Mathematical Analysis I

Solving equations and inequations (secondary education)

Basic definitions of real-valued functions of a real variable: polynomial functions, exponential and logarithmic functions, trigonometric functions; derivatives (secondary education)

Classes are taught in a theoretical-practical regime, in accordance with the curriculum plan.

In the theoretical part, the expository methodology is used to present the concepts, fundamental results and methods for solving the mahthematical problems under study.

The practical part is aimed at carrying out exercises to apply the concepts transmitted under the guidance of the teacher, but encouraging autonomous work or in small groups. Some of these exercises are based on real-life practical problems, allowing an interaction between theory and practice.

Goals:

– Provide knowledge of the wide variety of problems that can be solved using mathematical concepts and results.

– Foster logical/deductive reasoning and mental calculation.

– Encourage the use of analytical methods in solving concrete problems through the application of acquired knowledge.

Skills:

– Use some notions of Logic and Set Theory that support the algorithmic construction and research in databases.

– Ability to apply mathematical knowledge of matrix calculus, namely in solving systems of linear equations.

– Apply the knowledge of real functions of real variable in several problems such as function optimization, among others.

Chapter 1 – Logic

1.1 Propositions and logical operators

1.2 Propositional expressions and quantifiers

Chapter 2 – Sets

2.1 Generalities

2.2 Real number intervals

2.3 Operations with sets

2.4 Cartesian products, binary relations and functions

Chapter 3 – Matrix Calculus

3.1 Matrix definition. Generalities

3.2 Operations with matrices. Invertible matrices

3.3 Condensation of matrices. Gauss-Jordan elimination method

3.4 Systems of linear equations

3.5 Determinants

Chapter 4 – Real-valued functions of a real variable

4.1 Generalities

4.2 Composition of functions. Inverse function

4.3 Limits and Continuity

4.5 Exponential function and logarithmic function

4.6 Inverse trigonometric functions

4.7 Derivatives. Applications of derivatives

4.8 Study of functions

4.9 Taylor and Mac-Laurin polynomials

Students can choose between two assessment methods:

A) Periodic assessment - consists in the realization of two tests: Test 1 and Test 2.

Dates of realization: on dates to be announced by the curricular unit's responsable teacher.

Duration: Test 1 - 1h10m ; Test 2 - 50m.

Quotation: Test 1 - 12 values ; Test 2 - 8 values.

The final classification is obtained with the sum of the values obtained in Test 1 and Test 2.

Approval: with a minimum final classification of 9,5 values.

- The student has access to this evaluation as long as he registers a maximum of 6 absences throughout the semester. There is no place for any justifications for absences, and the student is excluded from this assessment regime if he/she exceeds the aforementioned number of absences.

- In case of missing an assessment moment, regardless of the reason, there is no place for a replacement assessment moment.

- In case of absence or withdrawal from one of the evaluation moments, the classification will be 0 (zero) values.

B) Exam evaluation - Consisting of a written test.

Duration: 2h.

Quotation: 20 values.

Approval: with a minimum classification of 9,5 values.

Observations:

1) The use of generative AI is recommended as a study support tool, contributing to a better understanding of the content taught. However, answering the questions in Test 1, Test 2, and the Exam must follow the methods taught in class; solutions using alternative methods will not be accepted.

2) In both assessment methods: In the case of well-founded suspicion of fraud detected during the exam correction phase, the student will be summoned to take an oral exam to verify the knowledge and skills demonstrated. This verification is carried out by a panel composed of the curricular unit professor and two other professors from the same area. If the student fails to prove that they possess the knowledge and skills demonstrated in the written exam, the exam will be annulled.

NAO