AIMS AND CONTENT

LEARNING OUTCOMES

The aim of this teaching is to introduce to the rigorous treatment of analysis while developing the methods of differential and integral calculus in the context of real functions of one real variable, with the purpose of acquiring logical rigor, attaining a good command of calculus, and knowing the main proof techniques.

AIMS AND LEARNING OUTCOMES

The expected learning outcomes are that the student knows how to handle the basic tools of analysis and calculus. It is expected that the student has understood proofs and knows how to set up and write demonstrations of simple statements.

TEACHING METHODS

Traditional: blackboard.

SYLLABUS/CONTENT

1. Real numbers. The natural numbers and the integers. Rational numbers and their geometric representation. The axiom of completeness and its consequences. The real straight line. Archimedes of the reals. Decimal alignments.

2. Functions. Relations, functions, domain, codomain, image and graph of functions. Composition of functions. Invertible functions. Operations on real functions. Monotone functions. Polynomials and rational functions. Other elementary functions. Trigonometric functions. The exponential function in the rational body.

3. Limits. Metric and topological properties of R. Definition of continuity. Operations with continuous functions. Limits and their properties. Operations on limits. Theorems of comparison. Limits of monotone functions. Limits of compound functions and changes of variables. Successions and their limits. Subsuccessions. Bolzano-Weierstrass theorem. Cauchy successions. Use of successions in the study of limits. Limits of successions defined by recurrence. Neper's e-number.

4. Global properties of continuous functions. Weierstrass theorem. Theorem of zeros. Theorem of intermediate values. Continuity and monotony. Continuity of the inverse. Uniform continuity. Heine Cantor's theorem. The exponential function in the real body.

5. Differential calculus, The derivative: definition and first properties. Differentiability. Algebraic properties of the differential. Derivation of compound functions and the inverse function. Derivatives of elementary functions. Derivatives of higher order. Rolle's, Lagrange's and Cauchy's theorems and their consequences. De l'Hopital theorem. Local comparison between functions. Infinities and infinitesimals. Taylor's formula. Study of the monotonic and convexity properties of a function through the signs of the derivatives. Convex functions. Newton's method. Iterative methods for solving equations.

6. The indefinite integral. Rules of integration. Integration of some elementary functions. Integration by parts and by substitution. Integration of rational functions.

8. Geometric, telescopic series. Convergence.Numerical series with nonnegative terms: comparison, root and ratio criteria; condensation, order and integral criteria. Series with alternate signs. Leibniz's theorem.

9. (FOR CdL FISICA - 8 hours) Differential equations. Differential equations with separable variables. First-order linear equations. Second-order linear equations with constant coefficients.

9. (FOR CdL MATEMATICA and SMID - 22 hours) Differential calculus of functions of several variables.

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

Exam Board

ERNESTO DE VITO (President)

TOMMASO BRUNO

GIOVANNI ALBERTI (President Substitute)

EMANUELA SASSO (President Substitute)

LESSONS

LESSONS START

See https://corsi.unige.it/corsi/11897/studenti-orario

Class schedule

The timetable for this course is available here: Portale EasyAcademy

EXAMS

EXAM DESCRIPTION

The exam, which is unique for both modules, consists of a written test and an oral test.

Written test: the test will include three/four exercises on the topics covered in the syllabus.

1. Two midterm written tests are held during the year. You may take the second test if you scored 15/30 or higher on the first. If you score 15/30 or higher on the second test and the average of the two scores is 18/30 or higher, you may proceed to the oral test.
2. Five written tests are scheduled during the exam sessions. If you score 17/30 or higher, you may proceed to the oral test.
3. You may sit any of the exam sessions; however, if you submit a written test, any previously submitted written tests are cancelled.

Oral test. During the oral test, the examination committee will ask questions on the entire syllabus, with the exception (for Physics students) of certain proofs, a list of which will be announced during the year. In particular, students' knowledge of the definitions of the main concepts, and of the statements and proofs of the most important results, will be assessed, and their ability to solve exercises will be verified. Five oral exam sessions are scheduled. You may take the oral test in a session different from the one in which you took the written test.

For students with SLD (Specific Learning Disorders), please refer to the "Further Information" section.

ASSESSMENT METHODS

The exam, which is unique for both modules, consists of a written test and an oral test. The written test consists of solving three or four structured exercises with questions of increasing difficulty, related to the topics covered during the theoretical lectures. The types of exercises proposed have been the subject of in-depth study and discussion during the practice sessions.

In the first part of each exercise, the questions aim to verify:

• knowledge of the fundamental concepts of analysis
• mastery of basic calculation techniques.

Correctly completing this section allows the student to achieve a passing grade.

The second part of the questions assesses:

• the ability to analyze critically and independently;
• the application of advanced results in solving the questions.

Successfully completing this section allows the student to obtain the maximum grade.

Oral test

The questions aim to verify the ability to:

1. clearly present the fundamental concepts of the course;
2. rigorously present the statements and proofs of the main theorems;
3. solve any supplementary exercises proposed to verify mastery of calculation techniques.

Final evaluation

The overall assessment:

• takes significant account of the outcome of the written test;
• is supplemented by the outcome of the oral test, which may raise or lower the grade achieved in the written test.

FURTHER INFORMATION

Exemption measures and compensatory tools are intended to enable students to achieve the same learning outcomes as their fellow students, not to make the exam easier. The use of compensatory tools and the application of exemption measures must be authorized in advance by the professor responsible for the course, in agreement with the SLD (Specific Learning Disorders) Officer of the School of Science, Inclusion Officer: Sergio Di Domizio - sergio.didomizio@unige.it

To make use of accommodations during the exam, you must fill out the request form for accommodations available on this page of the online services, which can be accessed using your UniGePass credentials. The request will be automatically sent by the system to the professor responsible for the course, to the Officer of your School/Area/Department, with the Office copied for information; you will receive a copy of the submitted request at your University email address.

The accommodations available to students are as follows:

• Additional time (+30% for SLD)
• Additional time (+50% for disability/impairment)
• Additional time during oral tests to organize the response
• Calculator (programmable and graphing calculators are not permitted)
• Concept maps
• Tables and/or formula sheets
• Taking the exam in written form
• Taking the exam in oral form
• Reader tutor (written tests only)
• Writer/scribe tutor (written tests only)

Requests for accommodations must strictly be submitted at least 7 working days before the scheduled exam date.

Further information at this link.