# 25971 - Calculus

## DESCRIPTION & CONTEXTUALISATION OF THE SUBJECT

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.

## COMPETENCIES/LEARNING RESULTS FOR THE SUBJECT

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.

## THEORETICAL/PRACTICAL CONTENT

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

## METHODS

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 later be asked to defend orally.

5. Written exams (C).

## TYPES OF TEACHING

Type of teaching | M | S | GA | GL | GO |
---|---|---|---|---|---|

Classroom hours | 75 | 30 | 15 | ||

Hours of study outside the classroom | 112,5 | 45 | 22,5 |

**Legend: M:** Lecture **S:** Seminario **GA:** Pract.Class.Work **GL:** Pract.Lab work **GO:** Pract.computer wo

**GCL:** Clinical Practice **TA: **Workshop **TI:** Ind. workshop **GCA:** Field workshop

## ASSESSMENT SYSTEMS

- Continuous assessment system

- Final assessment system

## TOOLS USED & GRADING PERCENTAGES

- Extended written exam 70%

- Practical work (exercises, case studies & problems set) 15%

- Team work (problem solving, project design) 15%

## ORDINARY EXAM CALL: GUIDELINES & DECLINING TO SIT

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.

For the ordinary call, three types of evaluation test will be used:

1) Six individual written exams, three in each semester: 70% of the qualification:

- First exam: subjects 1-4.

- Second exam: subjects 1-8.

- Third exam: subjects 1-10.

- Fourth exam: subjects 11-15 if the student passed the first semester; subjects 1-4 plus 11-15 if failed.

- Fifth exam: subjects 11-17 if the student passed the first semester; subjects 1-8 plus 11-17 if failed.

- Sixth exam: subjects 11-19 if the student passed the first semester; subjects 1-19 if failed.

2) Evaluation on classroom and out-of-classroom work: 10% of the final qualification. Classroom work consists of the

active participation in the tasks commissioned (problem solving, discussions, etc.). Out-of-classroom, which consists of

work given from the textbook, is divided into two types: first, the previous lecture and understanding of the commissioned

items; second, the solving of the commissioned exercises.

3) Group work and oral presentation: 20% of the qualification. Six group works, three each semester. All the group

members will attend to the lecturer's office in the fixed date in order to defend their work. The lecturer will choose a group

member to explain some aspects of the job done.

The details of each qualification are listed below:

FIRST EVALUATION (9% of the qualification)

SECOND EVALUATION (18% of the qualification)

THIRD EVALUATION (23% of the qualification)

1) Students that have passed the first semester

FOURTH EVALUATION (10% of the qualification)

FIFTH EVALUATION (17% of the qualification)

SIXTH EVALUATION (23% of the qualification)

2) Students that have failed the first semester

FOURTH EVALUATION (19% of the qualification)

FIFTH EVALUATION (35% of the qualification)

SIXTH EVALUATION (46% of the qualification)

IMPORTANT: for the last qualification it is compulsory that the contribution of the three written exams to be of 3 points

over 7. If this minimum contribution is not achieved, the final qualification will be exclusively the sum of the written exams.

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 %70 of the qualification. In this exam the

student must obtain at least 3 points over 7. 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 %10 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: any student that has not attended the official written exam of

ordinary call will have a “not qualified” academic record, both in the case of continuous and final

evaluation.

## EXTRAORDINARY EXAM CALL: GUIDELINES & DECLINING TO SIT

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 %70 of the qualification. In this exam the

student must obtain at least 3 points over 7. 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 %10 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: for those students that have chosen the modality of countinuous evaluation, proofs 2) and 3) are optional. That is

to say: if they do not attend proofs 2) and 3), their qualifications at those proofs will be the ones obtained thoughout the

year; if the attend them, their qualifications will be the ones obtained at those proofs.

PROCEDURE TO RENOUNCE TO THE EXTRAORDINARY CALL: any student that has not attended the official written

exam of the extraordinary call will have a “not qualified” academic record, both in the case of continuous

and final evaluation.

## COMPULSORY MATERIALS

-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.

## BIBLIOGRAPHY

**Basic bibliography**

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

Barragués, Arrieta y Manterola, Análisis Matemático con soporte interactivo en Moodle.Ed. 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.

Demidovitch, Problemas y ejercicios de Análisis Matemático, Ed. Paraninfo.

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

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.

**Useful websites**

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

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-analisimatematikoa/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/