27991 - Mathematical Analysis T-1

Learning outcomes

The student will be able to solve the typical problems of Mathematical Analysis concerning limits, continuity, derivability and integration.

Course contents

Introduction: Logic, properties of the real numbers and of N, Z, Q, R, induction principle.

Functions: Real-valued functions of one real variable; injectivity, surjectivity, invertibility, inverse function,
composition of function, monotone functions. Elementary functions: basic facts.

Complex numbers: Definiton of the field of the complex numbers. Algebraic form. Modulus and argument of a complex number.
Exponential form of a complex number. De Moivre's formula. Complex roots of a complex number. Algebraic equations in C.

Sequences: limits, convergence tests, indeterminate forms.

Limits: Accumulation point, definitions of limit; one/two-sided limits. Elementary properties of limits: unicity, locality.
Algebraic properties of the limit, comparison theorems, limits of monotone functions. Indeterminate forms.

Continuity: Definition of a continuous function of one real variable. The Weierstrass Theorem, the Bolzano Theorem and the Intermediate Value Theorem.
Continuity of the composition of two continuous functions.

Differential calculus: Definition of a differentiable function and of the derivative of a function. The algebra of derivatives.
The mean value theorems and their application in the study of the monotonicity of a function.
Higher order derivatives. Hopital's Rule. Taylor's formula.
Local maxima and minima of a function: definitions, necessary conditions, sufficient conditions. Convex functions.

Integration: Definition of the Riemann integral. Properties of the integral: linearity, additivity, monotonicity, the mean value theorem.
The fundamental theorems of the integral calculus. The theorems of integration by substitution and of integration by parts.

Improper integrals: basic definitions and the comparison theorem.

Series: Basic definitions; necessary condition for the convergence of a series; series of nonnegative real numbers: comparison theorems; the root and ratio criteria. Leibniz theorem.

Readings/Bibliography

Analisi Matematica I di C. Canuto and A. Tabacco, Springer.

Analisi Matematica 1 di M. Bramanti, C.D. Pagani and S. Salsa, Zanichelli.

Teaching methods

Frontal lectures.

Assessment methods

Written and oral examinations.


A detailed program for the oral part will be published in the institutional web-site VIRTUALE. The written part of the examination will check the knowledge of all the topics presented in the exercises, regularly published online. During the oral examination, the student will be asked theorems/proofs/examples/definitions, presented during the lectures.


Dates:


3 exams in January/February


1 in June, 1 in July, 1 in September


In order to take the written/oral examinations, students must register at least five days before the exam through the website AlmaEsami https://almaesami.unibo.it/
The written test remains valid for the oral exam in the same examination period.

Teaching tools

Additional material regularly upoloaded on Virtuale.

Office hours

See the website of Eugenio Vecchi

See the website of Filippo Morabito

See the website of Giovanni Eugenio Comi