Mathematical Analysis I

Code:

L.EIC002

Acronym:

AM I

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2024/2025 - 1S

Active?	Yes
Responsible unit:	Mathematics Section
Course/CS Responsible:	Bachelor in Informatics and Computing Engineering

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
L.EIC	393	Syllabus	1	-	6	52	162

Teaching language

Portuguese

Objectives

This course aims to acquaint students with the differential and integral calculus, in order to make them able to apply basic tools of mathematical analysis in problem solving related with subjects of Informatics and Computing Engineering. This course also aims to expand students’ knowledge, so that they can address new methodologies applied to engineering problems. At the end of the course, the learning outcomes are:
1. To solve derivatives of functions, draw graphics and study functions in general;
2. To solve integrals and use them in various engineering applications;
3. To use different integration techniques and differential equations;
4. To use and understand approximation concepts based on series and polynomials.

Learning outcomes and competences

As a result of this course, students should be aquainted with the following matters:
1. To study functions, solve derivatives and draw graphics
2. To solve integrals and use them in various engineering applications
3. To use differential equations and Laplace Transform
4. To understand approximation concepts using series and polynomials.

Working method

Presencial

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

Knowledge of pre-calculus at the level of the high school program of Math A.

Program

A. Differentiation in R
   A.1. The Derivative
   A.2. Physical Interpretation of the Derivative
   A.3. Derivative Rules
   A.4. The Product and Quotient Rules
   A.5. Chain Rule
   A.6. Derivatives of Trigonometric Functions
   A.7. Derivatives of the Inverse Function
   A.8. Mean Value Theorem or Lagrange Theorem
   A.9. Notion of Differential and Calculus Rules
   A.10. Cauchy's Theorem and L'Hôpital's Rule
   A.11. Polynomial Approximation
   A.12. Taylor Series with Limit of Taylor Polynomials
   A.13. Numerical Series

B. Riemann Integration in R
   B.1. Concept of Definite Integral
   B.2. Calculation of Areas and Mean Value Theorems for Integrals
   B.3. Fundamental Theorems of Calculus
   B.4. Integration by Substitution and by Parts
   B.5. Calculation of Volumes using Integrals
   B.6. Definition of Functions and Calculation of Areas using Polar Coordinates
   B.7. Other Primitive Methods

C. Additional Topics
   C.1. Hyperbolic Functions
   C.2. Improper Integrals
   C.3. Differential Equations
   C.4. Laplace Transform
   C.5. Fourier Series

Mandatory literature

Carlos A. Conceição António; Análise Matemática 1 - Conteúdo teórico e aplicações, AEFEUP, 2017. ISBN: 978-989-98632-3-1
Madureira, Luísa; Problemas de equações diferenciais ordinárias de Laplace . ISBN: 972-752-065-0
Madureira Maria Luísa Romariz; Problemas de integrais de linha e superfície e de séries de Fourier., Universidade do Porto. Faculdade de Engenharia, 2018. ISBN: 978-989-99559-2-9

Complementary Bibliography

Apostol, Tom M; Calculus. ISBN: 84-291-5001-3
Banner, Adrian; The calculus lifesaver: all the tools you need to excel at calculus., Princeton University Press, 2007
Roland E. Larson; Cálculo. ISBN: 85-86804-56-8 (v. 1)

Teaching methods and learning activities

Theoretical classes will be based on the presentation of the themes of the course unit. These classes are aimed to motivate students, where examples of application will be showed. Theoretical-practical classes will be based on the analysis and on problem solving by students, where they have to apply tools and mathematical concepts taught in theoretical classes. These classes are aimed to assess students’ understanding and dexterity of the themes of the course unit.

Attendance in practical courses is controlled and the student may not exceed the designated number of absences (25% of the scheduled hours) indicated by the professor for each practical course. If the student exceeds the specified number of absences, he/she will not be able to attend the course or take any exam in that course unless he/she has a special statute (see FEUP's pedagogical and evaluation rules).

keywords

Physical sciences > Mathematics > Mathematical analysis > Functions
Physical sciences > Mathematics > Mathematical analysis > Differential equations

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	92,00
Frequência das aulas	70,00
Total:	162,00

Eligibility for exams

Participation in 75% of the practical lessons (solving exercises)

Calculation formula of final grade

The final grade will be obtained through the average of the two tests (T1 and T2) – 50% of the grade of Test 1 (T1) plus 50% of the grade of Test 2 (T2).
Tests 1 and 2 will have essay questions.
T1 and T2 will be evaluated on a scale of 0-20 points.

The appeal exam (ER) will consist of a test that covers all the content covered throughout the academic semester. ER will be evaluated on a scale of 0-20 points.

Classification improvement

The student that has already obtained approval can attend the “appeal exam”.