Mathematical analysis 1 - mod.1 (2008/2009)

Learning outcomes

This course is devoted to the presentation of the basic notions of differential and integral calculus for functions of one real variable.
In the first unit, the real numbers are introduced together with the notions and main properties of limits, continuity and derivative.

The numeric sets N, Z, Q. The real line R: completeness axiom.
Maximum and minimum of a subset of R, supremum and infimum.
Real functions of one real variable: domain, codomain, image, graph.
Some simple manipulation of graphs, basic functions and their graphs. Trigonometric functions, inverse trigonometric functions, exponential and logarithmic functions.

Limits: from the naive idea to the rigorous definition. Infinite limits, limits
at infinity.

Sequences and their limits. Sequential characterization of limits of real functions. Limits of increasing sequences. Some fundamental limits.

Continuous functions. Basic theorems on continuous functions.

Slope of a graph at a point: intuitive and rigorous definition
of the derivative of a function. Applications of derivatives.
Derivation of elementary functions and derivation rules.
Convex functions and first/second derivatives of a function.

Written test and oral exam.