The course aims at providing the students with fundamental knowledge and understanding in Mathematical Analysis. A specific purposes is to let the students develop basic technical expertise, and critical thinking, that are needed in modelling and solving mathematical problems in different settings. Special attention will be paid to help the students develop communication skills necessary for teamwork. The course covers topics and encourages learning ability that are needed, or strongly suggested, to pursue studies in mathematics or in any scientific subject
Prerequisites
Basic algebraic calculus. Trigonometry. Logarithms. Polynomial, rational, irrational, logarithmic, exponential, trigonometric equations and inequalities. Elementary analytic geometry in the plane.
Teaching Methods
Lectures: presentation of the theory described in the course program, with teacher-student direct interaction, to ensure a full understanding of the subject.
Training sessions: training of the students to modelling and solving a wide selection of problems of mathematical analysis.
The training sessions are aimed at:
-- helping the students develop communication skills and apply theoretical notions;
-- improve their independence in evaluation.
Supplementary exercise sheets and samples of final examinations are distributed.
Type of Assessment
Intermediate and final written examination: A selection of exercises is proposed. The problems are designed to assess the ability of the students to apply their skills to problem modelling and solving. In the evaluation, special attention is paid to the correctness of the solution procedure, as well as to the originality and effectiveness of the methods adopted.
Oral examination: A number of questions are posed. The oral examination is designed to evaluate the degree of understanding of the theory presented in the course. In the assessment, special attention is paid to communication ability, critical thinking and appropriate use of mathematical language.
Course program
Real number. Axioms of the system of real numbers. Supremum of a set. Real powers, logarithms.
Sequences of real numbers. Limits of sequences. Fundamental theorems on limits. Monotone sequences and the Nepero number. Cauchy sequences. Limsup and liminf.
Real-valued functions of one real variable. Linking theorem between limits of sequences and limits of functions. Fundamental theorems on limits. Limits from the right and from the left. Monotone functions. Limsup and liminf.
Continuous functions. Continuity of elementary functions. Discontinuity points. Fundamental theorem on continuous functions. Theorems on continuous functions on intervals. Uniformly continuous functions, Lipschitz and Hoelder continuous functions.
Differential calculus. Derivative of a function. Derivatives of elementary functions. Differentiation rules. Fundamental theorems of differential calculus for functions on intervals. De l’Hopital rule. Taylor formula. Convex functions.
Integral calculus. Definition of Riemann integral. Elementary properties of integral. Criteria for the existence of integrals. Fundamental theorem of integral calculus. Indefinite integrals. Integrations by parts and change of variable formula. Integrals of rational functions.
Series. Convergence criteria for series with positive terms. Convergence criteria for series with terms of arbitrary sign.