Università degli studi di Modena e Reggio Emilia

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Marco PRATO

Overview

The course aims to provide the knowledge, skills, and tools necessary to tackle basic mathematical analysis problems, which are preparatory for different courses of the degree program.
For a more complete understanding of the training objectives, please refer to the reading of the expected learning outcomes following the completion of this training course.

Admission requirements

Basic knowledge of Mathematics principles acquired in secondary school. In particular, students will have to:
- know the sets and the main operations between the sets;
- know the sets of natural, integer, rational, real numbers and their main properties;
- knowing how to solve algebraic, exponential, logarithmic, and trigonometric equations and inequalities;
- know the principles of analytical geometry (Cartesian plane, equations of lines and conics).

Course contents

The scanning of the contents for CFU is to be understood as purely indicative. In fact, it may undergo changes during the course of teaching in light of the feedback from students.

1 CFU (8 hours)
Sets and numerical sets. Real numbers: supremum, infimum, maximum, and minimum.

1 CFU (8 hours)
Introduction to functions. Properties of functions and elementary functions. Numerical sequences.

2 CFU (16 hours)
Functions of a real variable: limits, continuity, asymptotes, elementary functions; monotone functions; compound and inverse functions; theorems on continuous functions; notable limits.

2 CFU (16 hours)
Differential calculus for functions of one variable: derivative and tangent line, rules of derivation and fundamental derivatives; Fermat and Lagrange theorems; De L'Hospital's rule; successive derivatives, Taylor's formula, convexity and concavity. Study of functions.

1 CFU (8 hours)
Integral calculus for functions of one variable: definition and properties of the definite and indefinite integral, mean theorem, fundamental theorem of integral calculus, integration methods, fundamental integrals.

1 CFU (8 hours)
Differential equations: solution of equations with separable variables, linear differential equations of the first order, linear differential equations with constant coefficients.

1 CFU (8 hours)
Numerical series: convergence criteria for series with positive terms, Leibnitz criterion for series with alternating signs, absolute convergence. Outline of functions of several variables.

Teaching methods

The course is delivered through face-to-face lectures and exercises that are carried out with the aid of a blackboard and audiovisual means (graphics obtained with the Matlab computing environment). Attendance to face-to-face lessons is not compulsory. The course is delivered in Italian.

Assessment methods

The exam will take place at the end of the course according to the official exam schedule. The test is written. The duration is 2 hours and 30 minutes. The exam includes 1 exercise consisting of 3 theoretical questions with a request for definitions, theorems and/or examples (score up to 15 points) and 7 practical exercises on limits (score up to 15 points), domains and derivatives (score up to 15 points), continuity and differentiability (score up to 10 points), integrals (score up to 10 points), differential equations (score up to 10 points), numerical series (score up to 10 points) and function study (score up to 20 points). These exercises are aimed at evaluating: - knowledge and understanding skills; - the application of knowledge and understanding; - communication skills; - autonomy of judgment. The grade reported in the exam is given by the sum of the points obtained in the test multiplied by 0.3 and rounded to the nearest integer number (60 points are equivalent to a final grade of 18, 100 points to 30, over 100 points to 30 cum laude) . The results will be communicated via email to individual students no later than one week after the written test.

Learning outcomes

1) Knowledge and understanding.
At the end of the course and through classroom lessons and individual study, it is hoped that the students will be able to orient themselves within the main concepts of mathematical analysis relating to the functions of one variable and to differential and integral calculus, to sequences and numerical series, recognizing and knowing how to rigorously describe the main definitions, properties and theorems seen in class.

2) Applied knowledge and understanding.
At the end of the course and through classroom exercises, support activities and individual work, it is hoped that the students will be able to model and solve mathematical problems using the techniques of mathematical analysis with accuracy.

3) Autonomy of judgment.
At the end of the course, it is hoped that the students will be able to:
a) verify their degree of learning and understanding of the concepts exposed thanks to the possibility of intervention in class;
b) reorganize the knowledge learned and implement one's own ability to critically and independently evaluate what has been learned;
c) mastering a methodological approach that leads to verifying the statements and methods presented by means of rigorous arguments.

4) Communication skills.
At the end of the course, it is hoped that the students will be able to:
a) express their knowledge correctly and logically, recognizing the required topic and responding in a timely and complete manner to the exam questions.
b) face a dialectical confrontation in a timely and coherent way, arguing with precision.

5) Learning skills
At the end of the course, it is hoped that the students will be able to:
a) acquire mathematical knowledge as one's own heritage, which can be used at any other moment of one's cultural path;
b) have developed an aptitude for a methodological approach that leads to an improvement of the study method with consequent deepening of the ability to learn.

Readings

Il testo di riferimento del corso sarà il seguente:

C. Canuto, A. Tabacco. Analisi Matematica I. Springer, 2014.

Sulla pagina del Portale DOLLY relativa all’insegnamento di Fondamenti di Analisi saranno proposti alla fine di ogni argomento degli esercizi di approfondimento.

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The reference text of the course will be the following:

C. Canuto, A. Tabacco. Analisi Matematica I. Springer, 2014.

On the page of the DOLLY Portal relating to the course of Fondamenti di Analisi, in-depth exercises will be proposed at the end of each topic.