Academic management

Contextualization

The course Infinitesimal Calculus forms part of the subject Mathematics of the basic learning module, which is generic for all Bachelor’s Degrees in Industrial Engineering. At the same time, it is similar to the course given under the same name in all other Bachelor’s Degrees in Engineering. Because of its fundamental nature, the material of this course is necessary in the other Bachelor’s Degree modules.

The aim of the course is to develop students’ ability to solve mathematical problems appearing in engineering, as well as their capability to transfer and apply the acquired knowledge and skills in different situations which might emerge in course of learning and the further professional activity.

Requirements

The student is supposed to be proficient in the high school courses Mathematics I and II. In case that another way of access to the University has been chosen, an equivalent previous mathematical preparation is required.

Special competences

Be able of solving mathematical problems appearing in engineering. Be capable of applying knowledge of: Differential and Integral Calculus (the competence denoted by CB1).

            General and transversal competences

Grades of Industrial Branch (Electricity, Industrial Electronics and Automatics, Mechanics, Industrial Chemistry and Industrial Technologies)

CG3: Proficiency in basic concepts and technologies, which would provide the capacity to master new methods and theories as well as the ability of adaptation to new situations.

CG4: Capability of being initiative in solving problems and taking decisions, creativity and critical thinking.

CG5: Be able to communicate and to transmit to any audience, orally or in written, the knowledge and skills of Industrial Engineering.

CG14: Honesty, responsability, ethical commitment and the spirit of solidarity.

CG15: Capability of the team work.

Results of learning

RA1: Operations with functions of one real variable and their representation, calculus of limits, analysis of continuity, calculation of derivatives and resolution of optimization problems.

RA2: Handling the concepts of sequences and series, application of power series in representation of functions.

RA3: To pose and calculate integrals of functions of one real variable, their application to solving problems related to engineering.

RA4: To know and to be able to apply the basic properties of functions of several real variables. Calculus of limits, analysis of continuity and differentiability, resolution of optimization problems.  

Contents:

BLOCK 1: REAL FUNCTIONS OF ONE REAL VARIABLE 

Topic 1: Sets and functions. Natural numbers. The method of induction. Real numbers. Absolute value of a real number. Elementary functions. Superposition of functions and the inverse function.

Topic 2: Limits and continuity. Definition of limit. Properties. Infinitesimals and infinites. Indeterminations. Asymptotes. Continuous functions. Properties of continuous functions: theorems of Bolzano, Darboux (the mean value) and Weierstrass.

Topic 3: Derivability. Properties of derivable functions. Derivatives of a function at a point.The derivative function. Differentiability and continuity. Properties of the derivative. The chain rule. Rolle’s theorem. The mean value theorem of Lagrange. L’Hôpitale’s rule.

Topic 4: Taylor’s polynomials. Sequential derivatives. Taylor’s polynomial. Taylor’s formula for the remainder.

Tema 5: Optimization. Analysis of local behavior of a function. Monotonicity, concavity and the inflection points. Absolute extrema. Graphical representation of functions. 

BLOCK 2: THE RIEMANN INTEGRAL

Topic 1: Calculation of primitives. Immediate integration. Methods of integration.

Topic 2: Definite integral. Basic concepts and geometrical interpretation. Integrable functions. The fundamental theorem of the integral calculus. Barrow’s rule. Applications.

Topic 3:  Improper integrals. Definition of the improper integral. The types of improper integrals. Application to the study of Euler’s integrals.

BLOCK 3: SEQUENCES AND SERIES. POWER SERIES.

Topic 1: Numerical sequences. Definition. Convergence. Calculus of limits.

Topic  2: Numerical series. Definition. Convergence and sum. Harmonic and geometrical series. Criteria of convergence.

Topic  3: Power series. The power series expansion. Definition. Radius of convergence. Expansion of a function into a power series: Taylor’s series. Frequently used expansions. 

BLOCK 4: FUNCTIONS OF SEVERAL VARIABLES

Topic 1: Euclidean space Rn. Basic concepts of topology in Rn. Real functions. Vector-valued functions.

Topic 2: Limits and continuity. The limit of a function at a point. Basic properties. Calculus of limits. Continuity of a function. Properties.

Topic 3: Derivability. Directional derivative. Partial derivative. Geometrical interpretation. Higher-order derivatives. Derivability and continuity.

Tema 4: Differentiation. Differential of a function at a point. Lineal approximation. Sufficient condition of differentiability. The gradient vector. Tangent plane. The chain rule.

Tema 5: Optimization. Local extrema without constraints. A necessary condition of existence. Absolute extrema. Local extrema with constraints. Multipliers of Lagrange.

Metodology and working plan

As an exception, if specific sanitary conditions require so, the non-presential teaching activities can be employed. In such a case, the students will be informed of these changes.

Evaluation of student learning

THE ORDINARY CALL.

(i) In course of the semester, two partial controls or one final exam will be held. The date and time of the first partial control will be announced well in advance. The second one will coincide with the final exam of the January call. The final exam will be held on the date fixed for the January call and will embrace the whole of the course material. The final score of the written controls will be either the arithmetic mean of the grades for the two assessments, or the grade of the final exam.

(ii) The evaluation of the classroom practices and Lab practices will be held during the corresponding practice sessions.

(iii) The final grade (FG) of the course will be

FG=0.15*LG+ max (0.15*PG+0.7*EG, 0.85*EG),

being LG the grade of the Lab practices, PG the grade of the classroom practices, and EG the grade of the written control (the arithmetic mean of the grades of the two partial controls, or the grade of the final exam). In case that a student's EG does not attain 3,5 points, the student fails the call with the final score EG.

To those students who were granted the right of the differentiated assessment, the following model of evaluation will be applied:

1. The students will participate in the written controls in the same way as the rest of the students, as stated in session (i).
2. An effort will be made to find out a Labs group these students could belong to in order to be evaluated.  If this is impossible, the students will be assessed of the Labs in a final Labs exam. 
3. To these students the score for the practices will not be applied, so that the weight of the controls of item (a) will be always 85% and the weight of the Labs will be always 15%. This means that the final grade will be

FG = 0.15*PA+ 0.85*EG,

where PA is the grade for the Labs and EG is the final grade for the written assessments, each of them of 10 points.

EXTRAORDINARY CALLS:

1. A written assessment will be done over the whole of the course material. The Lab practice will be evaluated on the day of the writen exam. A student has the choice to be newly evaluated for the Lab practice, or to conserve the grade for the Lab practice obtained during the lecture period. The students who decide to make the new assessment renounce the grade obtained during the lecture period and their grade for the Lab practice will be the obtained on the new assessment.
2. The final grade is calculated by means of the same formula as in the ordinary call, where now EG stands for the grade of the unique written assessment. In case that a student's EG does not attain 3,5 points, the student fails the call with the final score EG.

• The students who were granted the right of differentiated assessment will be evaluated acording to the same model than the others.
• As an exception, if specific sanitary conditions require so, the non-presential evaluation methods can be employed. In such a case, the students will be informed of these changes.

Exceptionally, and depending on public health conditions, remote online assessment methods could be used. In this case, the students will be informed of the changes.

Resources, bibliography and complementary material

Resources:

The classroom endowed with a PC and a projector for the lecturer.

The computer classrooms for the Labs.

Virtual Classroom of the University of Oviedo.

Basic bibliography in Spanish:

Bradley G. L.; Smith, K.J.  Cálculo de una variable y varias variables. (Vol.  I y II). Prentice Hall (4ª ed.), 2001.

García López, A y otros. Cálculo I: teoría y problemas de análisis matemático en una variable, CLAGSA (3ª ed.), 2007.

García López, A y otros. Cálculo II: teoría y problemas de funciones de varias variables. CLAGSA (2ª ed.), 2002.

Stewart, J. Cálculo de una variable y Cálculo multivariable. Paraninfo Thomson. (6ª ed.), 2009.

Complementary bibliography:

Bayón  L, Grau J. M., Suárez P.M. Cálculo. Grados en Ingeniería. Ediciones de la Universidad de Oviedo. EDIUNO,  2011

Burgos Román, J. Cálculo Infinitesimal de una variable y en varias variables. (Vol. I y II).  McGraw-Hill. (2ª ed.), 2008.

Larson, R. E. y otros.  Cálculo y geometría analítica. (Vol. I y II).  McGraw-Hill (8ªed.), 2005.

Marsden, J.; Tromba, A.Cálculo vectorial. Addison-Wesley Longman (5ªed.), 2004.

Neuhauser, Claudia. Matemáticas para ciencias. Pearson. Prentice Hall, 2004.

Tomeo Perucha, V. y otros. Problemas resueltos de Cálculo en una variable. Thomson, 2005.

Bibliography in English:

Trench,W.F. Introduction to real analysis, Pearson Education, 2003

http://ramanujan.math.trinity.edu/wtrench/texts/TRENCH_REAL_ANALYSIS.PDF

Piskunov, N. Differential and Integral Calculus, MIR, 1969
http://es.scribd.com/doc/132055961/N-piskunov-DifferentialAndIntegralCalculus1969mir

Strang, G. Calculus, R.R.Donnelley & Sons 1992
http://ocw.mit.edu/ans7870/resources/Strang/Edited/Calculus/Calculus.pdf

Craw, I. Advanced Calculus and Analysis, University of Aberdeen, 2000
http://es.scribd.com/doc/57488172/Advanced-Calculus-and-Analysis-Ian-Craw