Mathematical Analysis 1

Code:

L.EC002

Acronym:

AM1

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2025/2026 - 1S

Active?	Yes
Web Page:	https://moodle2526.up.pt/course/view.php?id=5224
Responsible unit:	Department of Civil and Georesources Engineering
Course/CS Responsible:	Bachelor in Civil Engineering

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
L.EC	237	Syllabus	1	-	6	58,5	162

Teaching language

Portuguese

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, trigonometric and inver trigonometric functions
1.2. Real functions of real variables: limits and continuity. L'Hôpital's rule. 
1.4. Differential calculus of functions of one variable: definitions and geometric interpretation. Rolle and Lagrange and 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. Brief notions of 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 15
2 50
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 theoretical lectures concepts and results are presented resorting to geometrical interpretation (when possible) and illustrative examples. Some constructive demonstrations are presented. In theoretical-practical lectures the contents introduced in theoretical classes are worked and exercised. 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	103,50
Frequência das aulas	58,50
Total:	162,00

Eligibility for exams

Approval for a course unit requires compliance with the attendance requirement. Students are deemed to  have met this requirement if, having been enrolled, they do not exceed the maximum number of  absences correspopnding to 25% of the scheduled in-person classes for each type. In addition to the cases statutorily provided for in FEUP's current rules, students who obtained a final grsde of 6 or higher in the course unit in the immediately academic year are exempt from the attendance requirement.

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)

The final marks (CF) are determined as follows:
CT1 - MT1 classification
CT2 - MT2 classification

CF = 0,5CT1+0,5CT2

The CTP component will be the average gradeof the small assessments carried out, if the student is classified in 4 or more of these assessments.
Students who, having frequency, do not get approval on the course unit, have access to the examination of appeal to this effect. 

Notes:

      1 - In both tests (MT1 e MT2) the student must have a minimum classification of 6 values.

      2 - Non-attendance to one of the Tests implies a mark of 0 in that Test. In particular, the non-attendance to one of the tests MT1 and/or MT2 implies that no classification will be given to the UC of Computation.

      3 - Students not approved may have access to the exam of supplementary season ("exame de recurso"). 

     4 - During the appeal period, students may only improve their CF grade, and they can choose to improve either the CT1 or CT2 grade, or take a global test corresponding to CF.

Examinations or Special Assignments

Not applicable

Internship work/project

Not applicable

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.

Observations

During any assessment moment, the possession of any electronic device (e.g., mobile phones, tablets, earphones, smartwatches, etc.) is strictly prohibited, except for those explicitly authorized by the teaching staff.
It is the student's responsibility to anticipate this situation before the start of the assessment.

Calculators are not allowed at any of the various components of evaluation.