This course is an introduction to some basic facts about real functions, real analysis and linear algebra with applications to finding solutions to linear systems.
Expected learning outcomes
The course is devoted to provide a basic knowledge of the main mathematical tools with specific attention to the concept of mathematical function, of limit, and of differential and integral calculus. Part of the course gives the students the main techniques to solve linear systems in many variables.
Lesson period: First semester
(In case of multiple editions, please check the period, as it may vary)
The course starts with a short recall of some basic facts about set theory, functions and logic. After this the first part of the course, the one which deals with real function with one variable and which covers about three quarters of the lessons, begins. We define the classical notion of domain and limit of a function. We talk about continuity, differentiability and higher regularity of a function. The main theorems related to the study of the graph of a function and those which gives useful techniques for the computations of limits are stated. Finally, we explain the theory of integration according to Riemann and some techniques to compute the primitive function of a given one. In the second part of the course we deal with matrices and vector (products between matrices, determinant and rank). Then we apply these techniques to the study of linear systems in any number (but finite) of variables. Theorems which tell whether a linear system has solutions or not and which tell how many solutions there are, are stated. Other results which enable the students to simplify a linear system and to arrive to its solution are the last topic presented.
Prerequisites for admission
Prior knowledge of some important basic mathematics concept such as: basic equations and inequalities with radicals, exponentials, logarithms and trigonometric functions, graphs of elementary functions.
The course (which is composed of 48 hours) is organised in classes (with both theory and exercises). During class a lot of attention is given to examples. Lessons are always followed by exercises although is not planned an explicit division between theoretical lessons and practical ones. Attendance in mandatory. A student can undergo the exam only if he has participated 70% of the lessons. Nevertheless, attendance to all the lessons is recommended.
Bibliography: Matematica per le scienze, A. Guerraggio, Pearson Matematica e Statistica, M. Abate, McGraw Hill. On the online platform Ariel of the University there are the notes "Matematica Assistita" which covers basically the whole program of the course (with some extra).
Assessement methods and criteria
Exam is a single written test. It consists of practical exercises (in general, one about the qualitative study of the graph of a function, one about integration and one about the study of a linear system, possibily with parameters) and of some questions related to the study of the theory. No partial tests are planned. During the exam it is forbidden to use notes, books, formulary and electronic devices. In addition to the ability to correctly solve the given exercises it will be evaluated also the ability of being able to use theoretical tools and the proficiency with the mathematical language. The mark is given by a number between zero and thirty. A mark of eighteen is the minimum value for passing the exam. The outcome of the exams is announced on the website of the professor and is thereafter recorded online giving to the student the possibility to refuse a mark (for a given time, according to the regulations of the University).