Ateneo - UPO

The course will cover the following topics:

* Functions
* Derivatives
* Integrals
* Descriptive statistics
* Probability
* Statistical tests
* Introduction to the R software

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

The aim of the course is to provide students with the mathematical knowledge and tools that are fundamental for the other courses in the Degree Programme. The course also introduces the numerical and graphical presentation and summarization of simple experimental datasets, the mathematical modelling of natural phenomena, and the fundamental methods of differential and integral calculus required to understand mathematical models. In addition, students will acquire the basic concepts of probability and statistics needed for the analysis of experimental data. To support data analysis activities, the course includes 1 ECTS credit of virtual laboratory sessions dedicated to learning the free and open-source software R.

Knowledge and understanding
Acquire fundamental mathematical knowledge and tools, with particular reference to functions, differential and integral calculus, probability, and statistics, and understand their role in the modelling of natural phenomena and the analysis of experimental data.

Applying knowledge and understanding
Apply the acquired knowledge and methods to the solution of simple mathematical and statistical problems, including in multidisciplinary contexts, using appropriate computational tools for data analysis.

Making judgements
Select and apply the most appropriate mathematical and statistical tools to address simple problems and critically interpret the results obtained.

Communication skills
Understand statements expressed in mathematical and statistical language and translate simple problems described in natural language into an appropriate mathematical formulation.

Learning skills
Develop the ability to deepen the topics covered independently by making effective use of textbooks, documentation, and other learning resources, including online materials.

Students are expected to possess the basic mathematical knowledge normally acquired in secondary school. In particular, they should be familiar with arithmetic operations, powers, the Cartesian coordinate system, equations of lines, parabolas and circles, angles and their measurement in degrees and radians, trigonometric functions, and the main formulas of plane and solid geometry.

Students who do not pass the initial knowledge assessment (minimum entry requirements test) are required to attend the preparatory course in Mathematics (introductory elements of mathematics and/or logic) and successfully complete the corresponding final assessment.

The course will be delivered primarily through face-to-face lectures and will include interactive activities based on real-time quizzes using the University’s Wooclap platform. The course also includes laboratory sessions introducing the use of the R software.

Interaction during the course will be encouraged through both in-class quizzes delivered via Wooclap and online quizzes available on the DIR platform, which students will be required to complete throughout the course. The DIR platform will also serve as a learning support environment by providing teaching materials, self-assessment activities, and tools for interaction between students and the instructor.

Course materials and further information are available on the DIR platform. The enrollment key will be provided during class.

Students with disabilities, Specific Learning Disorders (SLD), or Special Educational Needs (SEN) may request dedicated services and support by contacting the University's Student Services Office for Career Development and Student Support and by consulting the dedicated webpage: https://uniupo.it/it/servizi/servizi-studenti-disabili-e-dsa.

After contacting the University's Student Services Office, students may discuss with the course instructor the arrangements for examinations and any appropriate teaching accommodations.

Assessment will consist of a final computer-based examination and online quizzes administered during the course. The final examination will be divided into two separate sections, one covering Mathematics and the other Statistics. Students must obtain a passing grade in both sections in order to pass the examination.

The online quizzes will cover individual course topics, while the final examination may include questions covering the entire course syllabus. The final examination will include different types of questions designed to assess not only the knowledge acquired, but also the ability to understand and interpret the proposed questions, solve problems requiring an in-depth understanding of the subject, and select the most appropriate methods for their solution.

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. This assessment will be conducted either through the DIR platform or as an oral examination, according to the procedures established by the instructor, and may also include open-ended theoretical questions. If the examination is not completed within one semester after the online quizzes have closed, the quiz results will automatically be 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.

ECTS Credit 1 [Functions]

The concept of a function. Domain and codomain. Composition of functions. Invertible functions and inverse functions. Inverse trigonometric functions. Logarithms, exponential functions, and powers.

ECTS Credit 2 [Derivatives]

Introduction to derivatives and their geometric meaning. Geometric interpretation of the derivative. Graphical determination of the derivative (zoom method). Approximation of the derivative at a point using Newton’s difference quotient (difference quotient) and the three-point rule. Differentiation of functions defined by tabulated data. The derivative function.

Differentiation rules. Derivatives of sums, products, quotients, reciprocals, and composite functions. Derivative of the inverse function.

Applications of derivatives. Analysis of functions using derivatives. Increasing and decreasing functions. Local maxima and minima. Determination of stationary points using derivatives. Second derivative and the study of concavity and convexity. Introduction to the asymptotic behavior of functions.

ECTS Credit 3 [Integrals and Area]

Numerical integration. Definite integrals of positive functions over bounded intervals. Methods of evaluation. Numerical integration using the rectangle rule, the trapezoidal rule, and the Monte Carlo method.

Analytical integration. Fundamental Theorem of Calculus. Indefinite integrals and antiderivatives. Determination of antiderivatives. Area between two curves.

Extensions of the concept of integration. Integrals of functions with changing sign. Reversal of the limits of integration.

ECTS Credit 4 [Statistical Data]

Introduction. Experimental data and simulation. Simulation of repeated rolls of a fair die.

Descriptive statistics for a single variable. Statistical units, populations, samples, and variables. Representation of a single variable. Data organization: ordered data, absolute frequencies, and relative frequencies. Discrete and continuous variables. Graphical representations: bar charts, histograms, and boxplots. Sigma notation and its properties. Descriptive statistics: measures of central tendency (mean and median) and measures of dispersion (range, quartiles, sample variance, and standard deviation). Descriptive measures for grouped data.

ECTS Credit 5 [Bivariate Statistics]

Analysis of two variables. Representation of two variables. Scatter plot. Linear regression: scatter plots, regression lines, linear correlation, and covariance. Applications of linear regression. Power-law and exponential models. Variable transformations for data linearization.

ECTS Credit 6 [Probability]

Frequentist interpretation of probability. Complete systems of events. Union and intersection of events. Bayes’ theorem and its applications. Random variables and probability distributions for discrete and continuous variables. Expected value and standard deviation of a population. Formulas for expected value and variance. Normal distribution. Computation of the probability (P(a < X < b)) for a normally distributed random variable. Standard normal variable.

ECTS Credit 7 [Statistical Tests]

Parameter estimation. Reliability criteria. Confidence level. Point and interval estimation. Confidence intervals for the mean of a normally distributed population.

Statistical tests. Student’s t-tests: one-sample, paired two-sample, two-sample assuming equal variances, and Welch’s t-test. Chi-square goodness-of-fit and independence tests.

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, differential and integral calculus, probability, and statistics;
* apply these concepts and methods to the formulation and solution of simple quantitative problems, including the numerical and graphical analysis of experimental data;
* understand statements expressed in mathematical and statistical language and translate simple problems described in natural language into an appropriate mathematical formulation;
* select and apply the most appropriate mathematical and statistical tools to solve simple problems and interpret the results.