Analysis I Functions of one variable and sequences

The aim of this course is to clarify the concepts of limits of sequences and functions, to deepen the study of sequences and functions, to study the concepts of continuity and differentiability of functions, and 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.
• Basic operations on limits. Limits and inequalities.
• Upper terminal and lower terminal
• Convergence of increasing (resp. decreasing) sequences.
• Adjoining suites.
• Properties of the set of real numbers, links to rational numbers and decimals.

Limits of numerical functions

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

Continuity of digital functions

• Continuity at a point and on an interval. Sequential characterization.
• Operations on continuous functions. Intermediate value theorem and applications, bijection theorem (continuous monotonic applications)
• Limits and continuity of usual functions. Limits by "comparative growth."
• Boundedness theorem: a continuous function on a bounded closed interval is bounded and reaches its bounds (accepted).

Derivability

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

Asymptotes and convexity

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

The following standard functions will be presented: integer power functions and their reciprocals, nth 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 specialization in the junior year and a specialization in mathematics in the senior year or an additional mathematics option.

Recommended prerequisites:

High school mathematics program (particularly sequences and functions), ideally with a specialization in mathematics or even an advanced mathematics option.

Hourly volumes:

            CM: 24 hours

            Tutorials: 25.5 hours

            TP: 0

            Land: 0