|SCIENTIFIC DISCIPLINARY SECTOR||MAT/05|
Propedeuticità in uscita
The course "Mathematical Analysis I" aims to provide students with some basic mathematical tools, both theoretical and computational, useful for engineering and application-oriented topics of all the next courses.
The course will be focused on functions of one and several real variables, on the related differential and integral calculus, on the resolution of ordinary differential equations and numerical series.
AIMS AND CONTENT
The course introduces general mathematical notions and tools at the basis of engineering modeling, related to the study of the functions of one or more real variables. In particular, the concept of limit and continuity, the differential and integral calculus, also of functions of several real variables, the resolution of ordinary differential equations, the analysis of curves and surfaces, and the study of the convergence of numerical series.
AIMS AND LEARNING OUTCOMES
The "Mathematical Analysis I" course aims at giving basic mathematical tools necessary to the studies in the engineering field.
At the end of the lessons the student will have acquired sufficient theoretical knowledge:
- to identify, understand, formulate and solve general engineering problems related to mathematically modeled quantitie using appropriate analytical methods;
- to combine notions of theory into practice to solve basic model engineering problems;
- to be able to autonomously learning new mathematical tools useful for engineering applications, throughout the entire working life;
- to analyze and model geometric and physical objects related to functions of one or more real variables, and to calculate quantities associated with them;
- to apply mathematical resolution tools in the context of the differential calculation of the functions of one or more real variables;
- to apply mathematical resolution tools in the context of the integral calculation of the functions of one real variable;
- to compute the maximum and minimum unconstrained values of functions of one and several variables, useful in application areas of optimization;
- to analyze and model geometric objects related to curves, and calculate associated quantities;
- to understand and solve simple models related to ordinary differential equations, through which physical phenomena of engineering interest are represented;
- to know the concept of numerical series and to evaluate their convergence, useful in the approximate calculation of quantities in the numerical-computational field.
Elementary algebra: literal calculus, polynomials, equations and inequalities, trigonometry.
72 hours of theoretical lessons, 48 hours of classroom practices. During the theoretical lessons the definitions and the theorems will be presented with many examples and applications. During the other part of the course many exercises will be solved.
In addition, a tutor will solve some exercises in extra (optional) lesson hours.
The teaching program includes both theoretical study and practical resolution of exercises in the following topics:
- Sets, logic, real numbers, infimum and supremum
- Functions of one real variable, elementary functions, limits, infinitesimals and infinities, continuous functions, derivable functions, differentiable functions. Taylor’s formula, expansion of elementary functions.
- Primitives and indefinite integrals, methods of indefinite integration, definite integrals, fundamental theorem of integral calculus.
- Functions of several variables (scalar and vectorial fields), limits and continuity. Directional derivatives. Differentiable functions. Necessary and sufficient conditions for differentiability. Derivatives of composite functions. Derivatives of higher order, Schwarz Theorem and Taylor polynomial in several variables. Unconstrained maxima and minima of scalar fields, necessary and sufficient conditions, Hessian matrix.
- Differential equations of the first order, with separable variables, linear, homogeneous, Bernoulli and Riccati types. Existence and uniqueness theorem (hints) for the Cauchy’s problem. Linear differential equations. Linear differential equations of higher order with constant coefficients, homogeneous and non-homogeneous,
- Improper integrals of one variable.
- Numerical series. Convergence criteria for constant sign numerical series. Numerical alternating series and absolutely convergent series.
Handouts: "MATHEMATICS I" and "MATHEMATICS II" by prof. Maurizio Romeo, downloadable for free from the web page of the course.
Sheets containing links to web pages with different solved exercises, downloadable for free from the web page of the course.
Workbook: Laura Recine - Maurizio Romeo, Esercizi di analisi matematica - Volume II, Maggioli Editore.
P. Marcellini – C. Sbordone: Calcolo, Liguori Editore, Napoli, or any other good text of mathematical analysis.
M.Baronti – F.De Mari – R.Van Der Putten – I.Venturi: Calculus Problems, Springer
TEACHERS AND EXAM BOARD
CLAUDIO ESTATICO (President)
MARCO BARONTI (President Substitute)
ULDERICO FUGACCI (President Substitute)
L'orario di tutti gli insegnamenti è consultabile all'indirizzo EasyAcademy.
The final exam consists of a written test and an oral exam. The student must obtain an evaluation of at least 16/30 in the written test to access the oral exam.
Students with DSA certification ("specific learning disabilities"), disability or other special educational needs are advised to contact the teacher at the beginning of the course to agree on teaching and examination methods that, in compliance with the teaching objectives, take account of individual learning arrangements and provide appropriate compensatory tools.
The exam consists of a written test and an oral test.
The written test consists in solving exercises concerning the arguments of the course. The written test must be passed before attending the oral examination and can be taken both in previous sessions and in the same session in which the student intends to attend the oral examination.
Only students who have previously passed the written test with a grade greater than or equal to 16/30 can access the oral exam.
Attendance is not compulsory but strongly recommended to all students.