Universidad Pública de Navarra - Nafarroako Univertsitate Publikoa

Module/Subject matter

Basic formation module / Mathematics 

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Contents

Real functions in one variable. Limit. Continuity. Differentiation. Extrema and optimization. Taylor polynomial. Integration in one variable. Applications

Linear system of equations. Vector spaces. Orthogonality. Determinants. Eigenvalues and eigenvectors. 

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Descriptors

Single Variable Calculus. Linear Algebra.

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General proficiencies

• G1: Self-learning ability
  
• G2: Problem solving proficiency with personal initiative, decision making, creativity and critical reasoning. Ability to elaborate and communicate knowledge, abilities and skills in engineering
• G4: Knowledge of basic and tecnological subjects to have the ability to learn new methods and theories, and versatility to adapt to new situations
• G5: Analysis and synthesis ability

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Specific proficiencies

E1: Ability to solve mathematical problems in engineering. Ability to apply theoretical knowledge on linear algebra, differential geometry, calculus, differential equations, numerical methods, algorithms, statistics and optimization

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Learning outcomes

• Know and work with real and complex numbers. 
• Work and recognise elementary functions (polynomials, rational, trigonometric, logarithms, exponentials,...). Know their elementary properties.
• Become familiar with the concepts of limit, continuity and differentiablility.
• Know the fundamentals of integral calculus in one variable. Apply it to compute lenghts of curves, areas and volumes of solids. 
• Know and apply the concepts of vector spaces, linear systems,  matrices, determinants, inner (dot) products.
• Apply Calculus and Linear Algebra techniques to problems in Engineering
• Know selected algorithms of applied mathematics in these fields

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Methodology

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Languages

English

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Evaluation

There are two alternatives for passing this course:

1 Ordinary path:

• Two written mid-term exams, which will evaluate the knowledge of Calculus and Algebra separately. These exams will each count for 40% of the total grade.
• A homework assignment on selected topics in Algebra, which will count for the remaining 20%  

The student must achieve a score of 5 (over 10) or above and attain at least a score of 3.5 (again over 10) in the partial exams and assignment homework previously mentioned to pass the course. Otherwise, the final grade will be the minimum between 4.9 and the (weighted) average of these partial grades. 

2 Extraordinary examination call: If the student fails to pass the course following the ordinary path described above, he/she can apply for an extraordinary call examination which consists in a unique final exam. This exam is divided into two parts, the first on is focused on Calculus and the second part on Algebra. These parts will count for 40% and 60% respectively of the total grade.  

Students can use for the exams, notes and any book he/she considers appropriate. Programmable calculators and electronic devices such as laptops, tablets, and smartwhatchs are banned.

 (Updated information on timetable and venue for the exams can be found at http://www.unavarra.es/ets-industrialesytelecos/estudios/grado/grado-en-ingenieria-en-disenio-mecanico-campus-de-tudela/periodos-de-evaluacion?submenu=yes)

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Agenda

Lesson 0. Introduction 
Number sets. Complex numbers. Sets in R.  Absolute value function. Inequalities

First part 

Lesson 1. Functions in one variable, limits and continuity.  

Basic notions on real functions in one variable. Continuity: definition and local properties . Weierstrass, Bolano and Intermediate value Theorems

Lesson 2: Differential calculus

Derivative of a function at a point: definitions, interpretation and first properties. Derivative function. Derivatives of higher degree. Chain rule. Rolle and Mean Value Theorems. Applications: extrema, L'Hopital rule. Finding functions zeros. Taylor and Maclaurin polynomials. 

Lesson 3: Integral calculus 

Riemann integral: definition and properties. Integral mean value theorem. Fundamental Calculus Theorem. Barrow rule. Integration by parts. Change of variables.  

Second part

Lesson 1: Linear systems of equations 

Linear systems of equations. Gauss elimination. Gauss elimination with pivoting. Gauss-Jordan method. Matrix form of a linear system. Matrix definition and first properties. Matrix product. Inverse matix. Rank. Rouche-Frobenius theorem

Lesson 2: Vector spaces in Rn  

Null and Column space of a matrix. Vector subspaces.  Linear dependence and independence. Basis. Coordinates. Dimension of a subspace

Lesson 3: Inner product and Euclidean spaces 

Orthonormal basis. Gram-Schmidt algorithm. Orthogonal matrix. QR decomposition. Orthonormal basis. Orthogonal projection and least square approximation. Pseudoinverse.

Lesson 4: Determinants

Definition  and main properties. Cramer rule.

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Bibliography

Access the bibliography that your professor has requested from the Library.

Basic bibliography 

• S.L. Salas, E. Hille y Etgen:  Calculus: one variable. Wiley
• V. Domínguez, PALM. Notas de Álgebra Lineal y Matricial (in spanish)

Complementary bibliography

• G.L. Bradley, K.J. Smith: Cálculo de una variable. Prentice Hall.  
• J. B. Fraleigh, A. Beauregard, Linear Algebra, Pearson
• S. Lang,  Introduction to linear algebra. Springer
• G. Strang, Linear Algebra and its applications,  TBS

( Library Catalogue can be consulted at https://biblioteca.unavarra.es/abnetopac/abnetcl.cgi/O7164/ID7e647614?ACC=101 )

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Location

Classrom 

(Updated information on timetables and classrooms can be found at http://www.unavarra.es/ets-industrialesytelecos/estudios/grado/grado-en-ingenieria-en-disenio-mecanico-campus-de-tudela/horarios?submenu=yes )

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