## 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

APPLICATIONS OF DIFFERENTIAL CALCULUS

Consequences of the Mean value theorem: Derivative test for monotonic functions.

Convexity. Derivative test for local max/min.

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.

## Bibliography

Teoria

D. Addona, B. Gariboldi, L. Lorenzi: AM1 Analisi Matematica 1. Società editrice Esculapio 2012.

Esercizi

D. Addona, B. Gariboldi, L. Lorenzi: AM1 Analisi Matematica 1. Esercizi. Società editrice Esculapio 2013.

## 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

The exam of the course Analisi Matematica 1 consists of a written part and an oral part in different dates.

The written part is based on exercises (indicatively 3 or 4) and it is aimed at evaluating the skills of the student in applying the abstract results proposed during the course to some concrete situations. The maximum score of the written part of the exam is 30. The written part is successful if the student reaches a score non inferior to 15.

The oral part is aimed at evaluating 1) the knowledge of the abstract results seen during the course and their proofs 2) the correct use of the mathematical terms, 3) the knowledge of those arguments which have not been included into the written test. The final vote will be given by a

weighted average of the votes of the written and oral part of the exam.

## Other information

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