Mathematics II

Differential and integral calculus of real functions of a real variable.

Presentation and analysis of the subject of the course unit in theoretical classes. Resolution and discussion of exercises with guidance from the teacher in theoretical-practical classes.

Knowledge of mathematics essential for understanding the subjects taught in the remaining curricular units of the degree, namely knowledge of ordinary differential equations, real functions of several real variables and their derivatives, multiple integrals and series. Ability to apply this knowledge to solve problems. Development of critical thinking and reasoning skills.

1. Differential calculus in IR^n – Conics and quadrics surfaces. Notions of topology in IRn. Real functions of several real variables: domain; contour and graph of a two-variables function; limit and continuity; partial derivatives; gradient vector; differentiable functions; directional derivative; tangent plane and normal straight line; linear approximation; free extrema; extrema with constraints and Lagrange’s multipliers method.

2. Integral calculus in IR^n – Multiple integrals: double integral (definition, properties and geometric interpretation, calculation in Cartesian and polar coordinates, examples of application); triple integral (definition, properties, calculation in Cartesian, cylindrical and spherical coordinates, examples of applications).

3. Introduction to the study of ordinary differential equations – Introduction and motivation. Ordinary differential equations: first order linear equation; Bernoulli’s equation; separable variable equation; zero degree homogeneous equation.

4. Introduction to the study of series – Brief revision about numerical sequences. Numerical series: definition; nature; properties; examples (geometric series, Dirichlet’s series and telescoping series); necessary condition for convergence; series of non-negative terms; comparison tests; integral test, root and ratio test; conditional and absolute convergence; alternating series; Leibniz’s rule. Real power series: definition; radius and interval of convergence; properties; Taylor series expansions.

Students who attend classes may receive a bonus of up to 1 point in their final grade, proportional to their actual class attendance (e.g., 80% attendance results in 0.80 points). 

Continuous and periodic assessment:

• Test 1 - 8 points (8th week of classes);
• Test 2 - 9 points (last week of classes);
• Class questions  - 3 points.

The final mark for continuous and periodic assessment is the sum of of the results of the assessment tests and class questions, with the attendance bonus; the student obtains approval if the grade is equal to or greater than 10 points.

During the normal period of exams, the student can take one of the tests to increase the grade of the continuous/periodic evaluation, or take the comprehensive exam (20 points). In the appeal period of exams, the student has to take the comprehensive exam.

Students who obtain a grade higher than 18 points must defend the grade if the teacher requests it; in this situation, the minimum grade assigned is 18.

NAO