30122 - PRECORSO DI MATEMATICA / MATHEMATICS - PREPARATORY COURSE

The aim of the preparatory course in mathematics is to consolidate some topics in pre-undergraduate mathematics, in order to help students begin their university studies with comfort and competence. In university courses, these topics are considered as known and are not repeated. The preparatory course is a blended learning course, that is it is partly online and partly in person. The online part is accessible from the summer that precedes the first year of university studies. The classroom part deals with different topics with respect to the online part, consists of a 15 hours course and it is entirely delivered during the Welcome Week of the first year. It is preferable that students complete the online part before the beginning of the classroom part. The knowledge of the content delivered in both the online and classroom parts are integral in helping students earn high marks on the first exams in mathematics, which are part of their plan of study.

Online part:

• Sets. Operations with sets. Number sets. Representation of number sets on the line.
• Powers with integer exponents. Roots. Powers with rational exponents and with real exponents.
• Polynomial algebra.
• First and second degree equations. Higher degree equations. Fractional equations. Systems of equations.
• Cartesian coordinates in the plane. Straight lines. Parabolas. Other curves.
• Elements of trigonometry.
• First and second degree inequalities. Higher degree inequalities. Fractional inequalities. Systems of inequalities. Equations and inequalities with terms in absolute value.

Classroom part:

• Cartesian product. Relations. Equivalence relations, order relations. Functions. Surjective, injective, bijective functions. Internal operations, external operations. 
• Real functions of one real variable. Composite function. Inverse function. Power functions and their graphs. Exponential functions and their graphs. Introduction to logarithms. Properties of logarithms. Logarithmic functions and their graphs. Trigonometric functions and their graphs. Inverse trigonometric functions and their graphs. Piecewise defined functions. Absolute value function. Transformations of elementary functions. Positive part and negative part of a function. Graphical solution of equations. Bounded functions. Increasing, decreasing, monotonic functions. Global maxima, global minima. 
• Mathematics as an axiomatic system: axioms and theorems, primitive notions and definitions. Basic terminology on theorems. Examples of proofs and proving techniques. Conjectures: proofs and counterexamples. Implication, equivalence. Sufficient condition, necessary condition, necessary and sufficient condition. The negation of a proposition. Proofs by contradiction. Proofs by contrapositive. Proofs by induction.
• Elements of combinatorics. Permutations, combinations. Binomial coefficients, Pascal's triangle, binomial theorem.

• Face-to-face lectures
• Online lectures

    Online lectures: the first part of the preparatory course takes place online, on the Bboard teaching platform.

         Assessment is included in the first year mathematics courses, with the methods used for those courses.

• Online part: all teaching materials are available on the Bboard platform.
• Classroom part: we use the textbook BAI Preparatory Course in Mathematics, Guido Osimo, EGEA (2022), ISBN 978-88-6407-473-3.