Mathematical Analysis I

Code:

EC0001

Acronym:

AMAT1

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2018/2019 - 1S

Active?	Yes
Web Page:	https://moodle.up.pt/course/view.php?id=891
Responsible unit:	Mathematics Division
Course/CS Responsible:	Master in Civil Engineering

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
MIEC	248	Syllabus since 2006/2007	1	-	7	75	187

Teaching language

Portuguese and english

Objectives

OBJECTIVES:
Understand, handle and apply the concepts of integration of real variable functions and series. Provide a base set of math skills required for the proper functioning of other course curriculum units. Develop scientific-mathematician reasoning and the ability of being receptive to the application of mathematical concepts.

Learning outcomes and competences

EXPECTED SKILLS:

At the end of the term, students should, in each of the following strands, be able to:

Knowledge: Describe the main achievements in the area of basic training of mathematical analysis, notably in the field of differential and integral calculus, numerical series and polynomial approximation of real variable functions by Taylor Polynomials. Identify the techniques to use in solving proposed problems.

Understanding: Build suitable attitude and thought for solving engineering problems.

Application: Develop a solid foundation for subsequent course units, enabling the proper use of the techniques and the rigorous formulation of problems.

Working method

Presencial

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

Students should have basic knowledge about real functions of a real variable, derivation, polynomials and logic.

Program

1. Differential calculus of real functions of real variables:
1.1. Methodology of mathematics, Fundamentals of mathematical logic, axiomatic of real numbers, brief topological notions. Bolzano-Weierstrass theorem.
1.2. Trigonometric, inver trigonometric and hyperbolic functions
1.3. Real functions of real variables: limits and continuity. Intermediate values and Weierstrass theorems
1.4. Differential calculus of functions of one variable: definitions and geometric interpretation. Rolle, Lagrange and Cauchy theorems. Derivative of inverse and composed functions. Practical rules and applications of derivation.

2. Integral calculus of real functions of real variable:
2.1. Riemann integral, operations with integrals, indefinite integrals. Fundamental theorem of calculus, Barrow’s formula and mean value theorem.
2.2. Integration of rational functions. Integration by parts and substitution.
2.3. Improper integrals
2.4. Calculation of areas in the plan.

3. Numerical sequences and series:
3.1. Cauchy sequence, convergence analysis and sum of a series, simple convergence and absolute convergence.
3.2. Series of positive terms: criteria of comparison, d’ Alembert, Cauchy and the integral.
3.3. Series of alternating terms: Leibniz’s criterion

4. Sequences and series of functions:
4.1. one point convergence and uniform convergence
4.2. power series: radius and convergence range
4.3. polynomial approximation: Taylor’s polynomial and series.

DISTRIBUTION:
The estimated percentage distribution of scientific and technological content conforms to the following table:

Program item estimated %
1 25
2 40
3 20
4 15

DEMONSTRATION OF THE SYLLABUS COHERENCE WITH THE CURRICULAR UNIT'S OBJECTIVES:
This curricular unit is within the group of curricular units in the scientific area of mathematics, mainly focusing on providing students a solid education in the concepts of integration and the calculation of real functions of real variable and numerical and function series. The syllabus includes the differential calculus, integral calculus, numerical sequences and series of functions. These materials are the basis of mathematical calculation, and the concepts presented in class and the results using clear examples with possible use of appropriate software.

Mandatory literature

Apostol, Tom M.; Calculus
Stewart, James; Cálculo, Thompson. ISBN: 85-221-0479-4
James Stewart; Cálculo. ISBN: 85-221-0236-8 (vol. 2)
Apontamentos e colectânea de exercícios de apoio às aulas; Disponível na plataforma Moodle da UC

Complementary Bibliography

M. Spivac; Calculus, Volumes 1 e 2, Addison Wesley
Alves de Sá, Ana e Louro, Bento; Sucessões e Séries, Teoria e prática, Escolar Editora, 2009
Larson, Hostetler & Edwards; Cálculo, Volumes 1 e 2 (Oitava Edição), McGraw-Hill, 2006. ISBN: 85-86804-56-8

Teaching methods and learning activities

In lectures (theoretical and practical), concepts and results are presented resorting to geometrical interpretation (when possible) and illustrative examples. Some constructive demonstrations are presented. Strong appeal is made to the understanding of the concepts and calculation capacity. Throughout the course unit, students are reminded of the available computational tools, its capabilities and limitations. Classes are held with some demonstrations of the use of appropriate software and the student is guided along the selected problem solving.

DEMONSTRATION OF THE COHERENCE BETWEEN THE TEACHING METHODOLOGIES AND THE LEARNING OUTCOMES:
The presentation of the concepts and results using the geometric interpretation of illustrative examples aims to develop scientific and mathematical logical thinking and the ability to be aware for the application of mathematical concepts. In this way a proper attitude and thinking is developed to solve engineering problems and a solid basis for the subsequent curricular units is acquired, allowing the use of correct techniques and rigorous problems formulation.

keywords

Physical sciences > Mathematics > Mathematical analysis

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	145,00
Frequência das aulas	55,00
Total:	200,00

Eligibility for exams

Achieving final classification requires compliance with attendance at the course unit, according to the MIEC assessment rules. It is considered that students meet the attendance requirements if, having been regularly enrolled, the number of absences of 25% for each of the classes’ types is not exceeded.

Calculation formula of final grade

Components of the evaluation (ratings of all the components of assessment are expressed in the range 0 to 20):
• Two mini-tests (MT1 and MT2).
• One practical exercise, performed during lectures time (before 1st mini-test (EP1)).

The final marks (CF) are determined as follows:

EP1 – result of EP1
M1 – result of 1st moment of evaluation.
M1 = 0,3 Max {EP1, group I of MT1} + 0,7 (Group II of MT1).
M2 – result of 2nd moment of evaluation.

CF = 0,5 M1 + 0,5 M2

Students who, having frequency, do not get approval on the course unit, have access to the examination of appeal to this effect, and may opt to be assessed only on the matters concerning one of the mini-tests (1st or 2nd - proof up to 1h30m) or to the whole matter (2h30m). The calculation of the final ranking is done according to the formulas above for M1 and M2, replacing M1 and M2 (both or only one) by the results obtained in the facility.

Examinations or Special Assignments

Internship work/project

Special assessment (TE, DA, ...)

SPECIAL RULES FOR MOBILITY STUDENTS: Proficiency in Portuguese; attendance to secondary disciplines that underpin this course unit; evaluation components as determined for ordinary students.

Classification improvement

Students approved according to CF classification, may apply to a final exam, repeating M1 or M2 or all the matters of the curricular unit.

Observations

Estimated weekly working time outside lectures: 5 hours.

Calculators are not allowed at any of the various components of evaluation. Fraud in the realization of an examination - in any of the several modes -implies the respective cancellation (article 13 of Normas Gerais de Avaliação da FEUP).