The objective of this subject, together with Calculus II, is to provide the knowledge to apply differential and integral calculus and complex variables.

COMPETENCIAS GENERALES

G003 Knowledge in basic and technological subjects, which enable to learn new methods and theories, and provide versatility to adapt to new situations.

COMPETENCIAS TRANSVERSALES

T001 Ability to solve problems with initiative, decision making, creativity and critical reasoning, respecting the principles of universal accessibility and design for all people.

COMPETENCIAS ESPECÍFICAS

M01FB01 Ability to solve mathematical problems in engineering. Ability to apply knowledge of: linear algebra; geometry; differential geometry; differential and integral calculus; partial differential equations; differential equations; numerical methods; numerical algorithms; statistics and optimization.



RESULTADOS DE APRENDIZAJE-Titulación

RAG5 The graduate will be able to identify the concepts and methods related to mathematics that are applicable in the field of engineering.

RAT1 The graduate will be able to solve problems with initiative, decision making, creativity and critical reasoning.

Topic 01. Previous concepts: Elementary functions

Topic 02. Complex numbers and complex variable.

Topic 03. Real numerical series and sequences

Topic 04. Potential series and Taylor series.

Topic 05. Limits, Continuity, Derivability and Differentiability of real functions of a real variable.

Topic 06. Limits, Continuity, Partial Derivative and Differentiability of real functions of several real variables. Directional Derivatives and Gradient.

Topic 07. Composite functions.

Topic 08. Implicit Functions.

Topic 09. Extrema of real functions of several real variables.

Lectures:

The contents of the syllabus will be explained. They will be completed with examples and clarifying exercises. In general, no distinction will be made between lectures and classroom practices. Some short videos will be used to illustrate concepts of the subject. In the virtual classroom of the course, students will be able to access complementary material of the course.

In general and whenever possible, each topic will be introduced with an exercise or problem from everyday life related to biomedical engineering studies. Students will try to solve it, individually or collectively. The aim will be to make them aware of the real difficulty of the problem they are facing and the need to introduce new concepts that can help them to solve it. The teacher will give them indications to be able to solve the obstacles that are presented to them. Subsequently, the students will try to apply the concepts explained in classes to the resolution of the problem posed.



Seminars:

-The teacher will solve some exercises of special difficulty.

-Exercises and problems that the students will solve theoretically and/or using Mathematica will be proposed.

-Divided into groups, the students will solve exercises proposed by the teacher and will deliver the resolution at the end of the seminar.

-Presentation by the students of the different works and projects carried out individually or in groups.



Other applications of active methodologies:

-Problem-solving: it will consist of the resolution of exercises on the different topics by the students individually or in groups.

-Directed studies: they will consist of the realization by the student, individually or in group, of a practical study related to differential calculus, under the direction of the teacher. It may be necessary for the practical exposition of the work by the students.

• Continuous Assessment System
• Final Assessment System
• Tools and qualification percentages:
  • Written test to be taken (%): 90
  • The sum of the work and tests carried out throughout the course. These tests can be written, practical work (exercises, cases, or problems), individual work, teamwork, and/or their exposition of work. (%): 10

The continuous evaluation system is based on the following two blocks and criteria:



Block 1: Active methodologies.

- Resolution of problems and proposed exercises.

- Delivery and/or exposure to the proposed work and projects individually or collectively.

- Several written or oral tests of short duration, of theoretical and practical nature.

The result of this block will represent 10% of the final grade.



Block 2: Global evaluation of the acquired knowledge.

- The weight of this block will be 90% of the final grade. The following two mid-term exams (non-eliminatory) will be carried out with the following contents and values:

-Previous concepts, complex numbers, and complex variables (topics 1 and 2). Limits, continuity, derivability, and differentiability of real functions of a real variable (topic 5). Limits, continuity, derivability, and differentiability of real functions of a real variable (topic 5). Sequences and real numerical series. Potential series and Taylor series. Fourier series (topics 3 and 4): 35%.

-Limits, continuity, partial derivatives, and differentiability of real functions of several real variables. Directional derivatives and gradient. Composite functions (topics 6 and 7). Implicit functions. Extremes of real functions of several real variables (topics 8 and 9): 55%.

The final grade of this part will be the average obtained from the five partial exams.

The final grade will be expressed numerically as the result of the weighted average between blocks I and II.



Resignation: Single final evaluation.

Following the regulations of the university, students will be able to do a final evaluation through a single written exam with several theoretical and practical questions that guarantee that the student has acquired all the competencies described in this teaching guide. The score obtained in it will represent 100% of the final grade.

To do so, students must submit a written resignation of continuous assessment, for which they will have a period of 9 weeks from the beginning of the term, according to the academic calendar of the center.

There will be a written exam with several theoretical and practical questions to ensure that the student has acquired all the skills described in this teaching guide. The score obtained in this exam will represent 100% of the final grade.

Basic bibliography

– “Calculus - Early Transcendentals”, Stewart (Thomson, 6th ed, 2008)

- “Calculus”, Thomas_Finney

- "Calculus", Larson, Hostetler, Edwards. Houghton Mifflin Company.

- "Exámenes Resueltos de Cálculo Infinitesimal 1996-2005". Servicio editorial UPV/EHU.

In-depth bibliography

-"Calculus", Apostol. Ed Reverte
-"The Elements of Real Analysis" Bartle, R.G. Ed. Jhon Wiley and Sons.
-"Matemáticas Avanzadas para Ingeniería". Kreyszing. Ed. Limusa.
-"Matemáticas Avanzadas para Estudiantes de Ingeniería". Kaplan, Ed. Addison Wesley.
-"Problemas de Cálculo Infinitesimal e Integral". Bronte R.
-"Cálculo Infinitesimal de una Variable". Burgos J. Ed. Mc. Graw-Hill.