## Learning objectives

Knowledge and understanding.

At the end of the lectures, students should have acquired very good knowledge and understanding of the numerical field N, Q, Z and R, of the numerical sequences and series, of the differential and integral calculus for functions of one variable.

This course will contribute to making students able to understand advanced texts in Mathematics and to consult research papers in mathematics.

Applying knowledge and understanding.

By means of the classroom exercises students learn how to apply the theoretical knowledges to solve concrete problems, such as optimization problems.

Making judgements.

Students must be able to evaluate coherence and correctness of results obtained by themselves or by others.

Students should be also able to produce precise mathematical arguments clearly identifying the assumptions and the conclusions.

Communication skills.

Students must be able to communicate in a clear and precise way mathematical statements in the field of study, also in a context broader than mere calculus. Through the front lectures and the assistance of the teacher, the students acquire the specific and appropriate scientific vocabulary.

This course contributes to making the students able to work in groups and to work with a good degree of autonomy.

Learning skills.

The student who has attended the course Analisi Matematica 1, is able to deepen autonomously his/her knowledge of numerical sequences, differential calculus for functions of one variable, starting from the basic and fundamental knowledges provided by the course. He/She will be also able to consult specialized textbook, even outside the topics illustrated during the lectures. This to facilitate the learning of the other activities of the degree course in Physics, which use notions from Mathematical Analysis. This course contributes to furnishing a flexible mindset to the student, which helps him/her to easily enter into the labour market, being able to face new problems. Moreover, the course contributes, together with other courses of the bachelor program, to making the student able to acquire new knowledges in the mathematical/physics fields but also in the labour market field, through an autonomous study. Finally, the course contributes to making the student able to continue the studies in Physics or in other scientific disciplines with a high degree of autonomy.

## Prerequisites

Elementi di Matematica related to the whole course Analisi Matematica 1.

## Course unit content

The course aims at providing students with the fundamental notions on:

the numerical sets;

fundamental concepts of infinitesimal calculus for functions of one variable numerical sequences

integral of functions of one variable;

improper integral and numerical series;

complex numbers;

basic notions on ordinary differential equations

## Full programme

1. REAL NUMBERS

Axiomatic definition of real numbers, maximum, minimum, least upper and greatest lower bound; integer part and modulus of real numbers; powers, roots, n-th roots of non-negative numbers; rational and irrational numbers and their density in the set of all the real numbers; intervals, distance; neighborhoods, accumulation points, isolated points, interior points; closed sets, open sets, frontier. The principle of induction.

2. SEQUENCES OF REAL NUMBERS

The concept of numerical sequence, convergent and divergent sequences, uniqueness of the limit; infinitesimal sequences; subsequences, a criterion for the non existence of the limit of a sequence; limit of the sum, product, quotient of sequences, permanence of the sign, comparison theorems; monotone sequences; the Nepero’s number; sequences defined by recurrence; Cauchy sequences; Bolzano-Weierstrass theorem.

3. FUNCTIONS AND LIMITS

One to one, surjective and bijective functions; inverse functions; graphs; monotone functions; exponential, and logarithmic functions. Limits of functions with real values, uniqueness of the limit, limits of the restrictions; limit of the sum, product, quotient of two functions; permanence of the sign, comparison theorems; right and left limits; limits of monotone functions.

4. CONTINUITY

The concept of continuous function, restrictions of continuous functions, composition of continuous functions; sum, product, quotient of continuous functions; examples of continuous functions; discontinuity, examples of discontinuous functions; zeroes of continuous functions defined in an interval; continuity and intervals; continuity and monotonicity; continuity of inverse functions; Weierstrass theorem.

5. DIFFERENTIAL CALCULUS

Incremental ratio, derivatives, right and left derivatives; geometrical meaning of the derivative; derivation rules: derivatives of the sum, product, quotient of two functions; derivatives of composite functions and inverse functions; derivatives of elementary functions; relative maxima and minima; stationary points; Rolle's theorem and Lagrange's theorem and their geometrical interpretation, Cauchy's theorem and de l'Hopital's theorem. Taylor expansions of smooth enough functions with Peano's and Lagrange's form of remainder.

6. APPLICATIONS OF DIFFERENTIAL CALCULUS

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

Convexity. Derivative test for local max/min.

7. INTEGRAL CALCULUS

Primitives and indefinite integrals; rules of indefinite integration; definite integrals; Cauchy-Riemann integral; fundamental theorem of calculus; applications of integrals.

8. 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

9. 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

10. 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

The course schedules 5 hours per week of lectures and classroom exercises. During the lectures the fundamental properties of the numerical set, of the basic results of calculus for functions of one variable, of the integral calculus for functions of one variable will be analyzed and discussed. Students will be provided also with the basic results on sequences and series of real numbers, and on ordinary differential equations. The classroom exercises aim at showing how and where the abstract results can be applied to make the students understand better the relevance of what they are studying.

The didactic activities of the first half of the course are developed also with the help of a tablet PC which projects on a screen the notes the teacher is writing. A the end of each lesson, a pdf file with the notes of the lecture is uploaded on the elly website.

## 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 18.

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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