Analysis

Teacher(s)

Absil Pierre-Antoine; Glineur François;

Language

French

Prerequisites

This course assumes that the students already masters the skills in analysis (functions, derivatives and integrals) as expected at the end of secundary school.

Main themes

• understanding of mathematical tools and techniques based on a rigorous learning of concepts favored by highlighting their practical application,
• careful handling of these tools and techniques in the framework of applications.

• sets (intersection, union, difference)
• Order and equivalence,
• Interval, upper bounds, lower bounds, extremes,
• absolute value, powers and roots

• injective, surjective, bijective functions,
• algebraic operations on functions (including graphic interpretation)
• first order functions,
• exponential, logarithmic and trigonometric functions
• Composition of functions and inverse functions

• conditions to ensure that a limit exists,
• limits to infinity

Continuous functions

• fundamental theorems of continuous functions,

• derivative at a point (including graphical interpretation)
• The Hospital's theorem,
• linear approximation of a function,
• maximum and minimum,
• encreasing of decreasing function (sign study)
• concavity and convexity,
• Taylor's development

• primitive,
• definite integrals (including graphic interpretation)
• undefinite integrals

• notion and calculation of partial derivative
• graphical interpretation of the gradient
• interpretation and calculation of the Hessian matrix 
• Intuitive introduction to the use of the Hessian matrix and gradient for a 2-variable function to determine critical points and their nature
• concept and calculation of double integrals

For this last part, a mainly "tool" approach will be favored.

Learning outcomes

At the end of this learning unit, the student is able to :
1	Given the learning outcomes of the "Bachelor in Copputer science" program, this course contributes to the development, acquisition and evaluation of the following learning outcomes: S1.G1 S2.2 Students completing successfully this course will be able to Model real problems using the concepts of set, function, limit, derivative and integral; Solve real problems using computational techniques for limit, derivative and integral; Reason using correctly the mathematical notations and methods keeping in mind but exceeding a more intuitive understanding of the concepts; Model real problems using functions of 2 variables.

Content

• Sets and numbers
• Real univariate functions
• Limits and continuity
• Derivatives (computation and applications)
• Optimization
• Taylor polynomial
• Integration (computation and applications)
• Differential equations
• Functions of two variables

Teaching methods

Lectures in a large auditorium, supervised exercise (APE) and problem (APP) sessions in small groups, possibly supplemented with writing assignments and online exercises.

Evaluation methods

Assessments are carried out individually in writing, based on the learning outcomes listed above. A test is organized during the first term, shorter tests are organized during some exercise sessions, and a written exam is organized during each session.
For the January session, the final grade is awarded on the basis of the tests (5 points out of 20) and the exam (15 points out of 20). For the other two sessions, the grade is based on the exam only.

Online resources

https://moodle.uclouvain.be/course/view.php?id=2798

Bibliography

Mathématiques pour l'économie (5ème édition) par Knut Sydsæter, Peter Hammond, Arne Strøm et Andrés Carvajal, Pearson, 2020

Teaching materials

• Mathématiques pour l'économie (5ème édition) par Knut Sydsæter, Peter Hammond, Arne Strøm et Andrés Carvajal, Pearson, 2020
• Syllabus APE/APP fourni sur Moodle
• Recueil d'anciens examens fourni sur Moodle

Faculty or entity

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