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.