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.

1. Mathematical Logic:

• Designation and proposition. Logical value of a proposition. Conjunction and disjunction of propositions. Set, intersection, and union of sets.

2. Real Numbers:

• Rational and irrational numbers. Order and absolute value.

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).

• At the same time new ideas are presented to the students,  applications of such ideas or their geometric interpretation will be discussed.
• Students are encouraged to actively participate in the class and engage  in the discussion of the presented concepts. Illustrative examples and counterexamples are provided.
• In the theoretical-practical classes, students are invited to analyze and solve problems involving the concepts presented in the theoretical classes.
• The continuous assessment consists of:
  • 3 intermediate tests (A = average of the 3 intermediate tests) and
  • a final test covering the entire subject (B = grade of the final test).
• The final grade (NF) in the course is calculated as follows:
  • NF = 0.3*A + 0.7*B             if A > B and B >6
  • NF = B                                 if A < B or B <6
• In the 2nd and the 3rd stages of  evaluation, approval is obtained with a minimum grade of 9.5 points in the written test.

• 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