OVERVIEW

Language: Italian

AIMS AND CONTENT

LEARNING OUTCOMES

Introduction to the rigorous treatment of mathematical analysis, while developing at the same time the methods of differential and integral calculus in the context of real functions of a real variable.

PREREQUISITES

Elementary algebra; trigonometry

TEACHING METHODS

Traditional: blackboard.

SYLLABUS/CONTENT

 1. Real numbers. The axioms of ordered fields. Absolute value. Natural and integer numbers. Rational numbers and their geometric representation. Completeness and its consequences. Real numbers and the straight line. Archimedean property. Decimal representations.

2. Functions. Relations, functions, domain, codomain, image and graph of a function. Composition of functions. Invertible functions. Operations on real functions. Monotone functions. Polynomials and rational functions. Trigonometric functions. The exponential function on rational numbers.

3. Limits. Metric and e topological properties of R. Continuity. Operations with continuous functions. Limits and their properties. The algebra of limits. Comparison theorems. Limits of monotone functions. Limits of compositions and change of variables. Sequences and their limits. Sunsequences.  Bolzano-Weierstrass' theorem. Cauchy sequences. Sequences defined by recurrence and their limits. Neper's number e.

4. Global properties of continuous functions. Weierstrass' theorem. Zeroes of continuous functions. Intermediate value theorem.  Continuity and monotonicity. Continuity of the inverse function. Uniform continuity. Heine-Cantor's theorem. The exponential funcion on real numbers.

5. Differential calculus. The derivative: definition and elementary properties. Differentiability and the properties of the differential.  Derivative of compositions and inverse functions. Derivatives of elementary functions. Higher order derivatives. The classical theorems by Rolle, Lagrange and  Cauchy and their consequences. The theorem of de l'Hopital. Local comparison of functions. Vanishing and diverging functions. Taylor's formula. Convexity. Study of monotonicity and convexity by means of first and second derivatives.  Newton's method. Iterative procedures for the solution of equations.

6.The  indefinite integral. Integration techniques.  Integration of elementary functions.  Integration by parts and by substitution.  Integration of rational functions. 

7. The Riemann integral. Definition and properties of the definite integral. Integrability of continuous and monotonic functions. The oriented integral. The integral mean theorem. Relations between derivation and integration: integral functions, the fundamental theorem of calculus and its consequences. Improper integrals. Convergence criteria. 

8. Series.  Geometric and telescopic series. Convergence. Series with non negative terms: comparison, root and ratio criteria; condensation, order and integral tests.  Alternating series and Leibniz' theorem.

9. Differential equations. Separation of variables. Linear first order equations Second order linear equations with constant coefficients.

RECOMMENDED READING/BIBLIOGRAPHY

A.Bacciotti, F.Ricci - Analisi Matematica I - Liguori Editore

M. Baronti, F. De Mari, R. van der Putten, I. Venturi - Calculus Problems, Springer, 2016

Further readings will be posted on the web page (AULAWEB)

TEACHERS AND EXAM BOARD

LESSONS

LESSONS START

The class will start according to the academic calendar.

Class schedule

The timetable for this course is available here: Portale EasyAcademy

EXAMS

EXAM DESCRIPTION

The exam consists of a written test and an oral test.

Students enrolled in the course of study in Physics are not required to study the proofs of "the 6 theorems" described in the file published on Aulaweb.

Students with a certified DSA, disability or other special educational needs are advised to contact the lecturer at the beginning of the course in order to agree on teaching and examination methods that, while respecting the teaching objectives, take into account individual learning methods and provide suitable compensatory tools.

ASSESSMENT METHODS

1. Two intermediate written tests  will be provided during the year. If a student obtains an average mark greater than or equal to 18/30 and if he scores at least 15/30 in both, the average of the two marks counts as a written test. 

2. A written test with a score greater than or equal to 15/30 allows access to the oral test.

3. If a student submits a written test, the intermediate written tests will be considered canceled.

Oral tests. During the oral exam, the commission asks questions about the entire program. In particular, the knowledge of the definitions of the main concepts, and of the statements and proofs of the most important results will be evaluated, and the ability to carry out exercises will be verified.

Exam schedule

FURTHER INFORMATION

Teaching style: in presence.