Subject: MATHEMATICS I (A.A. 2023/2024)
Unit Matematica I
Mathematics, Information Technology and Physics (lesson)
in attesa di attivazione da parte del docente
This course essentially covers topics of Calculus. These techniques should enable students to treat properly some problems which are formulated in a mathematical language. These problems primarily involve the study of real functions of a single real variable.
The prerequisites are the knowledge of High School Mathematics, with focus on the following topics.
The main set-operations. The sets of natural numbers, integer numbers, rational numbers, real numbers and their main properties. The algebra of polynomials and rational functions. Plane analytic geometry. Algebraic equations and inequalities. Powers, n-th roots, exponentials and logarithms. Trigonometric functions.
1 CFU (8 hours) Real numbers and their subsets. Open and closed sets.
1 CFU (8 hours) Functions: the main definitions. Real functions of a single real variable: elementary functions and their properties. Numerical sequences.
2 CFU (8 hours) Limits of functions of one variable: the main theorems on limits. Tables of limits. Limits of monotone functions. Divergent functions, functions converging to zero.
Continuous functions of one real variable: definitions and main results.
2 CFU (8 hours) Differential calculus for functions of one real variable: the derivative and its geometric meaning. Differentiation rules. Derivatives of elementary functions. The mean value theorem. The connection between derivatives and monotonicity. Second derivatives, higher order derivatives. Convex functions, concave functions. Qualitative study of functions of one real variable. Zeroes of continuous functions.
1 CFU (8 hours) Integral calculus for functions of one variable. Integral function and primitives. The fundamental theorem. Methods of integration: by parts, substitution, decomposition.
1 CFU (8 hours) Complex numbers and their algebraic representation. Properties and operations with complex numbers. The module of a complex number. Trigonometric representation of a complex number, Euler's formula, powers and roots.
1 CFU (8 hours) Matrices and vectors. Scalar and vector products. Operations with matrices. Systems of linear equations, the Gauss-Jordan elimination method.
The course consists of formal lectures covering the basic theoretical features as well as the general strategies for the solution of meaningful exercises.
The final exam consists of a two-hour written test, followed by an oral interview. In the written test the candidate will be asked to solve correctly (at least 2 exercises out of 4) and with adequate motivations some standard exercises, which will be very similar to those that were presented in the classroom during the Course. The oral interview consists in the discussion of some of the topics that were covered in the lectures. The written test may well form a starting point for the oral discussion. The final score is obtained from averaging the outcomes of the written and oral parts.
- Knowledge and understanding:
through formal lectures the student should develop an understanding of the main topics in Calculus and become acquainted with the formulation of problems in mathematical terms, with a knowledge of the basic instruments required for their treatment.
- Applying knowledge and understanding:
through formal lectures which include exercises the student should be able to apply this knowledge to the solution of standard problems, exercises and questions requiring primarily techniques from Calculus.
- Making judgements:
by attending lectures and by studying the suggested course material, the student should be able
to recognize independently some approaches and solving techniques which are typical of Calculus and thus formulate deductions and logical links.
- Communicating skills:
at the end of the course the student should be
able to describe topics in Calculus in a concise and effective manner, with an appropriate technical language and a correct mathematical formalism.
- Learning skills:
the described activities should enable the student to gather independent learning skills and a methodology for carrying on with the study, even when topics in advanced mathematics should arise.
M. Bertsch, Istituzioni di Matematica, Bollati Boringhieri, 1994