Calculus - LEB


Biomedical Engineering

Curricular Unit (UC)


Mandatory  x
Scientific Area MAT Category  

Course category: B - Basic; C - Core Engineering; E - Specialization; P - Complementary.

Year: 1st Semester: 1st ECTS: 6.5 Total Hours: 172
Contact Hours T: 45 TP: 30 PL: S: OT:4
Professor in charge

 Filipe Cal

T - Lectures; TP - Theory and practice; PL - Lab Work; S - Seminar; OT - Tutorial Guidance.

  • Learning outcomes of the curricular unit

    After the student is approved, he should be able to:

    understand and use the concepts of differential calculus needed to the study of real functions of one real variable;

    know how to use the integration methods;

    understand and apply the notions of integral calculus and, in particular, the fundamental theorem of Calculus;

    know how to use series tests and obtain power series expansions of some functions;

    develop reflection, calculus and deductive reasoning capacities;

    develop analytical and critical capacity.  

  • Syllabus

    Real functions of a real variable. Topology. Limit and continuity. Derivative. Differentiation rules. The mean value theorem. Monotony intervals, maxima and minima. Taylor’s formula. Concavity and inflection points. L’Hospital’s rule and indeterminate forms. Asymptotes.

    Primitive functions. Direct integration and methods by decomposition, parts and substitution. Integration of rational functions.

    Integral calculus. The Riemann integral. Mean value theorem. Indefinite integral. The fundamental theorem of Calculus. Barrow’s formula. Integration by parts and substitution. Improper integrals.

    Infinite series. The geometric series. Telescoping series. The divergence test. The integral and Cauchy tests. The Dirichlet series. First and second comparison tests. Alternating series. Leibniz’s test. Conditional and absolute convergence. The ratio, root and Raabe tests.

    Power series. Interval and radius of convergence. Derivation and integration. Taylor series. Power series expansions of functions.

  • Demonstration of the syllabus coherence with the curricular unit's objectives

    The goals are met within contents of Chapters of the syllabus, in which analysis, algebra and deductive reasoning skills are widely developed.

    In addition to the applications studied in each chapter, the systematic use of applied and contextual problems yields increase of motivation, efficiency and spectrum of learning, since they enable:

    to convey the fact that the differential and integral calculus in IR is an indispensable tool in the study of engineering;

    to practice the mathematical formulation of problems, their solution and criticism;

    to enable a direct experience in mathematical formalization of problems and their solution;

    to formulate conjectures and to construct, evaluate, modify, and interpret physical models;

    to help students to recognize the concepts and techniques studied when they are met in the study of other courses.

  • Teaching methodologies

    Teaching methodologies:

    Theoretical lectures based on applied examples and theoretical-practical classes in which theoretical and practical problems are solved. Special emphasis is given to problems connecting the tools developed with concepts which are important in engineering-related courses. Exercises sheets are available for an effective monitoring and strengthen of the knowledge.


    The assessment comprises two alternative components: a continuous and an exam assessment. Continuous assessment consists of two partial written tests and the assessment by exam consists on one written examination.

    Continuous evaluation:

    Two partial written examinations (T1 e T2). The student is approved with an average grade (NF) of at least 10 values and with a minimum grade of 8 values at each partial examination. 

    NF = (T1 + T2) / 2 and NF >= 10, T1>=8 and T2>=8.


    Final exam evaluation:

    One final written examination (EF). The student is approved with a final grade (NF) of at least 10 values.

    NF = EF >= 10

  • Demonstration of the coherence between the teaching methodologies and the learning outcomes.

    Theoretical lectures are essential to a correct and comprehensive coverage of all topcis of the syllabus, while in-class solution of exercices allows for a successful application of the theorectical knowledge to practical problems.

    By their organization, contents and diversity in the degree of difficulty, the exercises sheets allow students to closely monitor all topics of the syllabus and are the main tool regarding

    individual study. The exercises that constitute them are suited for the development of algebra skills and deductive reasoning.

    Since the success in mathematics is not compatible with pre-assessment study on its own, it is essential to implement processes to avoid this inclination. The usage of exercises sheets requires students to closely monitor the progress of the syllabus. It is crucial to implement some processes other than a pontual study to have a sucessful mathematical study. The usage of exercises sheets contributes to follow better the topics of the syllabus. When confronted with less straightforward problems, students are led to question and deepen their knowledge while acquiring work and independence skills. This type of problems is also suitable for the development of analysis, reflection and criticism skills. Furthermore, group dynamics can encourage debate and support between students during lectures, which lead to better results than those achieved solely by individual study. Some control is made to the exercices sheets to improve their correct usage.

  • Main Bibliography

    J. C. Ferreira, Introdução à Análise Matemática, Fundação Calouste Gulbenkian, 8ª ed., 2005.

    T. Apostol, Calculus, volume I, Editorial Reverté,1994.