Introduction to Applied Mathematics

Code:

M1027

Acronym:

M1027

Level:

100

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2018/2019 - 2S

Active?	Yes
Web Page:	https://moodle.up.pt/course/view.php?id=423
Responsible unit:	Department of Mathematics
Course/CS Responsible:	Bachelor in Mathematics

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
L:M	102	Official Study Plan	1	-	6	56	162

Teaching language

Suitable for English-speaking students

Objectives

Application of mathematical concepts, namely the ones studied in other first-year courses, to the analytical and numerical treatment of mathematical models in Physics, Biology, Ecology, Economics, Medicine and other fields of knowledge.

Learning outcomes and competences

The student should be able to translate the proposed problems in mathematical language, classify them, propose an adequate model and test such model. 

Whenever possible, the student should solve the problem analytically as well as obtaining a graphic representation of it. He should also be capable of using the Maxima software for graphic representation and simulation of solutions to the problem. 

Working method

Presencial

Pre-requirements (prior knowledge) and co-requirements (common knowledge)

Prerequisites: Real Analysis I, Linear Algebra and Analytic Geometry I, Mathematical Laboratory.
Co-requirements: Real Analysis II.

Program

1) Discrete time mathematical modeling with classical examples of application: 

a) modeling in one dimension: discrete dynamical system and its variation, resolution of linear and affine model; fixed points, phase portrait and graph; models in Economics, Biology and Social Sciences; 

b) modeling in dimensions two and three: discrete dynamical system and its variation, resolution of the dynamical system in the linear case; fixed points, phase portrait (in dimension two) and graph; models in Ecology and Epidemiology.

2) Adapting a model to a set of points: transformation into an affine model, graphical method and least square method to determine an affine model. 

3) Continuous time mathematical modeling with classical examples of application: 

a) first order autonomous differential equation (o.d.e.): resolution in the linear and affine case as well cases where an explicit solution can be obtained; continuous time  models in Pharmacy, Physics and Biology; phase portrait of an autonomous o.d.e.: equilibria, monotonicity intervals, concavity; graph of the solutions obtained from the analysis of the phase portrait.

b) conservative systems with one degree of freedom, stability of equilibria, phase portrait on the plane, applications to Physics.

 

 

Mandatory literature

Giordano Frank R.; A first course in mathematical modeling. ISBN: 978-0-495-55877-4

Complementary Bibliography

Britton Nicholas F.; Essential mathematical biology. ISBN: 1-85233-536-X
Brauer Fred; Mathematical models in population biology and epidemiology. ISBN: 0-387-98902-1
Burghes D. N.; Modelling with differential equations. ISBN: 0-85312-286-5
Arnold V. I.; Equações diferenciais ordinárias

Teaching methods and learning activities

Theoretical classes: exposition of the theory and discussion of examlpes.
Proposed problems to be solved out of class and to be treated in practical classes. 
Practical classes, with evaluation: resolution of concrete problems with use of computer and adequate software for the resolution of problems in class time.

Software

Maxima

Evaluation Type

Distributed evaluation with final exam

Assessment Components

designation	Weight (%)
Apresentação/discussão de um trabalho científico	20,00
Exame	55,00
Trabalho prático ou de projeto	25,00
Total:	100,00

Amount of time allocated to each course unit

designation	Time (hours)
Estudo autónomo	104,00
Frequência das aulas	56,00
Trabalho escrito	2,00
Total:	162,00

Eligibility for exams

Participation in at least 4 practical classes.
Correct solving and adequate discussion, in the practical classses, of the practical exercises of at least 4 of the practical units.

Aproval of a  practical exercise implies that the exercise is completely correct.

New practical exercises of a unit may be done as many times as necessary to obtain aproval.

Each student only needs to be present in the number of practical classes necessary to obtain aproval in the  practical exercises.

Calculation formula of final grade

1. The final grade will be the sum T+P of the points obtained in two partial components: 
T - theoretical assessment in exam: 11 points, minimum 3.5 points.  
P - practical assessment using computer, in the practical classes: 9 points (1.5 point per complete and discussed exercise). 

2. The grading of component P will only take place during term time (see exception for working students below).

2.The component T is the result of the exam in either the first or in the  second call.

4. Exception: if the sum of the points obtained in the components T and P is greater than 17, then a (eventually oral) complementary assessment may be required, to take place in a date to fix with the student. The final grade can be 17, 18, 19 or 20 and will depend only on the student's performance in this complementary assessment.

 

 

 

Special assessment (TE, DA, ...)

Students that, due to special conditions, are exempted from presence in class may take a practical exame using the computer to obtain the partial grade P.

Classification improvement

The improvement of final grade for students who took the exame in the first call will be done in the exam in seond call.


For students who did the course in the previous year, the improvement will follow the same structure of the special evaluation, below.