Mathematical Analysis 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. Interpretation and exploration of concepts and resolution of exercises using MATLAB programming language and GeoGebra software in practical classes.

Classes presented in Portuguese.

Study of differential calculus and integral calculus of real functions of several real variables, numerical series and series of functions. Interpretation of concepts and resolution of exercises using the MATLAB programming language and GeoGebra software.

Rigor in the interpretation, use and description of the mathematical concepts studied. Analysis and resolution of problems using software.

1. Curves and surfaces
Conics, polar coordinates, quadric surfaces, cylindrical coordinates, spherical coordinates, parametric coordinates

2. Differential and integral calculus in IR^n
Brief notions of topology in IRn. Real functions of several real variables – domain, level set and graph of a two-variable function, limit and continuity, partial derivatives and gradient vector, partial derivatives of higher order, differentiable function, directional derivative, tangent plane and normal straight line, linear approximation, maxima, minima and saddle points, extrema with constraints, Lagrange’s multipliers. Double integral – properties, geometric interpretation and calculation, applications of the double integral, double integral in polar coordinates.

3. Numerical series and function series
Properties of numerical sequences. Numerical series – nature and properties, Dirichlet series, geometric series and telescoping series, necessary condition for convergence, series of non-negative terms, tests for convergence, integral test, root and ratio test, conditional and absolute convergence, alternating series, Leibniz’s rule. Real power series – radius and interval of convergence, properties of functions represented by power series, Taylor series, power series expansions. Fourier series.

Students will be able to choose two alternative assessment models: continuous/periodic assessment and assessment by final exam. The student will have to choose the model they want immediately after taking the 1st test.

Continuous/periodic assessment
Three tests will be carried out:

1st Test: quotation of 8 points, to be carried out in the middle of the semester on a date to be defined; This is a written test covering the subjects in Chapter I and part of Chapter II;

2nd Test: quotation of 8 points, to be carried out at the end of the semester, before the normal season, on a date to be defined; This is a written test covering the subjects of part of chapter II and chapter III;

3rd Test: quotation of 4 values, to be carried out at the end of the semester on a date to be defined; In this test, the computer and MATLAB software will be used.

To obtain approval for the subject, the sum of the classifications of the three tests must be greater than or equal to 9.5 points. The student who opts for continuous/periodic assessment waives the normal period.

Assessment by final exam

In the normal season there will be a test on all the subjects of the T and TP components, rated for 16 points; there will be no MATLAB test, which will be carried out before the start of this season (3rd Test).

At the time of the appeal, there will be a test on the entire subject of the T and TP components, rated for 16 points, and a MATLAB test rated for 4 points. At this time, students enrolled in continuous/periodic assessment may choose to take an exam on the entire subject or a test corresponding to the 1st test or the 2nd test and/or the 3rd test.

Students with a classification greater than or equal to seventeen points, or whose resolution of the tests raises doubts regarding their authorship, may be required to take an oral test to defend their grade.

NAO