Mathematics and Modelling

Recommended Prerequisites

Knowledge and mastery of the subjects taught in Mathematics in secondary education.     

Teaching Methods

In T classes relevant concepts and results are presented, accompanied by illustrative examples and motivating applications. In T/P classes, the instructor guides the students in solving exercises with various degrees of difficulty. The students are also confronted with simple problems in Biochemistry that can be modelled by differential equations. Active participation of students in class, individual and team work, and attendance of  the available office hours are strongly encouraged. The evaluation is periodic, with tests and, if possible, a team project involving modeling.

Learning Outcomes

The main objective is to acquire basic mathematical knowledge, fundamental to analyse and understand mathematical models used in Biochemistry. The main competences to be developed are: ability to calculate derivatives and primitives of real functions of one variable and to manipulate basic matrix operations, which are essencial tools to solve differencial equations and systems of differential equations; ability to formulate and solve problems; design, analyze and correctly use mathematical models; ability to work in teams; critical thinking. It is also expected that, at the end of the course, the students are capable of manipulating sequences and series, which are useful mathematical tools to understand further developments, namely stochastic models in the course Statistical Methods.

Work Placement(s)

No

Syllabus

Differential calculus

Real Functions of one real variable, limits, differentiation, extremes

Application to biological and chemical models

Integral calculus

Antiderivatives: integration of basic functions, by parts, by substitution

Definite integral: fundamental theorem of calculus and properties, applications

Improper integrals

Systems of linear equations

Matrix notations and operations, Gaussian elimination method

Least squares method and application to biological and chemical models.     

Differential equations and modeling

Separable variables, logistic equation, linear 1st order, systems of DE, predator-prey models

Applications to Biochemistry: biological growth, epidemiology problems, absorption of drugs, mixing problems, population dynamics, etc.

Numerical series, series of functions

Numerical sequences and series: limits, convergence criteria, sum of a series

Sequences and series of functions: uniform convergence (Weierstrass criterion), power and Taylor series

Head Lecturer(s)

António José Esteves Leal Duarte

Assessment Methods

Continous Assessment
Project: 20.0%
Frequency: 80.0%

Final Assessment
Exam: 100.0%

Bibliography

J. Stewart (2006)  Cálculo, vol. I e II,  6ª ed., São Paulo, Pioneira Thomson Learning.

J. Carvalho e Silva (1999) Princípios de análise Matemática Aplicada, McGraw-Hill.

A.P. Santana e J. F. Queiró (2009) Introdução à Álgebra Linear, Gradiva.

D. Zill (2011) Equações diferenciais com aplicações em modelagem, Cengage Learning Editions.

A. Araújo (2010) Biomatemática, Departamento de Matemática da FCTUC.

F. Leite e J.C. Petronilho (2013) Notas de equações diferenciais e modelação, Departamento de Matemática da FCTUC