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
Center Faculty of Mathematics
Call: Second Semester
Teaching: Sin Docencia (En Extinción)
Enrolment: No Matriculable (Sólo Planes en Extinción)
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 or Maple 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. (7h)
4. Higher order differentiation
Higher order derivatives. Concavity and convexity. Periodicity. Taylor Polynomial. Remainder formulas. Applications: approximate computations. Local study of a function. (6h)
In-library material, with reference:
Basic bibliography:
Bartle, R. G., Sherbert, D. R.. Introducción al Análisis Matemático de una variable. Limusa Wiley, 2010. (1202 196, 26 32)
Ballesteros, F. Ejercicios de análisis matemático. Autores 1994 (26 306)
de Burgos, J. Cálculo Infinitesimal de una variable, segunda edición. McGraw-Hill, 2007. (1202 381, 26 475, 26 424)
Complementary bibliography:
Ayres, F. Cálculo Diferencial e Integral. McGraw-Hill 1991 (1202 67)
Bradley, G. L. Cálculo de una variable. Prentice Hall 1998. (1202 318, 26 462)
Fernández Viña, J. A. Lecciones de Análisis Matemático I, Tecnos. (1202 17, 26 169)
Fernández Viña, J. A., Sánchez Mañes, E. Ejercicios y complementos de Análisis Matemático I, Tecnos. (1202 69)
Larson, R.E., Hostetler, R. P., Edwards, B. H. Cálculo. McGraw-Hill, 2006. (26 491)
M. Spivak. Cálculo infinitesimal. Reverté, 1994. (1202 95, 26 263)
On-line material:
• Aranda, Pepe. Cálculo infinitesimal en una variable. URL: http://www.iespppuquio.edu.pe/biblioteca/wp-content/uploads/2020/12/cal…
• Hardy, G. H. A Course of Pure Mathematics. Third Edition URL: https://www.gutenberg.org/files/38769/38769-pdf.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 or Maple 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.
No classes.
No classes. Final exam.
No classes. ECTS system recommends 150 hours of study.
No classes. No special recommendations
Maria Victoria Otero Espinar
- Department
- Statistics, Mathematical Analysis and Optimisation
- Area
- Mathematical Analysis
- Phone
- 881813170
- mvictoria.otero [at] usc.es
- Category
- Professor: University Professor
Fernando Adrian Fernandez Tojo
- Department
- Statistics, Mathematical Analysis and Optimisation
- Area
- Mathematical Analysis
- fernandoadrian.fernandez [at] usc.es
- Category
- Professor: University Lecturer
| 05.28.2026 10:00-14:00 | Grupo de examen | Classroom 06 |
| 07.06.2026 10:00-14:00 | Grupo de examen | Classroom 06 |