One-variable Calculus
Code: 104382Credits: 6
| Degree programme | Type | Course |
|---|---|---|
| Computational Mathematics and Data Analytics | FB | 1 |
Contact lecturer
- Name :
- Alberto Dayan
- Email :
- alberto.dayan@uab.cat
Teaching staff
- Jaume Capdevila Jové
- Ángel Lorenzo Martínez
Group languages
You can consult this information at the end of the document.
Prerequisites
Although there are no official prerequisites, it is recommended that students have good knowledge of basic Calculus: limits, continuity and derivability of real functions of one variable, notions of integral calculus and trigonometry. As well as the graphic representation of relatively simple functions of one variable. The most important requirement, however, is a great curiosity to understand and deepen the concepts that they will study.
Objectives
Solve the mathematical problems that can arise in the degree they are studying.
Understand the concept of sequences and the computation of limits.
Know and work intuitively, geometrically and formally the notions of limit, continuity, derivative and integral.
Understand and know how to make Taylor's developments of functions of one real variable.
Acquire basic notions of numerical series and power series. Know the construction of the integral, know how to solve integrals and
its applications to solving problems where the integral approach is necessary. Improper integrals will be also studied.
Learning outcomes
- CM01 (Work intuitively, geometrically and formally with the notions of limit, derivative and integral.) Work intuitively, geometrically and formally with the notions of limit, derivative and integral.
- CM03 (Contrast the use of calculus with the use of abstraction in algebra and analysis to solve a real problem.) Contrast the use of calculus with the use of abstraction in algebra and analysis to solve a real problem.
- CM04 (Explain ideas and concepts of fundamental mathematics, communicating one's own reasoning to others.) Explain ideas and concepts of fundamental mathematics, communicating one's own reasoning to others.
- KM01 (Identify the essential ideas of the proofs of some basic algebra and calculus theorems.) Identify the essential ideas of the proofs of some basic algebra and calculus theorems.
- SM01 (Write small mathematical texts (exercises, solving theoretical questions, etc.) in an orderly and precise manner.) Write small mathematical texts (exercises, solving theoretical questions, etc.) in an orderly and precise manner.
- SM02 (Handle inequalities, number sequences and derivatives and integrals of functions in one and several variables.) Handle inequalities, number sequences and derivatives and integrals of functions in one and several variables.
Contents
1. Sequences of real numbers.
Limit of a sequence and algebraic properties.
Monotone sequences.
Accumulation points. Subsequences.
Bolzano-Weierstrass theorem.
Cauchy sequences.
Computation of limits.
2. Real functions.
Domain of a function.
Elementary functions.
Limit of a function at a point. One-sided limits. Properties of the limits. Asymptotes. Limits of functions.
Continuity of a function.
Bolzano's theorem.
Mean value theorem and Weierstrass theorem.
3.Derivatives.
Derivatives of a function at a point.
Calculation of some derivatives.
Tangent line equation.
Chain rule. Inverse functions and differentiation. Logarithmic differentiation.
Absolute and relative extreme values of a function.
Rolle's theorem. Mean value theorem.
Hôpital Rule.
Newton's method for finding numerical solutions of functions.
4. Approximation by Taylor polynomials.
Order of contact between functions.
Taylor polynomial. Properties
Taylor's formula. Taylor's residue.
Approximate calculations. Application to the computation of limits.
Local study of functions.
5. Integration
Primitives of a function.
Immediate integrals. Integrals by change of variable. Integrals by parts.
Integration of rational functions. Integration of irrational functions.
The fundamental theorem of calculus.
Applications of integration: flat areas, length of a curve, areas and volumes of solids of revolution.
Improper integrals Convergence criteria. Absolute convergence
6. Numerical series and power series.
Numerical series. Necessary condition of convergence.
Criteria of: comparison, quotient, root, integral.
Alternate series. Absolute convergence
Power series.Radius of Convergence.
Derivation and integration of power series.
Learning activities and methodology
| Title | Hours | ECTS | Learning outcomes |
|---|---|---|---|
| Problem and practice sessions | 22 | 0.88 | CM01, CM04, SM01, SM02 |
| Office hours | 16 | 0.64 | CM03, CM04, KM01, SM01, SM02 |
| At home work | 70 | 2.8 | CM01, KM01, SM01, SM02 |
| Theory sessions | 27 | 1.08 | CM01, CM03, CM04, KM01 |
| Exam preparation | 15 | 0.6 | CM01, CM03, CM04, KM01 |
The theory sessions, problem sessions and practice sessions are undistinguishable, so we will alternate them according to the needs of the course and the students.
In principle, the theory teacher will give the main ideas on the various subjects. The student must solve the proposed problems.
The professors of problems and of practices will solve the doubts that appear in the sessions and will propose methods for solving them.
Throughout the semester the student must solve and deliver problems. These deliveries will be part of the continuous evaluation of the subject.
The use of AI is permitted only during those activities that are not evaluated. The use of AI during exams is strictly forbidden, and it will be considered an attempt of cheating.
Assessment
Continuous assessment activities
| Title | Weight | Hours | ECTS | Learning outcomes |
|---|---|---|---|---|
| First midterm exam | 40% | 0 | 0 | CM01, CM03, CM04, KM01, SM01, SM02 |
| Practice (Sage) | 20% | 0 | 0 | CM03, KM01, SM01 |
| Second midterm exam | 40% | 0 | 0 | CM01, CM03, CM04, KM01, SM01, SM02 |
During the semester, we will have a first midterm exam, a second midterm exam and activities with Sage.
At the end of the semester there will be a recovery exam, where you can re-take either one or both of the midterms, if needed. The score on the recovery exam will replace the score previously obtained. The part with Sage cannot be recovered.
Your final grade will be the weighted average between your score on the first midterm exam (40%), the second midterm exam (40%) and the practical activities with Sage (20%). In order to pass the course, such average must be greater than or equal to 5, and the grade of each midterm exam must be greater than or equal to 2,5.
Students who choose the single assessment option will take a single exam on the day of the second midterm, covering the entire course syllabus. This exam will account for 80% of the final grade. The remaining 20% will come from a Sage exam.
Bibliography
1.S.L. Salas, E. Hille. 'Calculus' Vol. 1, Ed. Reverté, 2002.
2.Bartle, R.G., Shebert, D.R. (1996) Introducci ́on al An ́alisis Matem ́atico de una variable. 2a ed. Limusa. ISBN: 978-968-18-5191-0.
3.Ortega Aramburu, J.M. (2002). Introducci ́o a l’An`alisi Matem`atica. 2a ed. Manuals de la Universitat Aut`onoma de Barcelona.
4. Zill, D.G., Wright, W.S. (2011). Cálculo de una variable. 4a edició. McGrawHill. ISBN: 978-607-15-0501-9.
Software
SageMath
Course groups and languages
The information provided is provisional until November 30. After this date, you will be able to consult the language of each group through this link. To access the information, you will need to enter the course CODE
| Type of teaching | Group | Language | Semester | Shift |
|---|---|---|---|---|
| (TE) Theory | 1 | Spanish | first semester | morning-mixed |
| (PLAB) Practical laboratories | 1 | Catalan | first semester | morning-mixed |
| (SEM) Seminars | 1 | Catalan | first semester | morning-mixed |