Calculus

Credits
7.5
Types
Compulsory
Requirements
This subject has not requirements , but it has got previous capacities
Department
MAT
Introduction to the basic calculus tools for functions of one variable (continuity, derivation, integration and series) as well as some basic notions of calculus for functions of several variables. The presentation is used to introduce some basic numerical calculation tools.

Teachers

Person in charge

  • Jordi Villanueva Castelltort (jordi.villanueva@upc.edu)

Others

  • Jaume Franch Bullich (jaume.franch@upc.edu)
  • Jose Tomas Lazaro Ochoa (jose.tomas.lazaro@upc.edu)

Weekly hours

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

Competences

Technical Competences

Technical competencies

  • CE1 - Skillfully use mathematical concepts and methods that underlie the problems of science and data engineering.

    • Transversal Competences

    Transversals

  • CT5 - Solvent use of information resources. Manage the acquisition, structuring, analysis and visualization of data and information in the field of specialty and critically evaluate the results of such management.
  • 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

  • CB1 - That students have demonstrated to possess and understand knowledge in an area of ??study that starts from the base of general secondary education, and is usually found at a level that, although supported by advanced textbooks, also includes some aspects that imply Knowledge from the vanguard of their field of study.

    • Generic Technical Competences

    Generic

  • CG2 - Choose and apply the most appropriate methods and techniques to a problem defined by data that represents a challenge for its volume, speed, variety or heterogeneity, including computer, mathematical, statistical and signal processing methods.

    • Objectives

    1. Elementary functions, continuity, limit and associated concepts
      Related competences: CT6, CB1,
    2. The derivative and its use as a basic calculation tool.
      Related competences: CE1, CT5, CT6, CG2, CB1,
    3. Calculation of primitives and definite integrals.
      Related competences: CE1, CT5, CT6, CG2, CB1,
    4. Discussion of the convergence of improper integrals, sequences and series and calculation of their limit in simple cases where it is approachable.
      Related competences: CE1, CT5, CT6, CG2, CB1,

    Contents

    1. Functions
      Rational and real numbers. Absolute value. Qualitative study of the most usual functions and their inverses. Limit and continuity. Theorem of Bolzano and theorem of the intermediate value.
    2. Derivation
      Derivative of a function. Direct applications of the derivative. Rolle's and mean value theorems. Rule of L'Hôpital. Taylor's formula and applications. Introduction to the functions of several variables. Numerical derivation. Numerical computation of zeros of functions.
    3. Integration
      Calculation of primitives. Definite integrals. Numerical integration. Improper integrals and their convergence criteria. Euler's Gamma function.
    4. Sequences and series
      Sequences and their limit. Numerical series and their convergence criteria. Power series. Taylor series.

    Activities

    Activity Evaluation act


    Midterm exam


    Objectives: 1 2
    Week: 10 (Outside class hours)
    Theory
    0h
    Problems
    0h
    Laboratory
    0h
    Guided learning
    0h
    Autonomous learning
    0h

    Final exam


    Objectives: 1 2 3 4
    Week: 15 (Outside class hours)
    Theory
    0h
    Problems
    0h
    Laboratory
    0h
    Guided learning
    0h
    Autonomous learning
    0h

    Functions



    Theory
    9h
    Problems
    7h
    Laboratory
    0h
    Guided learning
    0h
    Autonomous learning
    22h

    Differentiability



    Theory
    12.5h
    Problems
    10h
    Laboratory
    0h
    Guided learning
    0h
    Autonomous learning
    31h

    Integrability



    Theory
    11h
    Problems
    8h
    Laboratory
    0h
    Guided learning
    0h
    Autonomous learning
    26.5h

    Sequences and series



    Theory
    8h
    Problems
    5h
    Laboratory
    0h
    Guided learning
    0h
    Autonomous learning
    18h

    Teaching methodology

    Lecture classes introduce the necessary concepts, algorithms, and results to achieve a well-founded and clear understanding. These concepts are put into practice during problem-solving classes where, due to their structure, it is easier to encourage the active participation of students. We aim to introduce some problems in class designed to foster engagement with the more creative and transversal aspects of the subject. The continuous assessment grade is primarily intended to help students become aware of their level of assimilation of the subject throughout the course.

    Evaluation methodology

    Nota final = max(0.1*NAvCont + 0.3*NExParcial + 0.6*NExFinal, 0.1*NAvCont + 0.9*NExFinal, NExFinal)
    where
    [NAvCont]: class grade
    [NExParcial]: midterm exam
    [NExFinal]: final examen grade

    Bibliography

    Basic

    Complementary

    Web links

    • Primer dels tres volums dels divulgadors Edwin Herman i Gilbert Strang, professors del MIT, editats per OpenSatx sota llicència gratuïta. Gairebé 3000 pàgines amb una gran quantitat d'exemples, figures i problemes resolts. https://openstax.org/details/books/calculus-volume-1
    • Segon dels tres volums dels divulgadors Edwin Herman i Gilbert Strang, professors del MIT, editats per OpenSatx sota llicència gratuïta. Gairebé 3000 pàgines amb una gran quantitat d'exemples, figures i problemes resolts. https://openstax.org/details/books/calculus-volume-2
    • Tercer dels tres volums dels divulgadors Edwin Herman i Gilbert Strang, professors del MIT, editats per OpenSatx sota llicència gratuïta. Gairebé 3000 pàgines amb una gran quantitat d'exemples, figures i problemes resolts. https://openstax.org/details/books/calculus-volume-3
    • Material seleccionat i elaborat pel professor Rafael Ramírez amb multitud de vídeos de contingut matemàtic adaptat directament al temari de l'assignatura. https://web.mat.upc.edu/rafael.ramirez/ACcY/index.html

    Previous capacities

    Knowledge on basic calculus theory at level 2n Batxillerat