Learning objectives
The course aims to provide the student with the basic concepts of Mathematical Analysis, with particular reference to single variable integral calculus, numerical series, and rudiments of ordinary differential equations. These are some of the fundamental tools for a basic preparation in mathematics as well as for the understanding of parallel and subsequent teachings.
At the end of the course the student should:
1) Know how to solve exercises and problems on teaching topics, thus achieving the following operational skills: computation of indefinite and definite integrals; calculation of areas; evaluate convergence and, in some cases, calculate improper integrals; study some integral functions; evaluate convergence and, in some cases, calculate numerical series; know how to work with complex numbers, for example how to compute powers and roots of complex numbers; know how to solve separable first order differential equations, and constant coefficient second order equations.
2) Understand the concepts and be able to express them in a precise mathematical language; know the basic proof techniques of and be able to reproduce the proofs of the course.
Prerequisites
The contents of the first Module of Mathematical Analysis 1
Course unit content
Single variable integral calculus
Improper integrals and numerical series
Complex numbers
Ordinary differential equations
Full programme
INTEGRAL CALCULUS
Primitives and indefinite integrals; rules of indefinite integration; definite integrals; Cauchy-Riemann integral; fundamental theorem of calculus; applications of integrals.
IMPROPER INTEGRALS AND NUMERICAL SERIES
Definition of improper integral; a few simple examples; some convergence criteria; definition of numerical series; geometric series; telescopic series;
harmonic series; positive term series; convergence criteria: comparison, root and ratio; alternating series; comparison between improper integrals and numerical series
COMPLEX NUMBERS
Definition and first properties of complex numbers;
representations of complex numbers: Cartesian, trigonometric, exponential; powers and roots in the complex field; Fundamental theorem of Algebra
ORDINARY DIFFERENTIAL EQUATIONS
General definitions; initial value problem; first order equations: geometric interpretation (isocline); solving separable variable equations; second order differential equations with constant coefficients: homogeneous equation, equation with forcing term, method of variation of constants; applications: Malthusian growth, logistic growth, harmonic oscillator, damped oscillations.
Bibliography
Claudio Canuto, Anita, Analisi Matematica 1,
Ed. Pearson Italia (2021)
Teaching methods
Lectures at the blackboard held by the teacher at the blackboard, in which the theory is exposed and is applied to various examples and to the resolution of exercises.
Lecture notes, and further material for exercises will be provided through the Elly platform
Assessment methods and criteria
Final exam, consisting of a written test and an oral interview
Other information
- - -
