Ateneo - UPO

Basic mathematics, functions, derivatives, integrals. Some elements of descriptive statistics.

Sergio Invernizzi, Maurizio Rinaldi, Federico Comoglio, Moduli di Matematica e Statistica con l'uso di R, Zanichelli Editore, Bologna 2018

*Knowledge and understanding.
The course aims at providing student with the knowledge and the ability to manage the mathematical tools required to understand and follow the other courses of the Degree and at introducing her/him to the numerical and graphical presentation and synthesis of simple series of experimental data and to the modeling of natural phenomena. More specifically the course aims at providing students with the basic methods of differential and integral calculus needed to understand mathematical models.
*Applying knowledge and understanding.
Students are expected to be able to use the acquired skills even in a multidisciplinary context and they should be able to resort to mathematics and statistics even in situations external to the course.
*Making judgements.
At the end of the course students are expected to apply learned methods even in different situations and that they have acquired the tools needed to extend their knowledge by themselves.
*Communication skills.
At the end of the course students are expected to be able to express the learned concepts in a clear way.
*Learning skills.
During the course students should learn how to study by choosing their personal path and should become able to choose the appropriate resources, possibly online.

The student should have the basic knowledge of the subject that can be acquired in a normal high school. In particular: arithmetic operations, powers, Cartesian plane, equation of the line, parabola and circumference, angles, their measure in degrees and radians, circular functions, fundamental formulas of plane and solid geometry.

Lectures will be standard (in the classroom)

Reference material and further details will be provided on the DIR website. Enrolment key is provided during the lectures or given upon request.

Ongoing Quizzes and online Final Examination. The ongoing quizzes are assigned periodically at the end of a given subject. The Final focuses on all the course matters. Final evaluation gives a weight of 4/30 to the ongoing quizzes and 26/30 to the Final. Online quizzes can be replaced by a written test on DIR (or oral examination) which also contains theoretical questions. The online quizzes are IN ANY CASE canceled and replaced by the written theoretical test on DIR (or oral) if the exam is not completed by September 2024. The exams will take place in the computer classroom or in other suitable sclassroom on the personal computer. The teacher reserves the right to a short confirmatory oral interview consisting of a discussion on the written test.
Should any inconsistencies emerge, the teacher reserves the right to carry out an oral in-depth study and possibly cancel the mark of the written tests

0.1 Numbers 0.2. Equation and inequalities 0.3. Cartesian Plane. Straight lines and simple curves. Linear systems. 0.4 Angles and their measures: degree and radian. 1.1. Concept of mathematical functions. Domain and codomain. Image. Function composition. Invertible function and inverse function. 1.2. Cosine, Sine and tangent and their inverse functions. Logarithms and exponential functions. 2.1 Geometric meaning of the Derivative. Graphic computation of the derivative through repeated zooms. Approximation of the derivative at a point via Newton quotient. Three points rule. Derivative for tabulated functions. Derivative function. 2.2 Differentiation rules Derivative of the sum, of the product, of the composite function, of the reciprocal and of the quotient. Derivative of the inverse function. Applications of the derivatives 2.3 Increasing and decreasing functions, Minima and maxima of functions. Computation of maxima and minima with the help of the derivative. Second derivative and study of concavity-convexity of a graph. Netwon’s tangent method. 3.1 Numerical Integration Definite integral for positive functions over finite intervals, Computation. 3.2. Numerical integration with the rectangle methods and trapezoid method. Stochastic integration with the Monte-Carlo methods. Exact integration The Fundamental Theorem of Calculus. Indefinite integral and antiderivatives. Computation of antiderivatives. 3.3. Area between two curves. Extensions of the Integral Integrals for non positive functions. Inversions of integration endpoints. 4.1. Basic elements of Statistics. Statistical data and their representations. Population and samples. Variables and observations. Graphical representations: bar charts, histograms and box-plot. Numerical representations and statistical indices. Scatter plot. Linear regression. Variable transformations to reduce to a linear relationship.
The R software.

Knowledge of some basic concepts and methods of mathematics, in particular: real numbers and their representations, main elementary functions, derivative and its geometric interpretation, simple integrals, calculation of areas and basic elements of Statistics. Ability to apply these concepts and methods in the modeling of simple problems in which there are numerical and graphical presentations and synthesis of simple series of experimental data. Ability at communicating problem solutions in a clear and complete fashion. Ability in choosing the mathematical resources useful to deal with simple problems where it is required to interpret and use modeling of natural phenomena.