


Politecnico di Torino  
Academic Year 2007/08  
18ACFBK Mathematical analysis I 

1st degree and Bachelorlevel of the Bologna process in Automotive Engineering  Torino 





Objectives of the course
This course is a link between high school and university. Our students come from different types of high schools, so we aim to
homogenize the basic knowledge of students within a short time (almost two weeks). The basics, like inequalities, elementary graphs, trigonometry, are studied in depth and under a more general perspective. The aim of this course is to train students at the university system, and to teach them the basis of differential and integral calculus for variable functions, with applications on first order ordinary differential equations and on second order linear equations. Complex numbers are studied as well and they are applied to represent the solutions of second order ordinary differential equations. To representa ordinary differential equations it is necessary to know the concept of a partial derivative and the fondamental concepts on functions continuity of many variables and on partial derivatives. Many applications of differential and integral calculus are s hown. 
Expected skills
Skills to understand a logic chain; comprehension of basic properties of differential and integral calculus for a variable function.
Acquisition of good skills in calculation. 
Prerequisites
A good knowledge of basic algebra, basic geometry, trigonometry, calculus of logarithms studied at the high school is required.

Syllabus
Rudiments of sets.
Numerical sets. Real functions of a real variable: characteristics and elementary funtions. Limits: definition and basic limits. Continuity and global properties of continuous functions in an interval. Derivatives and differential: derivatives of elementary functions and derivative rules. Theorems of differential calculus and applications: critical points; global functions properties in an interval. Convex functions. De l'Hospital's theorems. Local functions comparison: infinity, infinitesimals and their classification. Taylor's formula and its applications. Primitives. Integral calculus: integral of a scale function. Integral of a limited function. Basic properties of an integral. Integral function and foundamental theorem of integralc alculus. Indefinite integral and foundamental formula of ain integral calculus. Integration methods and rules. Improper integrals. Basics of first and second order differential equations. Practicals: Practicals are on lectures' themes; they are carried out at the blackboard or at the PC and they are held by professors and carried out by students at their own PCs. Students are divided into two groups. Tutorship: Professros are available to explain the students some topics of the program and of practicals, and to check, in an informal way, the learning of the students. Students are divided into two groups. Didactic Material Didactic material referring to the she subjects studies during the lectures: Bacciotti A., Il calcolo differenziale e integrale, Celid Ed. C.Canuto, A.Tabacco, Analisi Matematica I, Springer Ed. Students will also have access to the execerises proposed and completed, as well as the exam papers, from the previous academic year. Supplementary texts available at the library: Apostol T. Calcolo. Volume I,Bollati Boringhieri Bacciotti A., Ricci F., Analisi Matematica I, Liguori Ed. Boieri P., Chiti G., Precorso di Matematica, Zanichelli Pandolfi L., Analisi Matematica I, Bollati Boringhieri Conti, Calcolo: Teoria e applicazioni, McGraw Hill Giublesi D., Tabacco A., Analisi matematica I. Raccolta di temi svolti, CLUT Exam Procedure The exam consists of a written paper and by an oral test. The consultation of texts is not permitted during the written test. 
