Description

The aim of this subject is to introduce the student in the basic techniques of Differential Calculus in one and several real variables and its applications.

At the end of the semester it is expected that students have achieved the understanding of the basic concepts, handle the usual differential operators of the mathematical physics and learn the techniques of differential calculus for the determination of extremes local approximation of functions and numerical solution of systems of equations. Besides, the student will learn to handle some computer programs of symbolic calculation and graphic representation.

Instructors

• Fernández Manin, Generosa
• Bajo Palacio, Ignacio
• Calvo Ruibal, Natividad
• Prieto Gómez, Cristina Magdalena

Contents

• Topic 1. Introduction
  Sets of numbers and functions of one variable.
• Topic 2. Continuity of Functions of One Variable
  Limit of a function at a point. One-sided limits. Continuity. Intermediate Value Theorem. Bolzano’s theorem. Bisection method.
• Topic 3. Continuity of Functions of Several Variables
  n-dimensional space. Inner product and norm. Vector product. Functions of several variables. Limits and continuity. Bolzano’s theorem.
• Topic 4. Derivatives of Functions of One Variable and Applications of the Derivative
  Derivatives at a point. Derivative function and successive derivatives. Properties of derivatives. Chain rule. Implicit differentiation. Derivative of inverse functions. Maxima and minima. Mean Value Theorem. L’Hôpital’s rule. Local analysis of the graph of a function. Taylor polynomials. Newton’s method.
• Topic 5. Differential of Functions of Several Variables
  Directional derivatives. Partial derivatives. Jacobian matrix. Chain rule. Higher-order derivatives. Differential operators.
• Topic 6. Applications of Differential Calculus
  Extreme values. Extreme values with equality constraints. Newton’s method.

Learning Outcomes

• CG3 – Knowledge of basic subjects and technologies that enables students to learn new methods and technologies, providing versatility to face and adapt to new situations.
• CG4 – Ability to solve problems with initiative, make creative decisions, and communicate and transmit knowledge and skills, understanding the ethical and professional responsibility of the Technical Telecommunication Engineer.
• CE1 / FB1 – Ability to solve mathematical problems in engineering, applying knowledge of linear algebra, geometry, differential geometry, differential and integral calculus, ordinary and partial differential equations, numerical methods, numerical algorithms, statistics and optimization.
• CT2 – Understanding engineering within a framework of sustainable development.
• CT3 – Awareness of the need for lifelong learning and continuous quality improvement, showing a flexible, open and ethical attitude toward different opinions and situations, particularly regarding non-discrimination based on sex, race or religion, and respect for fundamental rights and accessibility.

Recommended Readings and Tools

Subjects that Continue the Syllabus

• Physics: Analysis of Linear Circuits — V05G301V01108
• Mathematics: Calculus 2 — V05G301V01106
• Mathematics: Probability and Statistics — V05G301V01107
• Physics: Fields and Waves — V05G301V01202
• Digital Signal Processing — V05G301V01205
• Electromagnetic Transmission — V05G301V01207

Subjects Recommended to be Taken Simultaneously

• Mathematics: Linear Algebra — V05G301V01102

Planned Activities

• Lecturing – The teachers will present the theoretical contents of the subject.
  Competencies developed: CG3, CE1, CT3.
• Problem solving – Teachers will solve problems and exercises for each topic, and students will solve similar exercises independently.
  Competencies developed: CG3, CG4, CE1, CT2, CT3.
• Laboratory practical – Students will use computer tools (Maxima and/or Matlab) to solve exercises and apply the knowledge acquired in theoretical classes.
  Competencies developed: CG3, CG4, CE1, CT2, CT3.

Assessment Methods and Criteria

Evaluation Activities

• Problem and/or exercise solving – First session (1 hour)
  Topic 1.
  Weight: 10%
  Training and learning results: B3, B4, C1.
• Problem and/or exercise solving – Second session (1 hour)
  Topics 2 and 3.
  Weight: 20%
  Training and learning results: B3, B4, C1.
• Problem and/or exercise solving – Third session (1 hour)
  Topics 4 and 5.
  Weight: 30%
  Training and learning results: B3, B4, C1.
• Final exam
  Exam covering topics 5 and 6 of the subject.
  Weight: 40%
  Training and learning results: B4, C1.

Other Comments on the Evaluation

According to the guidelines of the degree programme, students may choose between continuous assessment or exam-only (global) assessment.

1. Continuous assessment

Continuous assessment consists of the three one-hour evaluation sessions described above and a final exam. If a student cannot attend one of the scheduled tests, that test will be considered missed.

The final grade is calculated as:

N = C + E

• C: grade between 0 and 6 obtained as the sum of the marks from the three one-hour sessions.
• E: grade between 0 and 4 obtained in the final exam covering topics 5 and 6.

The course is passed when N ≥ 5. The grades obtained in the continuous tests are valid only for the academic year in which they are taken.

On the day of the final exam, students may choose either continuous assessment or global assessment.

2. Global assessment and end-of-programme call

Students who do not choose continuous assessment will be graded through a final exam covering topics 1–6. This exam may differ from the exam for students in the continuous assessment track. The exam is graded on a 0–10 scale, with 5 as the passing mark.

3. Extraordinary exam

During the second examination call, students previously evaluated through continuous assessment may choose:

• To be graded only through the second final exam, or
• To keep their continuous assessment marks and apply the formula:

NR = C + D

• C: mark between 0 and 6 obtained from the continuous assessment sessions.
• D: mark between 0 and 4 obtained in an exam covering topics 5 and 6.

The course is passed when NR ≥ 5.

Students who choose to be graded only through the second final exam will take an exam covering topics 1–6, graded on a 0–10 scale, with 5 as the passing mark.

4. “No Presentado” (Not Presented)

A student will receive the qualification “No Presentado” if they do not attend the continuous assessment sessions and do not attend the final examinations.

5. Ethical behaviour

Students are expected to maintain ethical and responsible behaviour in all tests and examinations, ensuring that the results accurately reflect their knowledge and skills. Any dishonest behaviour (such as copying or using unauthorized material) will result in a grade of 0 in the affected test and the case will be reported to the relevant academic authorities.