Class Mathematics I

This subject comprises the fundamental methods in Calculus associated to real functions in one variable. It is a fundamental area in general engineering courses and other scientific areas. It basically comprehends the classical methods in differential and integral calculus

Not applicable

1. Sequences Bounded. Monotony. Limit 2. Series: Mengoli and Geometric.  3. Real Functions of a Real Variable: Generalities about functions. Polynomial and rational functions. Trigonometric functions. Exponential and logarithmic functions. 4. Limit: Definition and properties of function limits. Continuity of functions. 5. Derivative: Geometric interpretation. Rules of differentiation. Higher-order derivatives. 6. Global Study of a Function: Monotony and relative extrema. Concavity and inflection points. Asymptotes. Graphs. 7. Integration: Immediate primitive. Integration by substitution and by parts. Definite integral, fundamental theorem of calculus. Applications of integration (areas of planar figures).

In class, the ideas underlying the curriculum of this course are discussed, and multiple examples and application exercises are analyzed. For each topic in this course, a set of application exercises is presented. Students are encouraged to solve these exercises and to raise any doubts they may have. All supporting materials and relevant information will be shared with students through Moodle. The assessment includes a continuous component, which involves completing three 40-minute tests and a final exam (Final Exam or Make-up). The average of the three tests is denoted as A, and the Exam Grade is denoted as B. If A > 9.5, the student is approved for the course and can take the exam to improve the grade. In this case, the Final Grade = max(A, B). If A < 9.5, the student is not approved for the course and must take the exam to obtain approval. Students who achieve a final grade of at least 10 points are considered approved.

Lages Lima, E.; Análise Real, Vol.I (6ª ed.), Col. Matemática Universitária, IMPA, Rio de Janeiro, 2002. Sárrico, C.; Análise Matemática – Leitura e exercícios, Col. Trajetos Ciência 4, Gradiva, Lisboa, 1999. Apostol, Tom M.; Cálculo Vol.I (2ª ed.),  Reverté,  1994    ISBN 9788429150155 Guerreiro, J.S.; Curso de Análise Matemática, Escolar Editora     ISBN 9789725922224