Course: 2019/2020

Calculus I

(13966)

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

The student will be able to formulate, solve and understand mathematically the problems arising in engineering. To do so it is necessary, in this first course of Calculus, to be acquainted with the real functions of one variable, their properties of continuity, derivability, integrability and their graphic representation. The student will understand the concepts of derivative and integral and their practical applications. Also, he/she will manage sequences and series of real numbers and of functions that will apply to numeric approximation of functions and the resolution of equations.

Description of contents: programme

1. Functions
1.1 Numbers, functions and their graphs
¿ Real numbers
¿ Functions
¿ Graphs
1.2 Limits and their properties
¿ Evaluating limits analytically
¿ Infinite limits
¿ Limits at infinity
1.3 Funciones continuas
¿ Continuidad y límites laterales
¿ EL Teorema de los valores Intermedios
2. Differentiation
2.1 Definition and basic differentiation rules
¿ The derivative and tangent line
¿ Basic differentiation rules
¿ Product and quotient rules and higher-order derivatives
¿ The Chain rule
¿ Implicit differentiation
2.2 Applications
¿ Extrema on an interval
¿ Rolle¿s and mean-value theorems
¿ Increasing and decreasing functions
¿ Concavity
¿ Curve sketching
2.3 Optimization problems
2.4 Taylor polynomials
¿ Taylor polynomials
¿ Indeterminate forms and L¿Hôpital¿s rule
3. Integration
3.1. Primitives
¿ Antiderivatives and indefinite integration
¿ Area and definite integrals
¿ The Fundamental Theorem of Calculus
3.2. Integration techniques
¿ Basic integration rules
¿ Integration by substitution
¿ Integration by parts
¿ Partial fractions
¿ Improper integrals
3.3. Applications
¿ Area of a region between two curves
¿ Volume
¿ Arc length and surfaces of revolution
¿ Centers of mass, fluide pressure
4. Infinite series
4.1 Sequences
4.2 Series
¿ Real number series and convergence
¿ Alternating series
¿ Convergence criteria
4.3 Power series
¿ Representation of functions by power series
¿ Convergence radius
¿ Taylor series

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

- 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