Mathematics

Numbers: Positive integers, integers, rationals, operations and order. Introduction to the real numbers. Sup and Inf. Mathematical induction and combinatorics. Real functions of one real variable: Basic definitions, examples. Operations, composition. Bounded, monotone, symmetric and periodic functions. Inverse function. Basic functions. Limits: definitions and calculus. Indeterminate forms. Asymptotes. Comparison between infinitesimals and infinities. Sequences. Bolzano-Weierstrass Theorem. The Cauchy criterion. Monotone sequences, the Napery’s number. Recurrence sequence. Series. Continuity: Definition and basic properties. Discontinuities. Extreme Values Theorem. Intermediate Values Theorem. Differential calculus: derivatives, differen. Fermat’s Theorem, Rolle’s Theorem, Mean Value Theorem, Cauchy’s Mean Value Theorem, L’Hopital’s rule. Taylor’s Formula. Study of functions. Indefinite integrals: antiderivatives, integration by parts and by substitution. Integration of rational functions. Some other method of integration. Definite integrals: Definition and basic properties. Some classes of integrable functions. Mean Value Theorem for Integration. Fundamental Theorem of Calculus. Some application of the integral calculus. Improper integrals.

Basic knowledge on sets, operations among sets, the algebraic properties of real numbers, the equation of a line, a parabola and their representation in the Cartesian plane. Knowing how to solve inequalities and equations of first and second degree or involving rational functions, knowing how to solve simple systems. Knowing the properties of the powers, those of the logarithm function and the trigonometric functions.

G. Crasta, A. Malusa: Primo corso di Analisi Matematica, con prerequisiti ed esercizi svolti

70% lectures, 30% supervised exercises in class.

Attendance is optional and will not be verified, but it is highly recommended.

The exam aims to evaluate learning through a written test (consisting in solving problems of the same type as those carried out in the exercises and proposed in the exercise sheets available on the e-learning platform) and an oral test (consisting in the presentation of some concepts and results illustrated in class). The e-learning platform will provide detailed information on the topics covered in class and the applications that are expected to be known during the exam. The written test will last about three hours and can be replaced by two intermediate tests, both lasting two hours, the first of which will take place in the middle of the course and the second immediately at the end of the course. To pass the exam it is necessary to obtain a grade of not less than 18/30, obtainable by demonstrating knowledge of the basic calculation concepts and techniques. To achieve a score of 30/30 cum laude, the student must instead demonstrate to have acquired an excellent knowledge of all the topics covered during the course and be able to link them in a logical and coherent way.

Any book on first level analysis

70% lectures, 30% supervised exercises in class.

Numbers: Natural numbers, integers, and rational numbers; operations and ordering. Introduction to real numbers. Supremum and infimum. Induction and combinatorics. Real Functions of a Real Variable: Definition of function, examples. Operations between functions, composition. Bounded functions, monotone functions, symmetries, and periodicity. Injectivity, surjectivity, invertible functions. Basic functions: powers and roots, exponentials and logarithms, trigonometric functions. Limits: Finite limits at finite points. Tools for calculating limits. Infinite limits and/or limits at infinity, indeterminate forms. Asymptotes. Comparison between infinitesimals and between infinities, substitution principles. Sequences and Series: Limits of sequences. Bolzano-Weierstrass theorem. Cauchy criterion. Sequential characterization of limits. Monotone sequences, the number e (Napier's number). Recursively defined sequences. Numerical series. Series with non-negative terms: regularity, comparison test, asymptotic comparison test, root test, and ratio test. Simple and absolute convergence, Leibniz criterion. Continuity: Definition and basic properties. Classification of discontinuities. Weierstrass theorem, intermediate value theorem. Differential Calculus: Definition of derivative and calculation tools. Fermat's theorem, Rolle's theorem, Lagrange's theorem and its direct consequences, Darboux's property of derivatives, Cauchy's theorem and L’Hôpital's rule. Higher-order derivatives and Taylor's formula, convexity. Study of functions. Integration: Antiderivatives, integration by parts and substitution. Integration of rational functions. Other integration methods. Definite integrals. Definition and general properties. Classes of integrable functions. Mean value theorem for integrals. Fundamental theorem of integral calculus. Some applications of integral calculus. Improper integrals.

Arithmetic and Algebra: Operations on numbers, use of powers, roots, and logarithms, literal calculation (algebraic manipulation), polynomials (operations, factorization). Algebraic equations and inequalities of the first and second degree or reducible to them. Systems of first-degree equations. Rational fractional equations and inequalities. Geometry: Measurement and properties of segments and angles. Lines, planes, properties of the main plane and solid figures, and their related lengths, areas, volumes, and surface areas. Analytic Geometry: Cartesian coordinates, the intuitive concept of function, equations of lines, parabolas, circles, ellipses. Graphs and properties of elementary functions. Elements of Trigonometry: Graphs and geometric meaning of sine, cosine, and tangent. Main trigonometric formulas (addition, subtraction, double-angle, half-angle). Trigonometric equations and inequalities. Relationships among elements of a triangle.

E. Giusti. Analisi Matematica 1. Terza Edizione. Bollati Boringhieri

Attendance is optional but recommended

Passing the course requires both a written exam and an oral exam, both of which are mandatory. During the course, there will be in-progress tests, and passing these exempts the student from the written exam. Written exam: exercises similar to those done in class. Oral exam: discussion of some results included in the course syllabus and their application.

Classroom lessons.