Teachings
SM/0094  MATHEMATICAL ANALYSIS
Academic Year 2022/2023
Free text for the University
 Professor

MARIA CRISTINA CARRISI (Tit.)
 Period

First Semester
 Teaching style

Convenzionale
 Lingua Insegnamento

ITALIANO
Informazioni aggiuntive
Course  Curriculum  CFU  Length(h) 

[60/61] COMPUTER SCIENCE  [61/00  Ord. 2016] PERCORSO COMUNE  9  72 
Objectives
The course aims to provide the knowledge of abstract concepts and practical methods of differential and integral calculus for functions of one and more variables.
A) Knowledge and understanding
Know the tools of mathematical analysis for the study of functions in one and two real variables and the main methods for the solution of integrals of one real variable functions. Know the concepts of succession and numerical series and the main properties.
B) Ability to apply knowledge and understanding
The student must be able to understand and solve exercises related to the topics of the course and apply the methods studied to simple problems, in order to be able in the future to extend them to other study contexts and / or scientifictechnological applications (e.g., Artificial Intelligence).
C) Autonomy of judgment
Being able to identify the most suitable methodologies for solving problems.
D) Communication skills
The student, at the end of the course, should have acquired a minimum basis of mathematical language that allows him to communicate in a scientifically correct way.
E) Learning skills
Based on the knowledge and logicaldeductive skills acquired during the course, the student will be able to learn new methods and theories to tackle problems in different application contexts.
Theoretical lessons are supported by numerous exercises with the aim of teaching the student how to approach problem solving.
Prerequisites
Get used to the elementary operations between numbers. Be able to convert a ratio into a decimal number and vice versa. Be acquainted with the distinction between numerical sets (natural, integers, rational, real). Knowing, formal proprieties of the operations. ( commutative, associative, distributive ). Be able to calculate and handle expressions including powers. Be able to perform arithmetic operations on polynomials. Be able to simplify and handle rational algebraic expressions even with several variables. Be able to solve polynomial, fractional, irrational, exponential and logarithmic equations and inequalities in one unknown. Familiarity with the geometric proprieties of elementary plane figures is required, as well as the ability to calculate perimeter and area. Knowing Thales and Pythagoras theorems and be able to utilize them for solving elementary geometrical problems.
Contents
Sets. Elements of the axiomatic theory of real numbers. Supremum, infimum, maximum, minimum of numerical sets. The principle of induction. Functions of one real variable. Injective, surjective, and bijective functions. Composite functions, invertible functions, monotone functions. Elementary functions: linear functions, polynomial, absolute value, power function, irrational function, exponential function and logarithmic function, trigonometric functions.
Limits of sequences. Uniqueness, boundedness of the limit. Operations with limits. Indeterminate forms. Compare types of limits. Fundamental limits. Monotone sequences. The number of Neper.
Limits of functions and continuity. Limits for functions and sequences. Continuous functions. Discontinuity. Fundamental theorems for continuous functions. Monotonicity and continuity for functions.
Derivative. The geometrical and the physical meaning of the derivative. Operations with the derivative. The rule of derivation of composite functions and inverse functions. Derivatives of elementary functions.
Higher order derivatives. Applications of the derivatives. Theorems of Fermat, of Rolle, and the mean value theorem. Criteria for monotone functions. Taylor’s theorem.
Indefinite integral (antiderivative): definition, primitive. Immediate integrals and the basic properties. Methods for resolving indefinite integrals.
Riemann integral (definite integral). Definitions and notations. Geometrical and physical meaning of the definite integral. Basic properties of the definite integral. The first mean value theorem for integration. Fundamental theorem of calculus.
Multivariable differential calculus. Limits and continuity. Partial derivatives, gradient, differentiability, tangential plane, directional derivatives. Higher order partial derivatives, the theorem of Schwarz. Hessian matrix. Differentiability of vectorvalued functions. Jacobian matrix. Derivation of composition. Taylor’s theorem for functions of two variables.
Optimisation. Local maxima and minima for multivariable functions. Necessary condition (gradient zero), critical points . The study of maxima and minima via the Hessian matrix for twice differentiable functions of two variables.
Numerical series: convergence, criteria for convergence for series with positive terms. Alternating series. Power series. Taylor series of some elementary functions.
Examples and applications.
Teaching Methods
Teaching will be delivered in person. The lessons can be integrated with audiovisual materials and streaming.
Teacher provides students with course material and exercises to do at home that they will have corrected in the classroom. Didactical methods: interactive lessons with tutorials and individual supervisions. Didactical material at students' disposal: slides and teachers notes, homework exercises and texts available on the platforms recommended by the university. Language utilized: Italian
Verification of learning
The exam consists of a written and an oral test. Both are rated out of thirty and are considered passed with a rating greater than or equal to 18/30. To access the oral exam it is necessary to pass the written exam. If the oral test is not passed, the student must also repeat the written test. For the determination of the final grade, a weight of 40% will be attributed to the written test and 60% to the oral test.
The write test allows to verify the ability to apply the mathematical contents proposed in class to concrete cases as well as the ability to critically analyze the situations proposed to evaluate the truthfulness of statements concerning specific mathematical situations. It consists of a multiple choice test (5 answer options) in which the student must select the correct options (even more than one for each question). The test is delivered to the PC in one of the university's computer labs and lasts 2 hours. Wrong answers or not given do not give penalties. The student can only use white paper and pen. The result is visible immediately after sending the test and after a few hours the correct answers will also be available and will remain visible at least until the date of the oral tests.
Students will have sample tests available to familiarize themselves with the type of test and the delivery environment.
The oral exam takes place about 57 days after the write test and consists of an interview during which the student must demonstrate knowledge of the course contents (definitions of mathematical objects and their properties) and how they are connected, ability to use the process logicaldeductive to reproduce known demonstrations, expression and argumentation skills. Three main questions will be proposed but others can be added, aimed at ensuring that the student has mastered the contents (the number of interventions by the teacher depends on the level of preparation of the student and his or her expository skills). Clarifications on the written test may also be requested, especially in case of inconsistencies between the answers given in the written test or between the two tests The duration of the oral test is on average 30 minutes.
The final grade is assigned following the subsequent Docimological Table:
Insufficient: the student demonstrates that he does not know and / or has not understood the fundamental contents of the course.
1821: The performance of the exercises is affected by significant errors and / or inconsistencies. The oral presentation is unclear; the property of language is limited and knowledge is little more than sufficient.
2224: the student demonstrates that he has sufficiently understood and assimilated the topics. Oral exposure is affected by hesitations and language properties are limited. Knowledge is adequate.
2527: the student knows almost all the topics asked in the tests; demonstrates that he has understood and discreetly assimilated the arguments; possesses a good command of language and clarity in exposition.
2830: the student demonstrates that he has understood and assimilated the topics required for the exam very well; the property of language and clarity of presentation are definitely appropriate.
30 cum laude: the student perfectly knows all the topics asked during the exam; demonstrates that he has understood and assimilated all the topics in depth; has an excellent command of language and exposes the contents very clearly.
The dates of the exams (6 in an academic year) are set for the written test, while the schedule of the oral tests will be communicated after the written test and the analysis of the results (generally this takes place within the day following the written test).
Texts
Theory:
[1] Paolo Marcellini, Carlo Sbordone. Analisi matematica uno. Liguori editore.
[2] M. Bramanti, C.D. Pagani, S. Salsa. Analisi matematica vol.2, Zanichelli.
Exercices:
[3] Paolo Marcellini, Carlo Sbordone. Esercitazioni di Analisi Matematica. 1 volume, parte prima e seconda. Liguori Editore
[4] Paolo Marcellini, Carlo Sbordone. Esercitazioni di Analisi Matematica 2 prima parte e parte seconda. Zanichelli.
[5] Sandro Salsa, Annamaria Squellati: Esercizi di Analisi matematica 1, Zanichelli.
More Information
The material provided by the teacher (slides, exercises, selfassessment tests, etc.) will be made available on elearning.unica.it.
At the end of the course the detailed program will be published with precise indications on the various topics covered and on the demonstrations (cd = with demonstration, sd = without demonstration) required for the oral part of the exam.
Our University supports students with specific learning disabilities or disabilities. More information can be found at: https://www.unica.it/unica/it/ateneo_s01_ss02_sss06_02.page
All students can also take advantage of a psychological counseling service, useful, among other things, to overcome the difficulties of adapting to the new context that may arise especially in the first year of the course. For more information, you can visit the page https://www.unica.it/unica/it/counseling_psicologico.page