Mathematics I

The classes for this curricular unit are of a theoretical-practical nature.
Initially, the classes will have a more expository character, aiming to establish the theoretical foundations of the topics covered, followed by
the practical component, which involves solving proposed problems and exercises. The presentation and exploration of techniques and
concepts will be supported, whenever possible, by appropriate software to enhance understanding of the topics.
In parallel, applications of theoretical concepts to the fields of social and human sciences will be presented, with a particular focus on
economics and management.
ME1. Expository: Presentation of theoretical concepts with illustrative examples.
ME2. Participatory: Analysis, discussion, and resolution of proposed exercises.
ME3. Active: Solving tests, exams, and group work.
ME4. Autonomous Work: Reading and understanding the recommended bibliography and solving exercises provided by the instructor.

Specific learning objectives:
OA1. Understand how the concept of a limit arises in the solution of various problems, particularly the notion of a derivative as a special limit
and an essential tool for solving problems of rate of change, function approximation, and optimization.
OA2. Comprehend the theoretical foundation of the numerical methods under study; apply them to approximate roots and extrema of a
function, analyzing their accuracy and efficiency; acquire skills in solving application problems.
OA3. Understand the definite integral, a basic concept of integral calculus, and its connection to differential calculus; learn integration
methods and how to use the integral to calculate areas between curves or determine consumer surplus.
OA4. Recognize a differential equation and its importance in the mathematical modeling of real phenomena; know how to solve separable
and first-order linear differential equations and apply them to solve problems in the field of study.

1. Differential Calculus in IR
1.1. Preliminary concepts
1.2. Limits and continuity
1.3. Derivatives and rates of change, the derivative function
1.4. Chain Rule
1.5. Implicit differentiation
1.6. Linear approximation and differentials; Taylor polynomials
1.7. Applications of differentiation: extrema, Rolle’s Theorem, Mean Value Theorem, optimization problems
2. CP2. Numerical Methods
2.1. Bisection method: approximation of zeros and extrema
2.2. Newton’s method: approximation of zeros and extrema
3. Integral Calculus in IR
3.1. Introduction to integral calculus
3.2. Antiderivatives
3.3. Definite integral, Fundamental Theorem of Calculus
3.4. Techniques of integration
3.5. Numerical integration
3.6. Improper integrals
3.7. Applications of integrals: areas, applications to management and economics
4. Differential Equations
4.1. Differential equations with separable variables
4.2. First-order linear differential equations.

NAO