Università Roma Tre

Training objectives of the course
THE COURSE GIVES THE STUDENTS THE METHODOLOGICAL AND CONCEPTUAL TOOLS FOR UNDERSTANDING THE INFORMATION PROVIDED BY THE DEDUCTIVE AND FORMALIZED MATHEMATICAL LANGUAGE. THE AIM IS TO PROVIDE FOUNDATIONS OF MATHEMATICAL ANALYSIS AND PLANE GEOMETRY SUFFICIENT TO UNDERSTAND PHYSICAL-MATHEMATICAL MODELS. MAIN TOPICS ARE: DIFFERENTIAL AND INTEGRAL CALCULUS IN ONE VARIABLE; THE RELATED CONCEPTS, TOOLS, AND MODELLING FOUNDATIONS; LINEAR ALGEBRA FROM A GEOMETRIC POINT OF VIEW; ABSTRACT THEORY AND ITS GEOMETRICAL INTERPRETATION IN EITHER TWO OR THREE DIMENSIONS.

Course programme
THE MAIN EMPHASIS OF THE COURSE IS IN DEVELOPING THE MATHEMATICAL
TOOLS NEEDED BY THE STUDENTS TO SET UP A MATHEMATICAL MODEL OF THE
PRACTICAL PROBLEM THAT AROSE NATURALLY IN THEIR STUDIES/WORK AND SOLVE
THEM. WE DO THIS WITH THE AID OF THE SOFTWARE MATHEMATICA WHICH
PROVIDE THE MEAN FOR ANALYZING COMPUTATIONAL DIFFICULT PROBLEMS AND
VISUALIZE THEM. THE MAIN MATHEMATICAL TOPICS COVERED ARE THOSE OF A
CALCULUS COURSE, WITH AN SPECIAL ATTENTION TO THE CONCEPT OF
APPROXIMATION IN ITS VARIOUS GUISE. WHAT FOLLOWS IS A CONDENSE SUMMER
OF THE TOPICS COVERED: NATURAL AND RATIONAL NUMBERS, CONSTRUCTIONS OF
THE REAL NUMBERS;
NUMERABLE SET, NUMERABILTY OF THE RATIONALS, NON NUMERABILTY OF THE
REALS CANTOR DIAGONAL PROCESSES. ALGEBRA OF LIMITS, EXAMPLES AND THEOREMS.
CONTINUOUS FUNCTIONS, THEOREMS ON CONTINUOUS FUNCTION. THE CONCEPT OF
DERIVATE OF A FUNCTION AND ITS GEOMETRICAL INTERPRETATION. PROPERTIES
OF DIFFERENTIABLE FUNCTIONS. MAX AND MIN OF DIFFERENTIABLE FUNCTIONS,
WORD PROBLEMS. THEOREM ON DIFFERENTIABLE FUNCTIONS (ROLLE, LAGRANGE,
CAUCHY, DE L'HOPITAL).
TAYLOR POLYNOMIALS AND APPROXIMATIONS OF DIFFERENTIABLE FUNCTIONS BY
POLYNOMIALS. RIEMMAN'S INTEGRAL, THE FUNDAMENTAL THEOREM OF CALCULUS.
HOW TO COMPUTE ANTIDERIVATIVES. IMPROPER INTEGRALS.
PARAMETRIC CURVES, PARAMETRIZATION AND LENGHT OF ARCS OF CURVES.

G.B. THOMAS, R.L. FINNEY ELEMENTI DI ANALISI MATEMATICA E GEOMETRIA ANALITICA ED. ZANICHELLI
ROBERT A. ADAMS CALCOLO DIFFERENZIALE IED. CEA (CASA EDITRICE AMBROSIANA)
COURANT, ROBBINS "CHE COS' È LA MATEMATICA?" ED. BORINGHIERI

Quantifiers. Numbers: natural, integers, rational and real. Axioms of real numbers.
Cartesian coordinates in the plane. Points and vectors. Distance: formal definition. Absolute value. Density of Q in R.
Linear algebra: vector sum, scalar product. Matrices. Matrix operations of sum and product, determinant, rank of a matrix.
Matrix representation of linear transformations. Geometric meaning of the determinant.
Rotation matrices and omotethy. Parametric equation of the line. Orthogonality conditions.

Introduction to real functions. Graphs.
Working with graphics, absolute value of a graph. Exponential, logarithm of a function for which you know the plot.
Accumulation points. Limits. Operations with limits. Comparison theorem. Continuous functions. Theorems on continuous functions.

Asymptotes. Derivatives: definition, geometric meaning. Operations: sum, product, quotient, scalar product. Main rules of derivation. Equation of the tangent line at a point to the graph.

Derivative of a composite function and inverse functions. Stationary points.
Fermat's theorem. Theorems of Rolle and Lagrange. Monotony and sign of the first derivative.

Second derivatives, concavity, inflections. Plotting graphs of functions. Theorems of Cauchy and De l'Hopital. Word problems.

Taylor polynomial. Formula of the rest of Lagrange. Hyperbolic functions, conic sections as geometric loci. Classification of conic sections.

Eigenvalues and eigenvectors of symmetric matrices.

Introduction to the problem of calculating the area of a flat region. The fundamental theorem of calculus, definite integrals.
The theorem of the average. Integration by parts and substitution. Integration of rational functions.
Definition of parametric curve. From parametric to cartesian equations and viceversa.
Examples: circumference cycloid, conical. Vector and unit vector tangent vector and the unit vector normal. Length of a curve. Curvatue

G.B. THOMAS, R.L. FINNEY ELEMENTI DI ANALISI MATEMATICA E GEOMETRIA ANALITICA ED. ZANICHELLI
ROBERT A. ADAMS CALCOLO DIFFERENZIALE IED. CEA (CASA EDITRICE AMBROSIANA)
COURANT, ROBBINS "CHE COS' È LA MATEMATICA?" ED. BORINGHIERI