The EHEA (European Higher Education Area) has brought new ideas about the way the University must teach and about the skills and values it must provide. A crucial factor of EHEA is that the student must be able to solve problems. A problem is not a situation for which we already have an answer; for example, the calculus of the derivative of a function is not a problem, because we already have a rule to perform this task. A real problem means a new situation, perhaps not thoroughly defined, in which data might be missing, decisions to be taken undecided, techniques to be used unspecified, different ways of solution open, etc. That is precisely the expertise you will need in your work as an industrial engineer. Each of the subjects in your degree will help you to obtain the competences you will eventually need. In your first year you will study, besides Calculus, the subjects of Algebra, Chemical Fundamentals of Engineering, Fundamentals of Computer Science, Graphic Expression, Physical Basics of Engineering and Statistical Methods of Engineering. All these first-year subjects are the base you will need to study the specific disciplines of your degree.



The subject of Calculus will provide you with tools to analyse variables dependent on each other. It is customary in Engineering to have to analyse how variables change in relation to each other; variables such as speed, strength, charge, economic indexes, etc. Therefore, the tools that Calculus will provide will give you the ability of solving problems of your particular degree. We will try them on practical situations, so that you will see how the techniques can be applied to real and more complicate problems in your degree.

The Learning Results of the subject are a series of expertises your lecturer expects you to have at the end of the subject. Learning Results are proved performing activities; your lecturer will set out those activities for you and will evaluate to which extent you have attained each one of the Learning Results.



In the subject of Calculus we have the following Learning Results you must prove to have attained:



1. To develop the knowledge of the theoretical corpus of Mathematical Analysis that will allow you to recognise the concepts that can be applied to understand situations set out and to solve problems in the area of Engineering. This means that you will have to be able to choose which maths might be used to solve a problem, something that is not always clear in real situations.



2. To apply the procedures of Mathematical Analysis to solve specific problems of Engineering: to perform qualitative analysis, to use mathematical terminology and graphical language, to abstract, to pose hypothesis, to construct models, to apply mathematical results, to analyse the existence, uniqueness, properties and interpretation of solutions, to make generalizations and to construct demonstrations. This means that you will have to be able to find a solution for your problem and interpret what that solution means.



3. To use computer resources for the construction and operation of models based on concepts, results and procedures of Mathematical Analysis, with the aim of solving problems in Engineering. This means that you will have to be able to use the computer as a tool that helps you to find a solution for your problem and interpret what that solution means.



4. To explain justifiably the process you have followed to solve the problem using concepts, results and procedures of Mathematical Analysis (directly related to competence C4). This means that you will have to be able to explain the solution you have found for your problem.



5. To do team work successfully, integrating abilities and knowledge to adopt decisions in the development of the proposed tasks. This means that you will have to be able to work in a team, collaborating in the work.



6. To adopt a responsible attitude, tidiness in your work and readiness to learn, developing resources for an autonomous work. This means that you will have to be ever ready to learn new things and be able to work on your own.

BOE descriptors: Differential Calculus, Integral Calculus, Differential Equations.



1. Precalculus Review



2. Limits and Continuity



3. The Derivative; the process of Differentiation



4. The Mean-Value Theorem



5. Integration



6. Some applications of the integral



7. The Transcendental functions



8. Techniques of integration



9. Sequences; Indeterminate forms;



10. Infinite Series



11. The Conic Sections; Polar coordinates;



12. Vectors in three-dimensional space



13. Vector Calculus



14. Functions of several variables



15. Gradients; Extreme values



16. Double and Triple integrals



17. Line integrals and Surface integrals



18. Some differential equations



19. Additional differential equations

The types of tasks, both in the classroom (C) and outside the classroom (OC) will be the following:



1. Previous study of theoretical aspects or problems from the textbook (OC). The lecturer will commission, at the end of each class, the study of some section that will be the subject of the next class.



2. Solution of exercises of application from the textbook (C-OC). As a rule, the student himself will be commissioned to do the exercises; nevertheless, some will be explained in the classroom as examples.



3. Concept-tests (C-OC). The lecturer will propose questions to be debated and solved by the teams, some in the classroom (C) and some outside the classroom (OC).



4. Solution of problems using the computer (C-OC). In the laboratory class, the lecturer will commission the tasks to be done using the computer (OC). Each member of the team must write a report that can be asked later to defend orally.



5. Written exams (C).



Computer lessons will take place at the laboratories of the Applied Mathematics department, one hour each two weeks.

• Continuous Assessment System
• Final Assessment System
• Tools and qualification percentages:
  • Written test to be taken (%): 75
  • Realization of Practical Work (exercises, cases or problems) (%): 5
  • Team projects (problem solving, project design)) (%): 20

The evaluation is based on Chapter II of the normative "Graduko Titulazio Ofizialetako Ikasleen Ebaluaziorako Arautegia", according option a) Etengabeko ebaluazioa (Continous Evaluation) and the final mark will be calculated as has been stated in chapter 8.





1. WRITTEN EXAMS: 75% of the qualification (37.5 % per semester):



During the year, the student must do six written exams:



First semester:



EXAM 1: subjects 1-4.

EXAM 2: subjects 1-8.

EXAM 3: subjects 1-10.



Weight of exam 1: 4.5 %; weight of exam 2: 15 %; Weight of exam 3: 18 %. Total: 37.5 %





Second semester:



EXAM 4: subjects 11-15.

EXAM 5: subjects 11-17.

EXAM 6: subjects 11-19.



Weight of exam 4: 4.5 %; weight of exam 5: 15 %; Weight of exam 6: 18 %. Total: 37.5 %



2. EVALUATION OF TEAM WORK: 20% of the final qualification (10 % per semester)



Team work will be evaluated by means of several activities and reports, in teams and individually.



3. EVALUATION OF IN-CLASS AND OUT-OF-CLASS WORK: 5 % of the final qualification (2.5 % per semester):



In-class work will consist of the active participation in the tasks commissioned (problem solving, discussions, etc.). Out-of-class work will be of two types: previous study of theoretical aspects or problems from the textbook (OC), and solution of exercises of application from the textbook.



NOTE 1:



The student that in the written exams of the first semester has reached at least 1.5 points of the possible 3.75 obtainable by the written exams will liberate the subjects of the first semester. The student that has not reached 1.5 points of the possible 3.75 obtainable by the written exams must do again the exams of the first semester (exams 1, 2 and 3) during the second semester.





NOTE 2:



To pass the whole subject it is compulsory that the contribution of the three written exams to be of 3 points over the possible 7.5. If this minimum contribution is not achieved, the final qualification will be exclusively the sum of the written exams (over 10 points).



PROCEDURE TO RENOUNCE TO CONTINUOUS EVALUATION:



The student must notify the lecturer in writing, before 18 weeks starting on the first week of the school year, always according to the calendar of the College.



If the student has renounced to continuous evaluation, the final evaluation will consist of three proofs:



1. A written exam that will include all the subjects of the course, which will be 75 % of the qualification. In this exam the student must obtain at least 3 points over 7.5. With a lesser mark the student will not be qualified to do the rest of the proofs of the evaluation, and his/her qualification will be that of the written exam only (over 10 points).



2. A laboratory exam, which will be 20 % of the qualification. The student will have to solve exercises and problems using Geogebra software, similar to those of the program of computer classes.



3. An oral exam, which will be 5 % of the qualification. It will be similar to the team presentations that have been done throughout the year: the lecturer will pose questions about the exercises and theory of the subject.



PROCEDURE TO RENOUNCE TO THE ORDINARY CALL



1. In the case of continuous evaluation, to renounce to the call the student must notify the lecturer by writing, one month before the teaching of the subject is finished.



2. In the case of final evaluation, not attending the official written exam will be considered a renounce to the call and the student will have a "not qualified" academic record.

The evaluation will consist of:



1. A written exam that will include all the subjects of the course, which will be 75 % of the qualification. In this exam the student must obtain at least 3 points over 7.5. With a lesser mark the student will not be qualified to do the rest of the proofs of the evaluation, and his/her qualification will be that of the written exam only (over 10 points).



2. A laboratory exam, which will be 20 % of the qualification. The student will have to solve exercises and problems using Geogebra software, similar to those of the program of computer classes.



3. An oral exam, which will be 5 % of the qualification. It will be similar to the team presentations that have been done throughout the year: the lecturer will pose questions about the exercises and theory of the subject.



NOTE 1:



If the student has chosen the modality of continuous evaluation and has liberated the first semester but not the second, if he/she wishes, the student can keep the mark of the first semester and take only the exam of the second semester.



NOTE 2:



If the student has chosen the modality of continuous evaluation, proofs 2 and 3 are optional. That is to say: if he/she does not attend proofs 2 and 3, the qualifications at those proofs will be the ones obtained throughout the year; if he/she attends them, their qualifications will be the ones obtained at those proofs.



PROCEDURE TO RENOUNCE TO THE EXTRAORDINARY CALL



Not attending the official written exam will be considered a renounce to the call and the student will have a "not qualified" academic record.

*Book: Saturnino L. Salas, Einar Hille, Garret J. Etgen. Calculus: one and several variables 10th Edition.

*Work material included in the eGela virtual course by the lecturer.

*Exercises and other materials provided by the lecturer.

Basic bibliography

*Barragués, Arrieta y Manterola, Análisis Matemático. Ed. PRENTICE HALL (Pearson)

*Larson, Hostetler y Edwars, Cálculo, tomos I y II. Ed. Pirámide.

*Smith y Minton, Cálculo, tomos I y II. Ed. McGraw-Hill

*Zill, Ecuaciones diferenciales con aplicaciones. Ed. Grupo Editorial Iberoamérica.

*Piskunov, Cálculo diferencial e integral, tomos I y II. Ed. MIR.

*Piskunov. Kalkulu diferentziala eta integrala I eta II. Udako Euskal Unibertsitatea.

*Demidovich, Problemas y ejercicios de Análisis Matemático, Ed. Paraninfo.

*Manterola, eta besteak. Ingeniaritzarako Oinarri Matematikoak. Ariketa ebatziak. Elhuyar.

*Mijanjos. Ingeniaritzaren Oinarri Matematikoak. UPV/EHU

*Mártinez Sagarzazu. Ekuazio Diferentzialak. Aplikazioak eta ariketak. UEU.

*Viedma, J.A. "Métodos estadísticos". Ed. del Castillo

*Canavos, G. "Probabilidad y Estadística. Aplicaciones y métodos". Ed. McGraw-Hill

*Uña, I., Tomeo, V. y San Martín, J. "Lecciones de Cálculo de Probabilidades", Ed. Thomson.

*Parra, I. "Problemas de Inferencia Estadistica", Ed. Thomson.

*FERNANDEZ. Estatistikarako Sarrera. UEU.

*FERNANDEZ eta besteak. Estatistikarako Sarrera. Ariketak. UEU.

*FERNANDEZ eta besteak. Estatistika I eta Estatistika II. Ariketak. UEU

In-depth bibliography

*Kreyszig, Matemáticas avanzadas para Ingeniería, Vol I y II. Ed. Noriega-Limusa
*Martínez Salas, Elementos de Matemáticas. Ed. El autor.
*Bartle y Sherbert, Introducción al análisis Matemático en una variable, Ed. Noriega-Limusa.

Web addresses

http://ocw.ehu.es/ciencias-experimentales/fundamentos-matematicos-de-la-ingenieria-i/Course_listinghttp://ocw.ehu.es/saiakuntza-zientziak/aldagai-erreal-bat-eta-anitzeko-funtzio-errealen-analisi-http://ocw.ehu.es/ciencias-experimentales/fundamentos-matematicos-de-la-ingenieria-i/Course_listing
http://ocw.ehu.es/saiakuntza-zientziak/aldagai-erreal-bat-eta-anitzeko-funtzio-errealen-analisi-matematikoa/Course_listing
http://descartes.cnice.mec.es/indice_ud.php
http://www.wiris.con/demo/es/
http://www.slu.edu/classes/maymk/MathApplets-SLU.html
http://integrals.wolfram.com/index.jsp
http://www.geogebra.org/cms/
http://www.eecircle.com/applets/007/ILaplace.html
http://www.phy.ntnu.edu.tw/oldjava/sound/sound_s.htm
http://www.nst.ing.tu-bs.de/schaukasten/fourier/en_idx.html
http://www.dartmouth.edu/~chance/