# Analysis

linfo1111  2021-2022  Louvain-la-Neuve

Analysis
7 credits
45.0 h + 37.5 h
Q1
Teacher(s)
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
The course focuses on
• 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.
For most concepts, applications are selected from the other courses of the computer science program (eg economy).

Sets and Numbers
• sets (intersection, union, difference)
• Order and equivalence,
• Interval, upper bounds, lower bounds, extremes,
• absolute value, powers and roots
Real functions of one variable
• 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
Limits
• conditions to ensure that a limit exists,
• limits to infinity
Continuous functions
• fundamental theorems of continuous functions,
Differentiable 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
Integrals
• primitive,
• definite integrals (including graphic interpretation)
• undefinite integrals
Functions of two variables
• 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.
Aims
 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)
• 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
Evaluations are done individually in writing on the basis of the learning outcomes listed above. A test is organized during the first quadrimester, and a written exam during each session.
For the January session, the grade is awarded on the basis of the test (5 points out of 20) and the exam. During the other two sessions, the grade depends only on the exam.
Online resources
Bibliography
Mathématiques pour l'économie par Knut Sydsæter, Peter Hammond et Arne Strøm, Pearson, 2014
Teaching materials
• Mathématiques pour l'économie par Knut Sydsæter, Peter Hammond et Arne Strøm, Pearson, 2014
Faculty or entity

#### Programmes / formations proposant cette unité d'enseignement (UE)

Title of the programme
Sigle
Credits
Prerequisites
Aims
Bachelor in Computer Science

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