Mathematics I

Code:

EBE0001

Acronym:

MAT1

Keywords
Classification	Keyword
OFICIAL	Basic Sciences

Instance: 2017/2018 - 1S

Active?	Yes
Responsible unit:	Department of Civil Engineering
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	76	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- analyse linear dependence and independence of a group of vectors in R2 and R3.
2- determine the dimension and bases of R2 and R3 subspaces
3- carry out matrix operations.
4- calculate determinants of any order and be capable of using their main properties.
5- classify systems of linear equations regarding the type of solution and resolution of systems by using Gaussian elimination method.
6- determine eigenvalues and eigenvectors, as well as their eigensubspaces.
7- calculate the angle, the internal and the external product between 2 vectors.
8- determine vector, parametric and cartesian equations of lines and planes in R3.
9- obtain inverse trigonometric functions, as well as their derivatives. 10- calculate primitives by substitution and parts method.
11- calculate primitives of rational fractions.
12- calculate defined Riemann integrals using fundamental theorems. 13- obtain the areas of plane regions using defined integrals.

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; matrix rank; matrices condensation method.

III- Determinants: definition and properties; determinants calculation- Laplace theorem; application of determinants to the calculation of the inverse matrix and the matrix rank.

IV- Systems 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;  characteristic polynomial and determination of eigenvalues of a matrix; eigensubspaces associated to an eigenvalue.

VI- Analytical geometry: vector norm; angle of two vectors, collinear and normal 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 indeterminate forms; trigonometric functions and their inverse; hyperbolic function.

II- Derivation: definition and interpretation of the derivative; rules of derivation of composed and inverse functions; problems of application to the growth of function and determination of maximum and minimum; examples of exponential growth and logistic curve; l’Hôpital’s rule; notion of differential.

III- Primitives: definition of primitive or antiderivative; simple examples and elementary rules; methods of substitution and parts; decomposition and primitives of rational fractions.

IV- Riemann integral on an interval [a,b]:definition by Riemann sums; basic properties; fundamental theorem of calculus; application of integral to the calculation of areas; mean value and mean value theorem.

Mandatory literature

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

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 they will be based on the presentation of the theoretical concepts 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 without final exam

Assessment Components

Designation	Weight (%)
Participação presencial	0,00
Teste	100,00
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).

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) - 40%

- 2nd test (T2) - a mandatory test (date will be later announced) . It will not cover the themes of T1 - 60%; 

- 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= 0,4xT1+0,6xT2
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 answer some questions about concepts or methods already explained. 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.