Public University of Navarre

CastellanoEuskara | Academic year: 2019/2020 | Previous academic years:  2018/2019  |  2016/2017  |  2015/2016 
Bachelor's degree in Agricultural, Food and Rural Environment Engineering at the Universidad Pública de Navarra
Course code: 501101 Subject title: MATHEMATICS I
Credits: 6 Type of subject: Basic Year: 1 Period: 1º S
Department: Estadística, Informática y Matemáticas

Partes de este texto:


Module/Subject matter

Mathematics/ Mathematics I.



Differential and integral calculus of a single variable, linear algebra.

Area: Applied mathematics.


General proficiencies

  • CB1: Students are able to demonstrate they have acquired knowledge and understanding in a field of study based on the basic foundations gained within their general secondary education together with the support of advanced textbooks and aspects of the latest advances in the field.
  • CB5: Students can develop the necessary learning skills to undertake further studies with a high degree of autonomy.


Specific proficiencies

  • CE1: Ability to solve mathematical problems in engineering. Ability to apply theoretical knowledge on: linear algebra; geometry; differential geometry; differential and integral calculus; differential equations and partial differential equations; numerical methods, numerical algorithms, statistics and optimization.
  • CG2: Adequate knowledge of the physical problems, technologies, equipment, and water and energy supply systems, the limits imposed by budgetary factors and building regulations, the relationships between installations and/or buildings with farms, agro-food industries and spaces related to gardening and landscaping with their social and environmental surroundings, as well as the need to relate those surroundings from that environment with human needs and environmental protection.



Learning outcomes

  • (R1) Work with elementary functions (polynomials, rational, trigonometric, logarithms, exponentials,...), know their elementary properties and have an idea of the concepts of limit, continuity and differentiablility.
  • (R2) Work with analytic expressions and simplify and/or find apropriate bounds for them.
  • (R3) Solve approximation problems: Taylor, least squares, interpolation,...
  • (R4) Identify and compute the elementary forms of integrals in one variable; work with their physical and engineering applications and calculate volumes and areas.
  • (R5) Know the theoretical foundations and the direct algorithms for the resolution of algebraic linear systems.
  • (R6) Dominate the diagonalization of matrices: eigenvalues and eigenvectors.


    ENAEE-1: Knowledge and understanding of the scientific and mathematical principles underlying their engineering branch.





 Methodology - Activity  Attendance (Hours)  Self-study (Hours)
 A-1 Lectures  42  
 A-2 Practical lessons  14  
 A-3 Assignments    10
 A-4 Self-study    75
 A-5 Exams, evaluation tests  4  
 A-6 Office hours    5
Total      60  90



Relationship between formative activities and proficiencies


 Proficiency  Formative activity
 CE1, CG2, CB1, CB5 A-1 Lectures, A-2 Practical lessons, A-3 Assignments, A-4 Self-study, A-5 Exams, evaluation tests, A-6 Office hours




Basque, English and Spanish.




 Learning outcome   Evaluation system  Weight (%)   Possibility of resit 
R1, R2, R3, R4, R5, R6, ENAEE-1   Written exam on the calculus concepts.  At least 3 points out of 10  are required in this exam to pass the subject in the continuous evaluation.  43 %  YES
R1, R2, R3, R4, R5, R6, ENAEE-1   Written exam on the algebra concepts.  At least 3 points out of 10  are required in this exam to pass the subject in the continuous evaluation.  42 %  YES
R1, R2, R3, R4, R5, R6, ENAEE-1  Individual assignment.  15 %  NO


In case of not getting at least 3 points out of 10 in the written exams corresponding to the continuous evaluation, the student should go to the resit exam if wishing to pass the subject.

The resit evaluation is made through a unique written exam that corresponds to algebra and calculus. It represents the 85 % of the final grade. The students not passing the subject in the continuous evaluation are allowed to take this exam.

The students passing the subject in the continuous evaluation can improve their grade by taking the resit exam.

If a student takes at most one of the written exams (either algebra or calculus) will have the condition of "No presentado". 



Number sets. Basic operations with real and complex numbers.


Functions, limits, continuity, differentiability, integrability. Basic concepts on real-valued functions of a single real variable. Elementary functions. Limits. Continuity of a function at a point. Properties of continuous functions. Differentiation, aplications of derivatives.

Definite and indefinite integrals, applications of integrals.


Vector spaces on R: subspaces. Basis and dimension.


Euclidean space: dot product and Euclidean norm, orthonormal bases, Gram-Schmidt orthogonalization.


Matrix diagonalization: eigenvalues and eigenvectors. Fundamental subspaces. Least-squares approximation.


Matrices: inverse matrix, linear systems, Rouché-Frobenius Theorem.







Lesson 1. Introduction and real-valued functions of a real variable

Preliminaries on real numbers: inequalities; absolute value.

Complex numbers.

Functions and their graphs. 

Limits and continuity. 

Derivative function. 

Applications: Taylor polynomial; optimization.

Calculus of zeros of a function.


Lesson 2. Integrals of real-valued functions of a real variable

Definition and properties.

The fundamental theorem of calculus. 

Elementary integration methods. 




Lesson 3. Vectors and matrices 

Linear combination of vectors, linear independence, bases, dimension and coordinates. 

Matrices: rank, determinant and inverse matrix. 

Linear systems. 


Vector length and orthogonality. 

Orthogonal projection. 

Least squares method. 

Construction of orthogonal bases. 



Lesson 4. Matrix diagonalization

Eigenvalues and eigenvectors. 

Characteristic polynomial. 


Algebraic and geometric multiplicity. 

Diagonalizable matrices.

Polynomial functions of matrices. 

Quadratic forms. 

Diagonalization and classification of quadratic forms.


There are computer sessions.



Acceda a la bibliografía que su profesor ha solicitado a la Biblioteca.



Calculus: A Complete Course (9th edition), R.A. Adams and C. Essex, Pearson Education, Canada, 2017.

Calculus (11th edition), R. Larson and B.H. Edwards, Brooks/Cole, Cengage Learning, Boston, 2018.

Linear Algebra and Its Applications (5th edition), D.C. Lay, S.R. Lay and J.J. McDonald, Pearson Education Limited, Harlow (England) 2016.



Calculus With Analytic Geometry (8th edition), R. Larson and R.P. Hostetler, Houghton Mifflin Company, 2005.

Calculus for Biology and Medicine, Fourth Edition, C.Neuhauser and M. Roper, Pearson Education Limited, Harlow (England) 2018.

Calculus: Early Transcendentals (4th edition), J. Rogawski, C. Adams and R. Franzosa, W.H. Freeman Publishers, 2018.

Calculus: One and Several Variables (10th edition), S. Salas, E. Hille and G. Etgen, John Wiley & Sons Inc., United States of America, 2007.

Calculus With Analytic Geometry (2nd edition), G.F. Simmons, The McGraw-Hill Companies, Inc., New York, 1996.

Calculus (6th Edition), K.J. Smith, M.J. Strauss and M.D. Toda, Kendall/Hunt Publishing Co, Iowa, United States, 2014.

Thomas' Calculus: Early Transcendentals (14th edition), G.B. Thomas, Jr., J. Hass, C. Heil and M. Weir, Pearson, Boston, 2018.

Linear Algebra and Its Applications: A first course, D.H. Griffel, Ellis Horwood Ltd., New York 1989.

Introduction to Linear Algebra (2nd edition), S. Lang, Undergraduate Texts in Mathematics, Springer, New York, 1986.

Linear Algebra with Applications (7th edition), W.K. Nicholson, McGraw-Hill Ryerson Ltd., Toronto, 2013.

A course in Linear Algebra with Applications (2nd edition), D.J.S. Robinson, World Scientific Pub Co Inc., Singapore, 2006.

Linear Algebra and Its Applications (4th edition), G. Strang, Cengage Learning, United Kingdom, 2006.



Lecture rooms building.