Learning outcomes of the course unit
Knowledge and understanding: At the end of this course the student should know the essential definitions and results of analysis in a single variable, and she/he should be able to grasp how these enter the solution of problems. Applying knowledge and understanding: The student should be able to apply the aforementioned notions to solve moderately difficult problems, and to understand how they relate to concepts seen in different classes. Making judgments: The student should be able to evaluate coherence and correctness of the obtained results during the written test. Communication skills: The student should be able to communicate in a clear and in an enough precise way.
Elementary algebra; trigonometry; analytical geometry; rational powers; exponentials and logarithms; elementary functions
Course contents summary
Full proof is provided for a limited number of statements
Infimum and supremum (2 hours).
Real functions (4 hours): extremes of real functions; monotonic functions; even and odd functions; powers; absolute value; trigonometric functions; hyperbolic functions; real functions plots.
Sequences (8 hours): sequences and their limits; comparison and algebraic theorems; fundamental examples; Nepero’s “e”number; continuous Functions (10 hours): limit of a function; continuity; properties of continuous functions; continuous functions on an interval (roots, intermediate values); Weierstrass theorem; infinitesimal. Important limits.
Derivatives (22 hours): derivative definition and its relevant properties; algebraic operations on derivatives; derivatives and function local properties; Rolle, Lagrange, Cauchy theorems; indeterminate forms and de l’Hôpital theorem, Taylor formulas and different remainders, asymptotic expansion; functions qualitative study.
Integrals (20 hours): antiderivatives; integration techniques; improper integrals; rational functions integration.
Series (6 hours): series definition and properties; convergence criteria for series with non-negative terms; alternating series.
Both theoretical and practical lectures closely follow
E. ACERBI e G. BUTTAZZO: analisi matematica ABC- 1. funzioni di una variabile. Pitagora Editrice Bologna.
D. MUCCI: Analisi matematica esercizi vol.1, Pitagora editore, Bologna, 2004
Lectures are held in the classroom, encompassing both theoretical and applied aspects. Moreover, exercises are solved by students with the guidance of the Professor, so as to verify the degree of comprehension and knowledge of the students.
Assessment methods and criteria
The final exam consists in a written and an oral session. Students are admitted to the oral sessions only if they pass the written examination. In the written examination 2 open questions are asked. The students should exhibit calculus skills and mastery of different subjects taught in the course. Marks are given to each question, according to theoretical correctness, precision of execution, precision of exposition.
The oral examination consists of a discussion about the written examination and of questions to verify the level of comprehension of the theoretical parts of the course.