The purpose of this course is to provide solid bases needed by the students of scientific disciplines, to face easily the higher level courses in mathematics, computer science, physics, statistics, economy, finance, etc…Topics covered include: Real numbers ; sequences ; Function of a real variable, continuity, differential, Taylor's formula.
• Real numbers: Addition, multiplication, order, intervals, bounded set, upper-bound and lower-bound.
• Numerical sequences: Definitions, monotony, bounded sequence, notion of subsequence, convergent sequence, divergent sequence, Cauchy sequence, theorems of convergence, geometric and arithmetic sequences, adjacent sequences.
• Function of a real variable: Comparison, Composition, monotonic functions, inverse functions.
• Limit of a function: Definition of the limit in terms of the neighborhood and the sequences, theorems on the limits.
• Continuity: Fundamental Theorems on continuous functions, uniformaly continuous.
• Derivable Functions: Definition of the derivative, Geometric interpretation, derivative function, derivative of composed function, Rolle’s Theorem, Mean value theorem, graph of a function, successive derivatives, Leibniz formula.
• Usual functions: trigonometric functions and their inverses, Hyperbolic functions and their inverses.
• Expansion of a function: Taylor-Young and Mac-Laurin Theorems, definition, operations on the expansion, expansion of usual functions, Applications.