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.

NAO