Mathematics I

Code:

L.BIO002

Acronym:

MATI

Keywords
Classification	Keyword
OFICIAL	Basic Sciences (Mathematics, Physics, Chemistry, Biology)

Instance: 2023/2024 - 1S

Active?	Yes
Responsible unit:	Department of Civil Engineering
Course/CS Responsible:	Bachelor in Bioengineering

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
L.BIO	129	Syllabus	1	-	6	52	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- carry out matrix operations.
2- calculate determinants of any order and be capable of using their main properties.
3- classify systems of linear equations regarding the type of solution and resolution of systems by using Gaussian elimination method.
4- analyse linear dependence and independence of a group of vectors in R2 and R3.
5- determine the dimension and bases of R2 and R3 subspaces
6- determine eigenvalues and eigenvectors, as well as their eigensubspaces.
7- calculate the orthogonal projection, internal and external product between 2 vectors.
8- obtain inverse trigonometric functions, as well as their derivatives.
9- calculate primitives by substitution and parts method.
10- calculate primitives of rational fractions.
11- calculate defined Riemann integrals using fundamental theorems.
12- obtain the areas of plane regions using defined integrals.

Working method

Presencial

Program

A - Topics of Linear Algebra and Analytical Geometry

I- 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; matrices condensation method.

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

III- 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.

IV- Vector spaces: definition; Rn case; vector subspaces; linear dependence and independence; basis and dimension

V- Eigenvalues and eigenvectors: definition;  characteristic polynomial and determination of eigenvalues of a matrix; eigensubspaces associated to an eigenvalue.

VI- Analytical geometry: vector norm; orthogonal projection of a vector onto another; scalar or internal product; vector or external product and mixed product in R3.

B - Differential and Integral Calculus of real-valued 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

James Stewart; Cálculo. ISBN: 85-221-0235-X (vol. 1)
Cabral, I., Perdigão, C., Saiago, C.; Álgebra, Escolar Editora, 2009. ISBN: 978-972-592-239-2
Howard Anton; Elementary linear algebra. ISBN: 9781119666141
Howard Anton; Cálculo. ISBN: 978-85-60031-63-4

Complementary Bibliography

Giraldes,E., Fernandes,V.H., Santos,M.H.; Curso de Algebra Linear e Geometria Analitica, McGraw-Hill, 1994
Larson, Hostetler, Edwards; Cálculo, 8ª ed., vol1, McGraw-Hill, 2006

Teaching methods and learning activities

In theoretical-practical classes, the theoretical material will be presented with examples of application. In practical classes, students will solve proposed exercises, with guidance and monitoring by the teacher. Interaction with students during classes is maximised.

The curricular unit is complemented with a Moodle page where, in addition to all the pedagogical support material (acetates, notes, exercise sheets, resolutions, videos, etc.), online self-assessment activities are made available to enable the measurement of teaching / learning.

Software

Matlab

Evaluation Type

Distributed evaluation without final exam

Assessment Components

Designation	Weight (%)
Teste	90,00
Trabalho prático ou de projeto	10,00
Total:	100,00

Amount of time allocated to each course unit

Designation	Time (hours)
Estudo autónomo	106,00
Frequência das aulas	56,00
Total:	162,00

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

This course has distributed assessment without final exam. Thus, the final classification (FC) is defined by:

CF = maximum{0.45 x MT1 grade + 0.45 x MT2 grade + 0.1 MDL; 0.5 x MT1 grade + 0.5 x MT2 grade}

where:

- MT1: 1st Mini-Test,

- MT2: 2nd Mini-Test, (which will only include the subjects after the 1st mini-test);

- MDL: average of Moodle activity grades over the semester

If students do not pass with the distributed assessment or wish to improve their grade, they may take an appeal test related to a single part of the subject (Appeal of the 1st Mini-Test - MTR1 for the Calculus part or Appeal of the 2nd Mini-Test - MTR2 for the Algebra part) or take a global exam (ER).

Therefore, the final classification (CFR) of the appeal season will be:

CFR=ER grade (if a global exam is taken)

Or

CFR=0.5 x MTR1 grade + 0.5 x MT2 grade (if 'repeats' MT1 - Calculus)

Or

CFR=0.5 x MT1 grade + 0.5 x MTR2 grade (if 'repeats' MT2 - Algebra)

In any case, to obtain a grade higher than 18, the student must take a supplementary exam.

All assessment components are expressed on a scale of 0 to 20.

Special assessment (TE, DA, ...)

Final exam.

Classification improvement

Appeal exam.