ANALISI MATEMATICA I

Module MODULO B

1. Integral Calculus.
• Indefinite integrals. Antiderivatives of a function on an interval and indefinite integral. Rules of indefinite integration: linearity property, integration by parts, integrations by substitution. Integration of rational functions. Some integrals that can be transformed into integrals of rational functions. Trigonometric integrals. Integrals of irrational functions.
  
• Definite integrals. Riemann integral: lower sum and upper sum, definition of Riemann integrable function, condition of Riemann integrability. Classes of Riemann integrable functions. Properties of the Riemann integral. Integral extended to an oriented interval. Definition of integral average and its geometric meaning. Mean value Theorem. Fundamental Theorem of Integral Calculus. Integration by parts and integration by substitution for definite integrals.
• Improper integrals. Improper integrals on unbounded intervals. Improper integrals on bounded intervals. Convergence criteria: algebra of improper integrals, integrability of nonnegative functions, comparison test, absolute convergence test, asymptotic comparison test.
  
• Applications with MATLAB. Numerical integration: midpoint formula, trapezoidal formula, Simpson formula.
  
2. Numerical series.
• Basic definitions. Fundamental series: geometric series, p-series, telescopic series. General convergence criteria: adding, deleting and modifying a finite number of terms in a numerical series, algebra of numerical series, necessary condition for the convergence of a numerical series.
  
• Convergence criteria for nonnegative-term series. Comparison test, application of the comparison test to the study of the harmonic series, Asymptotic comparison test, root test. A test for positive-term series: ratio test. Cauchy condensation criterion and its applications to the study of p-series and Bertrand series. MacLaurin criterion and applications.
• Convergence criteria for alternating series. Leibniz's alternating series test and its consequences.
  
• Absolute converge test.
  
• Complements. Rearrangement of a series, product of series, Raabe criterion, Cauchy convergence criteria for numerical sequences and numerical series.
  
• Applications with MATLAB. Geometric series and fractals.
  
3. Ordinary differential equations.
• Basic definitions. 
• Methods for solving some types of ordinary differential equations. First-order differential equations with separable variables, first-order differential equations with non-constant coefficients, differential equations of order n with constant coefficients, Bernoulli first-order differential equations, homogeneous first-order differential equations.
• Applications with MATLAB. Euler methods and Crank-Nicolson method.
  

Recommended books for the Prerequisites

[P1] C.Y. Young, Algebra and Trigonometry. Fourth Edition, Wiley (2017).

[P2] C. Y. Young, Precalculus. Third Edition, Wiley (2018).

Recommended books for the course of Mathematical Analysis I

• Recommended books for the Theory:

[T2] R.A. Adams, C. Essex, Calculus. A Complete Course, Pearson (2021).

• Recommended books for the Exercices:

[E2] R.A. Adams, C. Essex, Calculus. A Complete Course, Pearson (2021).

During the period of delivery of the lessons, some self-assessment tests will be administered. These self-assessment tests have the task of guiding the student in the gradual learning of the contents displayed during the lessons. In addition, the self-assessment tests allow the teacher to quickly implement any additional activities aimed at supporting students in view of the exams.

Structure of the exam

The Mathematical Analysis I exam can be passed in two ways.

Mode 1: mid-term tests and oral test

There are two mid-term tests: the first focuses on the contents of Module A, while the second focuses on the contents of Module B. Once both written mid-term tests have been passed, there is a mandatory oral test which mainly focuses on the contents of the written intermediate tests. The oral exam calendar will be prepared by the Examination Commission which will take into account, where possible, any preferences expressed by the students.

In the case that the oral test is not sufficient, the Exam Commission will indicate to the Student a further date on which it will be possible to take this test again. It will not be necessary to take the intermediate written tests again. It is possible to take the second mid-term written test only if the first has been previously passed. The duration of each mid-term written test is 180 minutes.

There are three useful dates for the written mid-term test related to Module A: two within the First Exam Session and one during the period of suspension of teaching activities scheduled for the month of April 2024. The second mid-term test will be carried out at the end of the second period of training activity. All the dates of the aforementioned tests are regularly included in the Exam Calendar which can be consulted on the degree course web page.

The student who, despite having passed the first intermediate test, has not passed or taken the second intermediate test scheduled at the end of the second period of Educational Activity, will be able to take this test during one of the sessions of the Second Exam Session or the Third Exams session. Alternatively, the student can take the exam following Mode 2. The student who has not passed or taken the first mid-term test will have to take the exam following Mode 2.

Each mid-term test consists of a written test and an oral test. The oral exam is compulsory and can only be accessed after passing the written mid-term test. Each mid-term test is considered passed if and only if the interview relating to it has been passed, that is, if the student has obtained a score of at least 18/30. It is possible to take the second mid-term test only if the first has been previously passed. The duration of each written mid-term test is 120 minutes.

Mid-term tests dates

The dates of the written intermediate tests can be found on the degree course website.

Structure of the written mid-term tests

Each intermediate written test has the same structure. In each mid-term written test, three theory questions and four exercises will be proposed.

The maximum grade obtainable in each written mid-term test is 30/30. Each written mid-term test is passed if the student has achieved a score of at least 18/30. A passing grade (18/30) is obtained if and only if correctly answers two theory questions and correctly solves two of the four proposed exercises.

In formulating the final grade, the grades obtained in the two intermediate written tests and the evaluation of the oral test are taken into account.

Mode 2: complete written exam and compulsory oral exam

In this mode, only one written test is proposed which focuses on the contents of Module A and the contents of Module B and, once passed, the student will have to take the oral test. The written test lasts 180 minutes.

Exam dates

The dates of the exams can be found on the degree course website.

Structure of the written test

In the written test, five exercises will be proposed.

Evaluation of the written test

The maximum mark that can be obtained in the written test is 30/30. The written test is considered passed if the student has achieved a score of at least 18/30. A passing grade (18/30) is obtained if and only if the student correctly solves three of the five proposed exercises.

Oral test and final grade

The oral test covers all the topics of the course (see the "Course contents" section of the Syllabus relating to Module A and the Syllabus relating to Module B). When formulating the final grade, the grade obtained in the written test and the evaluation of the oral test are taken into account. The oral exam calendar will be prepared by the Examination Commission which will take into account, where possible, any preferences expressed by the students. In the event that the oral test is not sufficient, the Exam Commission will indicate to the Student a further date on which it will be possible to take this test again. It will not be necessary to take the written test again.

Note. Information for students with disabilities and / or SLD
To guarantee equal opportunities and in compliance with the laws in force, interested students can ask for a personal interview in order to plan any compensatory and / or dispensatory measures, based on the didactic objectives and specific needs. It is also possible to contact the referent teacher CInAP (Center for Active and Participated Integration - Services for Disabilities and / or SLD) of the Department of Electrical, Electronic and Computer Engineering.

Note. Verification of learning can also be carried out electronically, should the conditions require it. In this case, the duration of the written test may be subject to change.

All the topics mentioned in the program can be requested during the exam.

The attendance of the lessons, the study of the recommended texts and the study of the material provided by the teacher (handouts and collections of exercises carried out and proposed) allow the student to have a clear and detailed idea of ​​the questions that may be proposed during the exam.

An adequate exposition of the theory involves the use of the rigorous language characteristic of the discipline, the exposition of simple examples and counterexamples that clarify the concepts exposed (definitions, propositions, theorems, corollaries).

The main types of exercises related to the Module B of the course of Mathematical Analysis I are:

• Calculation of indefinite and definite integrals.
• Search the antiderivative of a function satisfying a condition.
• Study of the convergence of improper integrals and calculation of improper integrals.
• Qualitative study of an integral function.
• Study of the behaviour of a numerical series.
• Search the general integral of an ordinary differential equation.
• Search the integral of an ordinary differential equation satisfying a condition.