Mathematical Analysis I

Code:

EM0009

Acronym:

AM I

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2020/2021 - 1S

Active?	Yes
Web Page:	http://www.fe.up.pt
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
MIEM	230	Syllabus since 2006/2007	1	-	6	65	162

Teaching language

Portuguese

Objectives

Students should get solid theoretical and practical formation on the main concepts of differential and integral calculus of real functions of one real variable.

Learning outcomes and competences

Knowledge and Understanding-Theoretical and practical formation on differential and integral calculus of real functions of one real variable. Approximation for real-valued functions using Taylor's polynomial and error evaluation. Convergence of numerical series. Calculation of areas in cartesian and polar coordinates. Volume by integral calculus.
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 for one real variable functions.

Working method

Presencial

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

Program

A. Differential Calculus in R:
Definition of derivatives, chain rule and derivative of inverse function. Review of fundamentals of differentiation.
The mean-value theorem for derivatives.
Polynomial approximations to functions: Taylor's polynomials and Taylor’s formula with remainder.
The Taylor series. Numerical series: properties, convergence criteria, alternating series.
Reference of functional series and convergence interval.
B. Integral Calculus in R:
Riemann sums and the integral: Definition and properties. Mean-value theorem for integrals.
Fundamental theorems of calculus. Primitive functions and integration by substitution and by parts.
Areas of plane regions by integrals calculation. Polar coordinates and area calculation. 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 Alberto Conceição António; ANÁLISE MATEMÁTICA I - Conteúdo Teórico e Aplicações (edição aumentada), AEFEUP, 2017. ISBN: 978-989-99559-6-7
Ron Larson, Robert P. Hostetler, Bruce H. Edwards; Cálculo. ISBN: 85-86804-56-8 (vol. 1)
Larson, Hostetler & Edwards; Cálculo, McGraw-Hill Interamericana,, 2006. ISBN: 85-86804-82-7
Carlos C. António, Catarina F. Castro, Luísa C. Sousa; Exercícios propostos para as aulas práticas de AM I, 2013
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

The lectures consist of exhibitions supported by slides on the content of the course (UC). The presentation of each chapter includes examples of application. In practical classes are solved exercises.

keywords

Technological sciences > Engineering
Physical sciences > Mathematics

Evaluation Type

Distributed evaluation without final exam

Assessment Components

Designation	Weight (%)
Teste	100,00
Total:	100,00

Amount of time allocated to each course unit

Designation	Time (hours)
Estudo autónomo	97,00
Frequência das aulas	65,00
Total:	162,00

Eligibility for exams

To obtain frequence the student must: 1. Comply with Regulation specific assessment of students FEUP (Article 7 - Attendance), in particular with regard to attendance at lectures and practices of the course. 2. Carry at least one of the two written tests of Distributed Evaluation with a rating equal to or greater than 6 values.

Calculation formula of final grade

Valuation formula:
- 1st test (T1), mandatory and weighting 50% of the final grade and content of UC;
- 2nd test (T2) unique to students with a grade T1> = 6 (out of 20), only includes raw post-T1, weighting 50%;
- Final Exam (FE) about the whole matter, simultaneously with T2, mandatory for students to score - Examination of resource (ER) about the whole matter, for students without minimum grade tests, T1 or T2, or without averaging 10 values in both tests or in EF.
All the tests lasting 2 hours.
Final standings (CF): CF = (T1 grade + T2 grade ) / 2 since T1> and T2 = 6> = 6, or CF=EF grade or CF=ER grade.

Examinations or Special Assignments

N/A

Internship work/project

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 will be made only in next epochs after the conclusion of distributed evaluation process.

Observations

Attendance of students in the Mathematics Section of DEMEC (3rd floor, Building F) accordingly with the schedule timetable published by the UC teatcher.