COURS de Analyse MIP S1

L’objectif de ce module est de familiariser les étudiants avec le vocabulaire ensembliste, de donner des différentes méthodes de convergence des suites réelles et les différents aspects de l’analyse des fonctions d’une variable réelle.

COURS Les nombres réels COURS Les suites COURS les fonctions COURS Développements limités

PROGRAMME
Chp. 1- Les nombres réels.

L’ensemble des nombres rationnels Q
Propriétés de R
Densité de Q dans R
Borne supérieure

Relations d’ordre sur IR. Valeur absolue. Axiomes de la borne supérieure. Caractérisation de la borne supérieure. Borne inférieure. Caractérisation de la borne inférieure. Conséquences : la partie entière d’un nombre réel, densité des nombres rationnels et irrationnels.

Chp. 2- Suites de nombres réels.

Définitions
Limites
Exemples remarquables
Théorème de convergence
Suites récurrentes

Suites majorées, minorées, bornées. Suites convergentes. Suites adjacentes. Théorème des segments emboités. Sous-suites d’une suite. Propriétés des sous-suites. Théorème de Bolzano-Wierstrass. Notion de Suites de Cauchy

Chp. 3- Fonction d’une variable réelle

Notions de fonction
Limites Continuité en un point
Continuité sur un intervalle
Fonctions monotones et bijections
Logarithme et exponentielle
Fonctions circulaires
inverses Fonctions
hyperboliques et hyperboliques inverses
Dérivée
Calcul des dérivées
Extremum local,
théorème de Rolle
Théorème des accroissements finis

Chp. 4- Formule de Taylor, développements limités et applications

Formules de Taylor
Développements limités au voisinage d’un point
Opérations sur les développements limités
Applications des développements limités

COURS de Analyse MIP S1

Chapitre : Les nombres réels
Chapitre : Les suites
Chapitre : Fonctions réelles

Limites et fonctions continues
Dérivée d’une fonction

Chapitre : Développements limités

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