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Mathematics : Analysis 1

par:
Disponible
Pages 294
Dimensions 16*23,5
Publication 2025-11-19
Éditeur —
Papier : 1 500.00 DA Numérique: 750.00 DA

Easy is a magical word. For many students, Mathematical Analysis is often seen as difficult and abstract. Our main goal is simple: make analysis 1 EASY. We aim to help readers see that clarity and simplicity are not opposites of rigor, but its companions. This book is designed according to the curriculum framework of the first-year LMD license program. It is ideal for students in the common core tracks of Mathematics-Informatics (MI) and Science and Technology (ST), as well as for engineering students in preparatory schools. It can be used by students for revision purposes, but also by instructors as a resource for preparing tutorial sheets, tests, and/or exams. We advise students to read the recaps before solving the exercises. This manuscript consists of six carefully written chapters. Each chapter contains several sections, which include a brief course summary called ”Recap” followed by fully solved exercises with detailed, step-by-step solutions. The final chapter serves as a bonus summary of key formulas and identities that are worth remembering. For writing this book, we have used the L A T EX language. All figures are plotted using the Tikz package. If you have any questions or notice errors, please contact us at: ikram.bensaid@univ-mosta.dz and/or anes.mkh@gmail.com

Afficher la table des matières
I Preface
II Acknowledgment
III Notation
1 - Real numbers
System of numbers
Sigma Σ and Pi Π
Absolute value and the floor function
Expanding sets
The solution set
Principle of Mathematical Induction
Lower and upper bounds
2 - Numerical Sequences
Definition of limits
Monotonic sequences
Evaluation of some limits
Subsequences and Non-existence of some limits
Squeeze Theorem or Sandwich Theorem
Comparison Theorem for Sequences
Adjacent Sequences
Stolz-Cesàro Theorems
Cauchy Sequences
The Monotone Convergence Theorem
3 - Real Functions
Domain of a function
Graph of a function
Parity of a function
Bounded functions
Some properties of hyperbolic functions
Limits using definitions
Computing limits
Squeeze Theorem
Zero-Bounded Limit Theorem
4 - Continuity
Continuity at a Point
Graph of discontinuous functions
Operations on continuous functions
Continuity on an interval
Continuous Extensions
Intermediate Value Theorem
Uniform Continuity
k−Lipschitzian functions
5 - Differentiability
Graph of a non-differentiable function
Differentiability at a point
Operations on differentiable functions
Differentiability on an interval
Rolle’s Theorem
The Mean Value Theorem
Some corollaries of the MVT
L’Hopital’s Rule
6 Taylor Formulas and Power Expansion
The n-th derivatives
Leibniz Rule for derivatives
Tangent line approximation
Taylor and Macclaurin Polynomials
Taylor Expansion
Power expansion
Asymptote equation and its position
7 - Key Formulas
Lower and upper bounds
Sequences
Properties of the Exponential Function
Properties of the Natural Logarithm
Trigonometric Identities
Hyperbolic Identities
Limits
Continuity
Some derivatives
Usual Taylor Expansions
References

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