81853 - Mathematical Analysis 1A (M-Z)

Academic Year 2026/2027

  • Teaching Mode: In-person learning (entirely or partially)
  • Campus: Bologna
  • Corso: First cycle degree programme (L) in Mathematics (cod. 6649)

Learning outcomes

At the end of the course, students are familiar with the language of Mathematical Analysis and its fundamental concepts. They have acquired knowledge of the real number system, as well as the concepts of limits, continuity, and the foundations of differential calculus for real-valued functions of a real variable. Students are able to apply this knowledge to solve simple practical problems arising in the pure and applied sciences.

Course contents

  • The real number system; supremum and infimum. Natural numbers and the principle of mathematical induction.
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  • Elementary functions.
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  • Limits of sequences. Basic topology of the real numbers.
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  • Limits and continuity of functions of one real variable.
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  • Differential calculus for real-valued functions of a real variable. Differentiation rules, monotonicity criteria, Rolle's Theorem, the Mean Value Theorems of Lagrange and Cauchy, and Taylor's formula.
  • Readings/Bibliography

    E. Lanconelli, Lezioni di Analisi Matematica 1, ed. Pitagora

    P. Marcellini - C. Sbordone: Analisi Matematica 1, ed. Liguori

    E. Giusti, Analisi Matematica 1, ed. Boringhieri

    Per esercizi:

    M. Bramanti, Esercitazioni di Analisi Matematica 1, ed. Esculapio

    P. Marcellini - C. Sbordone: Esercitazioni di Matematica, volume 1, parte prima, ed. Liguori

    E. Giusti, Esercizi e complementi di Analisi Matematica, volume 1, ed. Boringhieri

    Teaching methods

    Lectures and classroom exercises.

    Assessment methods

    The examination consists of a preliminary written test and a written theory one before a colloquium.

    The written test consists of some exercises related to the arguments of the course, it lasts two and a half hours. In order to sustain the written test the student must register at least four days before the test through AlmaEsami [https://almaesami.unibo.it/] . The maximum grade is 15. The student is admitted to the written-colloquial subsequent step, that it will be hold in a different day, if the realized score is grater or equal to 8.

    The written test remains valid for the written-colloquial subsequent examination only in the same period. Namely only in January-February if the written exam has been held in January or February, or alternatively on June-July, if the written exam has been held in June-July, or alternatively on September, if the written exam has been held in September.

    The time for the written test before the colloquium is 45' and it takes place for all the admitted candidates at the beginning of the day in which is scheduled the session. The maximum score is 8. After it, there will be a colloquium only if the sum between the score in the first written exam is at least 15. Both the evaluations mainly concern the theoretical aspects of the course. The student must show to know the concepts explained during the course (in particular definitions, theorems and their proofs) and how to connect with each other. The maximum score at the colloquium is 12.

    How to determine the final grade: in case the score at the colloquium were les or equal to 3, then the candidate the exam has to be repeated because it is not sufficient, even if the sum of the three grades were grater or equal to 18.

    In case the colloquium were graded more or equal to 4, the the final grade is obtained by the sum of the three grades. Here below some examples:

    (8,7,3), then the student is rejected; (8,7,4), the student is promoted with 19.

    (10,7,3), rejected; (15,8,3), rejected; (15,8,4), grade 26; (8,7,12), grade 27. (15, 0, 12), grade 27.

    During the written examination and the written pre-oral test, the use of notes, books, or any kind of electronic device is not permitted. In particular, the use of mobile phones, smartwatches, and earphones is strictly prohibited.

    Furthermore, the use of artificial intelligence (AI) tools is prohibited in all examination assessments.

    During the oral examination, students are expected to demonstrate a thorough understanding of the concepts presented in the course (in particular, definitions, theorems, and their proofs) and to be able to establish meaningful connections among them.

    Teaching tools

    Lecture notes concerning the lessons on Virtuale (only in Italian) https://virtuale.unibo.it/. Tutor (if assigned).

    Office hours

    See the website of Eleonora Cinti