Course: 2020/2021

Calculus I

(13966)

Competences and skills that will be acquired and learning results. Further information on this link

By the end of this content area, students will be able to have:
1. Knowledge and understanding of the mathematical principles of differential and integral Calculus of one variable underlying their branch of Electrical engineering.
2. The ability to apply their knowledge and understanding to identify, formulate and solve problems related to differential and integral Calculus using established methods.
3. The ability to select and use appropriate tools and methods to solve problems of differential and integral Calculus .
4. The ability to combine theory and practice to solve problems of differential and integral Calculus.
5. The ability to understanding of mathematical methods of differential and integral Calculus and procedures, their area of ¿¿application and their limitations.
Evaluation of RAS
The first result from the student learning is evaluated through the implementation of approximated calculations, including error estimations, as obtained from the solution of optimization problems related to those found in the Electrical Power Engineering profession. These problems at the beginning are textually formulated (without the use of the mathematical notation). In a second step the Differential and Integral Calculus will be used to solve them. The 2th, 3th, 4th and 5th results from the student learning are evaluated in a systematic way through different partial and final exams because these learning results constitute essential parts in the formation of the mathematical way of thinking required to work as an Electrical Power Engineer.

Description of contents: programme

1. Functions o real variable
1.1 Sets of numbers. Real line, Mathemathical induction. Inequalities and absolute value.
1.2 Elementary functions, elementary trnasformations. Composition of functions, inverse function. Polar coordinates.
1.3 Limits of functions, definition, main theorems.
1.4 Continuous functions, properties and main theorems.
2. Differential Calculus
2.1 Diffentiation of functions, definitions, differentiation rules, differentiation of elementary functions.
2.2 Main theorems of differentiation, L'Hopital rule. Extrema of functions.
2.3 Local study of functions: Convexity and asymptotes. Graph of functions.
2.4 Taylor polinomial, definition, main theorems and known taylor expansions. Evaluations of limits with taylor polynomial.
3. Sequences and series.
3.1 Sequence of numbers, main notions, limits of sequences, recurrent sequences.
3.2 Series of numbers, main notions. Tests for convergence for series of positive numbers, absolute and conditional convergence. Leibniz's test. Sum of some series.
3.3 Taylor series, definitions, properties, convergence interval. Main examples.
4. Integration in one variable.
4.1 Integration, antiderivatives, integration by parts, substitution.
4.2 Definite integral. Fundamental theorem of Calculus and applications.
4.3 Application of integration: Areas, volumes and lengths.

Learning activities and methodology

The docent methodology will include:
- Master classes, where the knowledge that the students must acquire will be presented. To make easier the development of the class, the students will have written notes and also will have the basic texts of reference that will facilitate their subsequent work.
- Resolution of exercises by the student that will serve as self-evaluation and to acquire the necessary skills.
- Small groups classes, in which problems proposed to the students are discussed and developed.
- Office hours

Assessment System

- % end-of-term-examination 60
- % of continuous assessment (assigments, laboratory, practicals...) 40

Basic Bibliography

- PESTANA, D., RODRÍGUEZ, J. M., ROMERA, E., TOURÍS, E., ÁLVAREZ, V., PORTILLA, A.. "Curso práctico de Cálculo y Precálculo". Ariel. 2009
- R. Larson - B.H. Edwards. Calculus of a single variable. Cengeage Learning 9th ed.. 2009
- SALAS, S. L. , HILLE, E. , ETGEN, G. J.. "Calculus, one and several variables", Vol. 1,. Wiley. 2007

Additional Bibliography

- EDWARDS, C. H., PENNEY, D. E.. Calculus : with analytic geometry early transcendentals . Prentice Hall. 1998
- THOMAS, G. B.. Calculus and analytic geometry. Addison-Wesley. 1998