Subject

Calculus

  • code 12477
  • course 1
  • term Semester 2
  • type FB
  • credits 6

Module: FUNDAMENTALS

Matter: MATHEMATICS I

Main language of instruction: Spanish

Other languages of instruction: Catalan, English

Timetable
group M
 Sem.2  TU 12:00 14:00 P2A01
 Sem.2  TH 10:00 12:00 P2A03

Teaching staff

Head instructor

MsU Franco Fabian AMIGO - ffamigoa@uic.es

Other instructors

MsU Juan Luis MELERO - jmelero@uic.es

Office hours

Students can arrange a face-to-face meeting with the teacher by writing to dfernandezm@uic.es

Introduction

Calculus (infinitesimal) 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.

Objectives

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.

Competences / Learning outcomes of the degree programme

  • 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 of the subject

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 (in particular the statistical software R)

Syllabus

Topic 0. Preliminaries.

Review of derivatives and integrals in 1 dimension.

Topic 1. Differential calculus in several variables

1.1 Functions of several variables

1.2 Partial derivatives of the function of several variables

1.3 Derivatives of compound functions and derivatives of implicit functions

1.4 Directional derivatives. Gradient.

1.5 Extremes of a function of several variables

1.6 Advanced topics of practical interest: a) Lagrange multipliers b) least squares method

Topic 2. Comprehensive calculus in multiple dimensions

2.1 Preliminaries: integrals in a variable

a) integration methods b) engineering and geometric applications

2.2 Introduction to double integrals

2.3 Calculation of areas and volumes by double integrals

2.4 Change of variables in a double integral

2.5 Triple integrals

2.6 Change of variables in a triple integral

2.7 Engineering applications of multiple integrals (center of gravity, moments of inertia, density of matter)

2.8 Numerical integration

Topic 3. Differential equations

3.1 Preliminary: approach to the problem. Definitions.

3.2 Differential equations of the first order

3.3 Equations of separated and separable variables

3.4 Homogeneous first order equations

3.5 First order linear equations

3.6 Differential equations of second order that are reduced to the first order

3.7 Some differential equations of engineering importance

Teaching and learning activities

In person

Methodology

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 computerization, 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

The subject is approved making 4 deliveries and 2 written exams.

The 4 deliveries (homework) will involve problems to solve. Homework delivered with a delay of up to 3 days after the deadline will be evaluated with half the maximum grade. Otherwise (more than 3 days late) will not be evaluated. It will be obligatory to deliver the "homeworks" on paper or scanned and uploaded in the corresponding space of the intranet of the subject. The homeworks sent by email will not be evaluated.

The written exam will consist of two sessions held on different dates. The first exam will be done towards the middle of the course. The second exam will be held at the end of the course and will include the syllabus of the whole subject. The exams will not allow the consultation of books or notes.

The following weighting will be applied to evaluable activities:

1. Problem solving/deliveries (weighting: 20%)

2. Partial examination (weighting: 20%)

3. Final exam (weighting: 60%)

Second call

Students who fail the subject will have the opportunity to repeat the final exam. The notes of the first partial and of the deliveries will remain unchanged.

Re-sit

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

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.

 

 

Evaluation period

E: exam date | R: revision date | 1: first session | 2: second session:

  • E1 28/05/2020 10:00h
  • E2 23/06/2020 12:00h
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