Degree | Type | Year | Semester |
---|---|---|---|
2503740 Computational Mathematics and Data Analytics | FB | 1 | 1 |
Although there are no official prerequisites, it is recommended that students have good knowledge of basic Calculus: limits, continuity and derivability of real functions of one variable, notions of integral calculus and trigonometry. As well as the graphic representation of relatively simple functions of one variable. The most important requirement, however, is a great curiosity to understand and deepen the concepts that they will study.
Solve the mathematical problems that can arise in the degree they are studying.
Understand the concept of sequences and the computation of limits.
Know and work intuitively, geometrically and formally the notions of limit, continuity, derivative and integral.
Understand and know how to make Taylor's developments of functions of one real variable.
Acquire basic notions of numerical series and power series. Know the construction of the integral, know how to solve integrals and
its applications to solving problems where the integral approach is necessary. Improper integrals will be also studied.
1. Sequences of real numbers.
Limit of a sequence and algebraic properties.
Monotone sequences.
Accumulation points. Subsequences.
Bolzano-Weierstrass theorem.
Cauchy sequences.
Computation of limits.
2. Real functions.
Domain of a function.
Elementary functions.
Limit of a function at a point. One-sided limits. Properties of the limits. Asymptotes. Limits of functions.
Continuity of a function.
Bolzano's theorem.
Mean value theorem and Weierstrass theorem.
3.Derivatives.
Derivatives of a function at a point.
Calculation of some derivatives.
Tangent line equation.
Chain rule. Inverse functions and differentiation. Logarithmic differentiation.
Absolute and relative extreme values of a function.
Rolle's theorem. Mean value theorem.
Hôpital Rule.
Newton's method for finding numerical solutions of functions.
4. Approximation by Taylor polynomials.
Order of contact between functions.
Taylor polynomial. Properties
Taylor's formula. Taylor's residue.
Approximate calculations. Application to the computation of limits.
Local study of functions.
5. Integration
Primitives of a function.
Immediate integrals. Integrals by change of variable. Integrals by parts.
Integration of rational functions. Integration of irrational functions.
The fundamental theorem of calculus.
Applications of integration: flat areas, length of a curve, areas and volumes of solids of revolution.
Improper integrals Convergence criteria. Absolute convergence
6. Numerical series and power series.
Numerical series. Necessary condition of convergence.
Criteria of: comparison, quotient, root, integral.
Alternate series. Absolute convergence
Power series.Radius of Convergence.
Derivation and integration of power series.
The theory sessions, problem sessions and practice sessions are undistinguishable, so we will alternate them according to the needs of the course and the students.
In principle, the theory teacher will give the main ideas on the various subjects. The student must solve the proposed problems.
The professors of problems and of practices will solve the doubts that appear in the sessions and will propose methods for solving them.
Throughout the semester the student must solve and deliver problems. These deliveries will be part of the continuous evaluation of the subject.
Title | Hours | ECTS | Learning Outcomes |
---|---|---|---|
Type: Directed | |||
Problem and practice sessions | 23 | 0.92 | 2, 5, 4, 6, 22, 9, 11, 1, 16, 21, 20, 14, 15 |
Theory sessions | 30 | 1.2 | 2, 5, 4, 6, 22, 8, 9, 1, 21, 20, 19 |
Type: Supervised | |||
Doubt clearing sessions student-professor | 16 | 0.64 | 3, 23, 1, 16, 14, 18, 24, 15 |
Type: Autonomous | |||
At home work | 60 | 2.4 | 2, 5, 4, 9, 1, 16, 20, 14 |
Exam preparation | 15 | 0.6 | 9, 1, 21, 20 |
During the course, there will be two deliveries of exercises.
These can be doneindividually or in pairs.
The grades of these exercises will represent 20% of the final grade.
There will be an exam (Partial 1) before half semester in which the knowledge
of the first part of the course will be evaluated.
This exam will contribute 30% to the final grade. All students who do
This exam can no longer be rated as NON EVALUABLE.
Students who have taken this exam with a grade lower than 3.5,
must take a second-chance exam once the classes have ended, in
the date and time established by the Coordination of the Degree.
The student who has not taken this exam
will be considered NON EVALUABLE for academic purposes and will not have
the right to take the second-chance exam (except for cause
duly justified in which the second-chance examination will be allowed).
To be able to passthe subject, the grade of this exam
(or its second-chance version) can not be less than 3.5 and will represent 30%
of the final grade.
At the end of the semester there will be a second exam (Partial 2)
in which remaining part of the course will be evaluated.
The mark of this exam will provide another 30% of the final grade.
Students who have done
this exam with a grade lower than 3.5,
must do the second-chance exam once the classes are finished,
on the date and time established by the Coordination of the Degree.
The student who has not done
this examination will not have the right to do the second-chance version of it
(except for a justified cause in which it will be allowed to do
the second-chance exam). To be able to pass the subject, the grade of this exam
(or its second-chance version) can not be less than 3.5 and will represent 30% of the final grade.
Therefore, in order to pass the subject, it is essential to obtain a grade f not less than 3.5 in each of the
two partial exams or their second-chance versions.
There will be a practical exam with computer that will represent a 20% of the final grade.
The dates of delivery of exercises and partial exams will be published in the Virtual Campus (CV) and
they may be subject to possible programming changes due to possible incidents;
The CV will always contain all the information about
these changes since it is understood that the CV is the usual exchange mechanism
of information between teacher and students.
Title | Weighting | Hours | ECTS | Learning Outcomes |
---|---|---|---|---|
First delivery | 20% | 1 | 0.04 | 1, 16, 21, 20 |
First partial exam | 30% | 2 | 0.08 | 2, 5, 4, 6, 7, 22, 9, 11, 1, 13, 16, 20, 17 |
Second delivery | 20% | 1 | 0.04 | 1, 16, 19, 14 |
Second partial exam | 30% | 2 | 0.08 | 3, 23, 12, 8, 10, 1, 16, 20, 14, 18, 24, 15 |
1.S.L. Salas, E. Hille. 'Calculus' Vol. 1, Ed. Reverté, 2002.
2.Bartle, R.G., Shebert, D.R. (1996) Introducci ́on al An ́alisis Matem ́atico de una variable. 2a ed. Limusa. ISBN: 978-968-18-5191-0.
3.Ortega Aramburu, J.M. (2002). Introducci ́o a l’An`alisi Matem`atica. 2a ed. Manuals de la Universitat Aut`onoma de Barcelona.