ECTS credits ECTS credits: 9
ECTS Hours Rules/Memories Hours of tutorials: 2 Expository Class: 42 Interactive Classroom: 42 Total: 86
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)
Understand and apply the fundamental concepts of the differentiation of real-valued functions of a single variable, including its main rules, properties, and associated theorems (Rolle’s theorem, the Mean Value Theorem, L’Hôpital’s Rule, etc.).
Analyze the behavior of functions using higher-order derivatives, identifying relative extrema, inflection points, and producing graphical representations.
Understand the concept of the definite integral based on its formal construction, and apply its properties for the effective computation of integrals.
Relate differentiation and integration through the Fundamental Theorem of Calculus, and use techniques such as substitution and integration by parts to compute antiderivatives.
Apply integral calculus to geometric and analytical problems, including the computation of areas, lengths, volumes, surfaces of revolution, and improper integrals.
The course content is divided into two blocks:
Block 1: Differentiation of single-variable functions
Topic 1. Concept of derivative and properties (9h CLE)
Concept of derivative.
Chain rule and derivative of the inverse function.
Derivatives of elementary functions.
Rolle’s Theorem and the Mean Value Theorem.
Monotonicity and differentiation. L’Hôpital’s Rule.
Topic 2. Higher-order derivatives and properties (9h CLE)
Relative extrema.
Taylor polynomial. Remainder formulas.
Characterization of relative extrema.
Inflection points.
Graphical representation of real-valued functions of one variable.
Block 2: Integration of single-variable functions
Topic 3. Definite integral and properties (9h CLE)
Construction of the Riemann integral. Darboux sums. Riemann sums.
Integrable functions.
Properties of the integral.
Topic 4. Relationship between differentiation and integration (9h CLE)
Fundamental Theorem of Calculus.
Change of variables theorem.
Elementary antiderivatives. Integration by parts.
Computation of antiderivatives.
Topic 5. Applications of integral calculus (3h CLE)
Computation of planar areas.
Graph lengths.
Volumes and surfaces of revolution.
Topic 6. Improper integrals (3h CLE)
Improper integrals.
Convergence criteria.
Basic Bibliography
ABBOTT, S. (2015) Understanding Analysis. Springer (SpringerLink eBook Collection – Mathematics & Statistics, https://link-springer-com.ezbusc.usc.gal/book/10.1007/978-1-4939-2712-8)
APOSTOL, T. M. (1977) Análisis Matemático. Reverté.
BARTLE, R. G., SHERBERT, D. R. (1999) Introducción al Análisis Matemático de una variable (2ª Ed.). Limusa Wiley.
Complementary Bibliography
LARSON, R. HOSTETLER, R. P., EDWARDS, B. H. (2006) Cálculo (8ª Ed.). McGraw-Hill.
MAGNUS, R. (2020) Fundamental Mathematical Analysis. Springer (SpringerLink eBook Collection – Mathematics & Statistics, https://link-springer-com.ezbusc.usc.gal/book/10.1007/978-3-030-46321-2).
PISKUNOV, N. (1978) Cálculo Diferencial e Integral. Montaner y Simón.
SPIVAK, M. (1978) Calculus. Reverté.
The learning outcomes of this course are as follows:
Knowledge:
Con01: To understand the key concepts, methods, applications, and results of the main branches of Mathematics.
Con02: To know, comprehend, and use mathematical language in order to construct and understand proofs and to formulate mathematical models.
Con03: To be familiar with proofs of the most relevant theorems from different branches of Mathematics.
Con04: To assimilate the definitions of mathematical objects, relate them to one another, and be able to use them in different contexts.
Con05: To identify and abstract the essential properties and facts of a problem and to determine the appropriate mathematical tools to address it.
Skills and Abilities:
H/D01: To apply both theoretical and practical knowledge, as well as analytical and abstract reasoning skills, in the formulation and resolution of problems in academic and professional contexts.
H/D02: To make use of bibliographic resources and tools, both general and mathematics-specific.
H/D03: To organise and plan work effectively.
H/D04: To verify and assess arguments and reasoning, identify errors, and propose revisions or counterexamples.
H/D06: To read scientific texts in one’s native language and in other languages relevant to the scientific community.
H/D07: To construct mathematical proofs, formulate conjectures, and devise strategies to verify or refute them.
Competences:
Comp01: To gather and interpret relevant data, information, and results; to draw conclusions; and to produce reasoned reports on scientific, technological, or other problems requiring the use of mathematical tools.
Comp02: To communicate mathematical knowledge, procedures, results, and ideas effectively, both orally and in writing, to both specialised and general audiences.
Comp03: To study and learn autonomously new knowledge and techniques across various branches of Mathematics.
The general methodological guidelines established in the current Degree Program in Mathematics at the USC will be followed.
The lecture-based classes will mainly consist of lessons delivered by the instructor, focused on the presentation of theoretical content and the resolution of some problems or exercises. Sometimes, the model will resemble a traditional lecture, while at other times, a greater involvement of the students will be encouraged. The focus will primarily be on acquiring the knowledge Con01, Con02, Con03, Con04, and Con05.
The interactive classes will aim, in some cases, at the acquisition of practical skills and, in other cases, will serve for the immediate illustration of theoretical-practical content through the resolution of applications of the theory, problems, or exercises. The focus will primarily be on the skills H/D01, H/D04, and H/D07 and the competences Comp01 and Comp02.
The tutoring sessions will provide an opportunity to assist students in discussing specific issues related to assigned tasks or to resolve any doubts about the subject matter.
All student tasks (study, readings, exercises, practical work, etc.) will be guided by the instructors during the interactive classes and in very small group tutoring sessions. These tasks will serve to reinforce, in particular, the skills H/D02, H/D03, and H/D06 and the competences Comp01, Comp02, and Comp03.
The virtual course or the Teams platform will be used as a mechanism to provide students with the necessary resources for the development of the course (explanatory videos, notes, exercise sheets, etc.).
The teaching methodologies used in the course combine different approaches to promote a solid and applied understanding of the content.
In the lecture-based classes, the traditional lecture format will primarily be used, oriented toward the structured presentation of fundamental theoretical concepts.
The interactive classes will focus on problem-solving and case studies, encouraging active student participation and the practical application of the content.
In the tutoring sessions, students will be guided in the independent resolution of problems, promoting autonomous learning and the development of analytical and critical skills.
This methodological combination aims to facilitate both the acquisition of knowledge and the development of competences and skills.
The assessment of the course combines continuous assessment and final assessment, in a coordinated manner across the different lecture groups. The goal is to evaluate the degree of acquisition of the knowledge, competences, and skills defined for the course.
Final Assessment (FA)
The final assessment will consist of a written exam graded out of 10 points. It will assess the overall understanding of the theoretical content, formal reasoning ability, and the application of mathematical techniques to problem-solving. The exam is broken down as follows:
* Statement and proof of lemmas, propositions, theorems, and corollaries (25%): assesses theoretical knowledge (Con01, Con03), handling of formal language (Con02), and skills in constructing proofs (H/D07).
* Resolution of theoretical questions (25%): assesses conceptual understanding, reasoning ability, and the relationship between concepts (Con02, Con04, H/D04), as well as skills in constructing proofs (H/D07).
* Resolution of practical questions (25%): assesses mastery of specific techniques and their application (Con05, H/D01).
* Resolution of problems or exercises (25%): measures the ability to analyze and solve mathematical problems, also considering clarity in presentation (Comp01, H/D01, H/D03).
Continuous Assessment (CA)
The continuous assessment consists of two in-person intermediate tests, also graded out of 10 points. These will progressively cover the course content, and both the acquisition of knowledge and the development of skills and competences will be assessed. Each test will include:
* Theoretical questions (33%): assess the progressive assimilation of fundamental content and the correct use of mathematical language (Con01, Con02, Con04).
* Practical questions (33%): measure the ability to apply mathematical techniques in solving specific problems (Con05, H/D01, H/D04).
* Problems or exercises (33%): assess the competence to face new situations, argue solutions, and communicate results clearly (Comp01, Comp02, H/D01, H/D03, H/D07).
The continuous assessment grade will be calculated using the following formula:
CA = 1/2 * P1 + 1/2 * P2,
where P1 and P2 are the grades obtained in the two intermediate tests.
Final Grade (FG)
The final grade for the course will be calculated using the formula:
FG = max{FA, 0.3 * CA + 0.7 * FA}
This formula acknowledges the student’s continuous work and progress throughout the course, while also ensuring a rigorous and fair final evaluation.
A student will be considered as Not Presented if, at the end of the teaching period, they are not in a position to pass the course without taking the final exam and fail to appear for this exam.
Second Opportunity
The assessment in the second opportunity will follow the same scheme, with an equivalent final exam common to all groups, which will assess knowledge, skills, and competences in the same way as in the first exam session.
Additional Considerations
Although the continuous assessment activities and final exam may vary across different groups, they will be coordinated to ensure that all groups receive equivalent training.
In cases of fraudulent conduct during tests or exams (plagiarism or misuse of technology), the regulations regarding academic performance assessment and grade review will apply.
TOTAL HOURS
225 hours: 86 hours of face-to-face instruction and 139 hours of independent study.
FACE-TO-FACE TEACHING IN THE CLASSROOM
(CLE) Lectures (42 hours)
(CLIL) Interactive laboratory sessions (42 hours)
(TGMR) Small group tutorials (2 hours)
INDEPENDENT STUDY TIME
The hours of work will depend on the students. On average, 139 hours of independent study are estimated per student.
Take the course "Introduction to Mathematical Analysis".
Maria Victoria Otero Espinar
Coordinador/a- Department
- Statistics, Mathematical Analysis and Optimisation
- Area
- Mathematical Analysis
- Phone
- 881813170
- mvictoria.otero [at] usc.es
- Category
- Professor: University Professor
Francisco Javier Fernandez Fernandez
- Department
- Statistics, Mathematical Analysis and Optimisation
- Area
- Mathematical Analysis
- Phone
- 881813231
- fjavier.fernandez [at] usc.es
- Category
- Professor: University Lecturer
| Tuesday | |||
|---|---|---|---|
| 09:00-10:00 | Grupo /CLIL_04 | Galician | Classroom 07 |
| 09:00-10:00 | Grupo /CLIL_01 | Spanish | Classroom 08 |
| 10:00-11:00 | Grupo /CLIL_04 | Galician | Classroom 07 |
| 10:00-11:00 | Grupo /CLIL_01 | Spanish | Classroom 08 |
| 12:00-13:00 | Grupo /CLIL_06 | Galician | Classroom 07 |
| 12:00-13:00 | Grupo /CLIL_03 | Spanish | Classroom 08 |
| 13:00-14:00 | Grupo /CLIL_06 | Galician | Classroom 07 |
| 13:00-14:00 | Grupo /CLIL_03 | Spanish | Classroom 08 |
| Wednesday | |||
| 10:00-11:00 | Grupo /CLIL_05 | Galician | Classroom 08 |
| 11:00-12:00 | Grupo /CLE_01 | Spanish | Classroom 02 |
| 11:00-12:00 | Grupo /CLIL_05 | Galician | Classroom 08 |
| 12:00-13:00 | Grupo /CLE_02 | Galician | Classroom 06 |
| Thursday | |||
| 09:00-10:00 | Grupo /CLE_01 | Spanish | Classroom 02 |
| 10:00-11:00 | Grupo /CLE_01 | Spanish | Classroom 02 |
| 11:00-12:00 | Grupo /CLIL_06 | Galician | Classroom 08 |
| 12:00-13:00 | Grupo /CLIL_04 | Galician | Classroom 01 |
| 13:00-14:00 | Grupo /CLIL_05 | Galician | Classroom 05 |
| Friday | |||
| 10:00-11:00 | Grupo /CLIL_03 | Spanish | Classroom 09 |
| 11:00-12:00 | Grupo /CLIL_01 | Spanish | Classroom 08 |
| 12:00-13:00 | Grupo /CLIL_02 | Spanish | Classroom 07 |
| 06.03.2026 10:00-14:00 | Grupo /CLE_01 | Classroom 06 |
| 07.10.2026 10:00-14:00 | Grupo /CLE_01 | Classroom 06 |