ECTS credits ECTS credits: 6
ECTS Hours Rules/Memories Student's work ECTS: 99 Hours of tutorials: 3 Expository Class: 24 Interactive Classroom: 24 Total: 150
Use languages Spanish, Galician
Type: Ordinary Degree Subject RD 1393/2007 - 822/2021
Departments: Statistics, Mathematical Analysis and Optimisation
Areas: Mathematical Analysis
Center Faculty of Mathematics
Call: Second Semester
Teaching: With teaching
Enrolment: Enrollable | 1st year (Yes)
With the development of the contents of this course (which are basic for other courses in the Degree) the student will acquire knowledge regarding some of the principal concepts, results and techniques which are used in the study of functions of one real variable, which is the main goal of Mathematical Analysis.
The achievement of these objectives will imply the knowledge of the theoretical contents of the course as well as being able to relate them and apply them to specific problems of different kinds, sometimes with computer aid. We will use the software Maxima to illustrate the concepts of the course.
0. Topological preliminaries.
Open and closed sets, accumulation points, compact and connected sets in R (quick overview of the topological contents of Introduction to Mathematical Analysis and Topology of Euclidean Spaces). (2h)
1. Limits
Limit of a function at a point. Left-hand and right-hand limit of a function at a point. Infinite limits and limits at infinity. Computation of limits: indeterminate limits. (5h)
2. Continuity
Continuity of a function at a point. Sequential continuity. Continuous functions: properties. Weierstrass and Bolzano Theorems. Continuity of monotone functions and their inverses. Uniform continuity. Heine’s Theorem. Continuous Extension Theorem. Sufficient and necessary criteria for uniform continuity. (8h)
3. Derivability
Derivative, left-hand and right-hand derivative of a function at a point. Geometrical and Physical interpretations of the derivative. Computation rules of derivatives. Local behavior of derivable functions: critical points. Darboux’s Theorem. Mean Value Theorem. Monotonicity and derivation. L´Hôpital’s Rule: application to the computation of indeterminate limits. (6h)
5. Higher order differentiation
Higher order derivatives. Concavity and convexity. Periodicity. Taylor Polynomial. Remainder formulas. Applications: approximate computations. Local study of a function. (4h)
6. Periodicity
Diophantine linear combinations. Periodic functions. Existence of the minimal period. Periods of the sum and product of functions. (3h)
In-library material, with reference:
F. Ayres. Cálculo Diferencial e Integral. McGraw-Hill 1991 (1202 67)
R. G. Bartle, D. R. Sherbert. Introducción al Análisis Matemático de una variable. Limusa Wiley, 2010. (1202 196, 26 32)
F. Ballesteros. Ejercicios de análisis matemático. Autores 1994 (26 306)
G. L. Bradley, Cálculo de una variable. Prentice Hall 1998. (1202 318, 26 462)
J. de Burgos. Cálculo Infinitesimal de una variable, segunda edición. McGraw-Hill, 2007. (1202 381, 26 475, 26 424)
M. Contreras. Ejercicios resueltos y notas de clase de cálculo. Universidad de Sevilla (CD-ROM), 2005 (CD 34)
J. A. Fernández Viña. Lecciones de Análisis Matemático I, Tecnos. (1202 17, 26 169)
J. A. Fernández Viña, E. Sánchez Mañes. Ejercicios y complementos de Análisis Matemático I, Tecnos. (1202 69)
D. Jornet, V. Montesinos, A. Roca. Análisis Matemático, Universidad Politécnica de Valencia, 2003. (1202 390, 26 437)
R. Larson, R. P. Hostetler, B. H. Edwards. Cálculo. McGraw-Hill, 2006. (26 491)
E. J. Purcell. Cálculo diferencial e integral. Prentice-Hall 1988 (1202 126)
M. Spivak. Cálculo infinitesimal. Reverté, 1994. (1202 95, 26 263)
V. Tomeo, I. Uña, J. San Martin. Cálculo en una variable, Garceta, 2010. (1202 385)
Complementary:
A. D. Aleksandrov et al.: La Matemática: su contenido, métodos y significado. Alianza Universidad. 1985 (03 9 A)
Á. Gil. Introducción al cálculo infinitesimal. Volumen I, UNED, 2008. (1202 376)
J. R. Munkres. Topología, segunda edición. Prentice Hall, 2001. (1210 81, 54 185)
X.M. Masa, Topoloxía Xeral. Manuais Universitarios 1, USC, 1999. (1210 78)
A. J. Durán Guardeño: Historia del Cálculo con personajes. Alianza. 1996 (01 176)
On-line material:
• Acosta, María D. et al. Apuntes de Análisis Matemático. I URL: https://www.ugr.es/~jcabello/Analisismatematico.pdf
• Apóstol, Tom. Análisis Matemático, 2ª Ed. https://doku.pub/download/analisis-matematico-2da-edicion-tom-apostolpd…
• Aranda, Pepe. Cálculo infinitesimal en una variable. URL: https://openlibra.com/es/book/download/calculo-infinitesimal-en-una-var…
• Bonacina, Marta. Cálculo diferencial e integral. URL: https://openlibra.com/es/book/download/calculo-diferencial-e-integral
• Hardy, G. H. A Course of Pure Mathematics. Third Edition URL: https://www.gutenberg.org/files/38769/38769-pdf.pdf
• Hernández, Elsie. Cálculo diferencial e integral con aplicaciones. URL: https://openlibra.com/es/book/download/calculo-diferencial-e-integral-c…
• Larotonda, Gabriel. Cálculo y Análisis. URL: http://cms.dm.uba.ar/depto/public/Curso%20de%20grado/fascgrado3.pdf
• Nicolaescu, Liviu I., Introduction to Real Analysis. URL: https://www3.nd.edu/~lnicolae/Hon_Calc_Lectures.pdf
• Revilla, Fernando. Problemas resueltos de análisis matemático. URL: http://fernandorevilla.es/wp-content/uploads/2015/10/problemas-resuelto…
• Trench, William. Introduction to Real Analysis. URL: http://ramanujan.math.trinity.edu/wtrench/texts/TRENCH_REAL_ANALYSIS.PDF
During this course, the student will achieve, in different ways, all the competences gathered in the Plan of the Degree in Mathematics of the USC. In particular, the course will favor the acquisition of the following specific competences:
• To know the notions of limit, continuity, uniform continuity and differentiability of functions of one real variable.
• To express with precision and rigor, whether it is in oral or written form, the knowledge, procedures, results and ideas studied during the course.
• To identify errors in foul reasoning, proposing proofs or counterexamples.
• To recognize some of the problems for which the solving needs of the resources learned during the course (optimization problems etc.).
• To use the software Maxima as assistance in the development of those activities related to the contents of the course with the objective of improving concept understanding and the discovery and contrast of the results of the course.
In this and subsequent sections we will take into account the list of competences and references collected in the Plan of the Degree in Mathematics of the USC
http://www.usc.es/export9/sites/webinstitucional/gl/servizos/sxopra/mem…
In plenary lectures, we will develop the theoretical part of the course, illustrating it with examples in order to make it more comprehensible. Furthermore, we will reserve some time in order to solve exercises and sometimes we will pose questions in order to make students participate in a discussion. This dynamic is intended to work in competences CB1-CB5, CG1-CG5 and CE1-CE6.
In which respects the teaching in reduced groups, we intend to achieve more student participation, we will deal with problems and aspects of the course which are not treated during plenary lectures and we will analyze those matters of most difficult comprehension. In these sessions, we will work the competences CE7 and CE8.
Last, in lab lectures we will solve problems and, when they take place in the IT lecture rooms, we will deal with the computer software Maxima, in order to do calculations and graphic representations, which will be of use for the solution of problems and the understanding of the course material. In these lectures we will develop CE7-CE9 and CT1 and CT4.
The adaptation of the methodology to the other scenarios considered in the document “Directrices para o desenvolvemento dunha docencia presencial segura, curso 2020-21” appears in the section Comments.
Continuous assessment: it will consist of two examinations to be taking during lecture time. The exact date of the examination will be announced in advance. Each of the examinations will take place once each of the three main chapters of the course is finished: Continuity and Differentiability.
Calculation of the final grade: The numerical grade of the opportunity will be computed as max{E,0.4C+0.6E} where E is the grade of the final exam of the opportunity (which will take place at the dates indicated by the Faculty) and C is the average of the continuous assessment.
Those students who do not participate at the final exam of a given opportunity will be scored as “not presented” in that opportunity.
In those cases of fraudulent behavior regarding assessments the precepts gathered in the “Normativa de avaliación do rendemento académico dos estudantes e de revisión de cualificacións” will be applied.
The adaptation of the assessment system to the other scenarios considered in the document “Directrices para o desenvolvemento dunha docencia presencial segura, curso 2020-21” appears in the section Comments.
IN LECTURE ROOM TIME
Plenary lectures (28 h)
Reduced group lectures (10 h)
Lab lectures (5 h)
Non-IT lectures (10 h)
IT lectures (3 h)
Reduced group tutoring (2 h)
Total: 58 h
PERSONAL WORK: About 92 h depending on the person and her background.
• To have a good knowledge of sequences of real numbers and real line topology.
• Daily study using bibliographical material, solving the proposed problems, summarizing the concepts of the course, repeating the definitions, etc.
• To plan beforehand the study time, keeping the study of the course up to date.
• To visit the lecturer’s office to consult any doubts concerning the course.
Considerations and modifications depending on the scenario:
Scenario II: The lecturer will provide written material, of theoretical content or exercises, so the students can follow the course’s blended learning model. For the on-line sessions, the students will have videos available on-line at the course site. Tutoring will take place in Microsoft Teams. The assessment system will be the same as in Scenario I.
Scenario III: The lecturer will provide written material, of theoretical content or exercises, so the students can follow the course on-line. The students will have videos available on-line at the course site. Tutoring will take place in Microsoft Teams. Assessment will have the same structure as in the other scenarios, but it will be on-line.
Fernando Adrian Fernandez Tojo
Coordinador/a- Department
- Statistics, Mathematical Analysis and Optimisation
- Area
- Mathematical Analysis
- fernandoadrian.fernandez [at] usc.es
- Category
- Professor: LOU (Organic Law for Universities) PhD Assistant Professor
Lucia Lopez Somoza
- Department
- Statistics, Mathematical Analysis and Optimisation
- Area
- Mathematical Analysis
- lucia.lopez.somoza [at] usc.es
- Category
- Professor: LOU (Organic Law for Universities) PhD Assistant Professor
Daniel Cao Labora
- Department
- Statistics, Mathematical Analysis and Optimisation
- Area
- Mathematical Analysis
- Phone
- 881813174
- daniel.cao [at] usc.es
- Category
- Professor: Temporary supply professor to reduce teaching hours
| Monday | |||
|---|---|---|---|
| 11:00-12:00 | Grupo /CLE_01 | Spanish | Classroom 07 |
| 12:00-13:00 | Grupo /CLIS_03 | Galician | Ramón María Aller Ulloa Main Hall |
| Tuesday | |||
| 11:00-12:00 | Grupo /CLE_01 | Spanish | Classroom 07 |
| 12:00-13:00 | Grupo /CLIS_04 | Galician | Classroom 06 |
| 13:00-14:00 | Grupo /CLE_02 | Galician | Classroom 09 |
| Wednesday | |||
| 11:00-12:00 | Grupo /CLE_02 | Galician | Classroom 08 |
| 12:00-13:00 | Grupo /CLIS_01 | Spanish | Ramón María Aller Ulloa Main Hall |
| 13:00-14:00 | Grupo /CLIS_02 | Spanish | Classroom 02 |
| Thursday | |||
| 10:00-11:00 | Grupo /CLIL_01 | Galician, Spanish | Ramón María Aller Ulloa Main Hall |
| 13:00-14:00 | Grupo /CLIL_05 | Galician | Classroom 03 |
| Friday | |||
| 09:00-10:00 | Grupo /CLIL_02 | Galician, Spanish | Classroom 03 |
| 10:00-11:00 | Grupo /CLIL_06 | Galician | Classroom 02 |
| 11:00-12:00 | Grupo /CLIL_04 | Galician | Classroom 06 |
| 12:00-13:00 | Grupo /CLIL_03 | Spanish, Galician | Classroom 06 |
| 05.27.2021 10:00-14:00 | Grupo /CLE_01 | Classroom 02 |
| 05.27.2021 10:00-14:00 | Grupo /CLE_01 | Classroom 03 |
| 05.27.2021 10:00-14:00 | Grupo /CLE_01 | Classroom 06 |
| 05.27.2021 10:00-14:00 | Grupo /CLE_01 | Ramón María Aller Ulloa Main Hall |
| 07.08.2021 16:00-20:00 | Grupo /CLE_01 | Classroom 06 |