Course: 2022/2023

Calculus I

(18296)

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

Part I: Real Numbers and Functions
Chapter 1: The Real Line
1.1 Ordered Fields
1.2 Number Systems
1.3 Absolute value, bounds, and intervals
Chapter 4: Real Functions
2.1 Definition and basic concepts
2.2 Elementary functions
2.3 Operations with functions
Part II: Sequences and Series
Chapter 3: Sequences
3.1 Sequences of real numbers
3.2 Limit of a sequence
3.3 Number e
3.4 Indeterminacies
3.5 Asymptotic comparison of sequences
Chapter 4: Series
4.1 Series of real numbers
4.2 Series of nonnegative terms
4.3 Alternating series
4.4 Telescopic series
Part III: Differential Calculus
Chapter 5: Limit of a Function
5.1 Concept and definition
5.2 Algebraic properties
5.3 Asymptotic comparison of functions
Chapter 6: Continuity
6.1 Definition, properties, and continuity of elementary functions
6.2 Discontinuities
6.3 Continuous functions in closed intervals
Chapter 7: Derivatives
7.1 Concept and definition
7.2 Algebraic properties
7.3 Derivatives and local behaviour
Chapter 8: Taylor expansions
8.1 Asymptotic comparison of functions
8.2 Taylor¿s polynomial
8.3 Calculating limits
8.4 Remainder and Taylor¿s theorem
8.5 Taylor series
8.6 Numerical approximations
8.7 Local behaviour of functions
8.8 Function graphing
Part IV: Integral Calculus
Chapter 9: Primitives
9.1 Integration by parts
9.2 Primitives of rational functions
9.3 Change of variable
Chapter 10: Fundamental Theorem of Calculus
10.1 Riemann¿s integral
10.2 Properties of the integral
10.3 Riemann¿s sums
10.4 Fundamental theorem of calculus
Chapter 11: Geometric Applications of Integrals
11.1 Area of flat figures
11.2 Area of flat figures in polar coordinates
11.3 Volumes
11.4 Length of curves
Chapter 12: Improper Integrals
12.1 Improper integrals of the first kind
12.2 Improper integrals of the second kind

Learning activities and methodology

The methodology will be the usual one for classes in the classroom, writing on the blackboard, with the occasional help of some resources on-line to illustrate some graphic or computational aspects of the course. Also, the classroom notes will be uploaded in Aula Global at the end of each chapter, along with the problem sheets that will be solved and discussed in the small groups.

Assessment System

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

Basic Bibliography

- Adrian Banner. The Calculus Lifesaver: All the tools you need to excel at calculus. Princeton University Press. 2007
- H. Anton, I.C. Bevis & S. Davis. Calculus: Early Transcendentals Single Variable. Wiley. 2008
- J. Stewart. Single variable calculus: early transcendentals. Brooks-Cole. 1999
- R. Larson, R.P. Hostetler & B.H. Edwards. Calculus. Brooks-Cole. 2005
- S.L. Salas, G.J. Etgen & E. Hille. Calculus: One and Several Variables. Wiley. 2006

- Gilbert Strang & Edwin Herman · Calculus : https://math.libretexts.org/Bookshelves/Calculus/Book%3A_Calculus_(OpenStax)

Additional Bibliography

- T.M. Apostol. Calculus vol. 1. Wiley. 1991

(*) Access to some electronic resources may be restricted to members of the university community and require validation through Campus Global. If you try to connect from outside of the University you will need to set up a VPN

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

**More information: **https://aplicaciones.uc3m.es/cpa/generaFicha?&est=370&plan=455&asig=18296&idioma=1