Mathematics I

Presentation

Programme (detailed contents):

-       Complex numbers : definition of the set C of complex numbers, calculus in C and application to trigonometry and to the study of equations in C.

-       Polynomials and rational functions : Definitions, Euclidian division, link between zeros and factorization in terms of irreductible polynomials, definition of rational functions and their partial fraction decomposition.

-       Ordinary differential Equations : general form of the solutions in terms of the solution to the homogeneous equation plus a specific solution, notion of initial data.

-       Limit of a function. Application to the study of continuity and differentiability for a function at a point or on an interval.

-       Recurrent sequences defined via a map and the fixed point Theorem.

-        Main techniques and results in real analysis : usual functions, classical inequalities, intermediate value Theorem, Rolle’s Theorem and the mean value Theorem.

-        Notion of antiderivative, fundamental theorem of analysis and first integral calculus. Integration by parts.

To these Lectures, additional Exercice classes named «Outils Mathématiques » (and including self-tests exercices on Maple TA)  will enable the students to strenghten their abilities in calculus.

Organisation:

Lectures and exercices classes.

Main difficulties for students:

The requirement of writing clearly and precisely mathematical arguments is sometimes a difficult task for first year students.

Objectives

At the end of this module, the student will have understood and be able to explain (main concepts):

Complex numbers, polynomials and rational functions,

Ordinary differential equations of order 1 and 2 with constant coefficients, limits, continuity and differentiability for functions, recurrent sequences and the fixed point Theorem, fundamental techniques and results in real analysis and elements of integration.

The student will be able to:

Understand the concepts and to apply the main results relative to the previous topics.

Needed prerequisite

High-School science stream

Form of assessment

The evaluation of outcome prior learning is made as a continuous training during the semester. According ot the teaching, the assessment will be different: as a written exam, an oral exam, a record, a written report, peers review...