Ateneo - UPO

The course will cover the following topics: Basic mathematics, functions, derivatives, integrals, and elements of descriptive statistics.

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

The objective of the course is to provide students with the mathematical knowledge and tools preparatory to other courses in the Degree program. It also aims to introduce the numerical and graphical presentation and synthesis of simple series of experimental data and the modeling of natural phenomena. The course intends to provide the fundamental methods of differential and integral calculus necessary for understanding mathematical models. Knowledge and understanding: Acquire basic mathematical knowledge and tools, including the presentation and synthesis of experimental data and the modeling of natural phenomena. Understand the fundamental methods of differential and integral calculus for mathematical models. Applying knowledge and understanding: Utilize the acquired knowledge even in multidisciplinary contexts and apply mathematics to situations outside the course. Making judgements: Apply the learned methods even in different situations and possess the tools to independently extend their knowledge. Communication skills: Express the fundamental concepts learned in a simple, clear, and effective way. Learning skills: Develop the ability to study and learn by choosing their own path with originality and selecting useful resources, including online ones, to deepen their understanding of the subject.

Students must possess basic knowledge of the subject, acquirable in a normal high school. In particular: arithmetic operations, powers, Cartesian plane, equations of lines, parabolas, and circles, angles, their measure in degrees and radians, circular functions, and fundamental formulas of plane and solid geometry

Lectures will be held 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.

Learning will be assessed through: A final computer-based exam. Online quizzes during the course. The online quizzes will focus on individual topics, while the final exam may include questions on all course topics. The final evaluation assigns a weight of 4/30 to the online quizzes and 26/30 to the final exam. The online quizzes can be replaced by a written test on DIR (or an oral examination) which also contains open theoretical questions. The online quizzes are, in any case, canceled and replaced by the written theoretical test on DIR (or oral examination) if the exam is not completed by September 2024. The exams will take place in the computer classroom or on the student's personal computer in a suitable classroom (potentially the computer classroom itself). The teacher reserves the right to conduct a short confirmatory oral interview consisting of a discussion on the written test. Should any inconsistencies emerge, the teacher reserves the right to conduct a further oral in-depth study and potentially cancel the mark of the written test.

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.

At the end of the course, students should have acquired: Knowledge of some basic concepts and methods of mathematics, in particular: real numbers and their representations, main elementary functions, derivatives and their geometric meaning, simple integrals, calculation of areas, and basic elements of Statistics. Ability to apply these concepts and methods in the modeling of simple problems involving numerical and graphical syntheses of experimental data series. Ability to communicate the results of the problems addressed clearly and completely. Ability to choose useful mathematical resources to address simple problematic situations where the interpretation and use of natural phenomena modeling are required.