Base Knowledge
Differential and integral calculus of real functions of a real variable.
Teaching Methodologies
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.
Learning Results
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.
Program
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.
Curricular Unit Teachers
Internship(s)
NAO
Bibliography
Recommended bibliography
Cardoso, J. (2024). Cálculo Diferencial e Integral em IR^n. ISEC.
Guidorizzi, H.L. (2018). Um Curso de Cálculo (6ª ed., Vols. 2-4). LTC.
Larson, R., Hostetler, R.P., & Edwards, B.H (2006). Cálculo (Vol. 2). McGraw-Hill.
Rodrigues, R. (2022). Notas teóricas e exercícios de Análise Matemática. ISEC.
Santos, P. (2020). Apontamentos de MATLAB. ISEC.
Additional bibliography
MathWorks (2022). Getting Started with MATLAB.