Academic Year 2020/2021 - 1° Year

MAT/05 - 9 CFU - Annual Tuition

The aim of the course of Analisi Matematica I is to give the basic skills real and complex numbers,

differential and integral calculus for real functions of one real variable.

In particular, the learning objectives of the course, according to the Dublin descriptors, are:

**Knowledge and understanding**: The student will learn some basic concepts of Mathematical

Analysis and will develop both computing ability and the capacity of manipulating some common

mathematical structures, as complex numbers, limits, derivatives and integrals for real functions of

one real variable, numerical series.**Applying knowledge and understanding:**The student will be able to apply the acquired

knowledge in the basic processes of mathematical modeling of classical problems arising from

Engineering.**Making judgements**: The student will be stimulated to autonomously deepen his/her knowledge

and to carry out exercises on the topics covered by the course. Constructive discussion between

students and constant discussion with the teacher will be strongly recommended so that the

student will be able to critically monitor his/her own learning process.**Communication skills:**The frequency of the lessons and the reading of the recommended books

will help the student to be familiar with the rigor of the mathematical language. Through constant

interaction with the teacher, the student will learn to communicate the acquired knowledge with

rigor and clarity, both in oral and written form. At the end of the course the student will have

learned that mathematical language is useful for communicating clearly in the scientific field.**Learning skills:**The student will be guided in the process of perfecting his/her study method. Inparticular, through suitable guided exercises, he/she will be able to independently tackle new topics, recognizing the necessary prerequisites to understand them.

The course is divided into two parts. The first partdeals with the construction and the properties of the field of real numbers, complex numbers, basic topology notions, sequences and functions, differential

calculus and its applications. The second part deals with series and integrals. The lessons are

complemented by exercises related to the topic of the course and both the lessons and the exercises will

be carried out in frontal mode. Should teaching be carried out in mixed mode or remotely, it may be

necessary to introduce changes with respect to previous statements, in line with the program planned

and outlined in the Syllabus.

**Sets of numbers**. Real numbers. The ordering of real numbers. Completeness of R. Factorials and

binomial coefficients. Relations in the plane. Complex numbers. Algebraic operation. Cartesian

coordinates. Trigonometric and exponential form. Powers and nth roots. Algebraic equations.-
**Limits.**Neighbourhoods. Real functions. Limits of functions. Theorems on limits: uniqueness and

sign of the limit, comparison theorems, algebra of limits. Indeterminate forms of algebraic and

exponential type. Substitution theorem. Limits of monotone functions. Sequences. Limit of a

sequence. Sequential characterization of a limit. Cauchy's criterion for convergent sequences.

Infinitesimal and infinite functions. Local comparison of functions. Landau symbols and their

applications. **Continuity**. Continuous functions. Sequential characterization of the continuity. Points of

discontinuity. Discontinuities for monotone functions. Properties of continuous functions

(Weierstrass's theorem, Intermediate value theorem). Continuity of the composition and the

inverse functions.**Differential Calculus**. The derivative. Derivatives of the elementary functions. Rules of

differentiation. Differentiability and continuity. Extrema and critical points. Theorems of Rolle,

Lagrange and Cauchy. Consequences of Lagrange's Theorem. De L'Hôpital Rule. Monotone

functions. Higher-order derivatives. Convexity and inflection points. Qualitative study of a function.

Recurrences.**Numerical series**. Round-up on sequences. Numerical series. Series with positive terms.

Alternating series. The algebra of series. Absolute and Conditional Convergence.**Integrals.**Areas and distances. The definite integral. The Fundamental Theorem of Calculus.

Indefinite integrals and the Net Change Theorem. The substitution rule. Integration by parts.

Trigonometric integrals. Trigonometric substitution. Integration of rational functions by partial

fractions. Strategy for integration. Impropers integrals. Applications of integration.

1.Di Fazio G., Zamboni P., Analisi Matematica 1, Monduzzi Editoriale.

2. Di Fazio G., Zamboni P., Eserciziari per l'Ingegneria, Analisi

Matematica 1, EdiSES.

3. D'Apice C., Manzo R. Verso l'esame di Matematica, vol. 1 e 2,

Maggioli editore.