Universitat Internacional de Catalunya


Second semester
Main language of instruction: Spanish

Other languages of instruction: Catalan, English

Teaching staff

Office hours will be agreed at ccardo@uic.es


Calculus is the mathematical study of continuous change and is a matter of great importance for technical and scientific studies. It involves the learning of concepts such as differential calculus in multiple variables and integral calculus in several dimensions, among others. The infinitesimal calculus has wide applications in science and engineering and is used to solve problems for which algebra alone is insufficient. It also forms the necessary knowledge base for several subjects of the following courses of bioengineering studies.


This course aims to familiarize the student with the concepts of differential and integral calculus in multiple dimensions, as well as provide an introduction to differential equations.


  • CB2 - Students must know how to apply their knowledge to their work or vocation in a professional way and have the competences that are demonstrated through the creation and defence of arguments and the resolution of problems within their field of study.
  • CB5 - Students have developed the necessary learning skills to undertake subsequent studies with a high degree of autonomy.
  • CE1 - To solve the maths problems that arise in the field of Bioengineering. The ability to apply knowledge of geometry, calculate integrals, use numerical methods and achieve optimisation.
  • CE17 - To be able to identify the engineering concepts that can be applied in the fields of biology and health.
  • CE4 - To have spatial vision and know how to apply graphic representations, using traditional methods of metric geometry and descriptive geometry, as well as through the application of computer-assisted design
  • CG4 - To resolve problems based on initiative, be good at decision-making, creativity, critical reasoning and communication, as well as the transmission of knowledge, skills and prowess in the field of Bioengineering
  • CG5 - To undertake calculations, valuations, appraisals, expert reports, studies, reports, work plans and other similar tasks.
  • CT3 - To know how to communicate learning results to other people both verbally and in writing, and well as thought processes and decision-making; to participate in debates in each particular specialist areas.
  • CT5 - To use information sources in a reliable manner. To manage the acquisition, structuring, analysis and visualisation of data and information in your specialist area and critically evaluate the results of this management.
  • CT6 - To detect gaps in your own knowledge and overcome this through critical reflection and choosing better actions to broaden your knowledge.

Learning outcomes

After passing the subject, students will have acquired the following skills:

  • A good command of formal mathematical language
  • A good domain of differential calculation in multiple variables
  • A good domain of multiple integrals
  • Ability to analyze and synthesize the information obtained in the course
  • Ability to interpret and solve basic differential equations
  • Ability to formulate and solve optimization problems
  • Ability to use computer calculation programs



1.1 Introduction

1.2 Univaluated calculus. Derivatives. Series. Taylor theorem.

1.3 Calculus with several variables. Directional derivative. Gradient.


2.1 Optimization with one variable.

2.2 Optimization with several variables. Hessian Matrix. Matrix sign.

2.3 Constrained optimization. Method of Lagrange multiplier.


3.1 Riemann integration definition. Integrals immediatas

3.2 Integration methods. By parts. Change of variables. Rational and irrational integration.


4.1 Multiple integration definition. Rectangular domain. Fubini theorem.

4.2 Integration with non rectangular domains.

4.3 Change of variables. Jacobian matrix.


5.1 Differential equations, definition, examples and applications. Picard theorem.

5.2 Types and methods of resolution. Separated variables, homogeneous, linear and Bernoulli equations. Laplace transform.

5.3 Applications of differential equations.

Teaching and learning activities

In person


The subject will be taught face-to-face through theoretical classes and problem solving sessions. The theory of the subject will be exposed in a rigorous way avoiding, however, an excess of formalization, which could mask the true purpose of the subject: teach the basics of infinitesimal calculus to bioengineers. For this reason, the conceptual clarity and resolution of multiple examples will be emphasized using the R software. In addition, applications of calculation tools will be shown to problems of engineering interest such as calculation of the center of gravity, or moments of inertia.

Learning activities

Master Class and resolution of exercises and problems: 60 h (Presence: 100%)

Preparation and realization of evaluable activities: 30 h (Presence: 0%)

Autonomous work of study and performance of exercises: 60 h (Presence: 0%)

Evaluation systems and criteria

In person

During the course it will be necessary to present and carry out the following evaluation tasks:

E: four deliveries consisting of works of technical application of the theory or exercises.

P: partial exam of half of the syllabus and which will take place in the middle of the course.

F: final exam of the whole syllabus.


C: continuous evaluation is the weighted average consisting of C=20%E + 20%P + 60%F.

FINAL GRADE: the final grade in the first call will result from calculating the maximum between the continuous evaluation mark C and the mark of the final exam F. However, when was less than 4, F will be the final grade. 

In the second call will remain without altering the mark of the deliveries and the mark of the partial exam. Only final exam will be repeated and the same system of evaluation as in the first call will be applied. 


Students who repeat the subject will have to deliver the 4 "homeworks" again and perform both exams.

Important considerations:

  1. Plagiarism, copying or any other action that may be considered cheating will be zero in that evaluation section. Besides, plagiarism during exams will mean the immediate failing of the whole subject.
  2. In the second-sitting exams, the maximum grade students will be able to obtain is "Excellent" (grade with honors distinction will not be possible).
  3. Changes of the calendar, exam dates or the evaluation system will not be accepted.
  4. Exchange students (Erasmus and others) or repeaters will be subjected to the same conditions as the rest of the students.


Bibliography and resources

Main references

- "Cálculo diferencial e integral" (2015). Nikolai Piskunov. Limusa

Supplement references

- "Cálculo" (2009). Robert Adams (trad. esp. de Inés Portillo García), Addison Wesley, 6ª edición. Sistemas y criterios de evaluación.

- "Start R in Calculus" (2013). Daniel Kaplan. Project Mosaic.

- "Solving Differential Equations in R" (2012). Karline Soetaert, Jeff Cash, y Francesca Mazzia. Springer Science & Business Media.