The course aims to provide the basic knowledge of differential and integral calculus for real functions of a real variable, the main features of the theory of sequences and numerical series, the fundamental notions of complex numbers. Ample space will be given to examples and exercises.
Course Prerequisites
Graphs of elementary functions in one variable. Solving inequalities of various types. Trigonometry
Teaching Methods
Frontal and small group lessons and exercises
Assessment Methods
The exam consists of a written test and an oral test (optional and conditioned on the outcome of the written test or at the request of the teacher) on the topics of the course. The written test lasts 2 hours and consists of 9 questions with a score between 3 and 5 points for each question. For more detailed information see: http://matematica.unipv.it/rocca/
Texts
M. Bramanti, C.D. Pagani, S. Salsa, Mathematical Analysis 1, Zanichelli, Bologna, 2009. M. Bramanti, Mathematical Analysis Exercises 1, Ed. Esculapio, Bologna, 2011.
Contents
1. Main properties of numerical sets and in particular of the set of real numbers (totally ordered field, continuity axiom). Field of complex numbers. 2. Functions: definitions; generalities, graphs; invertible functions; even, odd, periodic functions; operations on functions; composite functions; elementary functions and their graphs. Limits of functions: definitions; operations on limits. Continuous functions; points of discontinuity and their classification; global properties of continuous functions. 3. Derivative of a function: definition and properties; applications to Geometry and Physics. Rules of derivation and calculation of derivatives. Fundamental theorems of differential calculus. Successive derivatives; Taylor's formula, search for extreme points, De L'Hopital's Theorem. 4. Numerical sequences; limits of sequences. Numerical series: definition; first properties and examples; series with positive terms (convergence criteria); absolute convergence and simple convergence. 5. Definite integrals: definition and main properties; applications to Geometry and Physics. Fundamental theorems of integral calculus. Techniques for integrating and calculating integrals. Improper integrals.
