# MATHEMATICAL ANALYSIS 1

Degree course:
Corso di First cycle degree in MATHEMATICS
Academic year when starting the degree:
2019/2020
Year:
1
Academic year in which the course will be held:
2019/2020
Course type:
Basic compulsory subjects
Credits:
9
Period:
First Semester
Standard lectures hours:
72
Detail of lecture’s hours:
Lesson (72 hours)
Requirements:

Basic knowledge of high school algebra, trigonometry and analytic geometry.

Final Examination:
Orale

The exam is divided into three parts:
-a written test consisting in three to five exercises covering the main topics studied in the course, which will test the ability of students in applying the computational techniques learned in class;
-a second written test covering the theoretical aspects of the course, and consisting in giving statements and proofs of a few theorems seen in class, which will test the understanding of the underlying theory and the ability to reproduce rigorous proofs, possibly of simple statements similar to but different from those seen in class, and to express themselves in correct mathematical language;
- an oral part, which follows immediately the second written test, consisting in the discussion of the two written tests, where the ability of express themselves in correct mathematical language, and to independently recognize the validity of mathematical reasonings will be assessed. Each part will be evaluated with a grade in the range 0 to 30, and the final grade will be the average, if greater than or equal to 18, of the grades of the three parts. Access to the second part of the exam is conditioned to having obtained at least 14/30 in the first part.

Assessment:
Voto Finale

The course aims at introducing students to fundamental methods and techniques of Mathematics, and in particular to the differential and integral calculus of one real variable and to sequences and series. A further goal is to train students in applying analytic techniques to other sciences.
Students will know the fundamental of differential and integral calculus of real functions of one real variable. In particular they will be able to study the qualitative graph of elementary function, to solve elementary integration problems, to determine the convergence of sequences and series; they will be able to state and prove the basics theorems of Analysis. Finally, students will be able to independently recognize the validity of mathematical reasonings, to produce proof of simple theorems similar to those seen in class, and to express themselves in correct mathematical language.

Number sets. Real and complex numbers. Topology of the real line. Limits of sequences and series.. Limits of functions and continuity. Continuous functions and topology. Differentiable functions and their properties. Differentiation rules. Fundamental theorems of the differential calculus: Fermat, Rolle, Lagrange e Cauchy, and their consequences. Taylor formula and second order conditions to determine the nature of a stationary point. Convexity. Study of the graph of a function. Riemann integral: definition of integrable function and properties of the integral. Classes of integrable functions. The fundamental theorem of calculus. Integration methods and computation of definite integrals. Improper integrals.

Number sets. Real and complex numbers. Topology of the real line. Limits of sequences and series.. Limits of functions and continuity. Continuous functions and topology. Differentiable functions and their properties. Differentiation rules. Fundamental theorems of the differential calculus: Fermat, Rolle, Lagrange e Cauchy, and their consequences. Taylor formula and second order conditions to determine the nature of a stationary point. Convexity. Study of the graph of a function. Riemann integral: definition of integrable function and properties of the integral. Classes of integrable functions. The fundamental theorem of calculus. Integration methods and computation of definite integrals. Improper integrals.

Lecture notes available at:
http://www.matapp.unimib.it/~demichele/libro.pdf
E. Giusti, Analisi Matematica 1, Boringhieri
E. Giusti, Complementi di Analisi Matematica 1, Boringhieri
W. Rudin, Principi di Analisi Matematica, McGraw Hill

Frontal lectures: 56 hours. Exercise sessions: 24 hours.

Office hours: by appointment (email the instructor)

## Professors

GALEAZZI MARISTELLA