Academic management

CONTEXT:

This subject is part of the Mathematical Foundations courses included in the Basic Training module of the computer software engineering degree and is similar to the subject with the same name taught in the rest of the engineering degrees. By its basic nature, its knowledge is essential for the development of other modules of the degree.

REQUIREMENTS:

The student needs only knowledge of the contents of Mathematics I and II of high school to follow the course.

General and cross-cutting skills:

CG3.Capacity for abstraction

CG4 Analysis and synthesis

CG6. Search, analysis and information management to transform it into knowledge

CG7. Skill in writing

CG9. Oral communication skills (direct or supported by audiovisual methods)

CG11. Teamwork ability

CG12. Leadership

CG16. Competence for self-criticism, essential for the professional and cultural development of the student.

CG18. Sense of responsibility

CG19. Effective work habits

CG20. Creativity

CG25. Critical reasoning

CG26. Ability to learn and work autonomously

Specific competences:

Bas. 1: Ability to solve mathematical problems that may arise in engineering. Ability to apply knowledge of: linear algebra, differential and integral calculus, numerical methods, numerical algorithms, statistics and optimization.

Learning outcomes:

RA.FM-7: Handling and plotting real functions of a real variable, obtaining their limits, determining their continuity, calculating derivatives and computing and solving problems of optimization.

RA. FM-8: Knowledge of the concepts of sequences and series. Use of power series to represent functions.

RA. FM-9: Setting out and calculating integrals of single variable functions and applying them to solving problems related to engineering.

RA. FM-10: The knowledge and application of basic properties of multivariable functions. Obtaining their limits, analyzing their continuity and differentiability and solving optimization problems.

Item 1: REAL-VALUED FUNCTIONS OF A REAL VARIABLE

1.1: Numerical Sets.  The natural numbers: Method of induction. The real numbers. Absolute value of a real number. Properties.

1.2: Functions of a real variable. Preliminary Notions. Elementary functions. Composition of functions and inverse function.

1.3: Limits of functions. Limits of functions. Properties. Infinitesimals and infinities. Indeterminate forms. Asymptotes.

1.4: Continuity of functions. Continuous functions. Properties of continuous functions: Bolzano’s Theorem. The Intermediate Value Theorem. Weierstrass’ Theorem.

1.5: Differentiation.  Properties of differentiable functions. Derivative of a function at a point. Derivative function. Differentiability and continuity. Properties of the derivative. The Chain rule.  Rolle's Theorem. The Mean Value Theorem. L'Hôpital’s Rule.

1.6: Taylor polynomial.  Higher-order derivatives. Taylor polynomials. Taylor's formula with remainder.

1.7: Optimization. Local study of a function. Monotonicity, relative extrema, concavity and inflection points. Absolute extrema. Graphing.

Item 2: RIEMANN INTEGRAL

2.1: Calculation of primitives. Immediate integrals. Integration methods.

2.2: The definite integral. Basic concepts and geometrical interpretation. Integrable functions. Properties of the definite integral. The Fundamental Theorem of Integral Calculus. Barrow’s Rule. Applications.

2.3: Improper integrals. Improper integrals. Application to the study of Eulerian integrals.

Item 3: SEQUENCES AND SERIES. POWER SERIES

3.1: Infinite sequences.Infinite sequences Convergence. Calculation of limits.

3.2: Infinite series. Infinite series. Convergence and summation of series. Harmonic series and geometric series. Convergence criteria.

3.3: Power series. Power series. Radius of convergence. Derivative and integral of a power series. Power series expansion of a function: Taylor Series. Expansion of commonly used functions.

Item 4: MULTIVARIABLE FUNCTIONS

4.1: The Euclidean spaceRn.  The Euclidean spaceRn. Basic notions of topology. Real-valued functions. Vector-valued functions.

4.2: Limits and continuity of functions of several variables. Limit of a function at a point and its properties. Calculation of limits. Continuity of a function and its properties.

4.3: Differentiability of functions of several variables. Directional derivative. Partial derivatives. Geometric interpretation. Higher-order derivatives. Differentiation and continuity.

4.4: Differentiation of functions of several variables. Differential of a function at a point. Linear approximation. Sufficient condition for differentiability. Gradient vector. Tangent plane. The Chain Rule.

4.5: Optimization without constraints. Relative extrema. Necessary condition. Sufficient condition. Absolute extrema.

4.6: Optimization with constraints. Constrained relative extrema. Lagrange multipliers.

6. Methodology and work plan:

Work plan:

Total volume of student work:

Exceptionally, if it is required because of the health situation, online activities could replace face-to-face teaching. In this case, the students will be informed of the necessary changes.

Assessment of student learning

There will be 2 exams . A midterm exam at a date and time that will be communicated in advance and a second on the official date for the ordinary evaluation. The arithmetic mean of the two scores weighs 80% of the final grade. There will be no final exam in the ordinary evaluation.

The computer practices will be evaluated in the practice sessions. This part will count 15% in the final grade. The score obtained is unchangeable and is retained for the extraordinary evaluations.

Attendance and participation in seminars counts 5% in the final grade. The score obtained in this section is unchangeable and is also kept for the extrordinary exams.

To pass the subject in the ordinary evaluation the arithmetic mean of the two exams taken must be greater than or equal to 4 (out of 10) and the weighted sum of the three scores (exams + computer labs + attendance) must be greater than or equal to 5. The students with an average mark lower than 4 in the exams will get 4.5 as the maximum possible final score.

To pass the subject in the extraordinary evaluations, the student must obtain a minimum of 4 (out of 10) in the official exam and the weighted sum of the three grades (official test+computer labs+attendance) must be greater than or equal to 5.

The special evaluation system differs from the previous evaluation in the following points:

1.The arithmetic mean of the marks obtained in the two exams weighs 85% of the final mark; the official exam resit has equal weighting.

2. No minimum attendance is required for the laboratory practices. A final practice test will be held in the last session. The practice score will count 15 % of the final grade. This mark will be kept for the official exam resit.

3. Attendance and participation in classroom practices are not evaluated.

4. The requirements to pass the subject are similar to those of the normal system, except that only two marks are considered instead of three.

Exceptionally, if it is required because of the health situation, online evaluation could replace face-to-face assesment. In this case, the students will be informed of the necessary changes.

Resources, basic references and further reading

Resources:

Classroom with a computer for the teacher and a projector.

Classrooms with computers for the computer practices

Virtual Campus of the University of Oviedo

Basic references:

Thomas G. B., Weir M. D. , Hass J.  Thomas’ Calculus (Single Variable)  Addison-Wesley Twelfth edition (2009)

Thomas G. B., Weir M. D. , Hass J.  Thomas’ Calculus (Multivariable)  Addison-Wesley Twelfth edition (2009)

Bradley G. L.; Smith, K. J.  Cálculo de una variable y varias variables. (Vol.  I y II). Prentice Hall (4ª ed.), 2001.

García López, A y otros. Cálculo I: teoría y problemas de análisis matemático en una variable, CLAGSA (3ª ed.), 2007.

García López, A y otros. Cálculo II: teoría y problemas de funciones de varias variables. CLAGSA (2ª ed.), 2002.

Stewart, J. Cálculo de una variable y Cálculo multivariable. Paraninfo Thomson. (6ª ed.) 2009.

 Further reading:

Larson R., Edwards B. H. Calculus Brooks/Cole  Ninth Edition (2010)

Bayón L., Grau J. M., Suárez P. M. Cálculo. Grados en Ingeniería. Universidad de Oviedo. 2011

Burgos Román, J. Cálculo Infinitesimal de una variable y en varias variables. (Vol. I y II).  McGraw-Hill. (2ª ed.), 2008

Marsden, J.; Tromba, A. Cálculo vectorial. Addison-Wesley Longman (5ªed.), 2004.

Neuhauser, Claudia. Matemáticas para ciencias. Pearson. Prentice Hall, 2004.

Stewart, J. Cálculo de una variable y Cálculo multivariable. Paraninfo Thomson. (6ª ed.) 2009.

Tomeo Perucha, V. y otros. Problemas resueltos de Cálculo en una variable.Thomson, 2005.