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)
- To introduce students with the essential support of examples and practice in the construction and understanding of the concept of Riemann integral of real bounded functions on compact intervals.
- To know and prove the main properties of the Riemann integral, and to check if a given function is integrable or not.
- To understand the relationship between the differential and the integral calculus, established by the Fundamental Theorem of Calculus. To obtain primitives and to calculate integrals by application of the rule of Barrow .
- To apply integral calculus in order to solve different geometrical problems.
- To use a package of symbolic calculation with application to the integral calculus.
1. The Concept of Riemann Integral of a limited Function in a Compact Interval: Equivalence Formulations. Examples of Riemann-Integral Functions (9 hours CLE).
Partitions of a Compact Interval.
Riemann Sums.
Concept of Riemann Integral of a Limit Function in a Compact Interval.
Intuitive Interpretation of the Integral.
Higher Sums and Lower Sums.
Higher Integral and Lower Integral.
Equivalence Formulations of the Integral Function Concept.
Examples of Integral Functions: Integrability of the Continuous Functions and the Monotonous Functions.
2. Properties of the Integral and Integral Functions (5 hours CLE).
Linearity of the Integral.
Additive Effect of the Integral in relation to the Interval of Integration.
Monotone of the Integral. Modular Bound.
Averages. The Theorem of the Mean Value of the Integral Calculus.
3. The Fundamental Theorem of the Calculus (5 hours CLE).
Concept of Primitive.
First Formulation of the Fundamental Theorem (generalization of the Barrow Rule).
The “Integral Function” of a Riemann Integral Function.
Second Formulation of the Fundamental Theorem.
Theorems of Change of Variable and Integration by Parts for a Riemann Integral.
4. Indefinite Integral (4 hours CLE).
Concept and Properties.
Primitive Calculus by Parts and by Change of Variable.
Methods of Calculus of Elemental Primitives.
5. Applications of the Riemann Integral (5 hours CLE).
Areas Calculus of some plane figures.
Volumes Calculus of Solids of Revolution.
Length Calculus of Graphics of Regular Functions.
Lateral Areas Calculus of Revolution Fields.
Basic Bibliography
- Bartle, R.G., Sherbert, D.R.: Introducción al Análisis Matemático de una Variable (2ª Ed.). Limusa Wiley.1999.
- Apostol, T.M.: Análisis Matemático. Reverté. 1977.
- Course notes
Complementary Bibliography
- Aleksandrov, A.D., Kolmogorov, A.N., Laurentiev, M.A. and others: La Matemática: su Contenido, Métodos y Significado. Alianza Universidad. 1985.
- Durán Guardeño, A.J.: Historia del Cálculo con Personajes. Alianza.1996.
- Larson, R., Hostetler, R.P., Edwards, B.H.: Cálculo (8ª Ed.). McGraw-Hill. 2006
- Piskunov, N.: Cálculo Diferencial e Integral. Montaner y Simón.1978.
- Spivak, M.: Calculus. Reverté. 1978.
In addition to contribute to achieve the general and transverse competences taken up in the memory of the degree, this subject will allow the student to get the following specific competences:
CE1 - To understand and use mathematical language;
CE2 - To know rigorous proofs of some classical theorems in different areas of mathematics;
CE3 - To devise demonstrations of mathematical results, formulate conjectures and imagine strategies to confirm or refute them;
CE4 - To Identify errors in faulty reasoning, proposing demonstrations or counterexamples;
CE5 - To assimilate the definition of a new mathematical object, and to be able to use it in different contexts;
CE6 - To identify the abstrac properties and material facts of a problem, distinguishing them from those purely occasional or incidental;
CE9 - To use statistical analysis applications, numerical and symbolic computation, graphical visualization, optimization and scientific software, to experience and solve problems in mathematics.
The general methodological indications established in the USC Mathematics Degree Title Memory will be followed.
Teaching is programmed in theoretical classes, small group practices, small group computer practices and tutorials. In the theoretical classes the essential contents of the discipline will be presented, and will allow the work of the basic, general and transversal competences, in addition to the specific competences CE1, CE2, CE5 and CE6. In the interactive sessions, problems or exercises of more autonomous realization will be proposed, which will make it possible to emphasize the acquisition of specific competences CE3 and CE4. Finally, the tutorials will be devoted to discussion and debate with the students, and to the resolution of the proposed tasks with which it is intended that the students practice and strengthen knowledge. In the computer classes, the MAPLE program will be used as a study tool, thus working on the specific competence CE9.
In a generic way to the three scenarios that will be detailed below, the virtual course will be used as the only mechanism to provide students with the necessary resources for the development of the subject (explanatory videos, notes, exercise bulletins, etc.). The fundamental difference between the three scenarios will be, on the one hand, the way in which the tutorials will be attended and, on the other hand, the type of teaching that will be taught: totally face-to-face in scenario 1, semi-presential in scenario 2 and totally virtual on scenario 3.
SCENARIO 1 (adapted normality):
The expository and interactive teaching will be face-to-face and will be complemented with the virtual course of the subject, in which the students will be able to find bibliographic materials, problem bulletins, explanatory videos, etc. Through the virtual course, students will also take self-assessment tests and participate in thematic work forums, as described in the corresponding section.
The tutorials will be face-to-face.
SCENARIO 2 (distancing):
Partially virtual teaching, according to the distribution organized by the Faculty of Mathematics. For this, the virtual classroom of the course will be used in which explanatory videos, bibliographic materials, problem bulletins, etc. will be uploaded, provided by the teaching staff and, if so established, synchronous virtual classes using the means provided for this purpose. Students will also take self-assessment tests and various activities for continuous assessment, and will also participate in thematic forums, as described in the corresponding section.
The tutorials will be attended electronically.
SCENARIO 3 (closure of the facilities):
Totally remote teaching through the virtual course of the subject and the means provided. For this, the virtual classroom of the course will be used in which explanatory videos, bibliographic materials, problem bulletins, etc., provided by the teaching staff and, if so established, synchronous virtual classes using the means provided for this purpose will be uploaded. Students will also take self-assessment tests and various tests for continuous assessment through the virtual campus and will participate in thematic work forums, as described in the corresponding section.
The tutorials will be attended electronically.
In general, there will be an assessment that combines a continuous formative assessment with a final test.
The continuous formative assessment will be based on the results obtained in the questionnaires that the students must do at the end of each of the topic of the subject, thematic work forums and various activities carried out in this area. It will allow verifying the degree of achievement of the specific competences previously determined, with special emphasis on the transversal competences CT1, CT2, CT3 and CT5. The continuous formative evaluation will be robust against possible changes in scenario. In this sense, use the tools of the virtual campus to replace assessment tests that require the presence of students in the classroom.
Regarding the final and second chance exam, the difference between the three scenarios consists of the way it is carried out: face-to-face in the case of scenario 1 and, in scenarios 2 and 3, telematics. Specific tests will measure the knowledge obtained by students in relation to the concepts and results of the subject, both from a theoretical and practical point of view, also assessing clarity, the logical rigor demonstrated in their exposition. The achievement of the basic, general and specific competences to which allusion is made in the Memory of the Degree in Mathematics of the USC and which were previously indicated will be evaluated.
SCENARIO 1 (adapted normality):
As previously mentioned, the evaluation will be carried out by combining a continuous formative evaluation with a final test.
The continuous formative assessment will consist, on the one hand, of carrying out tests that the students must do at the end of each of the topic of the subject and they will serve as a method of evaluation at different levels (student self-evaluation; assessment of work outside the classroom, etc.), on the other hand, in carrying out various activities (individual or group) that are They will take place throughout the course (in the classrooms or outside them) and, finally, in the participation in work forums in which the students will work on different problems proposed by the teacher.
With the mark of the continuous formative assessment (C), over 10 points, and the mark of the final face-to-face test (F), over 10 points, the final mark of the subject (NF) will be calculated using the following formula:
NF = max {F, 0.4 * C + 0.6 * F}
It will be understood as NOT PRESENTED who at the end of the teaching period is not in a position to pass the subject without taking the final exam and does not appear for said test.
In the second opportunity, the same evaluation system will be used but with the test corresponding to the second opportunity, which will be a test of the same type as the first.
SCENARIO 2 (distancing):
The same procedure as that described for scenario 1 will be used, with the only difference that both the activities that will be carried out throughout the course and the final test will be telematic.
SCENARIO 3 (closure of the facilities):
The same procedure as that described for scenario 1 will be used, with the only difference that both the activities that will be carried out throughout the course and the final test will be telematic.
Warning. In cases of fraudulent performance of tests or plagiarism (plagiarism or improper use of technologies), the provisions of the regulations for evaluating the academic performance of students and reviewing grades will apply.
TOTAL HOURS (150)
150 hours: 58 hours of presence work in the class and 92 hours of personal work of the student.
PRESENCE WORK IN THE CLASS (58 hours)
(CLE) Blackboard classes in big group (28 hours)
(CLIS) Interactive classes in reduced group (14 hours)
(CLIL) Interactive classes of laboratory/tutorials in reduced group (14 hours)
(TGMR) Small group tutorials or individualized (2 hours)
PERSONAL WORK OF THE STUDENT (92 hours)
Personal work will depend on the students. On average, 92 hours per student are estimated.
To have studied the subject "Introduction to the Mathematical Analysis" and to attend or have attended the subject "Continuity and Derivability of One Real Variable Functions".
Contingency plan:
Adaptation of the methodology to scenarios 2 and 3:
SCENARIO 2 (distancing):
Partially virtual teaching, according to the distribution organized by the Faculty of Mathematics. For this, the means provided for synchronous telematic teaching and the virtual classroom of the course will be used.
The tutorials will be attended telematically.
SCENARIO 3 (closure of the facilities):
Totally remote teaching through the means provided for this purpose and the virtual course of the subject.
The tutorials will be attended telematically.
Adaptation of the evaluation system to scenarios 2 and 3:
SCENARIO 2 (distancing):
The same procedure as that described for scenario 1 will be used, with the only difference being that the activities that will be carried out throughout the course as the final test will be telematic.
SCENARIO 3 (closure of the facilities):
The same procedure as that described for scenario 1 will be used, with the only difference being that the activities that will be carried out throughout the course as the final test will be telematic.
In the second opportunity, the same evaluation system will be used but with the test corresponding to the second opportunity, which will be an exam of the same type as that of the first.
Rosa Mª Trinchet Soria
Coordinador/a- Department
- Statistics, Mathematical Analysis and Optimisation
- Area
- Mathematical Analysis
- Phone
- 881813205
- rosam.trinchet [at] usc.es
- Category
- Professor: University Lecturer
Rosana Rodríguez López
- Department
- Statistics, Mathematical Analysis and Optimisation
- Area
- Mathematical Analysis
- Phone
- 881813368
- rosana.rodriguez.lopez [at] usc.es
- Category
- Professor: University Lecturer
Francisco Javier Fernandez Fernandez
- Department
- Statistics, Mathematical Analysis and Optimisation
- Area
- Mathematical Analysis
- Phone
- 881813231
- fjavier.fernandez [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
Cristina Lois Prados
- Department
- Statistics, Mathematical Analysis and Optimisation
- Area
- Mathematical Analysis
- Phone
- 881821048
- cristina.lois.prados [at] usc.es
- Category
- Ministry Pre-doctoral Contract
| Monday | |||
|---|---|---|---|
| 10:00-11:00 | Grupo /CLIS_01 | Galician, Spanish | Ramón María Aller Ulloa Main Hall |
| 13:00-14:00 | Grupo /CLE_02 | Galician | Classroom 07 |
| Tuesday | |||
| 10:00-11:00 | Grupo /CLIS_02 | Spanish, Galician | Classroom 06 |
| 11:00-12:00 | Grupo /CLE_02 | Galician | Classroom 09 |
| 12:00-13:00 | Grupo /CLE_01 | Spanish | Classroom 08 |
| Wednesday | |||
| 12:00-13:00 | Grupo /CLIS_03 | Galician | Classroom 03 |
| 13:00-14:00 | Grupo /CLIS_04 | Galician | Classroom 06 |
| Thursday | |||
| 09:00-10:00 | Grupo /CLIL_04 | Galician | Computer room 4 |
| 10:00-11:00 | Grupo /CLIL_06 | Galician | Computer room 3 |
| 12:00-13:00 | Grupo /CLE_01 | Spanish | Classroom 07 |
| 13:00-14:00 | Grupo /CLIL_01 | Galician, Spanish | Computer room 3 |
| Friday | |||
| 09:00-10:00 | Grupo /CLIL_05 | Galician | Computer room 3 |
| 10:00-11:00 | Grupo /CLIL_03 | Spanish, Galician | Computer room 2 |
| 12:00-13:00 | Grupo /CLIL_02 | Spanish, Galician | Computer room 3 |
| 06.01.2021 10:00-14:00 | Grupo /CLE_01 | Classroom 02 |
| 06.01.2021 10:00-14:00 | Grupo /CLE_01 | Classroom 03 |
| 06.01.2021 10:00-14:00 | Grupo /CLE_01 | Classroom 06 |
| 06.01.2021 10:00-14:00 | Grupo /CLE_01 | Ramón María Aller Ulloa Main Hall |
| 07.12.2021 10:00-14:00 | Grupo /CLE_01 | Classroom 06 |