Calculus I

Objectives

Linear Algebra and Differential Calculus play an important role in Mathematics and its applications. Besides its informative purpose, giving students ground foundations on the subject, the teaching of this discipline also has a formative purpose. It provides students the ability to present mathematical argumentations and proofs, as well its applications to various areas.

In Linear Algebra the student should be able to: solve systems of linear equations, perform matrix operations and calculate determinants with applications. Numerical Analysis will allow students to apply numerical methods in solving systems.

The chapter on Differential Calculus starts by revisiting some notions in the field of real functions of a real variable, in particular, algebraic, trigonometric, exponential, and logarithmic functions. It is intended that students understand the concept of derivative at a point and its geometric interpretation, calculate derivatives of functions, and some of its applications.

Program

Resolution of systems of linear equations: Gaussian elimination; matrix algebra, discussion of systems of linear equations, matrix inversion; determinants and its applications: matrix inversion, Cramer's rule; eigenvalues and eigenvectors; iterative methods for solving linear systems: Jacobi and Gauss-Seidle methods.

Functions: Mathematical models, linear models and rates of change; some function transformations; function composition.

Differential calculus: definition of derivative at a point and its geometric interpretation, derived function, derivative of trigonometric functions, higher order derivatives; Rolle, Lagrange, and Cauchy theorems, indeterminations; study and graphical representation of functions; Taylor’s Formula and differentials.

Teaching Methodologies

Practical classes aim at solving exercises, promoting teacher/student interaction.

Course materials are made available through the Moodle e-learning platform, boosting up asynchronous activities.

Bibliography

Lay, David C., Álgebra Linear e suas aplicações, LTC - Livros Técnicos e Científicos S.A., 2ª Edição, 1999.

Leon, Steven J., Álgebra Linear com aplicações, LTC - Livros Técnicos e Científicos S.A., 4ª Edição, 1999.

Monteiro, António, Pinto, Gonçalo e Marques, Catarina, Álgebra Linear e Geometria Analítica, Problemas e Exercícios, McGraw-Hill, 1997.

Lipschutz S, Álgebra Linear, Schaum's Outline Series, McGraw-Hill, 1994.

Barroso, Leônidas Conceição, Cálculo numérico com aplicaçöes, Ed. Harbra, Säo Paulo, 1987.

Apostol,T., Calculus, Vol 1, John Wiley and Sons, New York, 1976.

Swokowski, E. W., Cálculo com Geometria Analítica, McGraw-Hill, 1983.

Demidovitch B. ,Problemas e Exercícios de Matemática, Mir, 1986.

Piskounov, N. Cálculo Diferencial e Integral, Vol.I, Lopes da Silva Editora, 1988.

Howard Anton, Cálculo um novo horizonte, vol. 1, 6ª Edição, Kookman, 2000.

James Stewart, Cálculo, vol. 1, 5ª Edição, Thomson, 2008.