Computer engineering (Academic Year 2019/2020) - Information and communication technologies engineering (reserved for the students of Helwan University, Cairo, Egypt)

Calculus 1

Credits: 9

Content language:French

Course description

The course provides an introduction to the mathematical analysis and linear algebra. The course starts with the real numbers and the related one-variable real functions by studying limits, and continuity. Then it approach the core of calculus, differentatial and integral theory for one-variable real functions. The aspects of linear algebra are also included in the course: in particular by studying the linear spaces and the theory and calculus of matrices.

Prerequisites

Analytic geometry on the plane. Elementary functions. Algebraic, trigonometric, exponential and logarithmic equations and inequalities.

Objectives

• Calculus of limits; • Differentianting one-variable real functions, in particular elementary real functions; • Study of the behaviour of any one-variable real function; • Calculus of integrals.

Program

• Elementary logic. Sets, relations, functions. Transformations on graphics. Compositions of functions; inverse functions. • Limits and continuity. Calculus of limits. Discontinuities. Asymptotic. Sequences. Landau symbols. Basic results on limits and on global properties of continuous functions. • Derivatives and derivation rules. Second derivatives and convexity. Differential calculus results (Fermat, Rolle, Lagrange, Cauchy, De L’Hopital Theorems). Taylor approximations. • Primitives and definite integrals. Integration rules. Improper integrals. symbols. Basic results on limits and on global properties of continuous functions.

Derivatives and derivation rules. Second derivatives and convexity. Differential calculus results (Fermat, Rolle, Lagrange, Cauchy, De L'Hopital Theorems). Taylor approximations.

Primitives and definite integrals. Integration rules. Improper integrals.

Book

• Advanced Engineering Mathematics, A Jeffrey; Harcourt/Academic Press; 2002; • H Anton; Elementary Linear Algebra, Wiley; 1991; • R. Bartle & D. Sherbert, Introduction to Real Analysis, Wiley, 1982; • R. Haggerty, Fundamentals of Mathematical Analysis, Addison-Wesley, 1992; • Linear Algebra: S Lipschutz, McGraw-Hill • Dolciani, M. et al : Introductory Analysis , Houghton Mifflin , Boston , 1991. • Fouad Rajab: Differential and integral, knowledge house (Dar Al Maarfa), Al Cairo, 1972. • Sadek Bshara: Differential and integral calculus, Agency of Modern Publishing, Alexandrina Egypt 1962.

Exercises

The exercises copy the macro arguments on which the course is structured. However, there are few insights to better understand some little-intuitive arguments. In particular, given the broad class of knowledge and techniques related, contained in systematic qualitative study of a real function of one real variable, it presents a detailed method for how to correctly follow the study. The most common interaction teaching method for interacting with the teacher is the forum where you will find more specific information on the course.

Professor/Tutor responsible for teaching

Domenico Finco

Video professors

Prof. Assem Deif - University of Cairo (Cairo - Egypt)

Prof. Michael Lambrou - University of Crete (Heraklion/Crete - Greece)

List of lessons

• Lesson n. 1: Logique		Abdelilah Dahlane
• Lesson n. 2: Ensembles		Abdelilah Dahlane
• Lesson n. 3: Applications		Abdelilah Dahlane
• Lesson n. 4: Relations Binaires		Abdelilah Dahlane
• Lesson n. 5: Nombres Réels		Abdelilah Dahlane
• Lesson n. 6: Borne Superieure Borne Inferieure		Abdelilah Dahlane
• Lesson n. 7: Suites numériques: Généralités		Abdelilah Dahlane
• Lesson n. 8: Suites: Limites		Abdelilah Dahlane
• Lesson n. 9: Suites: Limites Particulières		Abdelilah Dahlane
• Lesson n. 10: Fonctions numériques: Généralités		Abdelilah Dahlane
• Lesson n. 11: Limites de fonctions numériques		Abdelilah Dahlane
• Lesson n. 12: Comparaisons Locales		Abdelilah Dahlane
• Lesson n. 13: Continuité		Abdelilah Dahlane
• Lesson n. 14: Fonctions usuelles		Abdelilah Dahlane
• Lesson n. 15: Derivation		Abdelilah Dahlane
• Lesson n. 16: Applications de classe Ck		Abdelilah Dahlane
• Lesson n. 17: Applications convexes		Abdelilah Dahlane
• Lesson n. 18: Développements limités		Abdelilah Dahlane
• Lesson n. 19: Intégrales de fonctions en escaliers		Abdelilah Dahlane
• Lesson n. 20: Intégrales de fonctions continues par morceaux		Abdelilah Dahlane
• Lesson n. 21: Primitive et intégrale d'une fonction continue		Abdelilah Dahlane
• Lesson n. 22: Compléments sur le calcul des primitives		Abdelilah Dahlane
• Lesson n. 23: Calcul approché des intégrales		Abdelilah Dahlane
• Lesson n. 24: Intégration de fonctions à valeurs complexes		Abdelilah Dahlane
• Lesson n. 25: Intégrales généralisées		Abdelilah Dahlane
• Lesson n. 26: Les nombres complexes		Youssef el From
• Lesson n. 27: Polynômes		Youssef el From
• Lesson n. 28: Vecteurs		Youssef el From
• Lesson n. 29: Espaces vectoriels		Youssef el From
• Lesson n. 30: Matrices		Youssef el From
• Lesson n. 31: Bases - Dimension		Youssef el From
• Lesson n. 32: Calcul du Rang		Youssef el From
• Lesson n. 33: Déterminants		Youssef el From
• Lesson n. 34: Systèmes d’équations		Youssef el From
• Lesson n. 35: Déterminants: applications		Youssef el From
• Lesson n. 36: Applications linéaires		Youssef el From
• Lesson n. 37: Matrices et applications linéaires		Youssef el From
• Lesson n. 38: Déterminants : Applications aux systèmes linéaires		Youssef el From
• Lesson n. 39: Diagonalisation		Youssef el From
• Lesson n. 40: Produit scalaire		Youssef el From
• Lesson n. 41: Espaces affines espaces euclidiens		Youssef el From
• Lesson n. 42: Droites dans le plan et dans l’espace		Youssef el From
• Lesson n. 43: Le cercle		Youssef el From
• Lesson n. 44: Coniques		Youssef el From
• Lesson n. 45: Ellipse		Youssef el From
• Lesson n. 46: Hyperbole		Youssef el From
• Lesson n. 47: Plans dans l’espace		Youssef el From
• Lesson n. 48: Sphères, cylindres et cônes de révolutions		Youssef el From
• Lesson n. 49: Rotations vectorielles		Youssef el From
• Lesson n. 50: Angles		Youssef el From

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Computer engineering (Academic Year 2019/2020) - Information and communication technologies engineering (reserved for the students of Helwan University, Cairo, Egypt)

Calculus 1