Mathematics I

Code:

M191

Acronym:

M191

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2013/2014 - 1S

Active?	Yes
Responsible unit:	Department of Mathematics
Course/CS Responsible:	Bachelor in Computer Science

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
L:CC	79	Plano de estudos de 2008 até 2013/14	1	-	7,5	-
MI:ERS	151	Plano de Estudos a partir de 2007	1	-	7,5	-

Teaching language

Portuguese

Objectives

With this course, it is intended that students will know and understand
some of the main classical results of Calculus. These results,  for their historical
importance in the explanation of scientific phenomena and on the resolution of scientific problems, should be known by every student of the area of science.

Learning outcomes and competences

The student knows, understands and is able to use concepts that explain scientific phenomena, as well as techniques therefrom derived for solving scientific problems. In the process, he develops his knowledge of the physical world and his ability and taste for analysis, in addition to his ability to understand, use and develop abstract structures.

Working method

Presencial

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

Basic techniques of calculus and basic knowledge of mathematics, which will be revised at the beginning of the course.

Program

1. Revision of some basic Mathematical concepts and techniques.
2. Systems of linear equations and matrices.
2.1 Solving systems of linear equations using Gauss's method and matrices.
2.2 Matrices: sum of matrices, scalar multiplication, multiplication of two matrices, inverse of a matrix, the rank of a matrix.
2.3 Determinants and Cramer's Rule.
3. Calculus - Preliminaries
3.1 Functions, domain, codomain and image ; injective, surjective and bijective functions; graphs of functions; composite functions; the inverse of a bijective function.
3.2 Trigonometric functions; properties. Inverse trigonometric functions. Exponential and Logarithmic functions.
3.3 Limits; definition and properties.
3.4 Continuous functions; definitions and properties, the Intermediate-Value Theorem, the Max-Min Theorem.
4. Differentiation
4.1 Definition of derivative, differentiation rules (sums and constant multiples, the product rule, the quotient rule, inverse functions). Derivatives of Trigonometric functions and of their inverse functions. Derivatives of exponential and logarithmic functions.
4.2 Rolle, Lagrange and Cauchy's theorem. Increasing and decreasing functions, maximum and minimum values.
4.3 Extreme-value problems.
4.4 Concavity and inflexictions.
4.5 Sketching the graph of a function.
5. Taylor polynomials.
6. Integration
6.1 The definite integral: definition and properties.
6.2 The fundamental Theorem of Calculus.The method of substitution, integration by parts. Integral of rational functions.
6.3. Improper integrals.
6.4. Areas and volumes of solids of revolution, length of  graphs of functions.
7. Sequences and series.
7.1. Limits of sequences.
7.2. Criteria for convergence of series: ratio, integral and Leibnitz criterion.

Mandatory literature

Spivak Michael; Calculus. ISBN: 0-914098-89-6
Stewart James; Calculus. ISBN: 978-0-495-38273-7
Anton Howard; Elementary linear algebra. ISBN: 0-471-66959-8
Monteiro António; Álgebra linear e geometria analítica. ISBN: 972-773-106-6

Teaching methods and learning activities

Lectures and classes: the contents of the syllabus will be presented in lectures, where numerous examples will be given to illustrate the concepts. There will also be practical classes, where exercises and related problems are solved. On the course site in the internet other supporting materials (texts, resolutions of the quizzes, etc..) will be available.

The first part of the course, reviewing the basic concepts and techniques of mathematics, will be supported by a program of e-learning specially created for this purpose, which will allow students to practice the basic techniques of calculus and simultaneously to self-assess their progress. This process will be supported in extra-curricular classes at the beginning of the course.

Evaluation Type

Distributed evaluation with final exam

Assessment Components

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

Eligibility for exams

For being accepted at the final exam, the students without previous "frequency" must be present in at least half of the practical classes.

Students who were considered acceptable for exam in 2010/11, 2011/12 or 2012/13 may be excused from attending TP classes. For this purpose, they must apply to one of the professors by email, until October 15.

Calculation formula of final grade

Final exam, where two groups of questions can be replaced, at student's will, independently of one another, by the scores obtained in two querys:

. The first one, worth 3/20,  will be held on computer and assesses the basic knowledge and mastery of basic techniques that are reviewed in the beginning of the course. In other words, it will correspond to chapters 1 and 3 of the syllabus.

. The second one, worth 5/20, will be held in November and will focus on the part of the syllabus after the initial revision hitherto taught.

Any special exam can be either an oral or a written exam.

Special assessment (TE, DA, ...)

Any special exam can be either an oral or a written exam.