Baccalaureate

I. Maths in Baccalaureate (final High school): Video course

Here are video lessons for mathematics at Final High school in English.  For French lessons, see Baccalauréat.

1 Sequences

PDFVideo
Maths lesson on « numerical sequences » (i.e. sequences of real numbers) for final High school year.

Outline:
1 Definition and properties of a sequence
    Increasing and decreasing sequences
    Bounded sequences
2 Arithmetic sequences
    Definition of an arithmetic sequence
    Sum of terms of an arithmetic sequence
3 Geometric sequences
    Definition of a geometric sequence
    Sum of terms of a geometric sequence
4 Limits of sequences
    Limit of a geometric sequence
    Sandwich theorem
    Limit of monotone bounded sequences
5 Induction principle 

2 Limit of a function

PDFVideo
Maths lesson on « Limit of a function » for final High school year.

Outline:
1 Defining the limit of a function
2 Operations on the limits
    Remove indeterminacy in the limit
    Limit of composition of functions
3 Sandwich theorem
4 Right and left limits
5 Asymptotes
    Horizontal and vertical asymptotes
    Oblique asymptote

3 Continuous function

PDF, Video
Maths lesson on « Limit of a function » for final High school year.

Outline:
1 Definition of continuity
2 Continuity basic properties
3 Intermediate value theorem
4 Continuous extension

4 Derivative function

PDF, Video
Maths lesson on « Derivative function » for final High school year.

Outline:
1 Definition of derivative
2 Derivative properties
    Sum, product and ratio of differentiable functions
    Derivatives to remember
    Composition of two differentiable functions
3 Table of variations
4 Tangent equation
5 Application of the derivative to calculate a limit

5 Logarithm function

PDFVideo
Maths lesson on « Logarithm function » for final High school year.

Outline:
1 Definition of logarithm
2 Properties of logarithm
3 Limits of logarithm
4 Base 10 logarithm

6 Exponential function

PDF, Video
Maths lesson on « Exponential function » for final High school year.

Outline:
1 Definition of exponential function
2 Properties of exponential function
3 Limits of exponential function
4 Relation of exponential function with the logarithm

7 Probability

PDF, Video
Maths lesson on « Logarithm function » for final High school year.

Outline:
1 Definition of a probability
2 Properties of a probability
    Uniform probability (or equiprobable)
    Independent events
3 Conditional probability
    Bayes formula
    Total probability formula
    Tree of probabilities

8 Convex function

PDF, Video
Maths lesson on « Convex function » for final High school year.

Outline
1 Definition of a convex function
2 Characterization of convexity of a function
3 Inflection point

9 Primitive and integral

PDF, Video
Maths lesson on « Primitive and integral » for final High school year.

Outline:
1 Primitive function
    Definition of the primitive
    Properties of the primitive
2 Integral
    Definition of integral
    Properties of the integral
    Integration by parts formula
    Relation of integral to surface
    Relation of integral to mean

10 Random variable

PDF, Video
Maths lesson on « Random variable » for final High school year.

Outline:
1 Definition of a random variable
2 Expectation and variance
    Expectation of a random variable
    Variance of a random variable
    Formula of expectation of a function of a random variable
3 Independence of random variables
4 Bernoulli random variable
5 Binomial random variable
6 Law of large numbers
    Markov and Chebychev inequalities
    Concentration inequality

11 Trigonometric functions

PDF, Video
Maths lesson on « Trigonometric functions » for final High school year.

Outline:
1 Cosine and sine functions
    Definition of cosine and sine functions
    Properties of cosine and sine
2 Remarkable values of cosine and sine
3 Tangent function
    Definition of the tangent function
    Properties of the tangent function
4 Solving equations with trigonometric functions

12 Complex numbers

PDF, Video
Maths lesson on « Complex numbers » for final High school year.

Outline
1 Set of complex numbers
    Definition of complex numbers
    Properties of complex numbers
2 Representations of a complex number
    Geometric representation of a complex number
    Trigonometric form of a complex number
    Exponential form of a complex number
3 Solving equations in C
    Second degree equation
    Equation of type z^n = a
4 Point transformations
    Transformation associated to an application in C
    Usual transformations

13 Differential equations

PDF, Video
Maths lesson on « Differential equations » for final High school year.

Outline:
1 Differential equations
    Definition of a differential equation
    Differential equation y′ = g
2 First order differential equations
    Homogeneous equation y′ = ay
    Equation with constant second member y′ = ay + b
    Equation with general second member y′ = ay + g (x)

14 Vectors in plane and space

PDF
Maths lesson on « Vectors in plane and space » for final High school year.

Outline:
1 Vectors in plane and space
    Construction of vectors
    Linearly independent or dependent vectors
    Direction vector
2 Vector plane
    Base and coordinates in the plane
    Cartesian and parametric equations of a line in the plane
3 Vector space
    Base and coordinates in the space
    Cartesian and parametric equations of a line in the space
4 Orthonormal basis and coordinate system
5  Affine or vector plane and space

15 Scalar product of vectors

PDF
Maths lesson on « Scalar product of vectors » for final High school year.

Outline:
1 Scalar product in the plane and in space
    Scalar product from coordinates
    Properties of the scalar product
    Remarkable identities of the scalar product
    Relation of the scalar product to geometry
2 Normal vector
    Vector normal to a line in the plane
    Vector normal to a plane in space
3 Intersections of lines and planes
    Intersection of two planes
    Intersection of a line and a plane

16 Arithmetic

PDF
Maths lesson on « Arithmetic » for final High school year.

Outline:
1 Divisibility in Z
    Definition and properties of divisibility
    Euclidean division
    Prime numbers
    Congruence
2 Greatest common divisor
    GCD of 2 integers
    Euclid’s theorem and algorithm
3 Coprime numbers
    Identity and Bézout’s theorem
    Gauss theorem
    Bézout-Euclide algorithm

17 Enumeration

PDF
Maths lesson on « Enumeration » for final High school year.

Outline:
1 Enumeration cases
    Lists
    Arrangements
    Combinaisons
    Permutations
    Summary table: order and repetition
2 Properties of binomial coefficients
    Pascal’s triangle formula
    Newton’s binomial formula

18 Matrices

PDF
Maths lesson on « Matrices » for final High school year.

Outline:
1 Introduction to matrices
    Definition of matrices and first properties
    Addition of matrices
    Multiplying a matrix by a number
    Product of matrices
    Inverse of a matrix
2 Power of a square matrix
    Definition and properties of the power of a matrix
    Diagonal matrix
    Diagonalization
3 Linear systems
4 Matrix associated with a linear application

19 Logic and sets

PDF
Maths lesson on « Logic and sets » for final High school year.

Outline:
1 Basics of logic
    Assertions and predicates
    Logical connectors
    Logical quantifiers
2 Basics of set theory
    Sets
    Operations on sets
    Finite sets

20 Isometries of the plane

PDF
Maths lesson on « Isometries of the plane » for final High school year.

Outline:
1 Definition and properties of an isometry
    Definition of an isometry of the plane
    Isometries and scalar product
    Inverse of an isometry
    Characterization of an isometry
2 Composition of isometries
    General properties of isometries composition
    Composition of two orthogonal symmetries
3 Isometries and invariant points
    Isometries having invariant points
    Isometries having no invariant point
    Isometries as compositions of symmetries

21 Displacements and antidisplacements

PDF
Maths lesson on « Displacements and antidisplacements » for final High school year.

Outline:
1 Definition and properties
    Classification of isometries
    Characterization of a displacement or an antidisplacement
2 Displacements
    Angle of a displacement
    Composition of displacements
    Representation of a displacement in C
3 Antidisplacements

22 Similarities

PDF
Maths lesson on « Similarities » for final High school year.

Outline:
1 Introduction to Similarities
    Homotheties and translations
    Definition of similarities
    Direct or indirect similarities
2 Direct similarities
    Characteristic elements of a direct similarity
    Reduced form and representation in C
3 Indirect similarities

Mohamed Kadhem KARRAY

My research activities at Orange aim to evaluate the performance of communication networks, by combining information, queueing theories, stochastic geometry, as well as machine and deep learning. Recently, I prepared video lectures on "Data science: From multivariate statistics to machine and deep learning" available on my YouTube channel. I also teached at Ecole Normale Supérieure, Ecole Polyetechnique, Ecole Centrale Paris, and prepared several mathematical books.