Mathematical Analysis I (2017/2018) - Departamento de Matemática
Description

This is an introdutory course to the main concepts, definitions and techniques required for the analysis of sequences and real functions of real variable.

Objectives

Domain of the basic techniques required for theMathematical Analysis of real functions of real variable.

The students should acquire not onlycalculus capabilities fundamental to the acquisition of some of the knowledge lectured in Physics, Chemistry and other Engineering subjects, but also to develop methods of solid logic reasoning and analysis.

Beinga first coursein Mathematical Analysis,it introduces some of the concepts which will be deeply analyzed and generalized in subsequent courses.

Syllabus
  1. Basic topology of the real numbers.
    1.1 Neighborhood of a Point. Interior, Exterior, Frontier, Isolated, Adherent and and Accumulation Point.
    1.2 Open, Closed, Limited and Compact Set.

    2. Mathematical induction and sequences
    2.1 Mathematical Induction.
    2.2 Limit of a sequence. Algebra of Limits. Subsequences and Sublimits. Theorem of Squeeze Sequences. Bolzano-Weierstrass theorem and other fundamental theorems. Cauchy''s sequences.

    3. Limits and Continuity in R
    3.1 Convergence according to Cauchy and according to Heine. Algebra of Limits.
    3.2 Continuity of a Function in a Point and in a Set. Prolonging a Function by Continuity. Bolzano''s theorem and Weierstrass''s theorem. Continuity of the Composite Function and Continuity of the Inverse Function. Inverse Trignometric Functions.

    4. Differential Calculus in R
    4.1 Derivative Definition. Physical and Geometric Interpretation. Differentiability. Algebra of Derivatives. Derivative of the Composite and Derivative of the Inverse Function. Derivatives of Inverse Trignometric Functions. Rolle''s Theorem, Lagrange''s Theorem. Derivative and Monotony. Darboux''s Theorem and Cauchy''s Theorem. Indeterminations and Cauchy’s Rule.
    4.2 Taylor''s theorem and applications to the study of extremes and concavities.


5. Integration in R

5.1 Primitives. Primitives by Parts. Primitives by Substitution. Primitives of Rational Functions. Primitives of Irrational Functions and Transcendent Functions.
5.2 Integral of Riemann. Theorem of the Mean Value. Fundamental Theorem of Integral Calculus. Barrow Rule. Integration by Parts and integration by Substitution. Application to the Calculus of Areas.

Bibliography

Adopted text

  1. Ana Alves de Sá e Bento Louro, Cálculo Diferencial e Integral em ℝ

Recommended Bibliography

  1. Alves de Sá, A. e Louro, B. - Cálculo Diferencial e Integral em ℝ, Exercícios Resolvidos, Vol. 1, 2, 3
  2. Anton, H. - Cálculo, um novo horizonte, 6ª ed., Bookman, 1999
  3. Campos Ferreira, J. - Introdução à Análise Matemática, Fundação Calouste Gulbenkian, 1982
  4. Carlos Sarrico, Análise Matemática, Leituras e Exercícios, Gradiva, 1997
  5. Larson, R.; Hostetler, R.; Edwards, B. - Calculus with Analytic Geometry, 5ª ed., Heath, 1994
  6. Figueira, M. - Fundamentos de Análise Infinitesimal, Textos de Matemática, vol. 5, Departamento de Matemática, Faculdade de Ciências da Universidade de Lisboa, 1996
Prerequisites

The student must master the mathematical knowledgelectured until the end of PortugueseHigh School.

Student work
  Hours per credit 28
  Hours per week Weeks Hours
Avaliação   4.5
Total hours 4.5
ECTS 6.0