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# Cours scientifiques - MAA102 : 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, Computer Science, etc).

## Objectifs pédagogiques

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.

4 heures en présentiel

4 heures de travail personnel estimé pour l’étudiant.

1/200

Numérique sur 20

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

Le rattrapage est autorisé (Note de rattrapage conservée écrêtée à une note seuil de 9)
L'UE est acquise si Note finale >= 9
• Crédits ECTS acquis : 5 ECTS

La note obtenue rentre dans le calcul de votre GPA.

## Pour les étudiants du diplôme Echanges PEI

Le rattrapage est autorisé (Note de rattrapage conservée écrêtée à une note seuil de 9)
L'UE est acquise si Note finale >= 9
• Crédits ECTS acquis : 5 ECTS

La note obtenue rentre dans le calcul de votre GPA.

### Programme détaillé

Week I - The Set of Real Numbers

• The construction of the set of Natural numbers
• The construction of the set of Rational numbers
• The construction of the set of Real numbers
• Upper bound and least upper bound properties

Week II - The Set of Complex Numbers

• Intervals
• The integer part of a real number
• The -th root of a real number
• The set of complex numbers
• Modulus and argument of a complex number
• The -th root of a complex number

Week III - Sequences

• Limite of sequence
• Comparison tests for sequences
• Monotone sequences

Week IV - Sequences

• Infinite limits
• Subsequences and accumulation points
• The Bolzano-Weierstrass Theorem

Week V - Infinite Series

• Finite and Infinite series
• Convergence Tests

Week VI - Infinite Series

• Convergence tests
• Further tests for convergence
• Absolute convergence of series
• Alternative series

Week VII - MIDTERM

Week VIII - Continuity of a Function of One Variable

• Subsets of
• Limits of functions
• Continuity of a function at a point

Week IX - Continuity of a Function of One Variable

• Continuous functions
• Intermediate Value Theorem
• Uniformly continuous function

Week X - Global Properties of Functions

• Global properties of continuous functions
• Monotone functions
• Periodic functions

Week XI - Differentiability of a Function of One Variable

• Differentiable functions
• The Inverse Function Theorem
• Higher order derivatives

Week XII - Variations of Functions

• Extrema of functions
• Rolle's Theorem and the Mean Value Theorem
• Convex functions

Week XIII - Classical Functions

• Trigonometric functions
• Exponential and logarithm
• Hyperbolic functions

Week XIV - Taylor Expansions

• Taylor-Lagrange's Theorem
• Taylor expansions of classical functions and Taylor series
• Manipulating Taylor expansions
• Applications of Taylor expansions