This subject presents the student with the basic topics of mathematical analysis that any engineering student requires to understand the concepts of the career, focusing on the field of differential and integral calculus of a single variable. The aim is for the student not to be limited to theoretical definitions and demonstrations, but to be able to apply this knowledge to problem solving.
Titular Professors
It is recommended to have basic knowledge of trigonometry, polynomials, exponentials and logarithms.
Students who take this subject acquire the knowledge and develop the skills indicated below:
- Achieve basic concepts of the analysis of real functions of a real variable and their applications.
- Achieve skills in calculating limits, studying functions, calculating integrals and solving applied problems.
- Understand and relate basic results and proofs.
- Ability to analyze and synthesize a given problem.
- Know how to use analytical and numerical tools to analyze real functions of a variable, in order to apply them in scientific and technical issues.
1. Numbers
1.1. Presentation of different types of numbers and their properties.
1.2. Real numbers. Inequalities 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 on intervals.
2.5. Asymptotes.
3. Differentiability
3.1. Definition and meaning. Differential
3.2. Derivation techniques
3.3. Theorems on differentiable functions on intervals.
3.4. Taylor polynomials
3.5. Growth and decrease. Relative extrema.
3.6. Concavity and convexity. Inflection points.
3.7. Graphical representation of functions.
3.8. Optimization
4 Calculation of primitives
4.1. Immediate integrals.
4.2. Integrals by change of variable and by 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 calculus (areas, lengths and volumes).
The course is structured around five 50-minute lecture sessions per week. Throughout the course, different types of sessions will be combined:
- Most sessions are dedicated to covering the conceptual content of the course through a combination of lectures and flipped classroom methods. The professor explains the key points of the material and works through exercises to reinforce the explanation.
- There will also be cooperative learning sessions where students solve problems to consolidate the material, under the professor's supervision.
- Practical sessions are primarily dedicated to Numerical Calculus exercises, where students work in pairs using laptops with Matlab software.
- Finally, some sessions will be dedicated to individual assessment through written tests or review sessions in preparation for exams.
To assess whether students have adequately achieved the course objectives, various assessments are used to gather data:
- Exams. Four main exams are administered throughout the semester: two in the first semester and two in the second.
- Quizzes given in class.
- Class participation and submission of assignments.
- Individual or group Matlab practical exercises.
The course is assessed through a continuous evaluation system supplemented by a midterm exam and a final exam. The final grade is calculated based on the following elements:
Individual exams: 70% Midterm quizzes: 18% Participation and assignment submissions: 3% Practical exercises: 9%
To pass the semester, a minimum grade of 3.5 out of 10 is required on the final exam. If this minimum is not achieved, the grade will not be averaged with the other assessments.
An extraordinary exam session is scheduled for July for students who fail the regular exam session. This resit will consist of an exam. The grade will be the higher of the exam grade and the grade obtained with a weighting of 70% for the resit exam and 30% for the continuous assessment activities obtained during the course.
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.
Cálculo Diferencial e Integral. Piskunov, N. Editorial Mir , 1983
Alfonsa García et al. (1994). Cálculo I. Teoría y problemas. Editorial GLACSA.
Tomeo, V. et al. (2007). Problemas resueltos de cálculo en una variable. Thomson.