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.

Assessment will consist of online quizzes administered during the course and a final computer-based examination. The quizzes will cover individual course topics, while the final examination may include questions covering the entire course syllabus.

The final grade will be based on a maximum of 4/30 from the online quizzes and 26/30 from the final examination.

Students may choose to replace the online quizzes with a supplementary assessment, conducted either orally or through the DIR platform, according to the procedures established by the instructor. The supplementary assessment may also include open-ended theoretical questions.

If the examination is not completed by the September 2027 examination session (inclusive), the quiz results will be automatically cancelled and replaced by the result of the supplementary assessment.

Examinations will be held either in the Computer Laboratory or, using the student’s own computer, in a suitable classroom.

The instructor may require a brief oral confirmation interview consisting of a discussion of the written examination. If circumstances arise that require further verification of the student’s actual mastery of the course content or of the authenticity of the submitted work, the instructor may require an additional oral examination. The outcome of the oral confirmation interview or the additional oral examination may result in confirmation or revision of the written examination grade or, where provided for by University regulations, invalidation of the examination.

0.1 Numbers.
0.2 Equations and inequalities.
0.3 Cartesian coordinate system. Lines and simple plane curves. Systems of linear equations.
0.4 Angles and their measurement: degrees and radians.

1.1 The concept of a function. Domain and codomain. Composition of functions. Invertible functions and inverse functions.
1.2 Sine, cosine, tangent, and their inverse functions. Logarithms, exponential functions, and powers.

2.1 Definition of the derivative. Graphical determination of the derivative (zoom method) and linearization. Approximation of the derivative at a point using Newton’s difference quotient (incremental ratio) and the three-point rule. Differentiation of functions defined by tabulated data. The derivative function.
2.2 Differentiation rules: sum, product, quotient, reciprocal, and composition of functions. Derivative of the inverse function. Higher-order derivatives.
2.3 Analysis of functions using derivatives. Increasing and decreasing functions. Local maxima and minima. Determination of stationary points using the derivative. Second derivative and the study of concavity and convexity. Newton’s method for the numerical determination of zeros.
2.4 Introduction to the asymptotic behavior of functions.

3.1 Definite integrals of functions over bounded intervals. Methods of evaluation.
3.2 Numerical integration using the rectangle and trapezoidal methods. The Fundamental Theorem of Calculus. Indefinite integrals and antiderivatives. Determination of antiderivatives.
3.3 Area between two curves. Integrals of functions with changing sign. Reversal of the limits of integration.

4.1 Elements of descriptive statistics. Populations and samples. Statistical variables and observations. Organization and presentation of data. Graphical representations (bar charts, histograms, boxplots, and scatter plots). Descriptive statistical measures. Linear regression and variable transformations for data linearization. Introduction to the use of the R software.

Upon successful completion of the course, students will be able to:

* understand and apply the fundamental concepts and methods of basic mathematics, with particular reference to real numbers, elementary functions, derivatives, integrals, and the basic principles of descriptive statistics;
* apply these concepts and methods to the formulation and solution of simple quantitative problems, including the numerical and graphical analysis of experimental data;
* iinterpret the results of simple mathematical and statistical analyses and present them using appropriate numerical and graphical representations.
* independently select and apply appropriate mathematical tools to interpret simple models of natural phenomena and analyze experimental data, including through the use of the R software.