MATHEMATICAL ANALYSIS I

Students will acquire the main concepts of mathematical analysis and they will be guided to connect them to concepts learned in other disciplines.

The course has the following objectives:

Knowledge and understanding: students will become familiar with theoretical notions of sets. The set of real numbers and its structure. At this point students will be able to understand the concept of limit and justify the main properties of limits, for a real variable functions. They will learn to recognize the main analytical properties of a function and to study a function. Linear Algebra concepts will be provided.

Applying knowledge and understanding: students will not only learn the individual concepts but they will have to think about structural properties. They can also exercise to use their knowledge. This will be done throughout classroom exercises homework assignment.

Learning skills: students will be led to acquire a method that allows them to recognize what are the necessary prerequisites.

Sets. Introduction. Terminology and symbols. Other symbols. Relations between sets. Boolean operations. Intersection. Union. Difference. Symmetric difference. Complement. The complement of Union. Boolean identity. Equivalences. Implications. De Morgan's law. Examples.

Numerical sets. Introduction. Operators. Natural numbers. Axioms of Real Numbers. Set of Natural Numbers.

Set of integers. Set of rational numbers. The operations on N, Z, Q. Operations on natural numbers. Operations on fractions. N, Z, Q do not satisfy the axiom of completeness. Representation of numerical sets. Representation of integers. Representation of fractional numbers. Intervals on the line. Bounded and unbounded intervals. Maximum and minimum of a numerical set. Upper and lower bounds of a numerical set. Cartesian product.

Matrices. Determinants and inverse matrices. Transposed of a matrix. Linear transformations. Linear equations. Distance between two points. Distance between point and line. Systems of a linear equations. Rouchè-Capelli theorem. Cramer's theorem.

Functions. Graph of a function. Injective, surjective and bijective functions. Functions even, odd, periodic. Monotone functions. Bounded functions. Maximum and minimum for a function. Extreme points. Examples of functions. The number of Nepier "e". Inverse of a function. Operations between functions.

Sequences. Monotone sequences. Limit of a sequence. Accumulation point. Infinitesimal sequneces and infinitely large. Theorems for limits. Subsequence.

Numerical series. Series of not negative terms. Criteria. Leibniz series.

Limits and continuity of functions. Operations on limits. Continuity at the point. Continuity in a set. Properties of continuous functions Points of discontinuity. Monotone functions. Infinitesimal and infinite. Order of the infinitesimal and infinite.

Derivative. Geometric interpretation of the derivative. Derivative rules. Derivatives of composite functions. Examples of derivatives of composite functions. Differential.
Limit functions of indeterminate forms. Hopital's rule - Examples. Asymptotes of increasing and decreasing. Higher order derivatives. Maxima and minima of a function. Absolute maximum and minimum of a function. Waiestrass's Theorem. Concavity. Convexity. Taylor's Formula.

Riemann's Integrales . Properties. Integral functions. Indefinite integrals. Methods of indefinite integration. Applications of the calculation of areas. Improper integrals.

G. Emmanuele - Analisi Matematica 1 - Pitagora Editrice, Bologna.

G. De Marco, C. Mariconda - Esercizi di Analisi 1 - Zanichelli, Bologna.

P. Marcellini, P. Sbordone - Esercitazioni di Matematica I - Liguori Editore, Napoli.