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).

• 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