Calculus

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Credits
6
Types
Compulsory
Requirements
This subject has not requirements, but it has got previous capacities
Department
MAT
The overall objective of the course is that at the end of the course, AI 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 ( )
  • Santiago Molina Blanco ( )

Weekly hours

Theory
2
Problems
0
Laboratory
2
Guided learning
0.033
Autonomous learning
7

Competences

Transversal Competences

Transversals

  • CT6 [Avaluable] - Autonomous Learning. Detect deficiencies in one's own knowledge and overcome them through critical reflection and the choice of the best action to extend this knowledge.

Basic

  • CB2 - That the students know how to apply their knowledge to their work or vocation in a professional way and possess the skills that are usually demonstrated through the elaboration and defense of arguments and problem solving within their area of ??study.
  • CB5 - That the students have developed those learning skills necessary to undertake later studies with a high degree of autonomy

Technical Competences

Especifics

  • CE01 - To be able to solve the mathematical problems that may arise in the field of artificial intelligence. Apply knowledge from: algebra, differential and integral calculus and numerical methods; statistics and optimization.
  • CE02 - To master the basic concepts of discrete mathematics, logic, algorithmic and computational complexity, and its application to the automatic processing of information through computer systems . To be able to apply all these for solving problems.

Generic Technical Competences

Generic

  • CG2 - To use the fundamental knowledge and solid work methodologies acquired during the studies to adapt to the new technological scenarios of the future.
  • CG4 - Reasoning, analyzing reality and designing algorithms and formulations that model it. To identify problems and construct valid algorithmic or mathematical solutions, eventually new, integrating the necessary multidisciplinary knowledge, evaluating different alternatives with a critical spirit, justifying the decisions taken, interpreting and synthesizing the results in the context of the application domain and establishing methodological generalizations based on specific applications.

Objectives

  1. Know how to solve linear, quadratic equations and inequalities and/or with absolute values.
    Related competences: CG2, CT6, CB2, CE01, CE02,
  2. Know and understand the basic concepts of elementary functions.
    Related competences: CT6, CB2, CB5, CE01, CE02,
  3. Know, understand and be able to use the approximation given by the Taylor Polynomial for functions of a variable.
    Related competences: CG4, CT6, CE01, CE02,
  4. Knowing and understanding the approximate calculation of definite integrals by the methods of trapezoids and Simpson.
    Related competences: CG2, CG4, CT6, CB5, CE01, CE02,
  5. Know and understand the basic concepts of successions and series
    Related competences: CG2, CG4, CT6, CB2, CE01, CE02,
    Subcompetences:
    • Know and understand the basics of power series and Taylor series.
    • Know and understand the basic concepts of sequences and series of real numbers.
  6. Know and understand the different distances in R^n.
    Related competences: CG2, CG4, CT6, CB2, CB5, CE01, CE02,
  7. Know and understand the basic concepts of domain, contour lines and continuity of functions of various variables.
    Related competences: CG2, CG4, CT6, CB5, CE01, CE02,
  8. Know, understand and be able to interpret the concepts of directional derivative, partial derivative, gradient vector and Jacobian matrix. Know and know how to find the optimal direction. Know and be able to use the chain rule.
    Related competences: CG2, CG4, CT6, CB2, CB5, CE01, CE02,
  9. Know how to find and classify the relative extreme of a scalar function of several variables.
    Related competences: CG2, CG4, CT6, CB2, CB5, CE01, CE02,
  10. Know, understand and know how to use the gradient descent method to optimize scalar functions of various variables.
    Related competences: CG2, CG4, CT6, CB2, CB5, CE01, CE02,

Contents

  1. Equations and inequalities with real numbers
    Know how to solve linear, quadratic equations and inequalities and/or with absolute values.
  2. Elementary functions
    Polynomial functions. Rational functions. Potential functions. Trigonometric functions. Exponential and logarithmic functions. Hyperbolic functions.
  3. Taylor polynomial for functions of a variable
    Taylor polynomials. Lagrange formula of the residue. Error propagation formula.
    Taylor polynomial approximation and error bounding.
  4. Approximate integration
    Trapeze rule and Simpson's formula for the approximate calculation of definite integrals. Dimension of the error.
  5. Sequences and series of real numbers
    Basic concepts of sequences and series of real numbers. Convergent, divergent and oscillating successions. Convergent, divergent and oscillating series. Calculation of succession limits and series sums.
  6. Powers series and Taylor series
    Basic concepts of power series. Basic concepts of Taylor series.
  7. The R^n space
    The space R^n. Norms and distances in R^n.
  8. Introduction to the functions of various variables
    Domain, contour lines and continuity of functions of several variables.
  9. Derivation of functions of several variables
    Directional derivatives and partial derivatives. Gradient vector and Jacobian matrix. Optimal direction. Chain rule.
  10. Relative extremes
    Critical points of a scalar function of several variables. Necessary condition. Sufficient condition. Calculation of relative extremes.
  11. Optimization
    Gradient descent method for optimization of scalar functions of several variables.

Activities

Activity Evaluation act


Equations and inequalities linear, quadratic and/or with absolute values


Objectives: 1
Contents:
Theory
0h
Problems
0h
Laboratory
2h
Guided learning
0h
Autonomous learning
5h

Elementary functions


Objectives: 2
Contents:
Theory
2h
Problems
0h
Laboratory
2h
Guided learning
0h
Autonomous learning
12h

Taylor polynomial


Objectives: 3
Contents:
Theory
3h
Problems
0h
Laboratory
2h
Guided learning
0h
Autonomous learning
12h

Approximate integration


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

Sequences and series of real numbers


Objectives: 5
Contents:
Theory
3h
Problems
0h
Laboratory
2.5h
Guided learning
0h
Autonomous learning
10h

Powers series and Taylor series


Objectives: 5
Contents:
Theory
3h
Problems
0h
Laboratory
2.5h
Guided learning
0h
Autonomous learning
10h

Theory
10h
Problems
0h
Laboratory
10h
Guided learning
0h
Autonomous learning
30h

Optimization in several variables


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




Teaching methodology

In theory classes the teacher will explain the topics accompanied by examples.


The worksho /laboratory classes are participatory sessions where students will be asked to solve problems. Students will solve problems under the supervision of the teacher; some of these problems will need to be prepared in advance. The teacher will explain some of the problems on the board.

Evaluation methodology

The grade of the subject is obtained from:

- Workshop mark (T): assesses the work and achievement of objectives with questionnaires in Athena.
- Mark of the mid-semester exam (P): it does a partial examination P to half semester that corresponds, approximately, to the part of Calculation in 1 variable.
- Final exam (F): a final exam is made in which the knowledge of all the syllabus of the subject is evaluated.

The final grade of the course (NF) is calculated according to:

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


CROSS-CURRICULAR COMPETENCE.

The mark of the autonomous learning competence will have grades: A (excellence), B (optimal), C (sufficient), D (not passed). This competence will be evaluated from the note of the subject.

Bibliography

Basic:

Complementary:

Web links

Previous capacities

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

Addendum

Contents

No es preveu cap canvi de continguts.

Teaching methodology

La metodologia docent s'adequarà a la situació del moment. Es faran de forma no presencial amb Meet o similar les activitats que no es puguin dur a terme de forma presencial.

Evaluation methodology

Està previst fer els exàmens de forma presencial. Si la situació no ho permetès, es farien de forma no presencial amb Meet i/o la plataforma Atenea, o qualsevol medi similar facilitat per la UPC.

Contingency plan

Si no fos possible dur a terme les activitats de forma físicament presencial, es faran tant les classses de teoria com de problemes, el seguiment dels estudiants i l'avaluació de forma no presencial.