Analysis I: Functions of a Single Variable and Sequences

The purpose of this unit is to clarify the concepts of limits of sequences and functions, to explore sequences and functions in greater depth, to examine the concepts of continuity and differentiability of functions, and to introduce the main “common” 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 Bound and Lower Bound
• Convergence of increasing sequences with an upper bound (and decreasing sequences with a lower bound, respectively).
• Adjoining suites.
• Properties of the set of real numbers; connections 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, on the right, on the left, ...
• Operations on limits. Limits and inequalities. Convergence of increasing functions with an upper bound (and decreasing functions with a lower bound).

Continuity of Numerical Functions

• Continuity at a point and on an interval. Sequential characterization.
• Operations on continuous functions. The Intermediate Value Theorem and its applications; the Bijection Theorem (continuous monotonic functions)
• Limits and Continuity of Common Functions. Limits by "Comparing Rates of Growth."
• Bound and Attained Theorem: A continuous function on a closed, bounded interval is bounded and attains its (given) bounds.

Derivability

• Rate of change, derivative, operations on derivatives. Tangent to the graph of a function at a point. Relationships between differentiability and continuity.
• Left-hand and right-hand derivatives. Derivatives of common functions: polynomials, rational functions, exponential functions, logarithmic functions, power functions, nth-root functions, trigonometric functions, and hyperbolic trigonometric functions.
• Rolle's Lemma, the Finite Increments Theorem. Applications: relationships between the sign of the derivative and monotonicity; justification for 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, and interpretation in terms of the relative positions of the graph and its chords. Characterization using the derivative or the second derivative.
• Arithmetic-geometric inequality. Relative position of the graph with respect to tangents or asymptotes.

The following common functions will be covered: integer powers and their reciprocals, nth roots; various logarithms, exponential functions, and non-integer powers; trigonometric functions: cos, sin, tan, arccos, arcsin, arctan; and hyperbolic trigonometric functions ch and sh.

High school mathematics curriculum (particularly sequences and functions), and at a minimum, the mathematics specialization in 11th grade and the mathematics specialization in 12th grade, or the supplementary mathematics elective.

Recommended prerequisites*:

High school mathematics curriculum (particularly sequences and functions), ideally the mathematics track, or even the advanced mathematics track.

Hourly volumes*:

            CM: 24 hours

            TD: 25.5 hours

            TP: 0

            Land: 0