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  • Target audience

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    These oriented works are intended for first-year Mathematics and Computer Science students.

  • Contact sheet

    • Course Instructor: Dr. Khelifa Daoudi, Institute of Sciences, Nour Bachir El-Bayadh University Center
    • Email: khelifa.daoudi@Gmail.com
    • Address: Nour Bachir El-Bayadh University Center, BP 900, 32000, Algeria.

    • Hall: D10.

    • Reception hour: Sunday, from 9:30 to 11:00 and Monday, from 9:30 to 11:00.
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  • General objectives

    The objective of this subject is to familiarize students with set theory vocabulary, to study the different methods of convergence of real sequences, and to explore various aspects of the analysis of functions of a real variable.

  • Prerequisites

    Mathematics at the level of the 3rd year of secondary education (scientific and technical stream).

  • Description

    This is a series of tutorials that addresses the following topics.

    Chapter I: The Field of Real Numbers

    • ℝ is a commutative field.
    • ℝ is a totally ordered field.
    • Mathematical induction.
    • ℝ is a valued field.
    • Intervals, upper and lower bounds of subsets of ℝ.
    • ℝ is an Archimedean field.
    • Characterization of upper and lower bounds.
    • The floor function.
    • Bounded sets, the extension of ℝ: Completed real line ℝ.
    • Topological properties of ℝ.
    • Open and closed sets.

    Chapter II: The Field of Complex Numbers

    • Algebraic operations on complex numbers.
    • The modulus of a complex number zzz.
    • Geometric representation of a complex number.
    • Trigonometric form of a complex number.
    • Euler's formulas.
    • Exponential form of a complex number.
    • n-th roots of a complex number.

    Chapter III: Sequences of Real Numbers

    • Bounded sequences, convergent sequences, properties of convergent sequences.
    • Arithmetic operations on convergent sequences, extensions to infinite limits.
    • Infinitesimal and infinitely large numbers.
    • Monotonic sequences, subsequences, Cauchy sequences.
    • Generalization of the concept of limit: upper and lower limits.
    • Recurrent sequences.

    Chapter IV: Real-Valued Functions of a Real Variable

    • Graph of a real function of a real variable.
    • Even and odd functions, periodic functions, bounded functions.
    • Monotonic functions, local maximum, local minimum.
    • Limits of a function, theorems on limits, operations on limits.
    • Continuous functions, discontinuities of the first and second kinds.
    • Uniform continuity.
    • Theorems on continuous functions on a closed interval.
    • Continuous inverse function, order of a variable - equivalence (Landau notation).

    Chapter V: Differentiable Functions

    • Right-hand and left-hand derivatives, geometric interpretation of the derivative.
    • Operations on differentiable functions.
    • Differentials - differentiable functions.
    • Fermat's Theorem, Rolle's Theorem, Mean Value Theorem.
    • Higher-order derivatives, Taylor's formula.
    • Local extrema of a function, bounds of a function on an interval.
    • Convexity of a curve, inflection points, asymptotes of a curve.
    • Construction of the graph of a function.

    Chapter VI: Elementary Functions

    • Natural logarithm, natural exponential function, logarithm of any base, power functions.
    • Hyperbolic functions, inverse hyperbolic functions.



  • The details

  • Some references

    1. Brannan, D. A. (2006). A first course in mathematical analysis. Cambridge University Press.
    2. Campuzano, J. C. P. (2019). Complex Analysis.‏
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