27991 - Mathematical Analysis T-1

Learning outcomes

At the end of the course the student will have the fundamental notions of differential and integral calculus for functions of a variable, and the fundamentals of linear algebra in Euclidean space. The classical tools of this course find useful applications in other disciplines

Course contents

Limits and continuity. Definition of succession of convergent and divergent real numbers. The theorems on the limits of successions: uniqueness of the limit, theorem of the two carabinieri. Monotone sequences: definition and their limits. The number of Nepero. Composition of functions, invertible functions. Definition of limit for real functions of a real variable. Definition of continuous function of a real variable. Continuity of the composition of two continuous functions. Weierstrass's theorems, zeros and intermediate values. Limits from the right and from the left. Monotone functions: definition and their limits.

Elementary functions: exponential, circular and their inverse.

Differential calculus. Definition of derivable and derivative function of a function. Geometric meaning. The theorems of the average value. Derivatives of higher order. Taylor's formula with the rest of Peano.

Local extremes: definitions, necessary conditions (Fermat's theorem), sufficient conditions. Convex functions.

Integral calculation: definition of Riemann integral. Properties of the integral: linearity, additivity, monotony, mean theorem. Classes of integrable functions. The fundamental theorems of integral calculus. Substitution integration theorem and Part integration theorem.

Generalized integrals. Definition of generalized integral for unrestricted functions or defined on unrestricted intervals. Definition of convergence of an integral. Basic examples: generalized integral of the functions of type 1/(x-c)^α on limited and not limited intervals. Criterion of comparison and asymptotic comparison. The absolute integrability and the relationship with the simple integrability.

Linear algebra The vector space R^n: operations between vectors. Linear combinations of vectors and linear independence. Vector subspaces and basic definition of subspaces. Euclidean space R ^ n: scalar product of two vectors. Orthogonal vectors. A vector's standard and property. Distance between two points. The vector product.Matrix: determinant, invertible matrices, rank of a matrix. Linear systems: Rouché-Capelli's theorem, Cramer's theorem.

Teaching methods

The course includes theoretical lessons alongside exercises that aim to help the student gain familiarity and mastery with the mathematical tools and methods introduced during the lessons

Assessment methods

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

More specifically, the final written test checks out the acquisition of knowledge and skills expected without the help of notes or books.

The exam is successful for students who will demonstrate mastery and operational skills in relation to the key concepts illustrated in the teaching. A higher score will be given to students who will demonstrate that they have understood and be able to use all the teaching content to solve even complex problems, showing good deductive skills.

Failure to pass the exam may be due to insufficient knowledge of the key concepts, the lack of mastery of the mathematical language.

Teaching tools

Exercises, notes and other online material on the page https://www.unibo.it/sitoweb/francesca.incensi3/didattica and at https://iol.unibo.it/

Office hours

See the website of Francesca Incensi