# Mathematical Analysis I A

## Objectives

The main goal of this discipline is to introduce the students to the universe of rigorous thinking. We start by the natural numbers and, in rigorous and construtive way, introduce all main elements of mathematical analysis up to the concept of differentiability.

## General characterization

##### Code

10969

##### Credits

9.0

##### Responsible teacher

Fábio Augusto da Costa Carvalho Chalub

##### Hours

Weekly - 6

Total - 72

##### Teaching language

Português

### Prerequisites

High school math.

### Bibliography

The main reference is:

Rudin, Walter. Principles of Mathematical Analysis, 3rd edition. Mcgraw-Hill.

An alternative bibliography is (in Portuguese):

Lima, Elon Lages. Análise Real, vol 1. Funções de uma variável real. Coleção Matemática Universitária. Instituto Nacional de Matemática Pura e Aplicada, Brasil.

See also free online textbooks:

Trench, William F.. Introduction to Real Analysis.

https://digitalcommons.trinity.edu/mono/7/

Hirst, Keith E., Calculus of One Variable, Springer

https://link.springer.com/book/10.1007%2F1-84628-222-5 (free download only on the university network)

A smooth introduction to part of the topics in this course can be find at

https://openstax.org/details/books/calculus-volume-1

### Teaching method

6hs per week, including theoretical classes and exercises.

### Evaluation method

**Presence in class of optional.**

Students are evaluted by two tests (T1 and T2); it is required minimum grade of 7 in the second test.

If T2<7, then MF=min(9,(T1+T2)/2); otherwise MF=(T1+T2)/2.

The final grade consists in the rounding of MF to the nearest integer (n.5 is rounded to n+1).

For students that to not obtain approval in tests ("continuous evaluation") it is possible to do a final exam ("recurso").

## Subject matter

1. Real numbers. Topological notions in IR. Mathematical induction.

2. Sequences of real numbers: Limits. Infinite limits. Limits at infinity. Monotone sequences. Convergent sequences. Subsequences. Upper limit and lower limit. Cauchy sequence. Completeness of IR.

3. Single real variable functions: limits and continuity. Properties of continuous functions; Bolzano’s theorem. Weierstrass theorem. Uniform continuity. Lispschitz continuous functions. Cantor’s theorem.

4. Differential calculus: Derivatives, physical and geometric interpretations and properties. Fundamental theorems: Rolle, Darboux, Lagrange and Cauchy. Cauchy rule. Taylor’s formula and applications. Extrema, concavity and inflection points.