Academic year 2013-14
Calculus and Numeric Methods
| Degree: | Code: | Type: |
| Bachelor's Degree in Computer Science | 21403 | Core subject, 1st year |
| Bachelor's Degree in Telematics Engineering | 21295 | Core subject, 1st year |
| Bachelor's Degree in Audiovisual Systems Engineering | 21592 | Core subject, 1st year |
| ECTS credits: | 8 | Workload: | 200 hours | Trimester: | 1st and 2nd |
| Department: | Dept. of Information and Communication Technologies |
| Coordinator: | Gloria Haro |
| Teaching staff: | Xènia Albà, Anna Carreras, Mariella Dimiccoli, Armin Duff, José Maria Esnaola, Àngel Garcia, Gloria Haro, Maria Oliver, Roberto Pérez, Jordi Taixés. |
| Language: | Xènia Albà (catalan), Anna Carreras (catalan), Mariella Dimiccoli (spanish), Armin Duff, José Maria Esnaola (spanish), Àngel Garcia (catalan), Gloria Haro(catalán), Maria Oliver (catalan), Roberto Pérez (spanish), Jordi Taixés (catalan). |
| Timetable: | |
| Building: | Communication campus - Poblenou |
This course , along with the course of Algebra and Discrete Mathematics provide students with the mathematical basis for the other courses in the engineering studies that are taught in parallel or subsequently.
The course starts from the concepts that students have studied in High School and consolidates and expands them.
The course consists of two parts, to be held respectively in the first and second trimesters. The first part is based on the study of functions of a single variable while the the second part is devoted to functions of several variables.
In the first part, corresponding to the first trimester, we review sets of numbers emphasizing the calculus with real variables and skills in working with algebraic inequalities and absolute value. The presentation of the concepts with scientific notation is emphasized, together with a rigorous analysis and problem solving, and formal presentation of the answer. The course consolidates the concepts of real variable functions introduced in High School and extends them: function definition, domain , basic properties , limits, study of continuity, differentiability and integration. It also introduces the numerical calculation by the method of Newton 1-D in order to find roots of functions. This first part ends with the introduction to sequences and series of real numbers and the application of these concepts to approximate sufficiently differentiable functions through power series (Taylor expansions).
The second part, corresponding to the second trimester, extends many concepts of the first part to functions of several variables. Specifically it deals with the following topics: definition of a function of several real variables, domain, image, curves and trajectories, surfaces, derivation, tangent subspaces, local approximation, integration in several variables . It also applies this knowledge to the resolution of optimization problems, namely the study of unconstrained and constrained extrema by using Lagrange multipliers. Regarding numerical methods, the second part deals with the extension of Newton's method to several variables and the method of descent/ ascent gradient for solving optimization problems and systems of nonlinear equations.
Strengthen basic math skills, introduce scientific language and work with rigorous reasoning are basic objectives of the course. Another objective is that the student learn the intrinsic meaning of the concepts of differential calculus and apply his/her knowledge to solve specific problems (which will be presented essentially in practical classes and seminars).
This course is related to many other courses of the Teaching Plan of Graduate Studies as Linear Algebra and Discrete Mathematics , Probability and Stochastic Processes, Signals and Systems , Waves and Electromagnetics , Infography, Differential Equations , Signal Processing , Computational Geometry, Synthetic Image, Communication Systems , Data Transmission and Coding, Radiocommunications, Speech Processing , Image Processing , Speech Coding Systems and Audio Systems, Image and Video Coding, Advanced Visualization, Video Processing, Audio Processing in Real Time, Acoustic Engineering, Optical Engineering, Perception and Audivisual Cognition, Architectural Acoustics, Electronic Circuits and Transmission Media, Sound and Music Processing, Computational Fundamentals of Audiovisual Systems, Analysis and Interpretation of Images, Pattern Recognition, 3D Audio, 3D Vision, Synthetic Image.
This course assumes a minimal mathematical background level of Bachillerato or Módulos Profesionales. In particular, notions and basic procedures of calculus and plane geometry. It is recommended that students have completed the course (or have the level of the course) of introduction to mathematics that ESUP offers in September.
Important note: In order to follow the course it is essential for students to know how to solve algebraic and transcendental equations and inequalities (exponential, logarithmic, trigonometric) and know the basic concepts related to the lines. At the beginning of the course we will briefly review these topics (relating them with the determination of the domain of functions). It is essential that students have an excellent skill in solving equations and inequalities and therefore we recommend high school books or check the website of the Descartes Programme of the Ministry of Education.
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Competencias transversales |
Competencias específicas |
|---|---|
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Instrumentals 1. Ability to understand and analyze mathematical statements. Interpersonals 5. Ability to work in teams to solve problems so as to deepen theoretical concepts. Systemic 7. Ability to work independently to solve a problem. 8. Ability to find the best solution according to the characteristics of each context.
9. Ability to infer mathematical notions. 10. Get used to check and interpret solutions, not forgetting the particular cases. |
1. Ability to identify identity and justify the application of suitable mathematical model to analyze a problem and find its solution. 2. Ability to express mathematical ideas and concepts orally and writing with precision.
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The final grade (NF) of the course will be the arithmetic mean of the two parts (trimesters) of the course (NT1 and NT2) . The formula to calculate the final grade is as follows :
- If NT1 > = 5, NT2 > = 5,
NF = 0.5 * ( NT1 + NT2 )
- If not,
NF = min ( NT1 , NT2 )
Here's how each part is evaluated .
Evaluation of each of the parts of the course :
There will be a final exam in December (or in March in the second trimester case) that is compulsory to pass with a score greater than or equal to 5 (on the maximum score of 10) to consider each part of the course as passed.
The grade of each trimester is calculated as follows, where Nc is the grade of the continuous assessment, and Final Ex is the grade of the final exam of the trimester:
- If Ex. Final > = 5,
NTX = max ( 5, Nc )
- If not,
NTX = Ex Final
(NT1 and NT2 denote, respectively, the first and second trimester grades)
where:
Nc = 0.65 * Ex Final + 0.05 * 0.05 * Test1 + Test2 + Partial + 0.2 * 0.05 * Lab
The different evaluated activities that are taken into account for computing the grade are:
Ex Final: Final exam of the trimester to be held during the December exam period (or Marchin case of the second trimester). It contains theoretical questions and exercises . The activity is recoverable in July.
Test1 and Test2: These two tests of 15 minutes each are performed in practical classes at weeks 4 and 9 (approximately). Each test will evaluate theoretical concepts and is based on both short-answer questions and multiple choice questions. These activities are not recoverable.
Partial: A partial exam around the mid period of the trimester. It examines the theoretical and practical preparation of students and consists of exercises. This activity is not recoverable.
Lab: lab on numerical methods. This activity is not recoverable.
Part One:
First part
Block 1.
Chapter 1.
Introduction to the course , sets , the concept of generic element of a set, mathematical notation, sets of numbers, real intervals, implications, quantifiers, proof by contradiction.
Chapter 2.
Functions : generalities, graphic, properties, algebraic operations, composition, inverse, elementary functions and their graphs, operations on graphs : translation, dilation, symmetries of the absolute value function, domain calculation (and review of algebraic and transcendental equations and inequalities).
Block 2
Chapter 3.
Historical Introduction to Calculus with motivations coming from the curvilinear geometry and mechanics, limits of functions, left and right limits, continuity, infinitesimals of higer and lower orders, indeterminate limits, asymptotic expressions, notable limits, calculation of polynomial limits; derivation, differentiation, their equivalence, differential, the concept of the local linearization, tangent to the graph of a function at a point, derivative of elementary functions.
Chapter 4.
Lagrange and l'Hôpital Theorems, infinite and infinitesimal orders, higher derivative , concavity and convexity, the Taylor expansion of a function, application to the calculation of limits in indeterminate form, study of the graph of a function, the concept of optimization and examples;
Block 3.
Chapter 5.
Newton's 1-D Method for determining zeros of functions, errors in iterative methods, application to the approximate determination of notable points of the graphic of a function.
Block 4.
Chapter 6.
Introduction to the integrals, Riemann integral, primitive, fundamental theorem of calculus and theorem and average integral. Calculation of integrals by immediate, semi-immediate, and substitution methods or by partial fraction decomposition, numerical approximation: the Cavalieri -Simpson formula, applications.
Block 5.
Chapter 7.
Sequences, numerical series, Taylor series and power series, a notable example : Euler 's series and complex exponential formula. Applications.
Second part:
Block 1.
Chapter 1.
Real Euclidean n-dimensional space, introduction to the functions and operations on several variables including component functions the concept of vector-valued functions;
Block 2 .
Chapter2 .
Curves as an example of vector-valued functions, smooth curves, tangent lines and velocity, length of a curve, line integral, Surfaces as an example of functions of two variables, level lines, conics and quadrics.
Block 3.
Chapter 3.
Differential calculus in several variables : partial derivatives, gradient, directional derivatives, gradient theorem and directions of maximum and minimum growth, tangent plane, Jacobian matrix, chain rule, higher derivatives, Hessian matrix, Taylor formula ;
Chapter 4.
Optimization in several variables: unconstrained and constrained extrema, technique of Lagrange multipliers .
Block 4.
Chapter 5.
Iterative numerical methods: Newton n-D method and gradient descent/ascent: Application to the approximate solution of nonlinear systems and optimization problems.
Block 5.
Chapter 6.
Integration in several variables : double and triple integrals, differential operators divergence and curl, vector fields, applications.
During each block of theoretical aspects, revision and consolidation exercises will be proposed. The resolution of these exercises helps the students to test their understanding of the arguments presented in class. The hours of study vary from person to person. During the hours of seminars students will be invited to present the solutions of the exercises and discuss with teachers the possible doubts or difficulties encountered during resolution of the exercises. We consider this interaction very important, therefore it is essential that students come with exercises previously done at seminars or, questions if they have had problems with their attempted solutions. Hours of practical exercises will be devoted mainly to numerical exercises and the modeling of applications and real problems.
Basic bibliography:
Apuntes de la primera y la segunda parte de la asignatura.
G. STRANG, Calculus, Wesley-Cambridge Press, 1992, disponible online:
http://ocw.mit.edu/ans7870/resources/Strang/strangtext.htm
T.M. APOSTOL: Calculus. Vol 1&2, 2a ed., Editorial Reverte, 1992;
M. SPIVAK: Calculo infinitesimal, 3a ed., Cambridge University Press, 2006.
J E MARSDEN, A J TROMBA: Cálculo Vectorial, 4ª Edición, Addison-Wesley Longman, México, 1998.
F GRANERO: Ejercicios y problemas de cálculo, Toms 1 i 2, Ed Tebar Flores, Madrid, 1991.
Complementary bibliography:
G. BARTLE i S. SHERBERT, Introducción al Análisis Matemático de una variable, Ed. Limusa, 1986.
S. THOMPSON, Calculus Made Easy, Macmillan, 1914 (sense copyright, disponible arreu)
R. COURANT and F. JOHN, Introducción al Cálculo y al Análisis Matemático, Ed. Limusa, 1990.
S. LANG, Introducción al Análisis Matemático, Addison-Wesley Iberoamericana, 1990.
DEMIDOVICH, B. Problemas y ejercicios de análisis matemático. Ed. Paraninfo 1993.
R. L. BURDEN, J. D. FAIRES, Análisis numérico, International Thomson, 1998.
J.M. ARNAUDIES et H. FRAYSSE, Analyse, Dunod, 1988.
C. MARTÍNEZ i M. SANZ, Análisis de una variable real, Ed. Reverté, 1992.
J. ORTEGA, Introducció a l'Anàlisi Matemàtica, Manuals de la UAB, 1990.
C. PERELLÓ. Càlcul infinitesimal, Biblioteca Universitària, 21. Enciclopèdia Catalana, 1994.
L V FAUSETT: Applied Numerical Analysis using Matlab, Prentice Hall, Upper Saddle River, New Jersey, 1999.
SALAS,S.L.; HILLE,E, ETGEN. Calculus Una y varias variables. Vol I i II, 4ª ed. Ed. Reverté, 2005.