Calculus I

Code:

EQ0059

Acronym:

AM I

Keywords
Classification	Keyword
OFICIAL	Physical Sciences (Mathematics)

Instance: 2013/2014 - 1S (of 01-09-2013 to 31-07-2014)

Active?	Yes
Responsible unit:	Department of Chemical Engineering
Course/CS Responsible:	Master in Chemical Engineering

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
MIEQ	89	Syllabus	1	-	6	63	162

Teaching language

Portuguese

Objectives

This course aims to endow students with fundamental knowledge on mathematics. It also aims to complete students’ education on IR defined functions, and develop the problem of primitives and function integration and study its representation

Objectives

- Understand, manipulate and apply the concepts of integration of functions of one variable and series.

- Provide a base set of mathematical fundamental to the proper functioning of other courses of integrated Masters.

- Develop scientific and mathematical reasoning and ability to apply mathematical concepts acquired.

Learning outcomes and competences

Ability to describe the main results in the field of differential and integral calculus, the series of real numbers and the polynomial approximation of functions of one variable by use of Taylor polynomials.

Ability to identify and properly apply the techniques to use in solving problems.

Obtaining a set of fundamental mathematical tools with direct application in other courses of the integrated MSc.

Working method

Presencial

Program

1. Functions, limits and derivates Revision of concepts studied in Secondary School about functions, limits, continuity and derivation 2. Functions and differential calculus New functions and their derivation; Inverse functions and their derivation; Linear and differential approximation of a function; Maximum and minimum of a function; Indeterminate forms and L’Hôpital’s rule 3. Primitive for a function Primitive of an elementary function; Stage primitive; Primitive by substitution or change of variable; primitive of rational functions; rational substitution; Primitive of trigonometric functions; Trigonometric substitutions 4. Integral Calculus Approximate calculation of areas; Riemann sum; Integral f(x) in (a,b); Basic properties of integrals; Assessment integrals of continuous functions; mean value of f(x) in (a,b); average value theorem; Integration of discontinuous functions; Geometric application of integrals; Deduction of integral formulas; Functions defined by integrals ; Improper integrals 5. Series of real numbers Series of real numbers; Study of series by its definition; Series of non-negative terms; Series of non-positive terms; alternate series; Series of positive and negative terms 6. Taylor’s and Maclaurian series: Function approximation by polynomial; Estimate of the rest of a Taylor polynomial; Taylor’s theorem and formula Taylor’s series (or Maclaurian’s) of f(x);

Mandatory literature

João Mendonça;Matemática I (Módulo de Análise Matemática) - Apontamentos das aulas, 2007/08

Complementary Bibliography

Edwards, C. Henry; Calculus. ISBN: 0-13-095006-8
Anton, Howard; Calculus. ISBN: 0-471-48237-4
Ana Alves de Sá e Bento Louro; Sucessões e Séries, Escolar Editora, 2009. ISBN: 978-972-592-222-4

Teaching methods and learning activities

General theoretical-practical classes are based on the presentation of the themes of the course, where examples are given. The theoretical-practical classes which are divided in groups, are based on problem solving and application of the themes that have been taught during theoretical-practical classes

Evaluation Type

Distributed evaluation with final exam

Assessment Components

Designation	Weight (%)
Exame	75,00
Teste	25,00
Total:	100,00

Eligibility for exams

Students cannot skip more than 3 theoretical-practical classes (Students who were only admitted on the second phase, cannot skip more than 2 classes) Students who are repeating the course, and who were admitted to exams in the previous year do not need to attend to classes.

Calculation formula of final grade

Final Mark will be based on one of the following formulas:

FM=max(0,25*MT + 0,35*T1 + 0,4*T2, 0.5*T1+0.5*T2) or FM = max(0,25*MT + 0,75*EF,EF)

MT- Mini-tests

T1- Test 1

T2-Test 2

EF- Exam

Students who were admitted to exams in the previous years and do not want to take part of distributed assessment, final mark will be calculated as follows:

FM = 0,5*T1 + 0,5*T2 or FM = EF

Examinations or Special Assignments

Not applicable

Internship work/project

Not applicable

Special assessment (TE, DA, ...)

An exam

Classification improvement

Students can improve their grade by attending to an exam at recurso season. Final mark will be calculated as follows:

FM = max(0,25*MT + 0,75*ER, ER).

Observations

The grade of mini-tests will be the average grade of all the mini-tests. If students do not attend to one mini-test, they will earn a 0. Students cannot use calculators on mini-tests and exams.