Calculus I

High school mathematics knowledge.

The curricular unit assumes an essentially formative character, with a primary focus on the systematic monitoring of the learning process, the provision of constructive and timely feedback, and the enhancement of the student’s academic development. The teaching methodologies employed extend beyond the exposition and critical discussion of programme content, privileging instead an exploratory and student-centred approach. This approach is operationalised through individual or pair-based work undertaken within the classroom setting, oriented towards the resolution of exercises and structured tasks under the close supervision and academic guidance of the lecturer. At specific stages, such tasks may necessitate the integration of digital technologies for purposes of computation, investigation, and representation, including calculators (notably graphical models) as well as specialised software applications such as GeoGebra.

By the end of this course unit, students should be able to:

Think and reason mathematically about real functions of one and two variables, understanding limits, continuity, derivatives and integrals, as well as their fundamental properties.

Model mathematically engineering phenomena using elementary functions (trigonometric, exponential, logarithmic, hyperbolic) and techniques of differential and integral calculus, applying them to problems such as the calculation of areas, volumes and rates of change.

Represent and manipulate mathematical entities both graphically and symbolically, through functions of one and two variables, derivatives, integrals and primitives, supported by the use of digital tools (GeoGebra, scientific calculator).

Pose and solve mathematical problems involving differentiation, integration, anti-differentiation and multivariable calculus, combining analytical techniques with practical applications in engineering.

Communicate in, with and about mathematics, presenting reasoning, calculations and interpretations clearly and systematically in academic and technical contexts.

Make use of digital and technological tools (GeoGebra and scientific calculator) critically and responsibly, exploring graphical and numerical simulations to validate results, support intuition and reinforce the connection between theory and practice.

1. Real function of a real variable
Properties of real functions of a real variable, limit and continuity, trigonometric functions and inverse trigonometric functions, exponential function, logarithmic function and hyperbolic functions.

2. Differential calculus
Derivative, properties, derivative of the composition function and the inverse function, theorems of Rolle and Lagrange, undetermined forms and Cauchy’s rule, polynomial approximation: differentials.

3. Primitive of real functions of a real variable
Techniques for calculating the primitive function.

4. Integral calculus
Definite Integral, properties, fundamental theorem of calculus, integration by parts and by substitution, applications of the definite integral: area of a plane region, volume of a solid of revolution and length of the arc of a curve; improper integrals: Integrals at unlimited intervals.

5. A introduction to calculus with real functions of two ​​real variables
Real function of two ​​real variables, domain, range, level set and graph, first order partial derivatives, double integral in type I and type II regions, applications of the double integral.

Continuous/Periodic Assessment

      The student may undertake, during class, 2 tasks under the lecturer’s supervision. These tasks will have a total maximum grade of 4 points;

       and

      The student may sit an examination at any of the scheduled examination periods listed in the examination calendar, worth 16 points.

      The first season examination shall consist of two parts, graded 6 and 10 points respectively.

      In the second season, the student may either sit the entire examination or only one of the two parts undertaken in the first season.

      The student is approved if he obtain a grade greater than or equal to 9.5.

Assessment by Final Exam

     Exam (Theoretical-Practical, written), rated for 20 points, to be taken on the exam dates especified in the ISEC school calendar. 

      The first season examination shall consist of two parts, graded 8 and 12 points respectively.

      In the second season, the student may either sit the complete examination or one of the two parts undertaken in the first period.

      The student is approved if he obtains a grade greater than or equal to 9.5.

      To sit the first season examination, it suffices for the student to be enrolled in the course unit.

      For the second season, registration in the Inforestudante system is compulsory.

Important Remark

     The teacher reserves the right to supplement the assessment  with an oral test, of which students will be notified at least 48 hours in advance.

     Grades above 18.0 values ​​may require an additional oral exam.

NAO