#
Mathematical Analysis 1

A.Y. 2018/2019

Learning objectives

Introduction to some basic concepts for the study of differential calculus in one and several variables. In particular, definition of number fields and metric spaces for the study of sequences. Study of series in finite-dimensional setting. Limits, continuity and differential calculus for real functions of a real variable.

Expected learning outcomes

Basic computational techniques.

**Lesson period:**
First semester

**Assessment methods:** Esame

**Assessment result:** voto verbalizzato in trentesimi

Course syllabus and organization

### Analisi Matematica 1 (ediz. 1)

Lesson period

First semester

**Course syllabus**

The real and complex numbers. The set of real numbers R and its characterization as an ordered field with the greatest lower bound property. Existence of n-th roots of positive real numbers. The extended real line. Euclidean spaces. The complex field C: operations, De Moivre's formula, n-th roots, logarithms. The fundamental theorem of algebra. Review of the elementary theory of sets, maps between sets, relations and functions. Sets of the same cardinality. Finite, countable and uncountable sets. The uncountability of R.

Metric spaces. Definition, examples and metric balls as neighborhoods. Open, closed, bounded, compact and connected sets. The extended real number system as a metric space. Sequences. Convergent sequences in metric spaces and their properties. Cauchy sequences. Subsequences. Sequences in R. Limits and operations on limits. Monotone sequences. The limit that defines Nepero's number "e" and applications.

Infinite series of real numbers. Series in R. Convergence, divergence and irregularity. Absolute convergence. Cauchy's criterion. Sufficient conditions for absolute convergence. Alternating series and Leibniz's criterion. Operations on series. Rearrangements.

Limits and continuity. Limits of functions. Reformulation using sequences. Continuity of functions between metric spaces. Pre-images of open sets. Relationships between continuity, compactness and connectedness. Continuity of composite and inverse functions. Uniform continuity. Real functions of one real variable. Existence of limits for monotone functions. Limits and algebraic operations. Indeterminate forms of limits. Asymptotes. Discontinuities.

Differential calculus for real functions of one real variable. Differentiability and the derivative function. Rules of differentiation: algebraic operations, composite and inverse functions. The theorems of Rolle, Cauchy and Lagrange. Applications of differential calculus to the study of functions: monotonicity, convexity and optimization (local and global). De L'Hospital's theorem. Taylor formulas and applications.

Metric spaces. Definition, examples and metric balls as neighborhoods. Open, closed, bounded, compact and connected sets. The extended real number system as a metric space. Sequences. Convergent sequences in metric spaces and their properties. Cauchy sequences. Subsequences. Sequences in R. Limits and operations on limits. Monotone sequences. The limit that defines Nepero's number "e" and applications.

Infinite series of real numbers. Series in R. Convergence, divergence and irregularity. Absolute convergence. Cauchy's criterion. Sufficient conditions for absolute convergence. Alternating series and Leibniz's criterion. Operations on series. Rearrangements.

Limits and continuity. Limits of functions. Reformulation using sequences. Continuity of functions between metric spaces. Pre-images of open sets. Relationships between continuity, compactness and connectedness. Continuity of composite and inverse functions. Uniform continuity. Real functions of one real variable. Existence of limits for monotone functions. Limits and algebraic operations. Indeterminate forms of limits. Asymptotes. Discontinuities.

Differential calculus for real functions of one real variable. Differentiability and the derivative function. Rules of differentiation: algebraic operations, composite and inverse functions. The theorems of Rolle, Cauchy and Lagrange. Applications of differential calculus to the study of functions: monotonicity, convexity and optimization (local and global). De L'Hospital's theorem. Taylor formulas and applications.

MAT/05 - MATHEMATICAL ANALYSIS - University credits: 9

Practicals: 44 hours

Lessons: 45 hours

Lessons: 45 hours

Professors:
Calanchi Marta, Tarallo Massimo Emilio

### Analisi Matematica 1 (ediz. 2)

Responsible

Lesson period

First semester

**Course syllabus**

The real and complex numbers. The set of real numbers R and its characterization as an ordered field with the greatest lower bound property. Existence of n-th roots of positive real numbers. The extended real line. Euclidean spaces. The complex field C: operations, De Moivre's formula, n-th roots, logarithms. The fundamental theorem of algebra. Review of the elementary theory of sets, maps between sets, relations and functions. Sets of the same cardinality. Finite, countable and uncountable sets. The uncountability of R.

Metric spaces. Definition, examples and metric balls as neighborhoods. Open, closed, bounded, compact and connected sets. The extended real number system as a metric space. Sequences. Convergent sequences in metric spaces and their properties. Cauchy sequences. Subsequences. Sequences in R. Limits and operations on limits. Monotone sequences. The limit that defines Nepero's number "e" and applications.

Infinite series of real numbers. Series in R. Convergence, divergence and irregularity. Absolute convergence. Cauchy's criterion. Sufficient conditions for absolute convergence. Alternating series and Leibniz's criterion. Operations on series. Rearrangements.

Limits and continuity. Limits of functions. Reformulation using sequences. Continuity of functions between metric spaces. Pre-images of open sets. Relationships between continuity, compactness and connectedness. Continuity of composite and inverse functions. Uniform continuity. Real functions of one real variable. Existence of limits for monotone functions. Limits and algebraic operations. Indeterminate forms of limits. Asymptotes. Discontinuities.

Differential calculus for real functions of one real variable. Differentiability and the derivative function. Rules of differentiation: algebraic operations, composite and inverse functions. The theorems of Rolle, Cauchy and Lagrange. Applications of differential calculus to the study of functions: monotonicity, convexity and optimization (local and global). De L'Hospital's theorem. Taylor formulas and applications.

Metric spaces. Definition, examples and metric balls as neighborhoods. Open, closed, bounded, compact and connected sets. The extended real number system as a metric space. Sequences. Convergent sequences in metric spaces and their properties. Cauchy sequences. Subsequences. Sequences in R. Limits and operations on limits. Monotone sequences. The limit that defines Nepero's number "e" and applications.

Infinite series of real numbers. Series in R. Convergence, divergence and irregularity. Absolute convergence. Cauchy's criterion. Sufficient conditions for absolute convergence. Alternating series and Leibniz's criterion. Operations on series. Rearrangements.

Limits and continuity. Limits of functions. Reformulation using sequences. Continuity of functions between metric spaces. Pre-images of open sets. Relationships between continuity, compactness and connectedness. Continuity of composite and inverse functions. Uniform continuity. Real functions of one real variable. Existence of limits for monotone functions. Limits and algebraic operations. Indeterminate forms of limits. Asymptotes. Discontinuities.

Differential calculus for real functions of one real variable. Differentiability and the derivative function. Rules of differentiation: algebraic operations, composite and inverse functions. The theorems of Rolle, Cauchy and Lagrange. Applications of differential calculus to the study of functions: monotonicity, convexity and optimization (local and global). De L'Hospital's theorem. Taylor formulas and applications.

MAT/05 - MATHEMATICAL ANALYSIS - University credits: 9

Practicals: 44 hours

Lessons: 45 hours

Lessons: 45 hours

Professors:
Salvatori Maura Elisabetta, Vesely Libor