In this subject, it is intended that the student can achieve the analysis of the basic concepts of real functions of real variable and its applications, as well as develop skills for the study of functions, calculation of limits, differentiation and integration, differential equations, applications of derivative and finding the convergence of numerical and functional series. It is emphasized that the student is able to understand and relate results and basic demonstrations, as well as to acquire the capacity for analysis and synthesis in the face of a problem. In addition, the basis of the numerical calculation is given by performing computer practices.
Titular Professors
Professors
Trigonometry, basic functions and derivation.
The course aims to provide students with a solid foundation in the fundamentals of single-variable differential and integral calculus, developing their ability to understand and work with functions, limits, derivatives, and integrals as essential tools for analysing continuous phenomena. It also seeks to foster rigorous and logical reasoning, mathematical modelling, and the application of calculus techniques to solve problems in engineering and applied sciences.
Furthermore, the course aims to help students develop proficiency in analytic techniques—such as differentiation, integration, and graphical representation—as well as in the interpretation of results, promoting a deep understanding of the fundamental concepts that underpin mathematical analysis. Finally, it introduces the basic principles of ordinary differential equations and their applications, contributing to the development of the competencies required for subsequent scientific and technological subjects that rely on calculus.
1. The numbers
1.1. Presentation of different types of numbers and their properties.
1.2. Real numbers in equations with absolute value.
1.3. Complex numbers.
2. Functions
2.1. Elementary Functions: Definition and properties.
2.2. Limits: Definition, properties and calculation.
2.3. Continuity: Definition, properties, types of discontinuities.
2.4. Basic theorems on continuous functions in intervals.
2.5 Asymptotes
3. Differentiation
3.1. Definition and meaning derivative
3.2. Referral techniques.
3.3. Theorems about derivable functions in intervals.
3.4. Taylor polynomials.
3.5. Growth and decrease, Relative extremes.
3.6. Concavity and convexity, Turning points.
3.7. Graphical representation of functions.
3.8. Optimization.
4. Integration
4.1. Immediate integrals
4.2. Integrals for change of variable and parts.
4.3. Integrals of rational functions.
4.4. Integrals of trigonometric functions.
4.5. Integrals of irrational functions.
5. The Riemann integral
5.1. Definition and properties. Geometric interpretation.
5.2. Fundamental theorem of Calculus.
5.3. Improper integrals: Definition and basic calculations.
5.4. Applications of integral calculation (areas, lengths and volumes).
6. Ordinary differential equations (EDO)
6.1. Classification.
6.2. Equation of variable separable.
6.2 Homogeneous equations of 1st order.
6.3. Linear equations of 1st order.
6.4 EDOs applications.
6.5. Linear equations of 2nd order.
The subject runs weekly with 2 100-minute class sessions. Throughout the course, different types of sessions will be combined:
- Most of the sessions are dedicated to working on the conceptual content of the subject through a combination of lectures and flipped classroom. The teacher explains the key points of the content worked on and solves exercises to complete the explanation.
- Cooperative work sessions in which students must solve problems to consolidate the subject, under the supervision of the teacher.
- Practical sessions that are mainly dedicated to Numerical Calculus practices where students work in groups of two with a laptop that has Matlab software.
- Finally, some sessions are dedicated to individual assessment through written tests or review sessions for exams.
In order to assess whether the student has adequately achieved the objectives pursued in the subject, different tests are used to obtain data from the student:
- Exams. During the course, 4 main exams are held: two in the first semester and two more in the second.
- Controls carried out in class.
- Participation in class and submission of exercises.
- Personal or group Matlab practices.
For each semester, the evaluation of the subject is carried out through a continuous assessment system complemented by an exam in the middle of the semester and another at the end. The final grade is obtained from the following elements:
Individual exams: 70% Partial controls: 18% Participation and Submission of exercises: 3% Practices: 9%
To pass the semester, it will be necessary to obtain a minimum grade of 3.5 out of 10 in the final exam. If this minimum is not reached, the average will not be calculated with the rest of the activities.
To pass the subject, a minimum grade of 5 is required in each semester.
An extraordinary call is planned in July to recover the semester/s not passed in the ordinary call. This recovery will consist of an exam for each semester. The semester grade will be the best between the exam grade and the grade obtained with a weighting of 70% of the recovery exam and 30% of the continuous assessment activities obtained in the corresponding semester.
The following will be assessed:
- The correct application of calculation methods in problem solving.
- The rigor and coherence in the development of mathematical reasoning.
- The ability to mathematically model basic technical situations.
- The accuracy of calculations and the correct interpretation of the results obtained.
Piskunov, N. (1981). Differential and Integral Calculus. Mir Publishers
Paul's Online Notes. https://tutorial.math.lamar.edu/
Alfonsa García et al. (1994) Cálculo I. Teoría y problemas. Editorial GLACSA.
Galindo Soto, F. et al. (2003). Guía práctica de cálculo infinitesimal en una variable real. Thomson.