Analysis I functions of one variable and sequences

The aim of this course is to clarify the notions of limits of sequences and functions, to deepen the study of sequences and functions, and to study the notions of continuity and derivability of functions, as well as to introduce the main "usual" functions.

Limits of numerical sequences, upper bound, real numbers

• Definitions of the limit (finite or infinite) of a sequence. Uniqueness of the limit.
• Elementary operations on limits. Limits and inequalities.
• Upper and lower terminals
• Convergence of increasing-major (resp. decreasing-minor) sequences.
• Adjoining suites.
• Properties of the set of real numbers, links to rationals and decimals.

Limits of numerical functions

• Define the limit of a function at a point or at infinity, uniqueness.
• Sequential characterizations. Zoology of boundaries: blunt boundaries, right, left, ...
• Operations on limits. Limits and inequalities. Convergence of increasing-major (resp. decreasing-minor) functions.

Continuity of numerical functions

• Continuity at a point and over an interval. Sequential characterization.
• Operations on continuous functions. Intermediate value theorem and applications, bijection theorem (monotone continuous applications).
• Limits and continuity of standard functions. Comparative growth" limits.
• Boundary reached theorem: a continuous function on a closed bounded interval is bounded and reaches its bounds (admitted).

Derivability

• Rate of increase, derivative, operations on derivatives. Tangent to the graph of a function at a point. Derivability-continuity links.
• Left-hand and right-hand derivatives. Derivatives of standard functions: polynomials, rational fractions, exponentials, logarithms, power and n-th root functions, trigonometric functions, hyperbolic trigonometry.
• Rolle's lemma, finite increase theorem. Applications: links between sign of derivative and monotonicity, justification of tables of variations.
• Study inverse trigonometric functions.

Asymptotes and convexity

• Lines asymptotes to a function graph: vertical asymptotes, oblique asymptotes. Higher-order derivatives, Leibniz formula.
• Introduction to convexity, definition, interpretation in terms of the relative position of the graph and its strings. Characterization by derivative or second derivative.
• Arithmetic-geometric inequality. Relative position of graph in relation to tangents or asymptotes.

The following common functions will be presented: integer power functions and their reciprocals, n-th roots; various logarithms, exponentials and non-integer powers; trigonometric functions: cos, sin, tan, arccos, arcsin, arctan; hyperbolic trigonometric functions ch and sh.

High-school mathematics program (including sequences and functions), and at least a première speciality and a terminale mathematics speciality or complementary mathematics option.

Recommended prerequisites* :

High school mathematics program (including sequences and functions), ideally with a mathematics specialization or even an expert mathematics option.

Hourly volumes* :

            CM : 24 h

            TD: 25.5 h

            TP: 0

            Land: 0