Calculus

Credits
6
Types
Compulsory
Requirements
This subject has not requirements , but it has got previous capacities
Department
MAT;UB
The aim of this subject is to bring the students to master the analysis of real functions of one or several variables. In particular, continuity, differentiation and integration will be covered, as well as the study of sequences, limits and series. Students must be able as well to determine criteria for extreme values of functions of one and several variables. Also, they must become familiar with differential equations and be aware of their applications in modelling in bioinformatics. This requires learning fundamental tools in both differential and integral calculus in order to apply them to specific problems related to the field of bioinformatics. With this aim, we will review concepts on basic functions (in particular exponentials, logarithms and trigonometric functions) as well as the main properties of complex numbers, derivatives and integrals. We will then focus our attention on the solution of differential equations, with special emphasis put on those equations arising in biological systems. We will also review calculus of functions in several variables and optimization methods for these functions (relative and absolute extrema).

Teachers

Person in charge

Others

Weekly hours

Theory
2
Problems
2
Laboratory
0
Guided learning
0
Autonomous learning
6

Competences

Learning Outcomes

Knowledge

  • K2 - Identify mathematical models and statistical and computational methods that allow for solving problems in the fields of molecular biology, genomics, medical research, and population genetics.
  • K3 - Identify the mathematical foundations, computational theories, algorithmic schemes and information organization principles applicable to the modeling of biological systems and to the efficient solution of bioinformatics problems through the design of computational tools.

    • Skills

  • S3 - Solve problems in the fields of molecular biology, genomics, medical research and population genetics by applying statistical and computational methods and mathematical models.

    • Competences

  • C3 - Communicate orally and in writing with others in the English language about learning, thinking and decision making outcomes.
  • C6 - Detect deficiencies in the own knowledge and overcome them through critical reflection and the choice of the best action to expand this knowledge.

    • Objectives

    1. Acquisition of the basic knowledge of differential and integral calculus.
      Related competences: C3, K2, K3, S3,
    2. Using calculus for solving biological problems.
      Related competences: C3, K2, K3, S3, C6,

    Contents

    1. 1.1. Real numbers and functions
      Real numbers; Real intervals; Real functions, domain, range, and graph; Function symmetries; Composite functions; Inverse functions
    2. 1.2. Usual functions
      Power functions; Polynomials; Exponential functions; Logarithms;Trigonometric functions
    3. 1.3. Limits
      Limit¿s properties ; Indeterminate forms ; Continuity
    4. 1.4. Derivatives
      Definition of derivative ; Derivatives of usual functions ; Leibniz notation ; Differentiability ; Product, Quotient, and Chain Rules Derivative of an inverse function
    5. 1.5. Applications of derivatives
      Increasing/decreasing functions ; L'Hôpital's Rule ; Extrema, concavity, inflections ; Linear approximation ; Taylor series
    6. 1.6. Integrals
      Riemann sums ; Fundamental Theorem of calculus ; Antiderivatives ; Integration by substitution ; Integration by parts
    7. 2.1. Differential equations
      Differential equations in separable variables; Logistic growth equation; Applications; First order linear differential equations; Second order linear differential equations with constant coefficients.
    8. 2.2. Multivariate calculus
      Functions of several variables; Partial derivatives; gradient vector, directional derivative.
    9. 2.3. Optimization.
      Critical points; relative extrema; absolute extrema ; extrema with constraints; Lagrange multipliers.

    Activities

    Activity Evaluation act


    Block 1: Theoretical expository lectures


    • Theory: Theoretical expository lectures
    Objectives: 1 2
    Contents:
    Theory
    13h
    Problems
    0h
    Laboratory
    0h
    Guided learning
    0h
    Autonomous learning
    22.5h

    Block 2: Theoretical expository lectures


    • Theory: Theoretical expository lectures
    Objectives: 1 2
    Contents:
    Theory
    13h
    Problems
    0h
    Laboratory
    0h
    Guided learning
    0h
    Autonomous learning
    22.5h

    Block 1: Problem solving sessions


    • Problems: Problem solving sessions
    Objectives: 1 2
    Contents:
    Theory
    0h
    Problems
    15h
    Laboratory
    0h
    Guided learning
    0h
    Autonomous learning
    20.5h

    Block 2: Problem solving sessions


    • Problems: Problem solving sessions
    Objectives: 1 2
    Contents:
    Theory
    0h
    Problems
    15h
    Laboratory
    0h
    Guided learning
    0h
    Autonomous learning
    20.5h

    Homework Block 1 Delivery (H1)


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

    Homework Block 2 Delivery (H2)



    Week: 12 (Outside class hours)
    Theory
    0h
    Problems
    0h
    Laboratory
    0h
    Guided learning
    0h
    Autonomous learning
    0h

    Mid-term exam (E1)

    Exam about the content of block 1 (E1)
    Objectives: 1 2
    Week: 7
    Theory
    0h
    Problems
    0h
    Laboratory
    0h
    Guided learning
    0h
    Autonomous learning
    0h

    Final Exam (E2)

    Exam about the content of block 2 (E2)
    Objectives: 1 2
    Week: 15 (Outside class hours)
    Theory
    0h
    Problems
    0h
    Laboratory
    0h
    Guided learning
    0h
    Autonomous learning
    0h

    Teaching methodology

    Lectures will be mainly of expository type. There will be also problem-based sessions and practical sessions

    Evaluation methodology

    The final grade will depend on the following assessment activities:

    - E1: first exam (contents from 1.1 to 1.6)
    - H1 : homework bloc 1
    - E2: second exam (contents from 2.1 to 2.3)
    - H2 : homework bloc 2
    - R: retake exam at the end of the course (if presented), with two parts, R1 and R2 corresponding to the two blocks, whose grade will substitut E1 and E2, respectively, in the formula below.

    Final score is given by the formula:
    Grade = max( 0.45 E1 + 0.45 E2 + 0.05 H1 + 0.05 H2 ; 0.5 E1 + 0.5 E2)

    Bibliography

    Basic

    Previous capacities

    Basic knowledge on real numbers, functions and calculus.