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Credits  Dept.  Type  Requirements 

9.0 (7.2 ECTS)  MAT 

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It is quite possible that computer engineers will encounter problems that involve making calculations at some point in their professional careers. Despite the fact that IT experts do not have to make these sorts of calculationsbut rather have them done for themthey still need to be able to implement them. For this reason it is useful for them to understand these calculations and to know where to look for relevant information. In some subjects in this course, students have to know how to perform certain calculations without starting from square one. Thus, the overall goal of the subject is that students in information technology come away with an understanding and ability to work with the fundamental concepts and techniques of mathematics. In particular, the subject aims for students to understand and be able to use the concept of functions with one or more variables.
Estimated time (hours):
T  P  L  Alt  Ext. L  Stu  A. time 
Theory  Problems  Laboratory  Other activities  External Laboratory  Study  Additional time 

T  P  L  Alt  Ext. L  Stu  A. time  Total  

8,0  8,0  2,0  0  0,5  18,0  0  36,5  
2.1 Defined integral: The problem of area. Riemann"s Integral Theorem Basic properties.
2.2 Approximate integration: The Trapezium Rule and error formula. Extrapolation. Simpson Method and error formula. 2.3 Indefinite integral: Functions defined by integrals. Fundamental Theorem of Calculus. Integral. Barrow"s Rule. 2.4 Calculation of integrals I: Immediate and rational numbers. 2.5 Calculation of integrals II: Change of variable and parts. 2.6 Improper integrals: Definition. Types and examples. Absolute and conditional convergence. 2.7 First Order improper integrals: Convergence criteria. Examples. 2.8 Second Order improper integrals: Convergence criteria. Examples.


T  P  L  Alt  Ext. L  Stu  A. time  Total  

5,0  5,0  1,0  0  0,5  12,0  0  23,5  
3.1 Numerical successions: Definition. Forms of expression. Limits of a succession. Algebraic properties. Indeterminations.
3.2. Succesions acotades: Convergence properties. Monotonous Convergence Theorem. The number e. 3.3 Numerical series: Infinite Sum Problem. Definition of a series. Convergence. Examples: Geometric and alternating series (Leibnitz criterion). 3.4 Convergence of numeric series: Convergence criteria: Series of nonnegative terms: Comparison, quotient, nroot, and integrals. Absolute and conditional convergence. 3.5 Sum calculation: Exact sum. Approximate sum: comparison, integral, and alternate methods.


T  P  L  Alt  Ext. L  Stu  A. time  Total  

4,0  4,0  1,0  0  0,5  10,0  0  19,5  
4.1 Taylor Polynomials: Polynomial approximation. Taylor Theorem and Lagrange Remainder. 4.2 Applications: Calculating extreme values. Local study of a function. Error propagation formula. 4.3 Powers series: Interval of convergence. Derivation and integration. Sum of a series. 4.4 Taylor Series: Convergence. Taylor Series associated with elementary functions.


T  P  L  Alt  Ext. L  Stu  A. time  Total  

12,0  12,0  2,0  0  0,5  30,0  0  56,5  
5.1 Topology at ndimensional space: Distance between two points. Boundary, interior and closure of sets. Open, closed, bounded, compact sets.
5.2 Curves and surfaces: Special curves and surfaces in 2 and 3dimensional spaces. 5.3 Functions of various variables: Definition. Domain and range. Level sets. 5.4 Partial derivatives: Definition. Geometrical interpretation. Gradient vector. Tangent plane and normal straight line to a surface at a point. 5.5 Directional derivatives: Definition. Geometrical interpretation. Optimal direction. 5.6 Implicit curves and surfaces: Composition derivatives. Implicit functions. Examples. 5.7 Taylor Polynomial: Higher order derivatives. Polynomial aproximation . Taylor formula and Lagrange residual term. 5.8 Local maxima and minima I: Definition. Critical points. Necessary condition for existence. 5.9 Local maxima and minima II: Sufficient condition for existence. 5.10 Conditioned maxima and minima: Lagrange multipliers, classification, Lagrange general method. 5.11 Absolute maxima and minima: Weierstrass theorem. Localization. 5.12 Applications: Geometrical, physical and computer science examples.

Total per kind  T  P  L  Alt  Ext. L  Stu  A. time  Total 
34,0  34,0  10,0  0  3,0  80,0  0  161,0  
Avaluation additional hours  5,0  
Total work hours for student  166,0 
Theory classes; these consist of the presentation of concepts, and the most basic Calculus techniques and methods.
Classes of problems; these expand on the examples covered in the theory classes and with the solution of problems.
Lab classes; the methods and techniques learnt in the theory classes will be applied to various problems with the aid of the Maple symbolic manipulator.
The final grade (N) for the course is calculated as follows:
N = max( 0.15*L + 0.25*P + 0.6*F, 0.15*L + 0.85*F )
where
L = Lab grade.
P = Part exam grade.
F = Final exam grade.
Lab: average grade from 5 practical sessions, assessed using a questionnaire submitted at the end of each session (15%).
Part exam: This consists of exercises and/or brief questions on theory (25%).
Final exam: This consists of a certain number of problems and/or theoretical questions (60%).
Any attempt of fraud during the course will entail the application of the UPC's general academic normative and the beginning of a disciplinary process.
Knowledge of integers and the properties of operations.
Operating with polynomials: addition, multiplication, division, factoring.
Have a basic grasp of the concept of function.
Know how to operate with exponential functions, logarithmic functions, and trigonometric functions.
The concept of continuous functions. Know how to calculate the limits of functions.
Understand the concept of derivable function. Know how to calculate the derivatives of functions.
The concept of an integral. Know how to calculate integrals.