Course: 2022/2023

Calculus I

(15489)

By the end of this content area, students will be able to have:
1. Knowledge and understanding of the mathematical principles underlying their branch of engineering.
2. The ability to apply their knowledge and understanding to identify, formulate and solve mathematical problems using established methods.
3. The ability to select and use appropriate tools and methods to solve mathematical problems.
4. The ability to combine theory and practice to solve mathematical problems.
5. The ability to understanding of mathematical methods and procedures, their area of application and their limitations.

Skills and learning outcomes

Description of contents: programme

UNIT 1: SEQUENCES AND SERIES OF NUMBERS.
1.1. The real line, intervals, inequalities, absolute value, sets in the real line and in the plane. Mathematical induction. 1.2. Sequences of numbers, main notions, limits of sequences, recurrent sequences. Stirling formula and Stolz test.
1.3. Series of numbers, main notions. Tests for convergence for series of positive numbers, absolute and conditional convergence. Leibniz test.
UNIT 2: LIMITS AND CONTINUOUS FUNCTIONS.
2.1. Elementary functions, composition of functions, inverse function. Polar coordinates and sketch of graphs of functions.
2.2. Limits of functions, definition, main theorems. Evaluation of limits.
2.3. Continuous functions, properties and main theorems.
UNIT 3: DIFFERENTIAL CALCULUS IN ONE VARIABLE
3.1. Differentiation of functions: definition, differentiation rules, interpretation.
3.2. Bernoulli-L'Hôpital rule. Main theorems on differentiation. Extrema of functions.
3.3. Optimization problems with constraints.
3.4. Convexity and asymptotes. Graph of functions.
3.5. Taylor polynomial and series: definition, main theorems. Evalution of limits with Taylor polynomial. Convergence domain for a Taylor series.
UNIT 4: INTEGRATION
4.1. Antiderivatives, integration rules, integration by parts and by decomposition in simple fractions. Integration by substitution and other methods to evaluate integrals.
4.2. Definite integral and the fundamental theorem of calculus. Applications of integration: areas, volumes and length. Physical applications of the definite integral.

Learning activities and methodology

The docent methodology will include:
- Master classes,
- Practical classes
- Selfevaluations.
- Partial controls.
- Tutorials.
- Final examination.

Assessment System

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

Basic Bibliography

- D. Pestana, J. M. Rodríguez, E. Romera, E, Touris, V. Álvarez y A. Portilla. Curso práctico de Cálculo y Precálculo. Ariel Ciencia. 2000
- Ron Larson y Bruce H. Edwards . Calculus I (single variable). Cengage Learning (9th edition).
- Salas/Hille/Etgen. Calculus. Una y varias varaibles (Volumen I).. Reverté, S. A.. Cuarta edición 2005

Additional Bibliography

- BURGOS, J. Cálculo infinitesimal de una variable. McGraw - Hill.
- EDWARDS, C. H., PENNEY, D. E.. Cálculo diferencial e integral. Prentice Hall.
- SPIVAK, M.. Cálculus. Reverté.
- STEWART, J.. Cálculo, conceptos y contextos. Thomson.
- THOMAS, G. B., FINNEY, R. L.. Cálculo una variable. Addison-Wesley.

Detailed subject contents or complementary information about assessment system of B.T.

The course syllabus may change due academic events or other reasons.