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Real Analysis I
| Code: | M1011 | Acronym: | M1011 | Level: | 100 |
| Keywords | |
|---|---|
| Classification | Keyword |
| OFICIAL | Mathematics |
Instance: 2020/2021 - 1S 
| Active? | Yes |
| Responsible unit: | Department of Mathematics |
| Course/CS Responsible: | Bachelor in Mathematics |
Cycles of Study/Courses
| Acronym | No. of Students | Study Plan | Curricular Years | Credits UCN | Credits ECTS | Contact hours | Total Time |
|---|---|---|---|---|---|---|---|
| L:B | 1 | Official Study Plan | 3 | - | 9 | 84 | 243 |
| L:G | 0 | study plan from 2017/18 | 2 | - | 9 | 84 | 243 |
| 3 | |||||||
| L:M | 109 | Official Study Plan | 1 | - | 9 | 84 | 243 |
| L:Q | 1 | study plan from 2016/17 | 3 | - | 9 | 84 | 243 |
Teaching language
PortugueseObjectives
Understanding and ability to use the basic concepts and results related to the subjects of the syllabus.Learning outcomes and competences
Upon completing this course, the student must know, understand and be able to use the basic notions and results about Real numbers (algebraic structure, order, completeness), functions (injectivity, overjectivity, image and reciprocal image of sets, composition, function inverse, monotony, maximums and minimums), successions as examples of functions, graphs, limits and continuity (definition and uniqueness of a function's limit at a point, lateral limits and limit arithmetic, limits at infinity, infinite and asymptote limits, continuous functions , Bolzano and Weierstrass theorems), derivatives (geometric motivation and definition, calculation of derivative elementary functions and algebraic properties, derived from the composite function of the inverse function, cancellation of the derivative at the local extremes, Rolle, Lagrange and Cauchy theorems, applications to determination of extremes of functions, monotony intervals and the drawing of graphs, second derivative and concavity; L'Hôpital's Rule), primitives and integrals (primitives of elementary functions, primitization by parts and substitution, concept of area, integral of a function limited in an interval, basic properties of integrals, Fundamental Theorem of Calculus and consequences, calculation of integrals , improper integrals: case of continuous functions defined in non-limited intervals and case of continuous functions not limited in an interval), poilnomial approximations (Taylor polynomial, tangency of degree n of a function to its Taylor polynomial of order n at a fixed point, rest, calculation of approximate values of functions).
Working method
PresencialProgram
1. Real Numbers: algebraic structure, order, completeness2. Functions: injectivity, surjectivity, image and reciprocal image of sets, composition, inverse function, monotony, maximums and minimums; sequences as examples of functions; graphs
3. Limits and Continuity: definition and uniqueness of the limit of a function at a point, lateral limits and limit arithmetic; infinite limits, infinite and asymptotes. Continuous functions; Bolzano and Weierstrass theorems
4. Derivatives: geometric motivation and definition; calculation of derivatives of elementary functions and algebraic properties; derivative of the composite function and inverse function; cancellation of the derivative at the local extremes; Rolle, Lagrange and Cauchy theorems; applications to the determination of function extremes, monotony intervals and the drawing of graphs; second derivative and concavity; undetermined, L'Hôpital's Rule
5. Primitives and Integrals: primitives of elementary functions; primitives by parts and substitution; concept of area, integral of a function limited in an interval; basic properties of integrals; Fundamental Theorem of Calculus and consequences, integral calculus; improper integrals: case of continuous functions defined in non-limited intervals and case of continuous functions not limited in an interval.
6. Poilnomial approximations: Taylor polynomial, tangency of degree n of a function to its Taylor polynomial of order n at a fixed point, remainder, calculation of approximate values of functions.
Mandatory literature
Michael Spivak; Calculus. ISBN: 0-914098-89-6Complementary Bibliography
Joseph W. Kitchen Jr.; CalculusRobert A. Adams; Calculus. ISBN: 0-201-39607-6
Teaching methods and learning activities
The contact hours are distributed in theoretical and theoretical-practical classes. In the first, the contents of the program are presented, using examples to illustrate the concepts dealt with and guide students. In theoretical-practical classes exercises and problems, previously indicated, are solved. Support materials are available on the course page. In addition to classes, there are weekly service periods where students have the opportunity to answer questions.
Evaluation Type
Distributed evaluation without final examAssessment Components
| designation | Weight (%) |
|---|---|
| Teste | 100,00 |
| Total: | 100,00 |
Amount of time allocated to each course unit
| designation | Time (hours) |
|---|---|
| Estudo autónomo | 159,00 |
| Frequência das aulas | 84,00 |
| Total: | 243,00 |
Eligibility for exams
No condition is required to obtain frequency.Calculation formula of final grade
Two tests will be carried out during the semester, worth respectively 8 and 12 points.
Four mini-tests will be carried out in TP classes, each one worth 1 point; the classification of each mini-test can be used to replace the classification of a corresponding question in one of the tests (if it is decided that TP classes will not be in person, the mini-tests will be canceled and not replaced).
To be approved it is necessary to have a minimum classification of 3 points in the first test and 5 points in the second.
The final classification will be the sum of the classification of the two tests.
If tests are not allowed during the semester, both tests will be done on the day for the exam of the normal season, the duration being in accordance with the rules imposed by FCUP.
There will be an exam at the time of appeal, accessible to any student who has not passed the normal time and who has obtained frequency.
Both in the normal season and in the appeal season, a complementary test may be required to assign ratings higher than 16 values.
Four mini-tests will be carried out in TP classes, each one worth 1 point; the classification of each mini-test can be used to replace the classification of a corresponding question in one of the tests (if it is decided that TP classes will not be in person, the mini-tests will be canceled and not replaced).
To be approved it is necessary to have a minimum classification of 3 points in the first test and 5 points in the second.
The final classification will be the sum of the classification of the two tests.
If tests are not allowed during the semester, both tests will be done on the day for the exam of the normal season, the duration being in accordance with the rules imposed by FCUP.
There will be an exam at the time of appeal, accessible to any student who has not passed the normal time and who has obtained frequency.
Both in the normal season and in the appeal season, a complementary test may be required to assign ratings higher than 16 values.
Special assessment (TE, DA, ...)
Any examination required under special statutes will consist of a written test which may be preceded by a previous oral or written test.Observations
Any student may be required to take an oral test to clarify any doubts that may have arisen regarding the tests or assessment work.
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Page created on: 2025-06-15 at 09:49:35 | Acceptable Use Policy | Data Protection Policy | Complaint Portal
Page created on: 2025-06-15 at 09:49:35 | Acceptable Use Policy | Data Protection Policy | Complaint Portal