Cálculo I

Trigonometry and elementary geometry; Study of functions and their inverses; Trigonometric Functions; Differential Calculus.

The lectures use the lecture method to provide an introductory explanation of the subject, with examples through the resolution of exercises, in order to acquire basic knowledge. The rest of the classes use shared, individual, and/or group resolution of exercises, which lead to understanding and application of the syllabus, as well as specific synthesis and analysis activities.

The main objective of this course is to promote the learning of mathematical concepts so that students develop reasoning skills and the necessary competencies to understand and apply mathematics as a support tool in the different subjects of the programme. By the end of the semester, students should be able to do the following in each of the following areas: Knowledge: Describe the main results in the area of basic training in mathematical analysis, particularly in the field of differential and integral calculus and numerical series, and identify the techniques to be used in problem solving; Understanding: Develop an appropriate attitude and mindset for solving engineering problems; Application: Develop a solid foundation for further subjects, enabling the correct use of techniques and the rigorous formulation of problems.

1. Real functions of real variable – Revision – Definition and generalities; Classes of functions; Composition of functions; Inverse of a function; Elementary functions (linear, quadratic, exponential, logarithmic and trigonometric). Limits and continuity.

2. Differential Calculus on IR – Revision – Definition of the derivative of a real function of real variable, properties and rules of derivation.

3. Indefinite integration (antiderivative) of real functions of real variable – Definition and properties; Immediate antiderivatives; Antiderivatives by decomposition.

4. Integral Calculus on IR– 4.1 Definite integral – Definitions and properties; Applications of the definite integral to the calculation of plane areas, volumes of solids of revolution and lengths of arcs of plane curves. 4.2 Improper integrals – Integrals over unbounded intervals and integrals of unbounded functions.

5. Techniques of indefinite integration– Antiderivatives by substitution; Antiderivatives by parts; Antiderivatives of trigonometric functions; Antiderivatives of rational functions.

6. Series – Definition of numerical series and convergence; Necessary condition for convergence; Special series; Convergence criteria. Power series and Taylor series.

Continuous/periodic assessment - Assessment consists of two tests and assignments. The student is not allowed to use a mobile phone, calculator, or any other reference material unless expressly authorized by the course lecturer.

Components of continuous/periodic assessment:

• Test 1 - Worth 8 points (minimum of 2.0);
• Test 2 - Worth 8 points (minimum of 2.0);
• Assignments - Individual or group assignments, during classes, worth 4 points.

The final mark for continuous/periodic assessment corresponds to the sum of the marks for the assessment components, except when the student does not achieve the minimum mark in any of the assessments tests, in which case the final mark will be NRC (Not Approved).

In the normal period of exams, students may take the full exam (20 points, no minimum) or continue with continuous/periodic assessment and take one of the tests, applying the above-mentioned minimums.

In the appeal period, students must take the full exam (20 points, no minimum). During this assessment period, students who achieve a final mark of 9.0 may be asked to take an oral examination.

A student with a mark of 18 or above may be required to take an oral or written examination to defend their mark, should the lecturer so request. Supplementary oral examinations will a form of assessment to be taken into account, whenever they are deemed necessary to clarify doubts and/or in cases where there is suspicion of cheating in the assessments carried out by the students.

NAO