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).

Self-assessment tests

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 three ways.

Mode A: mid-term tests and oral tests

There are two mid-term tests: the first at the end of the first teaching period and the second at the end of the second teaching period. The first mid-term test focuses on the contents of Module A, while the second focuses on the contents of Module B. 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.

Structure of the written mid-term tests

Each written mid-term test has the same structure. Five exercises will be proposed in each written mid-term exam.

Evaluation of the mid-term tests and final grade

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 pass is obtained (18/30) if and only if the student correctly solves three of the five proposed exercises. The student who, despite having passed the first mid-term test, has not passed or sustained the second mid-term test, can complete the exam following Mode B, thus supporting the partial written test relating to Module B as well as the oral test after having passed the the written test as a whole. Alternatively, the student can take the complete exam following Mode C. Each oral exam focuses on all the topics of the Module to which it refers. In the formulation of the final mark of each mid-term test, the score obtained in the written mid-term test and the evaluation of the oral test are taken into account. The final grade is the rounding up of the arithmetic average of the marks obtained in the two mid-term tests.

Mode B: written test in modules and oral test

The written test is divided into two partial tests. The first partial test focuses on the contents of Module A, while the second focuses on the contents of Module B. It is possible to take the second partial test only if the first has been previously passed. The duration of each partial test is 120 minutes. It should be noted that, following this procedure, the two partial tests cannot be sustained in the same appeal. After passing the first partial test, the student can take the second partial test in one of the following sessions and in any case no later than the third exam session. The oral exam is compulsory and can only be accessed after passing both partial tests.

Structure of partial tests

The two partial tests have the same structure. In each partial test five exercises will be proposed.

Evaluation of each part into which the written test is divided

The maximum grade obtainable in each partial test is 30/30. Each of the two partial tests is considered passed if the student has achieved a score of at least 18/30. In each partial test, passing (18/30) is obtained if and only if the student correctly solves three of the five proposed exercises. The final mark of the written test is the rounding up of the arithmetic average of the marks obtained in the two partial tests.

Oral exam and final grade

The grade with which the student is admitted to the oral test is the rounding up of the arithmetic average of the marks obtained in the two mid-term tests. The oral exam covers all the topics of the course. In formulating the final grade, the grade with which the student is admitted to the oral exam and the grade obtained in the oral exam are taken into account.

Mode C: complete written exam and compulsory oral exam

In this way, only one written test is proposed which focuses on the contents of Module A and on the contents of Module B and, after passing it, the student will have to take the oral test. The written test lasts 120 minutes.

Structure of the written test

Five exercises will be proposed in the written test. Evaluation of the written test. The maximum mark obtainable in the written test is 30/30. The written test is passed if the student has achieved a score of at least 18/30. A pass is obtained (18/30) if and only if the student correctly solves three of the five proposed exercises.

Oral exam and final grade

The oral exam covers all the topics of the course. In formulating the final grade, the grade obtained in the written test and the evaluation of the oral test are taken into account.

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.