Mathematics I

Code:

M195

Acronym:

M195

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2013/2014 - 1S

Active?	Yes
Web Page:	https://moodle.up.pt/course/view.php?id=252
Responsible unit:	Department of Mathematics
Course/CS Responsible:	Bachelor in Biology

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
L:AP	54	Planos de estudos a partir 2009	1	-	7,5	60	202,5
L:B	257	Plano de estudos a partir de 2008	1	-	7,5	60	202,5
L:BQ	104	Plano de Estudos a partir de 2007	1	-	7,5	60	202,5

Teaching language

Portuguese

Objectives

The aim of this course is that the student masters some basic techniques of differential and integral calculus of one variable (calculation of derivatives, primitives and integrals, solution of certain differential equations) and recognizes some of its applications. It is also an aim that the student masters some basic techniques of linear algebra (operations with matrices, computation of determinants, solving linear systems).

Learning outcomes and competences

Familiarity with basic techniques of differential and integral calculus, differential equations, and matrix theory, and their applications to allow the recognition of their importance in the students' areas of study.

Working method

Presencial

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

Prerequisites: basic knowledge of Mathematics acquired in the secondary education system.

Program

I. Calculus
1. Polynomial functions, exponentials, logarithms, trigonometric functions (review); inverse trigonometric functions, their derivatives; l'Hôpital's rule.
2. Primitivation by substitution, change of variable, and by parts; primitivation of rational functions.
3. Area and definite integral; Fundamental Theorem of Calculus; area of regions bounded by curves; improper integrals.
4. First order differential equations: separable or linear.
5. Examples of modelling by differential equations.

II. Linear Algebra
6. Real matrices; matrix operations.
7. Systems of linear equations; Gaussian elimination; chacteristic of a matrix; matrix inversion.
8. The area of parallelograma; determinants by Laplace expansion; volume of parallelpipeds; Cramer's rule.
9. Eigenvectors and eigenvalues; diagonalization of nxn matrices with n distinct eigenvalues.
10. Markov chains as a mathematical model; regular chains andstationary state vector.

Mandatory literature

J. Stewart; Cálculo - Volumes I e II, Pioneira Thomson Learning, 2006
W. Nicholson; Álgebra Linear, McGraw-Hill, 2006

Complementary Bibliography

F. Ayres e E. Mendelson; Schaum's Outline of Calculus, McGraw-Hill, 1999
G. Barker e H. Schneider; Matrices and Linear Algebra, Dover, 1989
M. Delgado e E. Mirra; Elementos de Matemática I, 2007

Teaching methods and learning activities

1. Lectures: presentation of the course material and of examples by the teachers.
2. Exercise sessions: solution of exercises by the students with the advice of the teachers; the exercises are published in advance to stimulate student work.
3. Regular office hours for student advice and clarification of doubts.
4. Besides the bibliography list, slides of the lecture notes are published online.

Software

sage (cf. http://www.sagemath.org)

keywords

Physical sciences > Mathematics > Mathematical analysis > Differential equations
Physical sciences > Mathematics > Mathematical analysis > Functions

Evaluation Type

Distributed evaluation with final exam

Assessment Components

designation	Weight (%)
Exame	100,00
Total:	100,00

Eligibility for exams

Course registration is the only requirement.

Calculation formula of final grade

There will be three optional midterm tests.

The final exam consists of three parts, corresponding to the tests, with equal weight. The classification of each part is the best between that of the test and that of the corresponding exam part. Analagous rule applies to all remaining exams, incluindo for grade improvement in 2013/2014.

For students approved in 2012/2013, grade improvement can be attempted only through examination.

Examinations or Special Assignments

(see the "Formula for the Calcultation of the Final Score")

Special assessment (TE, DA, ...)

Any type of special student evaluation may take one of the following forms: exclusively an oral examination; an oral examination plus a written examination, the student being required to pass both of them; only a written examination. The option for one of them is of sole responsibility of the professors in charge of the course unit.

Classification improvement

(see the "Formula for the Calcultation of the Final Score")

Observations

Article 13 of General Regulations for Student Evaluation at the levels of First Cycle, Integrated Masters, and Second Cycle at U.Porto, approved on May 19, 2010 (cf. http://www.fc.up.pt/fcup/documentos/documentos.php?ap=3&ano=2011): "Fraud committed during an exam, in any form, implies the annulment of the exam and the communication to the statutorily competent organ for possible disciplinary action."