The teaching is organized as follows:

Learning objectives

The course is divided into two modules. The Analysis 1 module aims to provide the fundamental concepts of single-variable mathematical analysis. The goal is to develop an understanding of the methods used, with a view to their applications in analysis. By the end of the course, students are expected to be able to apply basic techniques of single-variable mathematical analysis to problem-solving; apply these techniques to problems involving the concepts of function, derivative, integral, and series in various situations, including those outside strictly mathematical contexts; choose the most appropriate technique for the problem at hand; present the solution to a problem using correct terminology; and develop the skills necessary to expand their knowledge starting from the concepts learned.
In the Analysis 2 module, the concepts and techniques of differential and integral calculus for real functions of several real variables are developed. By the end of the course, students are expected to have knowledge and understanding of multivariable mathematical analysis techniques and the ability to use them to solve problems; to be able to choose which tools or theoretical results may be useful in solving a given problem; and to appropriately use the language and formalism of multivariable mathematical analysis in various engineering and modeling applications.

Program

Sets and real numbers: sets and set operations. Numerical sets. Field structure and ordered fields. Complete ordered fields. Majorities, minorities, upper and lower bounds, maximum and minimum. Principle of induction. Summations, factorials, and binomial coefficients.

Real functions of a real variable: definition of function, domain, and range. Injective, surjective, and bijective functions. Invertibility. Bounded functions. Even and odd, monotone, periodic, composite, and inverse functions. Study and graphs of the main elementary functions: powers, exponentials, logarithms, and trigonometric functions.

Numerical sequences: real sequences, boundedness, and monotonicity. Convergent and divergent sequences. Monotonicity theorem. Limit algebra. Sign permanence and comparison theorems. Remarkable and exponential sequences. Hierarchy of infinite numbers.

Limits of functions: definition of limit, neighborhoods, and accumulation points. Finite and infinite limits at the finite and at the infinite. Right and left limits. Nonexistence of limits. Horizontal, vertical, and oblique asymptotes. Special limits. Techniques for calculating limits. Continuity: continuity of real functions. Continuity of elementary functions and the algebra of continuity. Continuity of composite and inverse functions. Fundamental theorems: zero, intermediate value, and Weierstrass theorems.

Differentiation: incremental ratio and derivative. Geometric interpretation and tangent line. Derivatives of elementary functions and rules of differentiation. Chain rule. Points of non-differentiability. Fermat's and Lagrange's theorems. Study of monotonicity and local extrema. Convexity and second derivative. L'Hôpital's theorem. Study of functions.

Integration: definite integral and its properties. Integrability of continuous and monotone functions. Integral mean theorem. Primitives and antiderivatives. Fundamental theorem of integral calculus. Integration techniques: substitution, integration by parts, and integration of rational functions. Improper integrals.

Ordinary differential equations: first-order differential equations with separable and linear variables. Cauchy problem and growth models. Solution space. Second-order linear differential equations with constant coefficients. Structure of the general integral. Independence of solutions and the Wronskian.

Learning assessment procedures

The exam is written and includes a barrier test, designed to assess understanding of the course's fundamentals and mastery of basic techniques. Passing the barrier test is a prerequisite for admission to the next exam.