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. Real Numbers Rational and irrational numbers; algebraic operations with fractions. 2. Real functions of a real variable Generalities about functions (domain, range, reading domain and range from a graph, level curves). Polynomial and rational functions. Square root of a real number. Trigonometric functions. Exponential and logarithmic functions. 3. Limit Definition and properties of the limit of a function at a point. Indeterminate forms (0/0 and +∞–∞). Continuity of functions. 4. Derivative Geometric interpretation. Differentiation rules. Higher-order derivatives. Cauchy’s rule. 5. Global study of a function Monotonicity and relative extrema. Concavity and inflection points. Asymptotes. Graphs. 6. Integration Immediate antiderivative. Integration by substitution and by parts. Definite integral, fundamental theorem of calculus. Applications of integration (areas of plane figures).

All support material and relevant information will be shared with students through Moodle.   The lectures (and the notes published on Moodle) present the ideas that underpin the syllabus of this Course Unit (CU). For each topic of this CU, a set of application exercises is provided. Students are encouraged to solve these exercises as well as to bring up any questions they may have. Assessment consists of either 2 tests of 60 minutes each, or one test (Final Frequency, lasting 90 minutes), or an exam (Resit, lasting 90 minutes). (Average of the 2 tests) = A (Resit Exam grade) = B If A > 9.5, the student is approved in the CU and may sit the exam if they wish to improve their grade. In this case, the Final Grade = max(A, B). If A < 9.5, the student fails the CU and must take the exam in order to obtain approval in the CU.

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