Calculus

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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

  • Adrián Fernando Ponce Álvarez ( )

Weekly hours

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

Learning Outcomes

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: K2, K3, S3, C3,
  2. Using calculus for solving biological problems.
    Related competences: K2, K3, S3, C3, 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
2h

Homework Block 2 Delivery (H2)



Week: 12 (Outside class hours)
Theory
0h
Problems
0h
Laboratory
0h
Guided learning
0h
Autonomous learning
2h

Mid-term exam (E1)

Exam about the content of block 1 (E1)
Objectives: 1 2
Week: 7
Theory
2h
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
2h
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.