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# Cours scientifiques - FMA_1S005_EP : Integral and Differential Calculus

## Descriptif

Integrals and differential calculus (MAA105) develop students's skills in crucial analytical tools, in particular integration theory. The approach to integration employed in this course is Riemann's integral, a foundational mathematical theory. This course also introduces students to two important and related topics covered in the Bachelor program: Taylor expansions (a tool for function approximation) and differential equations, which are required to understand basic physical problems (trajectories, populations, etc.)

The first part of this course focusses on the notion of Riemann integral. After introducing the notion of Riemann integrable function, we briefly discuss the basic properties of such functions. Next we present the classical methods for computing integrals (integration by parts, integration by substitution, integration of rational fractions, elementary abelian integrals…).

The second part is dedicated to Taylor expansions. We review Taylor formulas for approximation of functions near a given point, then present the theory of Taylor expansion, giving all the tools for computing them in practice, as well as their direct applications.

The third part is the study of ordinary differential equations, mainly first order linear differential equations and linear systems of ODEs, with a special focus on linear differential equations with constant coefficients.

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

Vous devez avoir validé l'équation suivante : UE FMA_1F002_EP

Numérique sur 20

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

### Vos modalités d'acquisition :

The assessment will consist of two parts: a continuous assessment (coursework) based on short tests and homework assignments, as well as a final 3-hour written exam (closed book).
The final numerical grade will be the maximum between the grade of the final exam and the arithmetic mean of the grade of the coursework and that of the final exam, all of which will be on a scale from 0 to 20.
It will be converted to the official final letter grade according to a table published by the instructor at the beginning of the course.
Everything will be recalled at the begining of the moodle page of the course.

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

• Integration of functions of one real variable, Riemann sums, fundamental theorem of analysis
• Integration by parts, changes of variables, calculus of primitive functions
• Decomposition of rational fractions
• Taylor formula with integral remainder; Taylor-Young formula, Landau notations o and O, computations and applications.
• ODEs.