PRINCIPLES OF MATHEMATICS
Learning outcomes of the course unit
To provide students with the basic notions of mathematics used for applications, at the same time getting them accustomed to using the specific language correctly and precisely.
Prerequisites
The following subjects are expected to have been acquired during school-university studies:
ELEMENTS OF NAIVE SET THEORY: Sets and their representations. Relationships of equality and inclusion. Operations of union, intersection and difference. Universal and complementary sets
ELEMENTS OF REAL NUMBER THEORY: Language and axioms of real number theory. Natural, rational and irrational numbers and integers. Algebraic and ordinal properties of real numbers.
Operations and ordering of numerical sets. Absolute value.
Powers with natural and integer exponents. Arithmetical radicals. Powers with rational exponent.
Integer, rational and irrational equations and disequations and those with absolute value.
Exponentials and logarithms. Exponential and logarithmic equations and disequations.
Angles and their measurement. Elements of trigonometry. sine, cosine and tangent of an angle. Main trigonometric identities. Trigonometric equations and disequations.
ELEMENTS OF ANALYTICAL GEOMETRY : Geometric representation of real numbers: real line and Cartesian plane.
Distance between two points in R2. Linear functions and straight lines in plane.
Parabola, hyperbole, circumference, ellipse as locus of points:: canonical equations and Cartesian representation.
Course contents summary
CARTESIAN PRODUCT BETWEEN SETS. RELATIONSHIPS AND FUNCTIONS, SAGITTAL GRAPHS AND CARTESIAN GRAPHS. RELATIONSHIPS OVER A SET AND RELATIVE PROPERTIES. RELATIONSHIPS OF EQUIVALENCE AND CLASSES OF EQUVALENCE. ORDER RELATIONSHIPS. INJECTIVE, SURJECTIVE AND BIJECTIVE FUNCTIONS. COMPOSITE FUNCTIONS AND INVERSE FUNCTIONS.
REAL FUNCTIONS OF REAL VARIABLE: SYMBOLOGY AND NOMENCLATURE. GENERAL PROPERTIES: LIMITEDNESS, MONOTONICITY, SYMMETRIES, ABSOLUTE AND RELATIVE EXTREMES, CONVEXITY AND INFLECTIONS. ELEMENTARY FUNCTIONS AND THEIR GRAPHS: CONSTANTS, LINEARS, POWER, TRIGONOMETRIC AND THEIR INVERSE, EXPONENTIAL AND LOGARITHMIC. ALGEBRA OF FUNCTIONS AND GRAPHS BY POINTS. GRAPHIC RESOLUTION OF DISEQUATIONS.
HYPERREAL NUMBERS: INFINIESIMAL, FINITE AND INFINITE NUMBERS. THE HYPERREAL LINE. R* AXIOMS. HYPERREAL CALCULUS. THE RELATIONSHIP OF INFINITE NEARNESS IN R*; STANDARD PART. SIGNIFICANT INDETERMINATE AND HYPERREAL FORMS. THE HYPERREAL PLANE, THE NATURAL EXTENSION OF A SET AND OF A FUNCTION.. POWERS WITH REAL EXPONENT; THEOREM OF THE HYPERRATIONAL.
DERIVATIVE OF A FUNCTION IN ONE POINT AND ITS GEOMETRICAL INTERPRETATION. POINTS OF NON DERIVABILITY, CALCULATING TANGENTS AND SEMITANGENTS. THEOREM OF INFINITESIMAL INCREASE. SUCCESSIVE DERIVATIVES. THE DERIVATIVE FUNCTION; DERIVATIVE OF ELEMENTARY FUNCTIONS AND ALGEBRA OF DERIVABLE FUNCTIONS. DIFFERENTIAL AND ITS GEOMETRIC INTERPRETATION; DIFFERENTIAL NOTATION FOR DERIVATIVES. CRITICAL POINTS OF A FUNCTION AND OTHER CRITICAL SITUATIONS. ASYMPTOTE. THEOREM OF CRITICAL POINTS. LOCAL AND GLOBAL CONTINUITY. PROPERTIES OF CONTINUOUS FUNCTIONS IN [A, B]; PROPERTES OF CONTINUOUS FUNCTIONS IN AN INTERVAL. RELATIONSHIP BETWEEN MONOTONICITY AND FIRST DERIVATIVE, AND BETWEEN CONVEXITY AND SECOND DERIVATIVE. COMPLETE STUDY OF A FUNCTION. PRIMITIVES OF A FUNCTION AND RELATIVE THEOREM. DEFINITION OF AN INDEFINITE INTEGRAL. FUNDAMENTAL INDEFINITE INTEGRALS. INTEGRATION BY DECOMPOSITION INTO SUM, BY PARTS, BY SUBSTITUTION. RIEMANN’S FINITE AND INFINITE SUMS. DEFINITE INTEGRALS.
Recommended readings
Keisler, Elementi di Analisi Matematica, Ed. Piccin, PD
D. Monteverdi, D. Medici, Appunti ed esercizi Ed. Azzali, Parma
Texts for consultation
G. Bachelet, Matematica per Biologi, Ed. Piccin, PD
F. Bellissima, C. Crociani,Matematica di base Ed. Carocci, Rome
Teaching methods
Three written exercises during the course, or a written exam with the possibility of improving one’s grade with an oral exam in the case of a grade between 15 and 17 out of thirty.
