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

• Definitions of the limit (finite or infinite) of a sequence. Uniqueness of the limit.
• Elementary operations on limits. Limits and inequalities.
• Upper and lower terminal
• Convergence of increasing majorities (resp. decreasing minorities).
• Adjacent suites.
• Properties of the set of real numbers, links to rationals and decimals.

Limits of numerical functions

• Definition of the limit of a function at a point or at infinity, uniqueness.
• Sequential characterizations. Boundary zoology: 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 on an interval. Sequential characterization.
• Operations on continuous functions. Intermediate value theorem and applications, bijection theorem (continuous monotone applications)
• Limits and continuity of usual functions. Limits by "comparative growth".
• Boundary 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. Links derivability-continuity.
• Derivative on the left and on the right. Derivative of usual functions: polynomials, rational fractions, exponentials, logarithm, power and n-th root functions, trigonometric functions, hyperbolic trigonometry.
• Rolle's lemma, finite increase theorem. Applications: links between sign of the derivative and monotonicity, justification of tables of variations.
• Study of 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 the derivative or the second derivative.
• Arithmetic-geometric inequality. Relative position of the graph with respect to tangents or asymptotes.

The following usual functions will be presented: integer power functions and their reciprocals, n-th roots; different 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 first year specialization and a final year specialization in mathematics or additional mathematics option.

Recommended prerequisites*:

High school mathematics program (especially sequences and functions), ideally specializing in mathematics, or even expert mathematics.

Hourly volumes* :

            CM : 24 h

            TD : 25,5 h

            TP : 0

            Land : 0