Differential and Integral Calculus I

Objectives

Master concepts and techniques of differentiable and integral calculus in one variable. Develop analytic thinking, creativity and innovation capacity, through the application of those concepts and techniques in different contexts.

Program

Real numbers: algebraic, order and supremum axioms. Natural numbers and mathematical induction. Sequences: the concept oflimit; applications. Real functions of one real variable: limits and continuity; elementary functions. Global properties of continuous functions: intermediate value and Weierstrass theorems. The concept of derivative. Derivatives of elementary functions. Rolle, Lagrange and Cauchy theorems. L'Hôpital's rule. Derivatives of higher order. Inverse functions.
Primitives: parts, substitution, rational functions. Riemann's integral. Fundamental Theorem of Calculus. Barrow's rule. Applications: calculation of areas; definition of functions (ex.: logarithm, error and gamma functions); examples of separable differential equations of the form f(y) y’(t) = g(t). Taylor's polynomial. Numerical series. Convergence criteria. Simple and absoluteconvergence. Power series, convergence radius. Taylor series: definition, examples and convergence.

Teaching Methodologies

The teaching methodologies aim to promote learning based on problem solving and projects, reinforcing the practical component, active learning, autonomous work and student accountability. The assessment model incorporates exam/tests, possibly with minimum grade, complemented with continuous evaluation components and oral evaluation for grades above 17 (out of 20).

Bibliography

* Calculus, M. Spivak, 2006, 3rd Edition, Cambridge University Press;
* Introduction to Real Analysis, W. Trench, 2009, (free edition), Trinity University;
* Aulas teóricas de Cálculo Diferencial e Integral I, M. Abreu e R. L. Fernandes, 2014, DM-IST;
* Cálculo Diferencial e Integral I, M. A. Bastos e A. Bravo, 2010, (texto de apoio às aulas);
* Introdução à Análise Matemática, J. Campos Ferreira, 2018, 12ª edição, Gulbenkian;
* A First Course in Real Analysis, M. H. Protter e C. B. Morrey, 1993, Springer-Verlag;
* Calculus, J. Stewart, 2015, 8th edition.