Real Analysis I

Code:

M1011

Acronym:

M1011

Level:

100

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2019/2020 - 1S

Active?	Yes
Responsible unit:	Department of Mathematics
Course/CS Responsible:	Bachelor in Mathematics

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
L:B	4	Official Study Plan	3	-	9	84	243
L:G	0	study plan from 2017/18	2	-	9	84	243
			3
L:M	112	Official Study Plan	1	-	9	84	243
L:Q	0	study plan from 2016/17	3	-	9	84	243

Teaching language

Portuguese

Objectives

To master the basic concepts, results and techniques of the differential and integral calculus on one variable.

Learning outcomes and competences

The student should master the basic concepts of analysis of real functions in one real variable, namely: sequences, limits, series, continuity,  the derivative, primitive and integral operators, and Taylor polynomials and series.

It is also aimed that this unit provides the adequate environment for students to work rigorously with main concepts that so far have only been used in an intuitive way.

Working method

Presencial

Pre-requirements (prior knowledge) and co-requirements (common knowledge)

N/A

Program

0. THE SET OF REAL NUMBERS 

The set of real numbers: algebraic structure, ordering and completeness. 

1. LIMITS AND CONTINUITY

Sequences: definition, limit, uniqueness of its limit, monotone and bounded sequences, subsequences. Definition and uniqueness of the limit of a function at a point.  Heine’s characterization of limit. Lateral limits and the arithmetic of limits. Limits at infinity, and horizontal and oblique asymptotes. Infinite limits and vertical asymptotes. Continuous functions. 

2. DERIVATIVES AND ANTIDERIVATIVES

Geometric motivation and physical meaning of the notion of derivative of a real function at a point. Definition of derivative and lateral derivatives at a point. Anti-derivatives. Derivatives and anti-derivatives of elementary functions. Relationship between continuity and derivability. Squeeze theorem. Derivatives and anti-derivatives of sums and products by a scalar. Derivatives of products and quotients. Chain rule and corresponding anti-derivative rule. Derivative of the inverse function. Inverses of trigonometric functions and corresponding derivatives. Anti-derivative by substitution. Anti-derivative by parts. Anti-derivatives of rational functions.

3. INTEGRALS

Concept of area. Integral of a bounded function over an interval. Integrable functions. Basic properties of integrals. The area function. The Fundamental Theorem of Calculus and its consequences. Computation of integrals. Integration by substitution and integration limits. Improper integrals: the case of continuous functions defined on unbounded intervals and the case of continuous unbounded functions defined on an interval.

4. THE FUNDAMENTAL THEOREMS OF CALCULUS AND APPLICATIONS

Theorems of continuity: the permanence of sign in the neighbourhood of a continuity point; Theorem of Intermediate Values; Theorems of Bolzano and Weierstrass.

Theorems on differentiable functions: the derivative is zero in local extreme points (for functions whose domains are open); Mean Value Theorems (Rolle’s, Lagrange’s, Cauchy’s). Applications: determination of extremes of functions; proof that a function defined on an interval with nil derivative is constant; determination of monotonic intervals and concavity; classification of critical points. Indeterminate limits. L’Hôpital’s Rule. Convex functions whose domain is a closed interval. Possible discontinuities of a function which is the derivative of another inside an interval. Functions of class C^k.



If school hours are available:


5. POLYNOMIAL APPROXIMATION AND SERIES

Polynomial approximation of functions: Taylor polynomials; tangent of degree n of a function and its Taylor polynomial of order n at a given point; Lagrange’s formula for the remainder. Application: irrationality of Neper’s number.

Mandatory literature

Kitchen Jr. Joseph W.; Calculus
Adams Robert A.; Calculus. ISBN: 0-201-39607-6

Complementary Bibliography

Spivak Michael; Calculus. ISBN: 84-291-5139-7 (Vol. 1)

Teaching methods and learning activities

Exposition of the syllabus by the teacher. Slides for study of the theoretical classes are available, as well as exercise sheets, with previous indication of the problems to be actually discussed in the TP- classes to encourage autonomous work by the students. The webpages of the course contain other useful material.

keywords

Physical sciences > Mathematics > Mathematical analysis

Evaluation Type

Evaluation with final exam

Assessment Components

designation	Weight (%)
Exame	100,00
Total:	100,00

Amount of time allocated to each course unit

designation	Time (hours)
Estudo autónomo	161,50
Frequência das aulas	75,50
Trabalho escrito	6,00
Total:	243,00

Eligibility for exams

Attending classes is not compulsory: students are not required to attend the classes.

Calculation formula of final grade

The assessment consists on a mandatory exam, written or oral, lasting up to 3 hours.

Special assessment (TE, DA, ...)

The exams required under the special legal cases will be written, but may be preceded by na oral exam to establish if the student should be admitted to the written exam.

Observations

Article 13th of the General Regulation for Students’ Evaluation in the University of Porto, approved the 19th May 2010: "Any student who commits fraud during an exam or test fails that exam and will face disciplinary charges by the University."