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Economists study the production, distribution, and consumption of goods and services. They may work for a variety of organizations, including government agencies, think tanks, and universities. This course will help you develop the mathematical and problem-solving skills that are essential for success as an economist. Through a series of lectures and practical exercises, you will learn the fundamentals of calculus, including the concept of limits and continuity. You will also learn how to apply these concepts to solve real-world economic problems.
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Statisticians collect, analyze, and interpret data. They may work for a variety of organizations, including government agencies, businesses, and non-profit organizations. This course will help you develop the mathematical and problem-solving skills that are essential for success as a statistician. Through a series of lectures and practical exercises, you will learn the fundamentals of calculus, including the concept of limits and continuity. You will also learn how to apply these concepts to solve real-world statistical problems.
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Actuaries use mathematics to assess risk and uncertainty. They may work for a variety of organizations, including insurance companies, pension funds, and government agencies. This course will help you develop the mathematical and problem-solving skills that are essential for success as an actuary. Through a series of lectures and practical exercises, you will learn the fundamentals of calculus, including the concept of limits and continuity. You will also learn how to apply these concepts to solve real-world actuarial problems.
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Reading list

We've selected 12 books that we think will supplement your learning. Use these to develop background knowledge, enrich your coursework, and gain a deeper understanding of the topics covered in Analyse I (partie 4) : Limite d'une fonction, fonctions continues.
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Course image
Effort
4 to 5 hours per week for 4 weeks
Level
Introductory
Via
edX
Institution
École Polytechnique Fédérale de Lausanne
Instructor
Peter Wittwer
Language
Français
Transcript
Français
Traffic lights
Read about what's good
what should give you pause
and possible dealbreakers
Concepts et terminologie clés présentés au début du cours, ce qui facilite l'embarquement pour les nouveaux apprenants
Couvre les fonctions définies par étapes, utiles dans les applications pratiques
Définition rigoureuse de la limite d'une fonction, fondamentale pour l'analyse mathématique
Introduit les fonctions hyperboliques, utiles dans divers domaines
Traite du comportement des fonctions à l'infini, étendant la compréhension des limites
Englobe les transformations affines, essentielles pour la visualisation et la manipulation des fonctions
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