FONMAT - Mathematic Fundamentals

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1 Coordinating unit: ETSECCPB - Barcelona School of Civil Engineering Teaching unit: DECA - Department of Civil and Environmental Engineering Academic year: Degree: 2017 BACHELOR'S DEGREE IN GEOLOGICAL ENGINEERING (Syllabus 2010). (Teaching unit Compulsory) ECTS credits: 6 Teaching languages: Catalan Teaching staff Coordinator: Others: EUSEBIO JARAUTA BRAGULAT JAIME LUIS GARCIA ROIG, EUSEBIO JARAUTA BRAGULAT Opening hours Timetable: * Face: Tuesday 16 to 18 hours. Please arrange appointment in advance. * No face: attendance by whenever the student wants to use. Degree competences to which the subject contributes Specific: Ability to solve the types of mathematical problems that may arise in engineering. Ability to apply knowledge of: linear algebra; geometry; differential geometry; differential and integral calculus; differential equations and partial derivatives; numerical methods; numerical algorithms; statistics and optimisation Transversal: 591. EFFICIENT ORAL AND WRITTEN COMMUNICATION - Level 1. Planning oral communication, answering questions properly and writing straightforward texts that are spelt correctly and are grammatically coherent EFFECTIVE USE OF INFORMATI0N RESOURCES - Level 1. Identifying information needs. Using collections, premises and services that are available for designing and executing simple searches that are suited to the topic SELF-DIRECTED LEARNING - Level 1. Completing set tasks within established deadlines. Working with recommended information sources according to the guidelines set by lecturers. Teaching methodology The course is taught in 4 hours a week of classes in the classroom (large group). The sessions are classified as: - Theory: exposure to basic materials and concepts of matter, illustrated with examples of the application; - Classroom Practice: solving exercises and problems; - Laboratory Practices: performing calculations with free software application course. Used support material is available to students through campus Athena. Learning objectives of the subject Students willâ learn to perform differential and integral calculus of one variable and to apply these techniques to specific scientific and technical problems and to geological engineering in general. Upon completion of the course, students will be able to: 1. Use, derive and integrate trigonometric functions and analyse successions and series in engineering contexts; 2. Use differential calculus to solve maxima and minima problems related to simple engineering problems; 3. Solve integrals of one variable in relation to simple engineering problems. 1 / 5

2 Real numbers; Successions and calculation of limits; Numerical series and convergence; Theory of functions, including analysis of continuity and limits; Differential calculus of functions of a real variable, including maxima and minima problems in simple engineering problems and optimization; Integral calculus of functions of a real variable; Trigonometry Working the concept of mathematical modeling applied to engineering. Consolidate basic knowledge of algebra, trigonometry and geometry acquired in previous courses and deepen this knowledge. Achieving the proper method of reasoning applied mathematics and mathematical proof of the theorems and results. Practice with different methodologies show. Learn the basic numerical sets, their properties and characterization. To acquire basic knowledge in the field of linear algebra such as matrices, determinants, systems of linear equations and vector spaces. To acquire basic knowledge in the field of calculus: functions, limits, continuity, derivatives and applications of derivatives. Study load Total learning time: 150h Hours large group: 32h 30m 21.67% Hours medium group: 17h 30m 11.67% Hours small group: 10h 6.67% Guided activities: 6h 4.00% Self study: 84h 56.00% 2 / 5

3 Content 1. INTRODUCTION Learning time: 31h 12m Theory classes: 6h Practical classes: 4h Laboratory classes: 3h Self study : 18h 12m Mathematical Modelling in Engineering: concept, methodology and tools. Elements of a mathematical model. Examples. Introduction to Logic: definitions, propositions, theorems. Proof methods, examples. Basic Elements of the Theory of sets. Sets, subsets. Relations. Applications; types of applications. Operations. Algebraic structures. Sets numeric. Natural numbers, the principle of induction. Integers. Rational numbers. Real numbers. Absolute value. Properties of real numbers. Inequalities. Factors and developments. Powers. Complex numbers: operations and properties. Trigonometry. Angles, measurement of angles. Trigonometric relationships. Solving triangles. Cartesian geometry. Coordinates in the plane, distance between two points. Plane analytic geometry; line. Curves flat remarkable. Geometry of space: lines and planes. 1.5 Exercises and Problems Unit Topic 1 Lab 2. MATRICES. Determinants. Systems of linear equations Learning time: 24h Theory classes: 6h Practical classes: 3h Laboratory classes: 1h Self study : 14h Definitions and examples. Types of matrices. Matrix operations. Elementary transformations of matrices. Rank of a matrix. Regular Arrays: inverse of a regular array. Determinant of a square matrix: definition, calculation, properties and applications. Systems of linear equations: definitions, notations and examples. Systems. Solving systems: methodology and examples. Systems with Parameters. 2.4 Exercises Item Laboratory Item 2 3 / 5

4 3. Vector spaces and LINEAR APPLICATIONS Learning time: 31h 12m Theory classes: 7h Practical classes: 4h Laboratory classes: 2h Self study : 18h 12m The vector space Rn. Definition and properties. Other vector spaces. Linear combinations. Vector subspaces. Linear dependence and independence. Subspace generated. Bases and dimension; properties. Linear; definition and properties. Associated with a linear array: definition and properties. 3.3 Exercises and Problems Item 3 4. Real functions of real variables. Learning time: 57h 35m Theory classes: 13h 30m Practical classes: 6h 30m Laboratory classes: 4h Self study : 33h 35m Functions: definition. Operations with functions. Graphical representation. Basic properties of real functions of real variable. Elementary functions. Definition, properties and graphical representation. Limit of a function at a point; properties. Infinite limits. Limits at infinity. Forms unknowns or uncertainties. Functionally equivalent. Asymptotes of a function. Continuous functions: definition and properties. Discontinuities: definition and classification. Properties of continuous functions on closed intervals. Derivative of a function: definition, properties and calculation. Derivatives of elementary functions. Derivatives of higher order. Apply for calculating limits. Taylor formula. Local extremes. Extremes in a compact. Calculation of extreme. Problems extremes (optimization). Antiderivades or primitive. Indefinite integral. Definition, properties and calculation. 4.6 Exercises and Problems Item Laboratory Item 4 4 / 5

5 Qualification system AC: continuous assessment tests. They are two exams during the course and provide continuous assessment of the subject. AS: review synthesis. There is a REVIEW of synthesis at the end of the semester. HW: extra practice. Made during the course and serve as an indicator of student progress in achieving the knowledge, skills and abilities. Examination Weight: AC: ; AS: 0.4. The exercises how to improve the HW grade for the course. Criteria for re-evaluation qualification and eligibility: Students that failed the ordinary evaluation and have regularly attended all evaluation tests will have the opportunity of carrying out a re-evaluation test during the period specified in the academic calendar. Students who have already passed the test or were qualified as non-attending will not be admitted to the re-evaluation test. The maximum mark for the re-evaluation exam will be five over ten (5.0). The nonattendance of a student to the re-evaluation test, in the date specified will not grant access to further re-evaluation tests. Students unable to attend any of the continuous assessment tests due to certifiable force majeure will be ensured extraordinary evaluation periods. These tests must be authorized by the corresponding Head of Studies, at the request of the professor responsible for the course, and will be carried out within the corresponding academic period. Regulations for carrying out activities The evaluation of the course is obtained only as a result of the weighted continuous assessment tests and test synthesis. If a student can not make any of these tests cause documentadament must ask explicit permission to sit final extraordinary exam. Bibliography Basic: Estela, M.R. Fonaments de càlcul per a l'enginyeria. Barcelona: Edicions UPC, ISBN Jarauta, E. Anàlisi matemàtica d'una variable. Fonaments i aplicacions. Barcelona: Edicions UPC, ISBN Pelayo, I.M.; Rubio, F. Álgebra lineal básica para ingeniería civil. Barcelona: Edicions UPC, ISBN Complementary: Burgos, J. de. Cálculo infinitesimal de una variable. 2a ed. McGraw-Hill, ISBN Lipschutz, S. Álgebra lineal. 2a ed. McGraw-Hill, ISBN / 5

Mathematics. Mathematics

Mathematics. Mathematics Mathematics Program Description Successful completion of this major will assure competence in mathematics through differential and integral calculus, providing an adequate background for employment in