Mathematics
Code: 100745 ECTS Credits: 6| Degree | Type | Year |
|---|---|---|
| 2500250 Biology | FB | 1 |
Contact
- Name:
- Angel Calsina Ballesta
- Email:
- angel.calsina@uab.cat
Teaching groups languages
You can view this information at the end of this document.
Prerequisites
- Rational and real numbers, numerical approximation and exponential notation. Absolute value and inequalities.
- Elementary functions: linear, polynomial, rational, exponential, logarithmic and trigonometric.
Objectives and Contextualisation
This program of study has a double objective. The first is to give the student a basic mathematical training, focused on linear algebra and on one real variable calculus: derivation, integration and simple differential equations, which allows to understand the language of Science. The second is to introduce mathematical modeling of Biology, by means of simple examples that can be analyzed with the mathematical tools available to students.
With this idea in mind most of the contents will be presented motivated by scientific problems, usually from the field of Biology. In particular Population Dynamics and Ecology that are the most matematizable areas of Biology at an elementary level. Linear algebra will be addressed as the natural tool for the study of the linear growth and age-structured populations, while differential equations will be introduced as the fundamental tool for the study of the magnitudes that change with time continuously, biological populations, as well as concentrations of chemical substances, for example.
In short, the objective is that students see mathematics as a essential tool to describe most of the physical phenomena.
Learning Outcomes
- CM05 (Competence) Interpret relevant mathematical data that allow judgements to be made that include reflection on important social, scientific or ethical issues.
- KM08 (Knowledge) Describe natural phenomena in the field of biology through mathematics.
- KM09 (Knowledge) Carry out exponential, logarithmic and potential functions, applied to the resolution of biological problems.
- KM10 (Knowledge) Carry out vectors and matrices, recognising the simplification that this entails, in the resolution of problems of biological interest.
- SM05 (Skill) Apply the basic concepts of linear algebra, differential and integral calculus to the resolution and modelling of biological problems.
- SM06 (Skill) Apply classical mathematical models of population growth of different living organisms.
Content
1. Functions and derivatives
1.1 Sets of numbers. Inequalities and absolute value. Exponentiation and exponential notation.
1.2 Linear functions. Polynomial functions. Rational functions. Exponential functions. Inverse function. Logarithmic functions. Logarithmic scale. Graphics.
1.3 The derivative: tangent, velocity and rate of change of a magnitude.
1.4 Growth. Optimization. Graphics revisited.
2. Integral calculus
2.1 The integral. The fundamental theorem of calculus. Primitives. Applications
3. Linear algebra
3.1 Systems of linear equations. Matrices.
3.2 Eigenvalues and eigenvectors. Diagonalisation.
3.3 Discrete population dynamics: iteration. Dependence on age
6. Differential equations
6.1 Differential equations of separate variables. Exponential growth. Balance of matter. The logistic differential equation.
6.3 Geometric interpretation of differential equations. The problem of initial value.
6.4 The qualitative method: balances and stability.
Activities and Methodology
| Title | Hours | ECTS | Learning Outcomes |
|---|---|---|---|
| Type: Directed | |||
| Exercises | 15 | 0.6 | CM05, KM08, KM09, KM10, SM05, SM06, CM05 |
| Theory | 35 | 1.4 | CM05, KM08, KM09, KM10, SM05, SM06, CM05 |
| Type: Supervised | |||
| Tutoring | 5 | 0.2 | CM05, KM08, KM09, KM10, SM05, SM06, CM05 |
| Type: Autonomous | |||
| Exercises | 35 | 1.4 | CM05, KM08, KM09, KM10, SM05, SM06, CM05 |
| Study | 35 | 1.4 | CM05, KM08, KM09, KM10, SM05, SM06, CM05 |
| Tests | 15 | 0.6 | CM05, KM08, KM09, KM10, SM05, SM06, CM05 |
The student acquires the scientific knowledge of the subject by attending theory lectures and learns to use them in problem lectures. We must reinforce this knowledge through the personal study of the theoretical part to be able to apply it to the exercises.
The realization of exercises is one of the most important tasks of the study, they illustrate and motivate all the theoretical development. On the other hand, the objective of the subject is that students learn to use mathematics as a working tool and therefore learn to face different types of problems modeling it or turning them into a mathematical question that they can solve.
Theoretical lectures will be reinforced with as many applied examples as possible and in addition the student will be asked to give periodic exercises that will be focused on facing the student with these modeling tasks.
Annotation: Within the schedule set by the centre or degree programme, 15 minutes of one class will be reserved for students to evaluate their lecturers and their courses or modules through questionnaires.
Assessment
Continous Assessment Activities
| Title | Weighting | Hours | ECTS | Learning Outcomes |
|---|---|---|---|---|
| Exercises delivery | 20% | 2 | 0.08 | CM05, KM08, KM09, KM10, SM05 |
| First partial exam | 35% | 2 | 0.08 | CM05, KM08, KM09, KM10, SM05, SM06 |
| Recovery exam | 80% | 3 | 0.12 | CM05, KM08, KM09, KM10, SM05, SM06 |
| Second partial exam | 45% | 3 | 0.12 | CM05, KM08, KM09, KM10, SM05, SM06 |
The final grade for the subject will consist of different parts:
Two partial assessments of the subject (35%+45%). It will be compulsory to obtain a grade of at least 3.5 in the second partial test to avoid the recovery exam.
Individual assignments of exercises (20%).
Global exam/recovery of the entire subject (80%) (This exam is not mandatory and can be used both to raise the grade and to recover the grade obtained in the partials).
The student will obtain the qualification of "Not Assessable" when the assessment activities carried out have a weighting of less than 25% in the final qualification.
Repeating students will have to do the same assessment activities as new students. Honors Degrees can only be awarded to students who have obtained a final grade equal to or higher than 9. They can be awarded to a maximum of 5% of enrolled students. They will not be awarded for a grade obtained in the recovery exam.
The single assessment, if chosen, will consist of a global exam to be taken on the day of the second term, which will include a part related to the exercises released by the rest of the students. If this exam is not passed, a recovery exam will be taken on the day of the final exam for the rest of the students, with the characteristics described in the previous sentence.
Bibliography
There are no texts in the literature that adapts exactly to the content of the course. For this reason, three general-purpose texts are proposed that cover most topics and in which mathematical concepts are introduced intuitively and illustrated with many practical examples. These three texts are complemented by books that allow you to explore the most important topics of the course.
General bibliography
- Matemàtiques i modelització per a les Ciències Ambientals, Jaume Aguadé. (UAB, recursos electrònics http://ddd.uab.cat/record/158385)
- Curso práctico de Cálculo y Precálculo, Pestana i altres. (Ed. Ariel)
- Introducción al Álgebra Lineal, H. Anton (Editorial Limusa)
Complementary bibliography
- Calculus, Tomo I. S. Salas i E. Hille (Editorial Reverté)
- Aplicaciones del Álgebra lineal, Grossman, Stanley I. (Grupo Editorial Iberoamericano)
- Matemáticas básicas para biocientíficos, E. Batschelet (Editorial Dossat)
- Matemáticas para ciencias, C. Neuhauser (Editorial Prentice Hall)
- Mathematics for the Biological Sciences. J.C. Newby (Clarendon Press)
- Matemáticas para Biólogos, K.P. Hadeler, (Editorial Reverté)
Software
- Sagemath: https://www.sagemath.org
- Maxima: https://maxima.sourceforge.io
- WxMaxima: https://wxmaxima-developers.github.io/wxmaxima/index.html
Language list
| Name | Group | Language | Semester | Turn |
|---|---|---|---|---|
| (PAUL) Classroom practices | 111 | Catalan | first semester | morning-mixed |
| (PAUL) Classroom practices | 112 | Catalan | first semester | morning-mixed |
| (TE) Theory | 11 | Catalan | first semester | afternoon |