Mathematics I

Code:

EBE0001

Acronym:

MAT1

Keywords
Classification	Keyword
OFICIAL	Basic Sciences

Instance: 2015/2016 - 1S

Active?	Yes
Responsible unit:	Mathematics Section
Course/CS Responsible:	Master in Bioengineering

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
MIB	72	Syllabus	1	-	6	56	162

Teaching language

Portuguese

Objectives

This course unit aims to provide students with theoretical and practical knowledge in Linear Algebra and Analytical Geometry, as well as Differential and Integral Calculus of a real functions of a real variable.

Learning outcomes and competences

LEARNING OUTCOMES: Students should be capable of: 1- analysing linear dependence and independence of a group of vectors in R2 and R3. 2- determining the dimension and bases of R2 and R3 subspaces 3- carrying out matrix operations. 4- calculating determinants of any order and be capable of using their main properties. 5- classifying systems of linear equations regarding the type of solution and resolution of systems by using Gaussian elimination method. 6- determining eigenvalues and eigenvectors, as well as their eigensubspaces. 7- calculating the angle, internal and external product between 2 vectors. 8- determining vector, parametric and cartesian equations of lines and planes in R3. 9- obtaining inverse trigonometric functions, as well as their derivatives. 10- obtaining Taylor’s formula with the remainder of some simple functions. 11- calculating primitives by substitution and parts method. 12- calculating primitives of rational fractions. 13- calculating defined integrals using fundamental theorems. 14- calculating areas of plane regions using defined integrals. 15- Analysing functions in polar coordinates.

Working method

Presencial

Program

A- Topics of Linear Algebra and Analytical Geometry I- Vector spaces: definition; Rn case; vector subspaces; linear dependence and independence; basis and dimension II- Matrices: definition, dimension and operations; the special case of square matrices: triangular matrices, symmetric matrices and matrix transposition; inverse matrix and its properties; orthogonal matrices; power of a matrix; characteristic matrix; matrices condensation method. III- Determinants: definition and properties; determinants calculation- Laplace theorem; application of determinants to the determination of the inverse matrix and the characteristic matrix. IV- System of linear equations: homogeneous and non-homogeneous systems; vector space of solutions; matrix form of systems; discussion and resolution of systems- Gauss-Jordan method; Cramer rules. V- Eigenvalues and eigenvectors: definition; properties; characteristic polynomial and determination of eigenvalues of a matrix; eigensubspaces associated to an eigenvalue; algebraic multiplicity and geometric multiplicity; basis of eigenvectors. VI- Analytical geometry: vector norm; angle of two vectors, collinear and perpendicular vectors; orthogonal projection of a vector onto another; internal product and its properties; internal product, norm and distance in coordinates in a given basis; vector or external product and mixed product in R3; vector equation of a plane and a line; parametric and cartesian equations. B- Differential and Integral Calculus of real functions of a real variable I- Revision of some real functions of real variable: exponential and logarithmic function; its properties and graphs; Brief revision of the concepts of limits and continuity and its application to some functions; some indeterminates; trigonometric functions and their inverse; hyperbolic function; polar coordinates and their relationship with cartesian coordinates; functions in polar coordinates. II- Derivation: definition and interpretation of derivative; rules of derivation of composed and inverse function; problems of application of growth of function and determination of maximum and minimum; examples of exponential growth and logistic curve; l’Hôpital’s rule; notion of differential; polynomial approximation- Taylor polynomial and Taylor’s formula with remainder . III- Primitives: definition of primitive and antiderivative; immediate examples; elementary rules; primitive substitution; decomposition and primitives of rational fractions. IV- Riemann integral on an interval [a,b]:definition by Riemann sum; basic properties; fundamental theorem of calculation; application of integral to the calculation of areas; mean value and mean value theorem

Mandatory literature

Teresa Arede; Apontamentos de Matemática I (https://sigarra.up.pt/feup/pt/conteudos_adm.list?pct_pag_id=249640&pct_parametros=pv_ocorrencia_id=348705)
Cabral, I., Perdigão, C., Saiago, C.; Álgebra, Escolar Editora, 2009. ISBN: 978-972-592-239-2
Carlos A. Conceição António; Análise Matemática I - Textos de Apoio, AEFEUP, 2007-2008
Larson, Hostetler, Edwards; Cálculo, 8ª ed., vol1, McGraw-Hill, 2006

Complementary Bibliography

Neuhauser,C.; Calculus for Biology and Medicine, 2nd Ed., Prentice Hall, 2004
Giraldes,E., Fernandes,V.H., Santos,M.H.; Curso de Algebra Linear e Geometria Analitica, McGraw-Hill, 1994

Teaching methods and learning activities

Classes are theoretical and practical; therefore classes will be based on the presentation of the program and examples, along with problem solving by the students under professor supervision. The theoretical presentation of the program will be made by using the board and slides. Students will get notes to both the theoretical program and exercises.

Evaluation Type

Distributed evaluation with final exam

Assessment Components

Designation	Weight (%)
Exame	50,00
Participação presencial	0,00
Teste	50,00
Total:	100,00

Amount of time allocated to each course unit

Designation	Time (hours)
Estudo autónomo	102,00
Frequência das aulas	56,00
Total:	158,00

Eligibility for exams

To be admitted to exams, students cannot miss more than 25% of classes (according to the General Evaluation Rules of FEUP); thus students cannot miss more than 5 classes

Calculation formula of final grade

Final Grade will be based on the following components: - 1st test (T1) – a mandatory test (date will be later announced) - 2nd test (T2) - only for students who reached a minimum grade of 10 (out 20) in T1. It will not cover the themes of T1 and both tests are worth the same. - Final Exam (FE) – it will take place at the same time as T2 and it is a mandatory exam to students who did not reach a minimum grade of 10 (out of 20) in T1. However, any student can attend this exam, which will cover the entire program. - Recurso (resit) Exam- RE- Only for students who did not complete the course or want to improve their grades. Final Grade will be based on the following formula: FG= (T1+T2)/2 or FG= FE or FG= RE

Examinations or Special Assignments

Students will be asked to do some individual written exercises as homework. By doing this, they will show their interest and dedication regarding this course unit. During classes students may be asked to do some exercises on the board. This will also show students’ interest and dedication regarding this course unit.

Special assessment (TE, DA, ...)

According to General Evaluation Rules of FEUP

Classification improvement

Recurso (resit) exam