MATHEMATICS

The course is part of the tracks "Science" and "Economics" of the Foundation Year, a year-long propaedeutic programme that aims to provide students with the necessary requirements for enrolment at an Italian University. It bridges the gap for those students who do not have the number of years of schooling, or those students who do not have the exams or levels required for the enrolment.

Students at the end of the course will
-know the main theories required to take a first year university mathematics course;
-be able to solve exercises on all the covered topics and to correctly answer to multiple choice questions similar to those proposed in the university admission tests.

Basic knowledge of Mathematics as studied in Secondary Schools.

1. Terms and forms of mathematical language.
— Logic of propositions.
— Logical connectives.
— Logic of propositional functions.
— Quantifiers.
— Definitions.
— Axioms.
— Theorems.
— The summation symbol.
2. Numbers.
— Natural numbers, whole numbers and their properties. Need to extend the set of natural numbers for applications.
— Integer numbers and their properties.
— Rational numbers. Calculations with fractions. Decimal representations and related calculations. Numerical approximations.
— Real numbers and their properties.
3. Powers and logarithms.
— Powers and properties of powers.
— Why we need logarithms.
— Logarithms properties and calculations with logarithms.
— How to use pocket calculators for logarithms and exponentials.
4. Percentages.
5. Sets
— Elements of sets.
— How to write a set.
— Subsets.
— Operations between sets: union, intersection, difference, cartesian product, in particular the set R2.
— Special sets of real numbers and their representation.
6. Elementary algebra.
— Algebraic expressions and corresponding calculations.
— Factoring an algebraic expression. Special products.
— Simplifying algebraic fractions.
7. Functions.
— Definitions and examples. Examples from economics and other sciences.
— Real functions of one real variable.
— Composite and inverse functions.
— Injective, surjective, one to one functions.
— Monotone functions.
— Periodic functions.
— Even and odd functions.
— The graph of a function.
— Shifting graphs. The importance of units while plotting and comparing graphs.
— Graphs and properties of some elementary functions: linear functions, quadratic functions, the function of inverse proportionality, logarithmic and exponential functions.
— The absolute value and calculations with absolute values.
8. Equations and inequalities.
— Linear equations and inequalities in one or two unknowns.
— Systems of linear equations in two unknowns.
— Second degree equations and inequalities in one variable.
— Irrational equations and inequalities.
— Fractional equations and inequalities.
— Equations and inequalities with absolute values.
— Exponential and logarithmic equations and inequalities.
9. Analytic geometry.
— Cartesian coordinates in the plane and space.
— Distance between two points. Midpoint of a segment.
— The line in the cartesian plane and its various equations. The slope of a line.
— The vertical parabola or quadratic function.
— Conics: horizontal parabola, circumference, standard form of the ellipse and hyperbola.
— Intersection points between curves.
10. Basics of trigonometry.
— Angles and their measure: degrees and radians.
— The unit circle and the definition of the trigonometric functions: sine, cosine, tangent, cotangent.
— Trigonometric functions of the most important angles.
— Graphs of the trigonometric functions.
— Trigonometric relations: functions for the sum and difference of two angles, for the doubleangle and the half-angle.
— Right triangles and trigonometric functions.
— Simple equations and inequalities involving trigonometric functions.

Texts in pdf format freely available and detailed in the moodle page of the course.

The final grade will be based on mid-term evaluations, and a final exam. The student’s final grade will consider participation, mid-term evaluations and the final exam, distributed as follows.
a) Participation and attendance: 10%
b) Mid-term evaluations: 70%
c) Final exam: 20%
Both mid-term evaluations and the final exam have a written part with the resolution of exercises and on oral part.

Lectures by the teacher
Discussions
Excercises

written and oral