ANALISI MATEMATICA IModule MODULO B
Academic Year 2024/2025 - Teacher: ANDREA SCAPELLATOExpected Learning Outcomes
The aim of the course of Mathematical Analysis I - Module B is to give the basic skills on Integral Calculus for real functions of one real variable, numerical series and some types of ordinary differential equations.
In particular, the learning objectives of the course, according to the Dublin descriptors, are:
- Knowledge and understanding: The student will learn some basic concepts of Mathematical Analysis and will develop both computing ability and the capacity of manipulating some common mathematical structures, as integrals for real functions of one real variable, numerical series and some types of ordinary differential equations.
- Applying knowledge and understanding: The student will be able to apply the acquired knowledge in the basic processes of mathematical modeling of classical problems arising from Engineering.
- Making judgements: The student will be stimulated to autonomously deepen his/her knowledge and to carry out exercises on the topics covered by the course. Constructive discussion between students and constant discussion with the teacher will be strongly recommended so that the student will be able to critically monitor his/her own learning process.
- Communication skills: The frequency of the lessons and the reading of the recommended books will help the student to be familiar with the rigor of the mathematical language. Through constant interaction with the teacher, the student will learn to communicate the acquired knowledge with rigor and clarity, both in oral and written form. At the end of the course the student will have learned that mathematical language is useful for communicating clearly in the scientific field.
- Learning skills: The student will be guided in the process of perfecting his/her study method. In particular, through suitable guided exercises, he/she will be able to independently tackle new topics, recognizing the necessary prerequisites to understand them.
Course Structure
Required Prerequisites
Attendance of Lessons
Detailed Course Content
- 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. Techiniques of indefinite integration.
- 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.
- 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.
- Numerical series.
- Basic definitions. Fundamental series: geometric series, p-series,
telescopic series, logarithmic series and exponential 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. MacLaurin criterion and applications.
- Convergence criteria for alternating series. Leibniz's alternating series test and its consequences.
- Absolute convergence test.
- Abel-Dirichlet criterion.
- Ordinary differential equations.
- Basic definitions.
- Methods for solving some types of ordinary differential equations. First-order differential equations with separable variables, first-order linear differential equations, linear ordinary differential equations of second order, linear ordinary differential equations of order n with constant coefficients.
Textbook Information
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).
Course Planning
| Subjects | Text References | |
|---|---|---|
| 1 | Integral calculus | [T1, E1]: Ch. 10, 11; [T2, E2]: Ch. 5, 6, 7. |
| 2 | Differential equations | [T1, E1]: Ch. 13, 14; [T2, E2]: Ch. 19. |
| 3 | Numerical series | [T1, E1]: Ch. 11; [T2, E2]: Ch. 9. |
Learning Assessment
Learning Assessment Procedures
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 written and oral tests
- Mode 2: one written test and one oral test (see Module B)
Mode 1:
At the end of the lessons provided for in Module B, students will be offered a second intermediate test. It is possible to take the second written intermediate test only if the first has been previously passed (see Mode 1 - Module A) and two available dates are scheduled.
Dates of the intermediate test relating to Module B.
The dates of the written intermediate tests are available on the degree course website. The student can take this test also on the occasion of second exam sessions of the Second Exam Session or the first exam session of the Third Exam Session.
Structure of the written intermediate test relating to Module B.
Four exercises will be proposed in the second written intermediate test and the duration of the written intermediate test is 120 minutes. Evaluation of the intermediate tests. The maximum score obtainable in the written intermediate test is 30/30. The written intermediate test is considered passed if the student has obtained a score of at least 18/30. Each exercise will be assigned a score. The maximum score will be awarded if the work is done correctly, otherwise a partial score will be awarded and it will be determined based on the errors made. In the event that the total score is greater than or equal to 15 and less than 18, the Examination Commission may admit the student to the oral exam with reservations if the student demonstrates adequate argumentation skills.
Oral exam on Module B.
Students who have passed the written test and the oral test for module A, after having passed the written test for module B, will have to take an oral exam on the topics of module B by the end of September. Students who have passed the written test but not the oral test for module A, after having passed the written test for module B, will have to take an oral exam on all the topics of the program (see the “Detailed Course Contents” sections for Module A and Module B) by the end of September. Students who have not completed the aforementioned tests as required above, will have to take the exam again according to Method 2.
Mode 2: one written test and one oral test
In this mode, a single written test is proposed that focuses on the contents of Module A and the contents of Module B and, if passed, the student will have to take the oral test on the entire program. The schedule of the oral tests will be prepared by the Examination Commission. The written test lasts 120 minutes.
Dates of the exam sessions.
The dates of the exam sessions are available on the degree course website. Structure of the written test. Four exercises will be proposed in the written test.
Evaluation of the written test.
The maximum score obtainable 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 score will be assigned to each exercise. Each exercise will be assigned the maximum score expected if and only if it is carried out correctly. Otherwise, a partial score will be assigned that will be determined based on the errors made.
Oral exam.
The oral exam covers all the topics of the course. If the student does not pass the exam, it will be necessary to take the written test again.
Examples of frequently asked questions and / or exercises
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.