Descriptif
Introduction to analysis (MAA102) is an introductory-level mathematical analysis course that provides a well-balanced approach between calculus and foun- dational notions ; it is designed to equip students with the fundamental analytical tools required in all scienti c elds. In particular, this course covers sequences, series and function study. It also introduces students to important mathematical concepts which will be expanded upon later in the program : namely, limits, continuity and derivative.
Objectifs pédagogiques
Introduction to analysis (MAA102) is an introductory-level mathematical analysis course that provides a well-balanced approach between calculus and foundational notions; it is designed to equip students with the fundamental analytical tools required in all scientific fields. In particular, this course covers sequences, series and function study. It also introduces students to important mathematical concepts which will be expanded upon later in the program: namely, limits, continuity and derivative.
Diplôme(s) concerné(s)
Pour les étudiants du diplôme Programmes d'échange internationaux
Proof by induction. Basic notions about sequences. Manipulation of inequalities.
Pour les étudiants du diplôme Bachelor of Science de l'Ecole polytechnique
Proof by induction. Basic notions about sequences. Manipulation of inequalities.
Format des notes
Numérique sur 20Littérale/grade américainPour les étudiants du diplôme Programmes d'échange internationaux
Vos modalités d'acquisition :
Weekly Homework Assignments, written Midterm Exam (2 hour long) and written Final Exam (two hour long).
Le rattrapage est autorisé (Max entre les deux notes écrêté à une note seuil)- Crédits ECTS acquis : 5 ECTS
La note obtenue rentre dans le calcul de votre GPA.
Pour les étudiants du diplôme Bachelor of Science de l'Ecole polytechnique
Vos modalités d'acquisition :
Weekly Homework Assignments, written Midterm Exam (2 hour long) and written Final Exam (two hour long).
Le rattrapage est autorisé (Note de rattrapage conservée écrêtée à une note seuil de 11)- Crédits ECTS acquis : 5 ECTS
La note obtenue rentre dans le calcul de votre GPA.
Programme détaillé
Analysis (MAA 102) is an introductory-level mathematical analysis course which provides a well-balanced approach between foundational notions and calculus. It is designed to equip students with the fundamental analytical tools required to pursue studies in Mathematics and, more generally, in any scientific field (Physics, Mechanics, Economics, Engineering, etc).
The objective is to present fundamental notions and results regarding the set of real and complex numbers, real and complex-valued sequences, real and complex-valued infinite series and functions of one real variable.
With respect to the expected initial knowledge of the students, the Course follows a more systematic approach, providing a few insights on the roots of analysis and proving all important results. Though in the continuity of the students' previous studies in Mathematics, this course may also be a turning point towards more rigor and proofs.
The Course starts with the study of real and complex-valued sequences. This will be the opportunity to introduce mathematical quantifiers, explain how to work with mathematical statements and the rules of logic.
The Course then proceeds with the study of real and complex-valued infinite series. It also covers the analysis of functions of one real variable and in particular, the limit of functions at a point or at infinity, the notion of continuity and differentiability of a function. Finally, the Course culminates with the problem of the approximation of function of one real variable by Taylor series.
- Real numbers, sup and inf.
- Real and complex sequences : limits, comparison methods, infinite limits, subsequences.
- Real-valued functions of a real variable : limits, monotonic functions, continuous functions, global properties of continuous functions. Derivatives of functions.
- Variations of functions, convex functions, usual functions.
- Taylor-Lagrange theorem and applications.
- Real and complex-valued series : convergent and divergent series, series of non-negative numbers, convergence of real and complex-valued series.