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# Cours scientifiques - FMA_1F002_EP : Introduction to Analysis

## Descriptif

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.

In particular, this course covers many aspects of the theory of real valued and complex valued sequences, including the notion of subsequence, of accumulation points of a sequence and the Bolzano-Weierstrass Theorem. Building on this, the course also covers the theory of real valued and complex valued infinite series, including the study of absolutely convergent series and alternating sequences.

Next, the course also adresses the study of real valued continuous functions of one variable, starting from the definition up to global properties of continuous functions such as the Intermediate Value Theorem.

The third part of the course is concerned with the differentiability of real valued functions of one variable including higher differentiability. This leads to the Mean Value Theorem and Inverse Function Theorem. The study of classical functions (trigonometric functions, hyperbolic functions, etc) is also presented.

The course ends with an introduction to the theory of approximation of differentiable functions by polynoms and in particular the Taylor-Lagrange Theorem which provides the Taylor series of many interesting functions.

## 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.

49 heures en présentiel (14 blocs ou créneaux)

100 heures de travail personnel estimé pour l’étudiant.

## Pour les étudiants du diplôme Bachelor of Science de l'Ecole polytechnique

Students are expected to master the Proof by Induction, manipulation of inequalities and basic notions about sequences.

## Pour les étudiants du diplôme Programmes d'échange internationaux

Students are expected to master the Proof by Induction, manipulation of inequalities and basic notions about sequences.

Numérique sur 20

## Pour les étudiants du diplôme Programmes d'échange internationaux

### Vos modalités d'acquisition :

There will be a 2 hour long Midterm Exam graded on a scale from 0 to 20 and a 2 hour long Final Exam graded on a scale from 0 to 20.

The Coursework Grade will take into account the grades obtained for the different homework assignments (10% of the Coursework Grade) and the Midterm Grade (for 90% of the Coursework Grade).

The Final Grade will take into account the Coursework Grade (for at least 33%) and the Final Grade (for at least 33%).

Le rattrapage est autorisé (Max entre les deux notes écrêté à une note seuil)
L'UE est acquise si note finale transposée >= D
• 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 :

There will be a 2 hour long Midterm Exam graded on a scale from 0 to 20 and a 2 hour long Final Exam graded on a scale from 0 to 20.

The Coursework Grade will take into account the grades obtained for the different homework assignments (10% of the Coursework Grade) and the Midterm Grade (for 90% of the Coursework Grade).

The Final Grade will take into account the Coursework Grade (for at least 33%) and the Final Grade (for at least 33%).

Le rattrapage est autorisé (Note de rattrapage conservée écrêtée à une note seuil de 10)
L'UE est acquise si note finale transposée >= D
• 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.