Mathematics I

At the end of the Mathematics I course, students are expected to be able to solve qualitative and quantitative problems according to previously developed models, to recognize and analyze new problems, and to plan strategies to develop them.

1) Introduction to linear algebra and applications.
a) Matrices and determinants. Gauss method for solving systems of linear equations and calculating the inverse matrix.
b) Vector spaces and linear transformations. Matrices and orthogonal transformations.
c) Calculation of eigenvalues and eigenvectors. Diagonalization. Quadratic forms.

2) Differential calculation in one and several variables, and applications.
a) Functions of one and several variables. Basic concepts, review of elementary functions and their properties. Limits and continuity of functions of a variable.
b) Differential calculation of functions of a variable: derived from a function at a point and geometric interpretation. Derivation of elementary functions. Chain rule. Implicit derivation. Successive derivatives. Critical points. Taylor polynomial. Hospital Rule.
c) Differential calculation of functions of several variables: partial derivatives, tangent plane, gradient, Jacobian matrix, higher-order partial derivatives. Chain rule. Relative extremes of functions of two variables.

3) SageMath practices applied to the contents of the subject.

Basic bibliography:
- C. Neuhauser. “Matemáticas para Ciencias”. Pearson-Prentice Hall. 2004.
- G.B. Thomas. "Cálculo: Una variable". Volumen I. 12ª edición. Addison-Wesley, 2010.
- G.B. Thomas. "Cálculo: Varias variables". Volumen II. 12ª edición. Addison-Wesley, 2010.
- Notes prepared by the teachers of the subject and provided to the students through the virtual campus of the subject.

Complementary bibliography::
- G. A. Anastassiou e R. A. Mezei. "Numerical Analysis Using Sage". Springer. 2015.
- D. C. Lay. “Álgebra lineal y sus aplicaciones”. 3ª edición. Pearson-Prentice Hall. 2007.
- E. Steiner. “Matemáticas para las ciencias aplicadas”. Reverté. 2005.

Basic and general:
CB1 - That the students have demonstrated to possess and understand knowledge in a study area that starts from the general secondary education, and is usually found at a level that, although supported by advanced textbooks, also includes some aspects that involve knowledge from the forefront of their field of study.
CG3 - That they can apply both the theoretical-practical knowledge acquired and the capacity for analysis and abstraction in the definition and approach of problems and in the search for solutions in both academic and professional contexts.
CG5 - That they are able to study and learn autonomously, with organization of time and resources, new knowledge and
techniques in any scientific or technological discipline.

Transverse:
CT1 - Acquire analysis and synthesis capacity.
CT4 - Be able to solve problems.
CT10 - Acquire critical reasoning.
CT12 - Acquire autonomous learning.

Specific:
CE14 - Be able to solve qualitative and quantitative problems according to previously developed models.
CE15 - Be able to recognize and analyze new problems and plan strategies to solve them.
CE25 - Be able to relate Chemistry with other disciplines.

A) Expository classes (in large group):
In these classes the theoretical contents, problems and general examples of the subject are exposed, for which you can count on the support of audiovisual and computer media. The bibliography contained in the "Basic Bibliography" section is used.

B) Interactive seminar classes (in small group):
These classes cover exercises, problems, and applications of the theory. The student has to solve the exercises of the bulletins proposed by the teacher; These bulletins are published on the virtual page of the subject. The teacher can propose carrying out small exercises.

C) Interactive classes with computer (in small group):
These classes take place in the computer room. In them the student learns to use the mathematical software SageMath applying it to the theoretical-practical contents of the subject. To do this, you must do three guided practices (previously provided by the teacher through the virtual campus) where this software is used to illustrate examples or solve problems posed in the lectures or seminars. In the last session, an individual telematic questionnaire is carried out regarding the activity carried out throughout these classes.

D) Whiteboard tutoring (in a very small group):
Tutorials scheduled by the teacher and coordinated by the Faculty; they will take 1 hour for each student in the semester. This tutorial tries to clarify doubts about the theory, exercises or other proposed tasks.

E) Tutorials: weekly tutorial hours are published on the University website. They may be carried out entirely telematically (MS TEAMS platform) by appointment with the teacher.

The student is entitled to a call consisting of two opportunities. The qualification in the first and the second opportunities is done through continuous evaluation and taking an exam. The final numerical grade of the student is the maximum of the following grades: the exam grade and the grade obtained by weighing this with that of the continuous assessment, giving the latter a weight of 30%.

The final numerical grade is calculated as follows:

Final Numerical Note = Maximum {Note A, 0.7 x Note A + 0.3 x Note B},

where

Note A is the grade of the in-person final exam (out of 10).
Note B is the mark of the continuous evaluation (out of 10).

Note B (of the continuous evaluation) is calculated from the following activities:

1. Two questionnaires through the virtual campus related to the different blocks of the subject (maximum 7 points).
2. A questionnaire through the virtual campus regarding computer practices (maximum 3 points).

In this context, for cases of fraudulent performance of exercises or tests, the provisions of the Regulations for the evaluation of the academic performance of students and the review of grades of the USC will apply.

The student who obtains a suspense grade at the first opportunity, if presented at the second, has the maximum of the two final grades obtained.

The grade will be "not presented" if the student, having not carried out any evaluable academic activity, does not appear for the exams at the first and second opportunities.


All repeating students must submit to the same regime as ordinary students, except in relation to computer practices: the pass in the computer practices (grade equal to or greater than 1.5 in computer practices) in the 2023-24 academic year will be kept during the course 2024-25.


The proposed evaluation tools allow 100% evaluation of the set of basic, general, specific and transversal skills previously described. Specifically, the following table shows the skills assessed in the different training activities:

Competency Assessment | Seminar Classes | Practices with a computer | Tutoring Classes | Final exam

_______ CB1 _________ | _________ x_______ | _______ x_________ | ______ x_______ | ____ x______

_______ CG3 _________ | _________ x_______ | _______ x_________ | ______ x_______ | ____ x______

_______ CG5 ________ | _________ x_______ | _______ x_________ | ______ x_______ | ____ x______

_______ CT1 ________ | _________ x_______ | _______ x_________ | ______ x_______ | ____ x______

_______ CT4 ________ | _________ x_______ | _______ x_________ | ______ x_______ | ____ x______

______ CT10 ________ | _________ x_______ | _______ x_________ | ______ x_______ | ____ x______

______ CT12 ________ | _________ x_______ | _______ x_________ | ______ x_______ | ____ x______

______ CE14________ | _________ x_______ | __________________ | ______ x_______ | ____ x______

______ CE15________ | _________ x_______ | _______ x_________ | _______________ | ____ x______

______ CE25________ | _________ x_______ | __________________ | _______________ | ____ x______

Hours of classroom work: 34 (expository classes "E") + 10 (seminars "S") + 6 (practices with computer "P") +1 (Tutorial "T").

Hours of personal work of the student: 99

1. Be aware of communications through the virtual campus of the subject.

2. Actively follow the different classes.

3. Dedicate to the study of the subject a time regularly distributed throughout the semester.

4. After completing the study of a topic, it is useful to summarize the important calculation procedures, highlighting the basic formulas to remember.

5. Check the degree of assimilation of the concepts and acquisition of the basic calculation techniques by solving the exercises proposed in the class and in the problem bulletins.