# PRINCIPLES OF MATHEMATICS

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Versione italiana
2022/2023
Teacher
MASSIMILIANO DANIELE ROSINI
Credits
12
Didactic period
Primo Semestre
SSD
MAT/05

#### Training objectives

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Modulo: 55650 - ISTITUZIONI DI MATEMATICA (I PARTE)
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The main knowledge will be:
- basics of propositional logic;
- the concepts of real number and real function of one real variable;
- the concepts of lower extreme (infimum) and upper extreme (supremum) for sets and functions;
- the induction principle and proof techniques;
- the concept of limit for a sequence of real numbers and for a function;
- some methods for the calculation of limits.

The main skills that students should acquire (that is to say, the abilities to apply their knowledge) will be:
- to be able to check the properties of real numbers and determine specific sets of real numbers;
- to be able to determine the lower bound (infimum) and the upper bound (supremum) of sets and functions;
- to know how to apply the induction principle;
- to be able to verify the properties of real functions and recognize the main elementary functions (power, logarithm, exponential, trigonometric, ...);
- to be able to check by definition the limit of numerical sequences and real functions;
- to know how to calculate the limit of numerical sequences and real functions by various methods;
- to be able to carry out operations with complex numbers and use their representation properties;
- to be able to check the continuity properties of real functions of one real variable.

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Modulo: 55651 - ISTITUZIONI DI MATEMATICA (II PARTE)
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The main knowledge provided by the course will be:
- some notions on numeric series;
- the concept of derivability for a real function and the methods for the calculation of derivatives;
- the concept of Riemann integrability for a real function;
- the main methods for the calculation of primitives and definite integrals;
- the concept of complex number, its properties and the operations in the set of complex numbers;
- the concept of ordinary differential equation (ODE);
- some methods for the solution of ODE of the first and second order.

The main skills that students should acquire (that is to say, the abilities to apply their knowledge) will be:
- to be able to recognize the properties of numerical series and real functions;
- to be able to carry out operations with complex numbers and use their representation properties;
- to be able to determine the derivability of real functions of one real variable;
- to know how to compute derivatives of every order of derivable real functions of one real variable;
- to know how to study a real function of one real variable and draw an accurate qualitative plot;
- to be able to determine the Riemann integrability of real functions of one real variable;
- to know how to compute the primitives and the definite integral of Riemann-integrable real functions;
- to be able to solve simple first order ODEs (linear and nonlinear) and second order linear ODEs with constant coefficients.

#### Prerequisites

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Modulo: 55650 - ISTITUZIONI DI MATEMATICA (I PARTE)
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Just some basic knowledge of elementary calculus is needed, which is usually taught in high schools: sets, numbers, algebraic operations, representations in the Cartesian plane, coordinate systems, solution of first and second order algebraic equations, inequalities, radicals, operations with algebraic polynomials (sum, difference, multiplication, quotient), factorization of algebraic polynomials (relevant products, Ruffini rule, relevant polynomials).

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Modulo: 55651 - ISTITUZIONI DI MATEMATICA (II PARTE)
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The knowledge seen in PART I is required: the basics of propositional logic, the concepts of real number and real function of a real variable, the concepts of lower and upper bound for sets and functions, the induction principle and demonstration techniques , the concept of limit of a sequence of real numbers and of a function, techniques to evaluate limits.

#### Course programme

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Modulo: 55650 - ISTITUZIONI DI MATEMATICA (I PARTE)
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0 Basic notions of calculation
1 Real numbers: introduction, properties of real numbers, second degree equations, complex numbers, main subsets of R, subsets of R given by inequalities, upper and lower extremes, further properties of real numbers, principle of induction
2 Functions: definitions and general properties, some elementary functions, power function with natural exponent, exponential function, power function with real exponent, modulo function, integer part function, graphs deductible from that of the function f
3 Trigonometry: introduction, trigonometric identities, inverse trigonometric functions, trigonometric equations, trigonometric inequalities
4 Powers of complex numbers, complex numbers given in trigonometric form, complex numbers in exponential form
5 Limits and continuity of functions: point limit and continuity, properties of limits and continuous functions, some elementary continuous functions, right and left limits, limit to infinity, indeterminate forms
6 Sequences: introduction, limits of sequences, Bolzano-Weierstrass theorem, bridge theorem, calculation of sequence limits, asymptoticity, convergence test

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Modulo: 55651 - ISTITUZIONI DI MATEMATICA (II PARTE)
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7 Series: finite sums, infinite sums, convergence tests
8 Continuous functions in an interval: introduction, applications to the study of the extremes of a set
9 Derivatives: definition and first properties, geometric meaning of the derivative, rules of derivation
10 Applications of derivatives: local maxima and minima of a function, derivatives of higher order, concave and convex functions, study of the graph of a function, De L’Hôpital rules, Taylor's polynomial
11 Riemann integral: general definitions and properties, indefinite integrals, integrals for rational functions, integration by substitution, integration formula by parts, generalized integrals
12 Differential equations: general definitions and properties, linear first order differential equations, separable variable differential equations, second order differential equations, linear, homogeneous and constant coefficients, second order differential equations, linear and constant coefficients
13 Further considerations on series: series and Taylor's polynomial, series and integrals

#### Didactic methods

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Modulo: 55650 - ISTITUZIONI DI MATEMATICA (I PARTE)
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The first part of the course consists of 48 total hours of lessons, roughly divided as follows:

- Basic calculation notions - 4 hours;
- Real numbers - 8 hours;
- Functions - 6 hours;
-Trigonometry - 8 hours;
-Power of complex numbers - 6 hours;
- Limits and continuity of functions - 8 hours;
-Sequences - 8 hours.

This hourly division is indicative and may undergo variations, depending on the average abilities of the students attending the lessons.

The lessons will be held in rpesence alternately by the two teachers throughout the entire duration of the course thus ensuring uniformity of approach and didactic continuity for the entire duration of the course.
The course includes theoretical lessons, accompanied by exercises on the blackboard carried out by the two teachers on all the topics covered. About half of the lesson hours are devoted to examples and exercises.

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Modulo: 55651 - ISTITUZIONI DI MATEMATICA (II PARTE)
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The second part od the course consists of 48 total hours of lessons, roughly divided as follows:

-Series - 6 hours;
-Continuous functions in an interval - 6 hours;
-Derivate - 8 hours;
-Applications of derivatives - 8 hours;
-Riemann integral - 8 hours;
- Differential equations - 8 hours;
-Further considerations on the series - 4 hours.

This hourly division is indicative and may undergo variations, depending on the average abilities of the students attending the lessons.

The lessons will be held in presence alternately by the two teachers throughout the entire duration of the course thus ensuring uniformity of approach and didactic continuity for the entire duration of the course.
The course includes theoretical lessons, accompanied by exercises on the blackboard carried out by the two teachers on all the topics covered. About half of the lesson hours are devoted to examples and exercises.

#### Learning assessment procedures

The verification of learning of the course contents (module 1 + module 2) takes place through an exam consisting of two written tests: one of exercises and one of theory. The student must present himself with an identification document. During the written exercise test it is allowed to use ballpoint pens, pencils, discolors, eraser, ruler, calculator and a handwritten form (1 A4 sheet front / back) to be delivered at the end of the test.

- In the written exercises the student is asked to solve some problems and exercises related to the topics covered during the course. The estimated time for the test is 2 hours. A score out of thirty which varies between 0 and 25 is assigned to the performance of the written exercise. In order to access the written theory, it is necessary to obtain a score of at least 11/30 points in the written exercise. The score of the written exercises is considered valid for the entire duration of the academic year. If the student positively supports several written exercises during the same academic year, the one with the highest score will be considered.

In the case of partial tests, the final grade of the written test is given by the arithmetic average of the marks obtained in the two partial tests. If this average is greater than or equal to 11/30, it will constitute the final grade of the written year. Note that this grade cannot be greater than 25/30. If the average of the partial tests is less than 11/30, then it will be necessary to take one of the written tests of total session exercises. Those who have an average of at least 11/30 partial tests can still participate in the total exercises written during the same academic year: in this case the best grade will be considered.

The final grade of the written exercise, however acquired, remains valid until the last exam session of the academic year in which the course to which the written refers took place.

- In the theory paper, the student will be asked to answer questions about the theory. The estimated time for the second test is 20 minutes. The written theory is assigned a score expressed in thirty which varies between 0 and 7. The test is passed with at least 3 points. A negative result of the theory paper does not cancel the score of the exercise paper, but it will be necessary to take another theory paper.

The final mark, expressed out of thirty, will take into account the sum of the scores obtained in the two written tests.

Passing the exam is proof of having acquired the knowledge and skills specified in the educational objectives of the course.

#### Reference texts

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Modulo: 55650 - ISTITUZIONI DI MATEMATICA (I PARTE)
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Main references:
- U. Massari: Istituzioni di Matematica. Dispense disponibili online sul sito di Ateneo
- M. Amar, A.M. Bersani: "Analisi Matematica I, Esercizi e richiami di teoria", Edizioni La Dotta
- G. Buttazzo, G. Gambini, E. Santi: "Esercizi di Analisi Matematica I", Pitagora Editrice, Bologna
- B. Demidovic: "Esercizi e problemi di Analisi Matematica", Editori Riuniti
- C. Marcelli: "Analisi Matematica 1, Esercizi con richiami di teoria", Pearson
- S. Salsa, A. Squellati: "Esercizi di Matematica", volume 1, Zanichelli

- M. Bramanti, C.D. Pagani, S. Salsa: "Matematica", Zanichelli, 2000.
- F. Rosso, L. Fusi: "Matematica per le lauree triennali" (possibly, but not necessarily, with the volume "Esercizi di Matematica per le lauree triennali"), CEDAM, Trento, 2013;
- P. Marcellini, C. Sbordone: "Analisi Matematica I", Liguori Editore, Napoli, 1998;
- M. Bertsch, R. Dal Passo, L. Giacomelli, "Analisi matematica", seconda edizione, McGraw-Hill Italia, Milano, 2011.

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Modulo: 55651 - ISTITUZIONI DI MATEMATICA (II PARTE)
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Main references:
- U. Massari: Istituzioni di Matematica. Dispense disponibili online sul sito di Ateneo
- M. Amar, A.M. Bersani: "Analisi Matematica I, Esercizi e richiami di teoria", Edizioni La Dotta
- G. Buttazzo, G. Gambini, E. Santi: "Esercizi di Analisi Matematica I", Pitagora Editrice, Bologna
- B. Demidovic: "Esercizi e problemi di Analisi Matematica", Editori Riuniti
- C. Marcelli: "Analisi Matematica 1, Esercizi con richiami di teoria", Pearson
- S. Salsa, A. Squellati: "Esercizi di Matematica", volume 1, Zanichelli