Estudia
- Artes y humanidades
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Ingeniería y arquitectura
- Doble Grado en Ingeniería Civil e Ingeniería de los Recursos Mineros y Energéticos
- Doble Grado en Ingeniería en Tecnologías y Servicios de Telecomunicación / Grado en Ciencia e Ingeniería de Datos
- Doble Grado en Ingeniería Informática del Software / Grado en Matemáticas
- Doble Grado en Ingeniería Informática en Tecnologías de la Información / Grado en Ciencia e Ingeniería de Datos
- Grado en Ciencia e Ingeniería de Datos
- Grado en Ingeniería Civil
- Grado en Ingeniería de los Recursos Mineros y Energéticos
- Grado en Ingeniería de Organización Industrial
- Grado en Ingeniería de Tecnologías Industriales
- Grado en Ingeniería de Tecnologías Mineras
- Grado en Ingeniería Eléctrica
- Bachelor´s Degree in Industrial Electronics and Automatics Engineering
- Grado en Ingeniería en Geomática
- Grado en Ingeniería en Tecnologías y Servicios de Telecomunicación
- Grado en Ingeniería Forestal y del Medio Natural
- Grado en Ingeniería Forestal y del Medio Natural (En extinción)
- Grado en Ingeniería Informática del Software
- Grado en Ingeniería Informática en Tecnologías de la Información
- Grado en Ingeniería Mecánica
- Grado en Ingeniería Química
- Grado en Ingeniería Química Industrial
- Grado en Marina
- Grado en Náutica y Transporte Marítimo
- Información, acceso y becas
Cálculo
- Clases Expositivas (28 Hours)
- Prácticas de Aula/Semina (21 Hours)
- Prácticas de Laboratorio (9 Hours)
The course Calculus is part of the subject Mathematics included in the basic training module of the Mining and Energy Resources Engineering Degree, the Forestry and Natural Environment Engineering Degree and the Topography and Geomatics Engineering Degree. At the same time it is similar to the course given under the same name in all other Bachelor's Degrees in Engineering. By its basic nature, its knowledge is essential in the development of other modules of the degree.
The student needs only knowledge of the contents of Mathematics I and II of high school to follow the course.
BOE specific competence:
Ability to solve mathematical problems that may arise in engineering. Ability to apply knowledge of: linear algebra, geometry, differential geometry, differential and integral calculus, differential equations and partial differential equations, numerical methods, numerical algorithms, statistics and optimization.
General and cross-cutting skills:
Ability to apply theoretical and practical methods to the analysis and solution of engineering problems. Written and oral communication skills. Teamwork ability. Sense of responsibility, critical thinking and effective work habits.
Learning outcomes:
RA1:Handling and plotting real functions of a real variable, obtaining their limits, determining their continuity, calculating derivatives and computing and solving optimization problems.
RA2: Knowledge of the concepts of sequences and series. The use of power series to represent functions.
RA3: Setting out and calculating integrals of single variable functions and applying them to solve engineering problems.
RA 4: The statement 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. Natural numbers: Method of induction. 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 antiderivatives. 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: MULTIVARIABLE FUNCTIONS
3.1: The Euclidean space Rn. The Euclidean space Rn. Basic notions of topology. Real-valued functions. Vector-valued functions.
3.2: Limits and continuity of functions of several variables. Limit of a function at a point and properties. Calculation of limits. Continuity of a function and its properties.
3.3: Differentiability of functions of several variables. Directional derivative. Partial derivatives. Geometric interpretation. Higher-order derivatives. Differentiation and continuity.
3.4: 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.
3.5: Optimization without constraints. Relative extrema. Necessary condition. Sufficient condition. Absolute extrema.
3.6: Optimization with constraints. Constrained relative extrema. Lagrange multipliers.
Item 4: SEQUENCES AND SERIES. POWER SERIES
4.1: Infinite sequences. Infinite sequences Convergence. Calculation of limits.
4.2: Infinite series. Infinite series. Convergence and summation of series. Harmonic series and geometric series. Convergence criteria.
4.3: 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.
Lab sessions:
1.- Introduction to Matlab (1,5 hours).
2.- Functions of one variable: Limits, continuity and graphic representations (1,5 hours).
3.- Functions of one variable: Derivative (2 hours).
4.- Functions of one variable: Integration (2 hours).
5.- Multivariable calculus (2 hours).
Lab sessions:
1.- Introduction to Matlab (1,5 hours).
2.- Functions of one variable: Limits, continuity and graphic representations (1,5 hours).
3.- Functions of one variable: Derivative (2 hours).
4.- Functions of one variable: Integration (2 hours).
5.- Multivariable calculus (2 hours).
Lab sessions:
1.- Introduction to Matlab (1,5 hours).
2.- Functions of one variable: Limits, continuity and graphic representations (1,5 hours).
3.- Functions of one variable: Derivative (2 hours).
4.- Functions of one variable: Integration (2 hours).
5.- Multivariable calculus (2 hours).
Lab sessions:
1.- Introduction to Matlab (1,5 hours).
2.- Functions of one variable: Limits, continuity and graphic representations (1,5 hours).
3.- Functions of one variable: Derivative (2 hours).
4.- Functions of one variable: Integration (2 hours).
5.- Multivariable calculus (2 hours).
Lab sessions:
1.- Introduction to Matlab (1,5 hours).
2.- Functions of one variable: Limits, continuity and graphic representations (1,5 hours).
3.- Functions of one variable: Derivative (2 hours).
4.- Functions of one variable: Integration (2 hours).
5.- Multivariable calculus (2 hours).
We will start each unit reviewing basic knowledge necessary to understand it. Then, with the support of the screen (available to the student on course's moodle) and using the blackboard, the theory and exercises are explained. Student's voluntarily participation will be sought, entrusting them with several tasks.
Exceptionally, if sanitary conditions require it, distance learning activities may be included. In which case, students will be properly informed about changes.
Work plan:
Class work | Home work | |||||||
Items | Total hours | Lectures and evaluation sessions | Seminars | Computer practices | Total | Group work | Individual work | Total |
Item 1: Real-valued functions of a real variable | 39 | 8 | 6 | 4 | 18 | 6 | 15 | 21 |
Item 2: Riemann integral | 29 | 5 | 5 | 2.5 | 12.5 | 5 | 11.5 | 16.5 |
Item 3: Multivariable functions | 52 | 10 | 7 | 2.5 | 19.5 | 8.5 | 24 | 32.5 |
Item 4: Sequences and series. Power series | 30 | 5 | 3 | 0 | 8 | 7 | 15 | 22 |
Total | 150 | 28 | 21 | 9 | 58 | 26.5 | 65.5 | 92 |
Total volume of student work:
TYPES | Hours | % | Total | |
Class work | Lectures and evaluation sessions | 28 | 18,67% | 58 |
Seminars | 21 | 14% | ||
Computer practices | 9 | 6% | ||
Home work | Group work | 26.5 | 17.66% | 92 |
Individual work | 65.5 | 43,67% | ||
Total | 150 |
i) There will be a midterm written exam of an hour and a half to evaluate the ability of the students in handling basic concepts of differentiation and integration. It counts for 20% of the final grade.
ii) Attendance and active participation in classroom practices count for 10% in the final grade.
iii) The score corresponding to the computer practice sessions is obtained from a continuous evaluation process that counts for 10% in the final grade. The score obtained in this section is saved for extraordinary examination sessions.
iv) A final written exam of two hours, that weighs 60% of the final grade, will take place at the end of the semester. In this exam, the assimilation of the global contents of the course and their application in solving practical problems are tested.
v) The extraordinary evaluations will consist in a written exam that represents 90% of the final grade. The remaining 10% corresponds to the score obtained in section iii).
Exceptionally, if sanitary conditions require it, distance assessment methods may be included. In which case, students will be properly informed about changes.
Resources:
Classroom with a computer for the teacher and a projector.
Classrooms with computers for the computer practices
Virtual Campus from Oviedo University
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:
Burgos Román, J. Cálculo Infinitesimal de una variable y en varias variables. (Vol. I y II). McGraw-Hill. (2ª ed.), 2008.
Larson, R. E. y otros. Cálculo y geometría analítica. (Vol. I y II). McGraw-Hill (8ªed.), 2005.
Marsden, J. ; Tromba, A. Cálculo vectorial. Addison-Wesley Longman (5ªed.), 2004.
Neuhauser, Claudia. Matemáticas para ciencias. Pearson. Prentice Hall, 2004.
Tomeo Perucha, V. y otros. Problemas resueltos de Cálculo en una variable. Thomson, 2005.