
Course unit
MATHEMATICS AND STATISTICS
SCN1036952, A.A. 2016/17
Information concerning the students who enrolled in A.Y. 2016/17
ECTS: details
Type 
ScientificDisciplinary Sector 
Credits allocated 
Basic courses 
MAT/02 
Algebra 
3.0 
Basic courses 
MAT/03 
Geometry 
2.0 
Basic courses 
MAT/05 
Mathematical Analysis 
3.0 
Basic courses 
MAT/06 
Probability and Mathematical Statistics 
2.0 
Basic courses 
SECSS/01 
Statistics 
4.0 
Course unit organization
Period 
Annual 
Year 
1st Year 
Teaching method 
frontal 
Type of hours 
Credits 
Teaching hours 
Hours of Individual study 
Shifts 
Practice 
5.0 
80 
45.0 
No turn 
Lecture 
9.0 
72 
153.0 
No turn 
Examination board
Board 
From 
To 
Members of the board 
8 MATEMATICA E STATISTICA 20192020 
01/10/2019 
27/11/2020 
PRELLI
LUCA
(Presidente)
MARAGONI
LORENZO
(Membro Effettivo)
POLESELLO
PIETRO
(Supplente)

7 matematica e statistica 20182019 
01/10/2018 
30/11/2019 
PRELLI
LUCA
(Presidente)
COSSU
LAURA
(Membro Effettivo)
MARAGONI
LORENZO
(Supplente)

6 MATEMATICA E STATISTICA 2017/2018 
01/10/2017 
25/11/2018 
PRELLI
LUCA
(Presidente)
FINOCCHIARO
CARMELO ANTONIO
(Membro Effettivo)
MARAGONI
LORENZO
(Supplente)

5 MATEMATICA E STATISTICA 20162017 
01/10/2016 
30/11/2017 
COLPI
RICCARDO
(Presidente)
ADIMARI
GIANFRANCO
(Membro Effettivo)
BAZZONI
SILVANA
(Supplente)

Prerequisites:

Natural numbers: arithmetic operations and their properties. Division with remainder. Prime numbers. Greatest common divisor and least common multiple. Numerical fractions: operations and ordering. Relative integers. Relative rational numbers. Representation of numbers as alignments; alignments with comma, finite or periodic. Intuitive idea of the real numbers. Inequalities and related calculus rules. Absolute value. Powers and roots. Arithmetic average and geometric average of two positive numbers. Logarithms and their properties. Elements of formal calculus, use of brackets. Polynomials. Division with remainder between polynomials. Ruffini's Theorem. Algebraic fractions. Identity and equations: solution knowledge. Algebraic equations of first and second degree. Relations between roots and coefficients in a quadratic equation. Linear systems of two equations in two variables. Set elementary language; belonging, inclusion, intersection, union, complement, empty set. Notion of function and composition of functions. Graphs of the most important functions (powers, roots, exponential, logarithmic, cos, sin, tang). Implication. Sufficient conditions, necessary conditions. Euclidean plane geometry: incidence, parallelism. Existence and uniqueness of the parallel and perpendicular to a point at assigned line. Length of a segment (distance between two points); one to one correspondence between the points of a line and real numbers. Angle width: measured in degrees. Length of the circumference. Measure of angles in radiants. Sum of the angles of a triangle. Relations between the angles formed by two parallel lines cut by a transversal. Elementary knowledge of area. Area of the circle. Relations between areas of similar figures. Notion of locus and loci significant (segment axis, angle bisector, circumference etc.). Properties of plane figures: criteria of congruence of triangles. Highlights of the triangles (centroid, incenter, circumcenter, orthocenter). Parallelograms. Theorems of Thales, Euclid, Pythagoras. Criteria of similarity of triangles. Property, Segment and angle of the circle (ropes, secants, tangents, arc subtended by an angle). Corners to the center and circumference. Geometric transformations of the plan: symmetry with respect to a straight line and at a point, translation, rotation, similarities, and their compositions. Cartesian coordinates: equations of lines and circles. Equations of simple geometric places (parabolas, ellipses, hyperbolas) in suitable reference systems. Trigonometry: sin, cos, tang of an angle. Fundamental trigonometric identity cos2α sin2α + = 1, addition formulas. Euclidean geometry of space (do not require formal knowledge, only intuitive) mutual positions of two lines, two planes, a line and a plane (angles, parallelism, squareness). Symmetries. Sphere, cone, cylinder. Boxes, pyramids, prisms. Intuitive idea of volume of solids. Formulas for the calculation of volume and area of the surface of a parallelepiped, pyramid, prism, cylinder, cone and sphere. Relations between areas and between volumes of similar solids. 
Target skills and knowledge:

The course is a cultural mathematical basis which should be well known by any student who attends a course in any scientific area. The purpose of the course is double. On the one hand, it aims to train the student to adopt some major guidelines for a rigorous analysis of problems and for searching their logic solutions. On the other hand, it is responsible to provide some tools to concretely deal with mathematical problems, also extremely practical ones. For this purpose will be addressed and solved some examples having a physical and/or biological nature. Moreover, the course provides natural prerequisites for the subsequent courses of Statistics, Physics, Chemistry and Genetics. 
Examination methods:

The course develops during both the semesters of the first year. At the end of each semester written tests are provided. The final examination takes into account the individual tests carried out on the programs of the first and of the second semester, and a possible final oral test is provided. 
Assessment criteria:

The acquisition by the student of an intellectual maturity with respect of deductive logic  based methodologies, tools and content taught in class is verified. Next to verification of an understanding of the theoretical content of the course, the student is asked to demonstrate an appropriate capacity in solving new problems formulated in the language of mathematical modeling base. The student must then demonstrate that it is able to: understand the problem, find the correct interpretation by means of mathematical and quantitative methods, develop an appropriate calculation context, understand the answers derived from the method and its inferences . 
Course unit contents:

Elements of mathematical logic. Elementary set theory. Functions acting between sets: graphic of a function; composed functions, injectivity, surjectivity, inverse of a function. Real functions in a real variable. Monotony and invertibility. Inverse of exponential functions, inverse trigonometric functions. Complex numbers and basics on functions in a complex variable. Neighbourhoods, accumulation points, limits for functions and their properties. Infinite and infinitesimal functions. Continuous functions and their properties. Derivability of a function. Theorems of Rolle, Lagrange and their applications to the study of monothony of differentiable functions. Theorem of L'Hopital. Higher order derivatives. Study of functions and of their graphic. Search for branches asymptotic. Comparing infinitesimal (resp. Infinite). Order of infinitesimal (resp. Infinite). Approximation of functions, Taylor formula and property of the remainder. Approximate calculation. Indefinite integrals and methods of integration of continuous functions. Definite integral. Theorem of the integral average, theorem of Torricelli and applications to calculus. Study of integral functions. Calculation of planar areas and of volume of solids obtained by revolution. Calculus of the work done by a force (electrical or mechanical) and potential energy. Generalized integration. Differential equations of the first order. Physical and biological examples. Solution methods in the linear case and in case of separation of the variables. Some cases of second order. Analysis of some concrete problems of physical and/or biological solvable through the study of differential equations. Elements of linear algebra: matrices and determinants, matrix addition; the problem of diagonalization, eigenvalues and eigenvectors, similarity of matrices; algebraic and geometric multiplicity of an eigenvalue; real symmetric matrices. Elements of vector calculus: vectors, their coordinates and elementary operations; scalar, vector and mixed products, their meaning and calculation using coordinates. Elementary geometry of the space: planes and straight lines, their Cartesian and parametric equations, coordinates of a point on a straight line; mutual positions of lines and planes, distances between points, lines and planes; symmetries. Real functions in several real variables: limits, continuity; partial derivatives, differentiability, gradient and directional derivatives; transformations of R^n and the Jacobean. Maximum and minimum for functions in several variables, both free and in the presence of constraints: Hessian matrix, method of Lagrange. Multiple integration and coordinate changes (in particular polar, cylindrical and spherical). Applications to the determination of volumes, masses and centers of gravity.
Descriptive and inferential statistics. Simple and double distributions. Conditional distributions. Indices of position and variability. Relations between variables: dependence in distribution and in the media. Regression line. Elements of Probability. Some discrete and continuous random variables. Law of large numbers and central limit theorem. Estimators and their properties. Confidence intervals. Theory test: system hypothesis, test statistic, critical region, significance level, power of the test. Test on average. Test on the difference between average. Chisquare test of independence. Inference on proportions. 
Additional notes about suggested reading:

In addition to the suggested books, the teachers provide on the web some additional educational materials, such as lecture notes, texts and solutions of assigned tests. In particular, a page of the course is activated in the ELearning platform of the Department of Mathematics, which shows the contents of the lessons carried out and any additional teaching materials, as well as other information related to the course (scheduling and results of the examinations, scheduling meetings and materials relative to teaching support, etc.). 
Textbooks (and optional supplementary readings) 

Giuliano Artico, Istituzioni di Matematiche. Padova: Libreria Progetto, .

R. A. Adams, Calcolo Differenziale 1. : Ambrosiana, .

R. A. Adams, Calcolo Differenziale 2. : Ambrosiana, .

M. Pagano, K. Gauvreau, Fondamenti di Biostatistica. : Guido Gnocchi, .

G. Cicchitelli, Statistica – Principi e Metodi. : Pearson Education, .


