Mathematical Analysis I

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.

In some classes, it is intended to integrate the use of technology, using suitable software, to improve mathematical understanding and facilitate problem solving.

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;- provide students with some numerical methods for solving problems that are not always solved directly (analytically).

Skills:

– use some notions of logic and set theory that support programming and database research;

– apply mathematical knowledge of matrix calculus, namely in solving systems of linear equations;

– apply the knowledge of real functions of real variables in several problems such as the optimization of functions, among others;- solve some problems using numerical methods and interpret the numerical results obtained.

1 Logic

1.1 Propositions and logical operators

1.2 Propositional expressions and quantifiers

2 Sets

2.1 Generalities

2.2 Real number intervals

2.3 Operations with sets

2.4 Cartesian products, binary relations and functions

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

4 Real-valued functions of a real variable

4.1 Generalities

4.2 Composition of functions. Inverse function

4.3 Limits and Continuity

4.4 Exponential function and logarithmic function

4.5 Derivatives: definition, geometric interpretation and derivation rules

4.6 Applications of derivatives

4.7 Nonlinear Equations: Bisection method and Newton’s method

NAO