Università di Pavia - Offerta formativa

MATHEMATICAL ANALYSIS 1

Enrollment year

2021/2022

Academic year

2021/2022

Regulations

DM270

Academic discipline

MAT/05 (MATHEMATICAL ANALYSIS)

Department

DEPARTMENT OF ELECTRICAL,COMPUTER AND BIOMEDICAL ENGINEERING

Course

BIOENGINEERING

Curriculum

PERCORSO COMUNE

Year of study

1°

Period

1st semester (27/09/2021 - 21/01/2022)

ECTS

9

Lesson hours

83 lesson hours

Language

Italian

Activity type

WRITTEN AND ORAL TEST

Teacher

MAZZOLENI DARIO CESARE SEVERO (titolare) - 9 ECTS

Prerequisites

Those required for the enrollment in the Faculty of Engineering.

Learning outcomes

The course is aimed at providing the basic knowledge of sequences, series, calculus (differential, integral) for real functions of one real variable, together with an introduction to ordinary differential equations. Lectures will be mainly focused on the comprehension of notions (definitions, results), although some proofs will still be detailed. Many examples and exercises will be presented througout the course. By the end of the course the students are expected to be able to correctly handle limits, derivatives, function graphs, integrals, series, differential equations, and the corresponding theoretical results.

Course contents

1. Preliminaries.

Recalls and complements on: set theory, mathematical logic, real numbers. Complex numbers: algebraic, trigonometric, and exponential form. Operations on complex numbers; algebraic equations on the complex field.

2. Functions, Limits, Continuity. Sequences and Series.

Functions: definitions, graphs; invertible functions; odd and even functions; monotone functions; periodic functions; operations on functions; nested functions. Elementary functions and corresponding graphs. Limits of functions: definitions, operations on limits. Continuous functions. Discontinuity points and their classification. Global properties of continuous functions. Limits of real sequences. Real series: definitions and basic examples; series with positive terms (and convergence tests); absolute and simple convergence.

3. Differential Calculus in one real variable and Applications.

Derivative of a function: definition and properties, applications in Geometry and Physics. Derivation rules and calculus. Fundamental theorems of differential calculus. Primitives and indefinite integrals. Successive derivatives. Function study: extrema, monotonicity, convexity. De l'Hopital rules.

4. Integral Calculus.

Definite integrals: definitions and basic properties, applications in Geometry and Physics. Fundamental theorems of integral calculus. Integration techniques. Improper integrals.

5. Ordinary Differential Equations.

Introduction to ordinary differential equations. The Cauchy problem. Separation of variables. Linear ordinary differential equations of the first order. Linear ordinary differential equations of the second order with constant coefficients.

Recalls and complements on: set theory, mathematical logic, real numbers. Complex numbers: algebraic, trigonometric, and exponential form. Operations on complex numbers; algebraic equations on the complex field.

2. Functions, Limits, Continuity. Sequences and Series.

Functions: definitions, graphs; invertible functions; odd and even functions; monotone functions; periodic functions; operations on functions; nested functions. Elementary functions and corresponding graphs. Limits of functions: definitions, operations on limits. Continuous functions. Discontinuity points and their classification. Global properties of continuous functions. Limits of real sequences. Real series: definitions and basic examples; series with positive terms (and convergence tests); absolute and simple convergence.

3. Differential Calculus in one real variable and Applications.

Derivative of a function: definition and properties, applications in Geometry and Physics. Derivation rules and calculus. Fundamental theorems of differential calculus. Primitives and indefinite integrals. Successive derivatives. Function study: extrema, monotonicity, convexity. De l'Hopital rules.

4. Integral Calculus.

Definite integrals: definitions and basic properties, applications in Geometry and Physics. Fundamental theorems of integral calculus. Integration techniques. Improper integrals.

5. Ordinary Differential Equations.

Introduction to ordinary differential equations. The Cauchy problem. Separation of variables. Linear ordinary differential equations of the first order. Linear ordinary differential equations of the second order with constant coefficients.

Teaching methods

Lectures (hours/year in lecture theatre): 45

Practical class (hours/year in lecture theatre): 38

Practicals / Workshops (hours/year in lecture theatre): 0

Practical class (hours/year in lecture theatre): 38

Practicals / Workshops (hours/year in lecture theatre): 0

Reccomended or required readings

M. Bramanti, C.D. Pagani e S. Salsa. Analisi Matematica I. C.E. Zanichelli, Bologna, 2008-2009.

Assessment methods

The exam consists of a written examination and an optional oral examination: the written examination is further divided into two parts: exercises (first part) and theoretical questions (second part). Written and optional oral examinations should be passed within the same session. The oral examination is based on definitions, examples and counterexamples, theorems (some with proofs).

Further information

The exam consists of a written examination and an optional oral examination: the written examination is further divided into two parts: exercises (first part) and theoretical questions (second part). Written and optional oral examinations should be passed within the same session. The oral examination is based on definitions, examples and counterexamples, theorems (some with proofs).

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