Mathematics II

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Credits
7.5
Types
Compulsory
Requirements
This subject has not requirements, but it has got previous capacities
Department
MAT
Web
https://web.mat.upc.edu/fib/matematiques2/
The overall objective of the course is that at the end of the course, students are able to know and master, from the point of view of users, the concepts and fundamental techniques of mathematical calculus. More specifically, the course focuses on the understanding and use of the concept of function of a single and several variables.

Teachers

Person in charge

  • Mónica Sanchez Soler ( )

Others

  • Aitor Sort Nadal ( )
  • Andreu Bellés Roca ( )
  • Anna Rio Doval ( )
  • Eixarc Escaramis Babiano ( )
  • Eloy Cabezas Cardenas ( )
  • Fernando Martínez Sáez ( )
  • Guillermo González Casado ( )
  • Joaquim Soler Sagarra ( )
  • Maria Isabel Gonzalez Perez ( )
  • Montserrat Maureso Sánchez ( )
  • Roberto Gualdi ( )
  • Victoria Graffigna ( )

Weekly hours

Theory
3
Problems
0
Laboratory
2
Guided learning
0
Autonomous learning
7.5

Competences

Technical Competences

Common technical competencies

  • CT1 - To demonstrate knowledge and comprehension of essential facts, concepts, principles and theories related to informatics and their disciplines of reference.
    • CT1.2A - To interpret, select and value concepts, theories, uses and technological developments related to computer science and its application derived from the needed fundamentals of mathematics, statistics and physics. Capacity to solve the mathematical problems presented in engineering. Talent to apply the knowledge about: algebra, differential and integral calculus and numeric methods; statistics and optimization.
    • CT1.2C - To use properly theories, procedures and tools in the professional development of the informatics engineering in all its fields (specification, design, implementation, deployment and products evaluation) demonstrating the comprehension of the adopted compromises in the design decisions.

Transversal Competences

Autonomous learning

  • G7 [Avaluable] - To detect deficiencies in the own knowledge and overcome them through critical reflection and choosing the best actuation to extend this knowledge. Capacity for learning new methods and technologies, and versatility to adapt oneself to new situations.
    • G7.1 - Directed learning: perform the assigned tasks in the planned time, working with the indicated information sources according to the guidelines of the teacher or tutor. To identify the progress and accomplishment grade of the learning goals. To identify strong and weak points.

Objectives

  1. Understand real numbers and their properties.Solve linear equations and inequalities, with quadratic and / or absolute values.
    Related competences: G7.1, CT1.2A, CT1.2C,
  2. Understand the basic concept of sequences, calculate the limits of sequences and identify between convergent, divergent and oscillating sequences.
    Related competences: G7.1, CT1.2A, CT1.2C,
  3. Understand the basic theorems for continuous functions of one variable and know how to apply them to problems such as finding zeros for functions.
    Related competences: G7.1, CT1.2A, CT1.2C,
  4. Understand the basic theorems of differentiable functions of one variable and understand and know how to use Taylor polynomial approximations
    Related competences: G7.1, CT1.2A, CT1.2C,
  5. Understand the basic concepts of the integration of functions of one variable: geometric interpretation, calculation of areas, approximate calculation of definite integrals, etc.
    Related competences: G7.1, CT1.2A, CT1.2C,
  6. Understand the basic concepts of topologies in R^n.
    Related competences: G7.1, CT1.2A,
  7. Understand and know how to interpret the concepts of directional derivative, partial derivative and gradient vector.
    Related competences: G7.1, CT1.2A, CT1.2C,
  8. Locate and classify outliers in a function with several variables in a domain.
    Related competences: G7.1, CT1.2A, CT1.2C,
  9. Work with functions of several variables.
    Related competences: G7.1, CT1.2A, CT1.2C,

Contents

  1. Real numbers
    Equations and inequalities with real numbers. Absolute value. Intervals.
  2. Numerical sequences
    Definitions. Convergent, divergent and oscillating sequences. Convergence criteria. Recurring sequences. Monotone sequences. Monotone convergence theorem.
  3. Theorems for continuous functions of one variable
    Definitions. Sign theorem. Bolzano's theorem. Weierstrass theorem. Mean value theorem. Bisection and secant methods approximating zero in functions.
  4. Theorems for derivatives of functions of one variable
    Definitions. Rolle's theorem. Lagrange theorem. Cauchy's theorem. L'Hôpital's rule. Iterative methods for approximating zero in functions. Newton-Raphson method.
  5. Taylor formula for functions of one variable
    Taylor polynomial. Lagrange remainder formula. Error propagation formula. Using Taylor polynomials and bounding error.
  6. Integration of functions of one variable
    Definitions. Fundamental theorem of calculus. Barrow's rule. Definite integrals: areas and volumes. Approximated integrals: Trapezoidal rule and Simpson's rule.
  7. Functions of several variables
    Basic definitions of topology. Functions of several variables: domain, graphics, level sets, geometric interpretation. Continuous functions.
  8. Partial and directional derivatives. Gradient vectors
    Directional derivatives. Partial derivatives. Gradient vectors. Geometric interpretation. Planes tangent to a surface.
  9. Taylor polynomials in several variables.
    Higher order partial derivatives. Hessian matrix. Taylor polynomial. Lagrange remainder formula.
  10. Optimization of functions of several variables
    Definitions. Weierstrass theorem. Lagrange multiplier method. Outlier calculation: relative, conditional and absolute.

Activities

Activity Evaluation act


Real numbers


Objectives: 1
Contents:
Theory
0h
Problems
0h
Laboratory
2h
Guided learning
0h
Autonomous learning
7h

Numerical successions


Objectives: 2
Contents:
Theory
5h
Problems
0h
Laboratory
4h
Guided learning
0h
Autonomous learning
11h

Basic theorems of functions of a real variable


Objectives: 3 4
Contents:
Theory
11h
Problems
0h
Laboratory
6h
Guided learning
0h
Autonomous learning
20h

Integration of functions of one variable


Objectives: 5
Contents:
Theory
6h
Problems
0h
Laboratory
4h
Guided learning
0h
Autonomous learning
11h

Functions of several variables


Objectives: 6 7 9
Contents:
Theory
11h
Problems
0h
Laboratory
6h
Guided learning
0h
Autonomous learning
20h

Optimization variables


Objectives: 6 7 8 9
Contents:
Theory
10h
Problems
0h
Laboratory
6h
Guided learning
0h
Autonomous learning
15h


Mid-semester exam (P)

Exercise-based open-answer exam on learning objectives 1 to 5, referring to content for topics 1 to 6.
Objectives: 1 2 3 4 5
Week: 9 (Outside class hours)
Type: problems exam
Theory
0h
Problems
0h
Laboratory
0h
Guided learning
0h
Autonomous learning
7h

Workshop Exam

Exercise-based open-answer exam on all the learning objectives of the course referring to the problem-solving workshop session content.
Objectives: 1 2 3 4 5 6 7 8 9
Week: 14
Type: lab exam
Theory
0h
Problems
0h
Laboratory
2h
Guided learning
0h
Autonomous learning
4.5h

Final examination

Exercise-based open-answer exam on all learning objectives referring to content for topics 1 to 10.
Objectives: 1 2 3 4 5 6 7 8 9
Week: 15 (Outside class hours)
Type: theory exam
Theory
2h
Problems
0h
Laboratory
0h
Guided learning
0h
Autonomous learning
12h

Teaching methodology

Theory classes:
- lectures developing the theoretical aspects of the subject.
- lectures and participatory classes aimed at applying theory to problems.


Workshop/laboratory classes:
- participatory workshop sessions in which students solve problems in groups or individually.
- participatory laboratory sessions in which students complete problems individually or in groups using mathematical software.

Evaluation methodology

Technical and transferable competencies account for 80% and 20% of the subject, respectively. The transferable competency mark will be calculated on the basis of Atenea activities and from the note of the subject.
- Workshop mark (T): it evaluates the student's performance and achievement of objectives in workshop / laboratory sessions and Atenea.
- Mark of the mid-semester exam (P).
- Mark of the final exam (F).


The final mark is calculated as:

Note = 0.2 * T + max (0.3 * P+0.5 * F,0.8 * F)

Not taking the final exam means having a NP of M2 grade.

Bibliography

Basic:

Complementary:

Web links

Previous capacities

Students are expected be competent in mathematics to upper secondary level.