Description

The matter of Calculus II of the Degree in Engineering of Technologies of Telecommunication provides basic and common training to the branch of the telecommunication. Such as it figures in the memory of the degree, students should be able to formulate, to solve and to interpret mathematically problems within engineering of telecommunication at the end of the lectures. For this, they should know how to calculate integrals of functions of one and several variables and its meaning and they should handle the basic numerical methods of approximation for this kind of integrals. On the other hand, they should become familiar with the developments of functions in Fourier series. Also, they will have to know how to solve differential equations of first and second order. Finally, they should know to handle the Laplace transform in order to solve differential equations. All of these contents are notable for several matters that they must to study simultaneously or later in the degree.

Instructors

• Álvarez Vázquez, Lino José
• Martínez Varela, Áurea María

Contents

• Subject 1: Integral calculus in ℝ
  • The Riemann integral: integrable functions
  • Fundamental theorems of integral calculus
  • Computation of primitives: integration by parts and change of variable
  • Improper integrals
• Subject 2: Numerical methods for the approximation of integrals
  • Quadrature rules of interpolating polynomial type
  • Properties and interpolation error
  • Particular cases: Poncelet, Trapezoidal and Simpson rules
  • Composite quadrature rules
• Subject 3: Fourier series and Fourier transform
  • Orthogonal functions
  • Fourier series
  • Fourier series expansions for odd and even functions
  • Convergence
  • The Fourier transform
• Subject 4: Multiple integration
  • Double and triple integrals in elementary regions
  • Change in the order of integration
  • Theorems for change of variables
  • Applications
• Subject 5: Laplace transform
  • Definition of the Laplace transform
  • Properties
• Subject 6: Ordinary differential equations
  • General concepts: solution, families of curves and orthogonal trajectories
  • First-order differential equations: existence and uniqueness of solutions, exact equations, separable variables, homogeneous equations and linear equations
  • Second-order differential equations: existence and uniqueness of solutions for linear equations, application of the Laplace transform, method of undetermined coefficients, variation of parameters, Cauchy–Euler equation.

Learning Outcomes

• B3 (CG3) – The knowledge of basic subjects and technologies that enables the student to learn new methods and technologies, giving them versatility to face and adapt to new situations.
• B4 (CG4) – The 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.
• C1 (CE1/FB1) – The 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.
• D2 (CT2) – Understanding engineering within a framework of sustainable development.
• D3 (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: Fields and Waves — V05G301V01202
• Subjects recommended to be taken simultaneously
  • Physics: Analysis of Linear Circuits — V05G301V01108
  • Mathematics: Probability and Statistics — V05G301V01107
• Subjects recommended to have taken before
  • Mathematics: Linear Algebra — V05G301V01102
  • Mathematics: Calculus 1 — V05G301V01101

Planned Activities

• Problem solving – During these sessions, the professor will solve problems related to each topic and introduce new solution methods not covered in the lectures, from a practical perspective. Students will also solve problems proposed by the professor in order to apply the knowledge acquired.
  Competencies developed: B3, B4, C1, D2, D3.
• Laboratory practical – In these sessions, the computer tool MATLAB will be used to study and apply the numerical methods for the approximation of integrals described in Topic 2 of the subject.
  Competencies developed: B4, C1, D2, D3.
• Lecturing – The professor will present the theoretical contents of the subject during these classes.
  Competencies developed: B3, C1, D2, D3.

Assessment Methods and Criteria

Evaluation Activities

• Problem and/or exercise solving – The assessment includes three one-hour evaluation sessions and one final exam.

  • 1st session: Topics 1, 2 and 3 – weight 20% (2 points)
  • 2nd session: Topic 4 – weight 20% (2 points)
  • 3rd session: Topics 5 and 6 – weight 20% (2 points)
  • Final exam: weight 40% (4 points)

  Total weight: 100% (individual assessment).
  Training and learning results: B3, B4, C1, D2, D3.

Other Comments on the Evaluation

The evaluation system is preferably based on continuous assessment. A student is considered enrolled in this system if they attend any evaluable session. Once enrolled, it is not possible to withdraw from continuous assessment.

Continuous assessment tests cannot be repeated. If a student cannot attend a scheduled test, it will not be rescheduled. Before each test, the approximate publication date of the marks and the procedure for grade review will be communicated.

The marks obtained in the evaluation activities are valid only for the academic year in which they are obtained.

During the continuous assessment tests, students will solve problems and exercises related to the topics of the course.

The schedule for intermediate exams will be approved by the Academic Committee of the Degree (CAG) and published at the beginning of each academic semester.

1. Continuous Assessment

The final grade for students following continuous assessment is calculated as:

N = C + E

Where:

• C: grade obtained by adding the scores of the three evaluation sessions covering topics 1–6.
• E: grade obtained in the final exam covering topics 4, 5 and 6.

Students pass the course when N ≥ 5.

2. Global Assessment

Students who do not follow continuous assessment may take a final exam covering all the topics of the course on the same date as the final exam of the continuous assessment system.

The exam is graded on a 0–10 scale, and the course is passed when the score is ≥ 5.

3. Extraordinary Exam

Before the extraordinary exam, students who followed continuous assessment may choose to take a recovery exam covering topics 4, 5 and 6.

The final grade is calculated as:

NR = C + ER

Where:

• C: grade obtained from the three evaluation sessions covering topics 1–6.
• ER: grade obtained in the recovery exam covering topics 4, 5 and 6.

Students pass the course when NR ≥ 5.

If students do not choose this option, they will take an exam covering all topics of the course, graded from 0 to 10, with 5 as the passing mark.

4. “Not Presented” Grade

A student will receive the grade “Not Presented” if they are not enrolled in continuous assessment and do not attend any of the examinations. Otherwise, the student will be considered presented.

5. End-of-Program Exam

In the end-of-program call, students will be evaluated through an exam covering all topics of the course.