Mathematical Analysis I

Code:

EM0009

Acronym:

AM I

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2011/2012 - 1S

Active?	Yes
Responsible unit:	Mathematics Section
Course/CS Responsible:	Master in Mechanical Engineering

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
MIEIG	87	Syllabus since 2006/2007	1	-	6	70	160
MIEM	239	Syllabus since 2006/2007	1	-	6	70	160

Teaching language

Portuguese

Objectives

1- BACKGROUND
Almost every aspect of professional work in the world involves mathematics. A solid knowledge of mathematical analysis is required for any engineering degree namely mechanics and industrial management.
2- SPECIFIC AIMS
Development of the reasoning capacity of the students and understanding essential mathematical concepts. Students should get solid theoretical and practical formation on the main concepts and results of sequences, differential and integral calculus real functions of one real variable.
3- PREVIOUS KNOWLEDGE
High school math. Functions and graphs. The limit concept, the concept of continuity at a point and the derivative of a function. Differentiation rules.
4- PERCENT DISTRIBUTION
Scientific component:75%
Technological component:25%
5- LEARNING OUTCOMES
Knowledge and Understanding- Theoretical concepts and practical formation on differential and integral calculus real functions of one real variable. Polynomial approximation for real-valued functions using Taylor's polynomials and the error concept. Calculation of areas in cartesian and polar coordinates.Volume calculations.
Engineering analysis-Application of differentiation and integral calculus of one real variable function.
Engineering design- Engineering problems of one real variable.
Investigations- Practical formation on the main concepts and results of differential and integral calculus in R.
Engineering practice- Applications in Physics and Mechanics.
Transferable skills- Knowledge of differential and integral calculus of one real variable.

Program

A. Differential Calculus in R:
Review of fundamentals of differentiation.
Increments, differentials and linear approximations. The mean-value theorem for derivatives.
Polynomial approximations to functions: Taylor's polynomials generated by a function.
Taylor’s formula with remainder. Estimates for the error in Taylor’s formula.
The Taylor series as a limit of Taylor polynomials.
Numerical series: properties, convergence criteria, alternating series.
Reference of functional series. Concept of convergence interval.

B. Integral Calculus in R:
Riemann sums and the integral. Integrability of bounded monotonic functions. The integrability theorem for continues functions. Properties of the integral. Mean-value theorem for integrals.
The derivative of an indefinite integral. The first fundamental theorem of calculus.
Primitive functions and the second fundamental theorem of calculus.
Integration by substitution. Integration by parts.
Areas of plane regions. Polar coordinates. Area calculation in polar coordinates. Volume calculations by the method of cross sections.
Integration by rational partial fractions. Rational trigonometric integrals. Integrals containing quadratic polynomials.

C. Additional topics:
Hyperbolic functions.
Improper integral.
First order differential equations.

Mandatory literature

Carlos A. Conceição António; Análise Matemática 1, Texto de apoio, AEFEUP, 2007/2008
Larson, Hostetler & Edwards; Cálculo, McGraw-Hill Interamericana, 2006. ISBN: 85-86804-56-8
Larson, Hostetler & Edwards; Cálculo, McGraw-Hill Interamericana,, 2006. ISBN: 85-86804-82-7
Carlos C. António, Catarina F. Castro, Luísa C. Sousa, M. Graça Pinto; Exercícios propostos para as aulas práticas de AM I, AEFEUP, 2007/2008
Ana Alves de Sá e Bento Louro; Sucessões e Séries, Teoria e Pática, Escolar Editora, 2008. ISBN: 978-972-592-238-5
Ana Alves de Sá, Bento Louro; Sucessões e séries. ISBN: 978-972-592-222-4

Complementary Bibliography

Apostol, Tom M.; Cálculo, N. ISBN: 84-291-5015-3 (vol.1)
B. Demidovitch; Problemas e exercícios de análise matemática. ISBN: 978-972-592-283-5
Spivak, Michael; Calculus, N. ISBN: 0-914098-89-6
Edwards, C. Henry; Calculus. ISBN: 0-13-736331-1
J. Campos Ferreira; Introdução à Análise Matemática, Fundação Calouste Gulbenkian, 8ª edição

Teaching methods and learning activities

Theoretical-practical classes are based on written and oral expositions. Application examples are frequently presented, especially at the end of each unit. Students are encouraged to solve specific exercises during class.

Evaluation Type

Distributed evaluation without final exam

Assessment Components

Description	Type	Time (hours)	Weight (%)	End date
Attendance (estimated)	Participação presencial	65,00
Examinations	Exame	7,00		2012-02-09
	Total:	-	0,00

Amount of time allocated to each course unit

Description	Type	Time (hours)	End date
Study time for examinations	Estudo autónomo	21	2012-02-09
Study time following classes	Estudo autónomo	69	2012-02-09
	Total:	90,00

Eligibility for exams

To attain frequency, the student must:

1) Not exceed the absence limit allowed in Article 4 of General Evaluation Rules of FEUP; and
2) To attend at least one of the two tests with minimal grade of 6 out of 20.

Calculation formula of final grade

Final Mark (FM) is based on the provisional final mark (FM*) according to the following rules:

A. Provisional Final Mark (FM*)
FM* is based on two written tests during continuous assessment. They will last 2 hours.
At the end of the semester, it will take place a re-assessment test which will last 3 hours. This test will be divided into two parts, in which each of them will correspond to the previous tests. Students can do one part or the whole test.

Important: Students can only attend to the re-assessment test, if they do not miss more classes than allowed by the rules, or if they do at least one of the assignments and achieve a minimum mark of 6 out of 20.

FM* depends on students’ achievements during the semester:
A1. Students who attended to the two tests:
FM* will be based on the average mark of the two tests

A2. Students who attended the re-assessment test:
If the mark achieved in each of the parts of this test is higher than tests, it will be considered the mark of the re-assessment test.
FM* will be based on the average mark of the best classifications.

B. Final Mark (FM)
B1. Students’ FM will be the same as FM*, if they achieve an 18 out of 20 or a lower mark.
B2. Students who achieve a higher mark than 18 out of 20 have to attend to an oral exam. If they do not attend to the oral exam, they will earn an 18.
B3. Students who achieve an equal mark 9 out of 20 have to attend to an oral exam. If they do not attend to the oral exam, they will earn an 9.

Examinations or Special Assignments

N/A

Special assessment (TE, DA, ...)

This evaluation will exclusively be the result of the written exam according to General Evaluation Rules of FEUP.

Classification improvement

Classification improvement will exclusively be the result of the written exam according to General Evaluation Rules of FEUP. This exam wil be made only in next epochs after the conclusion of distributed evaluation process.

Observations

Language of instruction: Portuguese