Course: 2024/2025

Calculus I

(15320)

The student should acquire the background in calculus needed to understand and apply concepts and techniques for the solution of problems arising in the different areas of aerospace engineering.
SPECIFIC LEARNING OBJECTIVES:
- To acquire the basic concepts related to real functions and their graphical representations.
- To understand the formal definition of limit and to learn how to compute indeterminate limits.
- To learn and apply the basic numerical root-finding methods.
- To understand the concepts of continuity and differentiation.
- To understand the Taylor expansion technique and its applications.
- To understand the concepts of local and global approximation of functions and to be able to solve interpolation problems.
- To understand the formal definition of integral and to learn basic integration techniques.
- To be able to apply integration methods to compute lengths, areas, and volumes.
- To understand the concept of ordinary differential equation and to know basic solution techniques for first order equations.
- To learn complex numbers and to be able to operate with complex numbers.
SPECIFIC ABILITIES:
- To be able to handle functions given in terms of a graphical, numerical or analytical description.
- To understand the concept of differentiation and its practical applications.
- To understand the concept of definite integral and its practical applications.
- To understand the relationship between integration and differentiation provided by the Fundamental Theorem of Calculus.
GENERAL ABILITIES:
- To understand the necessity of abstract thinking and formal mathematical proofs.
- To acquire communicative skills in mathematics.
- To acquire the ability to model real-world situations mathematically, with the aim of solving practical problems.
- To improve problem-solving skills.

Skills and learning outcomes

Description of contents: programme

1. Introduction: sets, numbers, the real line, absolute value, intervals, mathematical induction.
2. Sequences: convergence, limits, indeterminate forms, introduction to series.
3. Functions, limits and continuity: elementary functions, algebraic operations and composition, inverse function, limits, continuity, intermediate value theorem.
4. Differentiation: derivative, algebraic operations and chain rule, Rolle's theorem, mean value theorem, L'Hôpital's rule, extrema, convexity, derivative of an inverse function, polynomial approximation, Taylor's theorem.
5. Integration: Riemann's integral, properties, fundamental theorem of calculus, integration by parts, changes of variables, improper integrals.
6. Series: series of non-negative terms, alternating series, absolute and conditional convergence, convergence tests, power series, radius of conergence, Taylor series.

Learning activities and methodology

Theory lectures (3 credits).
Problem-solving seminars (3 credits).

Assessment System

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

Calendar of Continuous assessment

Extraordinary call: regulations

Basic Bibliography

- Michael Spivak. Calculus, 3rd ed. Cambridge University Press. 1994
- Tom M. Apostol. Calculus, Vol. 1, 2nd ed. John Wiley & Sons. 1967

Additional Bibliography

- J. Stewart. Calculus. Thomson Brooks/Cole. 2009
- Juan de Burgos Román. Cálculo Infinitesimal de una variable. McGraw-Hill. 1994
- R. Larson, R. Hostetler, B. Edwards. Calculus. Houghton-Mifflin. 2006

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