# Mathématics (01_XBMAT)

**Coefficient :**7**Hourly Volume:**153h (including 81h supervised)- CTD : 81h supervised (and 13.5h unsupervised)
- Out-of-schedule personal work : 58.5h

### AATs Lists

## Description

Master the basic mathematical tools needed to train ENIB engineers.

- Sets and applications :
- Operations on sets (intersection, union, product sets).
- Direct and reciprocal images of a set by an application.
- Notion of bijective application (bijection and intermediate value theorems).

- Trigonometry
- Common functions and elementary graphical transformations of curves :
- Elementary functions (logarithms, exponentials, powers, trigonometrics).
- Hyperbolic functions.
- Reciprocal circular trigonometric functions (arctan, arcsin and arccos).
- Simple transformations from one curve to another.

- Elementary calculations in the complex field:
- Complex numbers in algebraic and exponential form.
- Modulus and argument.
- Linearization.
- Ninth roots of a complex number.
- Second-degree equations with complex coefficients.

- Global study of a numerical function:
- Limit of a function (fundamental theorems and classical limits).
- Continuity and extension by continuity.
- Derivation (definition, main formulas, geometric interpretation).
- Concavity, convexity, inflection point and search for extrema.

- Integration (high school revision) :
- Notion of primitive and usual primitives
- Integration by parts

- Polynomials and rational fractions :
- Euclidean division, notions of roots (or zeros) of a polynomial, irreducible polynomial
- Decomposition of a rational fraction into simple elements on $\mathbb{R}$ and on $\mathbb{C}$.

- Local study of functions :
- Taylor-Young formula, usual limited developments in 0.
- Analytical and graphical applications.

- Parametric curves
- Discrete probabilities
- Conditional probability, total probability, Bayes formula.
- Discrete real random variables, expectation, variance, independence of two discrete random variables
- Usual laws (Bernouilli, binomial, geometric, Poisson)

## Learning Outcomes AAv (AAv)

AAv1 [heures: 25, B2, B3] (trigonometry): At the end of this course, students will be able to use the relevant trigonometric tools to model an engineering situation (e.g. building height, navigation, astronomy, tidal studies, waves, sounds, etc.), and then solve the problem.

AAv2 [heures: 30, B2, B3] (complex): At the end of this course, each student knows how to write a complex number in algebraic and exponential forms and solve certain equations with complex solutions (second degree equations with complex coefficients or reverting to it, equation reverting to the search for an nth root).

AAv3 [heures: 35, B1, B2, B3] (functions): At the end of this course, each student knows how to implement fundamental analysis techniques concerning the study of numerical functions of the real variable (limits , derivation, basic integration) to solve optimization problems with a real variable in particular. He knows how to apply these techniques to solve simple concrete problems.

AAv4 [heures: 15, B3] (DLs): At the end of this course, each student knows how to use limited expansions (DL) to approximate and locally represent a function.

AAv5 [heures: 12, B1, B2, B3] (parameterized curves): At the end of this course, each student knows how to study a parameterized curve and draw it. He also knows how to configure an elementary curve (segment, circle, etc.). Precisely :

AAv6 [heures: 19, B2, B3] (polynomials): At the end of this course, students will be able to factor certain polynomials and decompose a rational fraction into simple elements in $\mathbb{R}(X)$ or $\mathbb{C}(X)$.

AAv7 [heures: 17, B1, B2, B3] (probabilities): At the end of this course, students will be able to model and solve an elementary problem involving discrete probabilities.

## Assessment methods

The AAVs will be validated by:

- a long continuous assessment
- the average of several short continuous assessments

## Key Words

Complex numbers, trigonometry, usual functions (logarithmic, exponential, powers, trigonometric, reciprocals), parameterized curves, rational fractions, limited expansions, discrete probabilities.

## Prerequisites

The notions of the first and final year mathematics specialty

## Resources

- Les mathématiques en licence : cours et exercices résolus. Tome 1 / Azoulay, Avignant, Auliac
- Introduction à l'analyse : cours & exercices corrigés. Licence 1 mathématiques / Aebischer
- Mathématiques : 170 fiches-méthodes, 560 exercices corrigés, formulaire / EL Kaabouchi
- Site Bibm@th (cours, exercices, quizz) https://www.bibmath.net
- Site Exo7 http://exo7.emath.fr/