Ateneo - UPO

Functions
Derivatives
Integrals
Descriptive Statistics
Probability
Statistical hypothesis testing
R basics

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 is intended to provide students with the knowledge and the ability to manage the mathematical tools required to understand and follow the other courses of the Degree. More specifically the course aims to provide students with basic methods of differential and integral calculus plus the basic notions of probability and statistics needed to understand mathematical models and to undertake statistical analysis of experimental results.
*Applying knowledge and understanding
Students are expected to be able 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 high school. In particular: arithmetic operations, powers, Cartesian plane, the line equations, parabola and circumference, angles, measured in degrees and radians, circular functions, fundamental formulas of plane and solid geometry.
In case of failure in the test of basic knowledge the student should follow the course on basic mathematics and/or logic and pass the final test.

The lessons will be delivered in classroom but interactions with students (real-time quizzes using the Wooclap resource made available by the University) and laboratory moments introducing the use of the R software are planned. Discussion among students is favored by both classroom quizzes and online quizzes that students will face outside the classroom and constantly present on the DIR website. The possibility of addressing the forum of the DIR site and discussing it also on other platforms should also favor the development of expository skills.

Reference material and further details are provided on the DIR website.
Enrolment key is provided during the lectures. Students with physical disabilities, Learning Disabilities or Special Education Needs can request specific services and tools via the Staff Sviluppo e Coordinamento Carriere e Servizi alle Studentesse e agli Studenti, consulting the University webpage: https://www.uniupo.it/en/services/services-students-physical-or-learning-disabilities
Students with disabilities, learning disabilities or special education needs, once they have contacted the University Staff, can refer to the tutor in charge of the course to define the examination modalities, concerning academic aspects.

Ongoing Quizzes and online Final Examination. Final exam on computer and online quiz during the course. The online quizzes focus on individual topics, the final exam may contain questions on all the topics of the course.
The final exam contains different types of questions in which we will also try to evaluate the ability of text understanding and the ability to deal with problems that require a non-superficial understanding of the subject and the ability to choose the appropriate tools for resolution.
The final evaluation assigns a weight of 4/30 to the online quizzes and of 26/30 to the final test. The online quizzes can be replaced by a written test on DIR (or an oral one) which also contains open theoretical questions. The results of the online quizzes are removed and replaced by the written exam on DIR (or oral) if the exam is not completed within one semester from the closing of the quizzes. The exams will take place in the computer classroom or in other suitable classroom 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

Credit 1 [Functions]
Mathematical concept of functions. Domain and codomain (target), Image. Exponential and power functions. Function composition. Invertible function and inverse function. Inverse circular functions. Logarithms.
Credit 2 [Derivatives]
Meaning of Derivative
Geometric meaning of the derivative. Graphic computation of the derivative through repeated zooms. Approximation of the derivative at a point via Newton quotient. Three point rule. Derivative for tabulated functions. Derivative function.
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.
Application of the derivatives
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.
Credit 3 [Area and Integrals]
Numerical Integration
Definite integral for positive functions over finite intervals, Computation. Numerical integration with the rectangle methods and trapezoid method. Stochastic integration with the Montecarlo methods.
Exact integration
The Fundamental Theorem of Calculus. Indefinite integral and antiderivatives. Computation of antiderivatives. Area between two curves.
Extensions of the Integral
Integrals for non positive functions. Inversions of integration endpoints.
Credit 4 [Statistical Data]
Experimental data and simulated data. How to simulate a fair dice.
Single Variable. Statistical Units. Samples, populations, variables. Single variable statistics. Data presentations: sorting, absolute frequencies, relative frequencies. Continuous and discrete variables. Bar charts, histograms and boxplot. Summation (the Sigma symbol). Statistical parameters, Measures of centrality (mean, median) and measures of dispersion (extensions, quartiles, sample variance and sample standard deviation). Statistical Indices for repeated data.
Credit 5 [Double variables statistics Probability.]
Double variables
Two variables representations. Scatter plot. Covariance, Linear regression (linear regression over X and over Y). Application of linear regression. Power and exponential laws. Variable transformations to reduce to a linear relationship.
Credit 6 [Probabilities]
Frequentist definition of probability. Complete systems of events. The Union and intersection of events. Bayes rule. Probability computations using Bayes rule. Random variables and probabilities densities for discrete and continuous variables. Expected values and standard deviation of a population.
Computation of Expected values and variance. Normal variables. Computation of the probability P{a < X < b} for a normal variable. Standardized variables.
Credit 7 [Statistical Tests]
Estimations of statistical parameters
Reliability criterions. Confidence level and reliability. Point estimates and interval estimates. Interval estimate of the mean of a normal population.
Statistical tests.
Student’s t-test with one sample and two samples (paired data, equal variance data). Welch test. chi-square-test.

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 at applying these concepts and methods in the modelisation 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 modelling of natural phenomena.