Real Analysis I

Code:

M1011

Acronym:

M1011

Level:

100

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2016/2017 - 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:M	98	Official Study Plan	1	-	9	84	243
L:Q	0	study plan from 2016/17	3	-	9	84	243

Teaching language

Portuguese

Objectives

Acquiring knowledge of 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: sequence, series, derivative, primitive, integral, and Taylor series. It is also intended that this unit allows students to work rigorously with concepts that, thus far, have been introduced only in an intuitive way.

Working method

Presencial

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. Lateral limits and the arithmetic of limits. Limits at infinity and horizontal and oblique asymptotes. Infinite limits and vertical asymptotes. Heine’s characterization of limit.


2. DERIVATIVES AND ANTIDERIVATIVES

Geometric motivation and physical significance of the notion of derivative of a real function at a point. Definition of derivative and lateral derivatives at a point. Antiderivatives. Derivatives and antiderivatives of elementary functions. Continuous functions. Relationship between continuity and derivability. Squeeze theorem. Derivatives and antiderivatives of sums and products by a scalar. Derivatives of products and quotients. Chain rule and associated integration rule. Derivative of the inverse function. Inverses of trigonometric functions and corresponding derivatives. Leibniz’s differential notation. Integration by substitution. Integration by parts. Integration of rational functions.

3. INTEGRALS

Concept of area. Integral of a bounded function over an interval. Integrable functions. Basic properties of integrals. Average value of a function. 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, Theorem of Weierstrass.
 Theorems on differentiable functions: the derivative is zero in local extreme points (for functions with open domains), Mean Value Theorems (Rolle’s, Lagrange’s, Cauchy’s). Applications: determination of extremes, proof that a function defined on an interval with nil derivative is constant, determination of monotony intervals and concavity, classification of critical points.  Indeterminate forms. L’Hôpital’s Rule. Functions of class $c^k$.

5. POLYNOMIAL APPROXIMATION AND SERIES

Polynomial approximation of functions:

Taylor polynomials, tangent of degree $n$ of a function and its Taylor polynomial of degree $n$ at a given point, Lagrange’s formula for the remainder. Application: irrationality of Neper’s number.

Numerical series:

Series of real numbers, sequence of partial sums, convergence. Geometric series and the harmonic series. Leibniz’s criterion for alternate series. Relationship between absolute convergence and convergence. The comparison criterion. Ratio and integral criteria for series of positive terms.

Mandatory literature

000097905. ISBN: 978-0-495-38273-7
000098594. ISBN: 85-221-0479-4 (Vol. I)

Complementary Bibliography

Spivak, Michael; Calculus, Houston : Publish or Perish, 1994

Teaching methods and learning activities

Exposition of the theory by the teacher. Slides for study and support of the theoretical classes are available. Exercise sheets with previous indication of the exercises to be actually discussed in the theoretical-practical classes in each week are available, to stimulate previous work by the students. The webpage of the course contains other materials.

Evaluation Type

Distributed evaluation with final exam

Assessment Components

designation	Weight (%)
Exame	50,00
Teste	50,00
Total:	100,00

Eligibility for exams

The students are not required to attend the classes.

Calculation formula of final grade

- The assessment consists on mandatory test (50%) and a mandatory  final exam (50%). 

- To be admitted to the final examthwe student has to get a rating equal to or greater than 2 values in test

- A special complementary evaluation is required to obtain final grades greater than 17.

Special assessment (TE, DA, ...)

The exams required under the special cases previewed in the law will be written, but may be preceded by na oral exam to establish if the student should be admitted or not 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 in an exam or test fails that exam and will face disciplinary charges by the University.''