Mathematics I

Scholar Year: 2018/2019 - 1S

Courses

Head Teacher Responsability Anabela das Neves Pereira Head Paula Cristina Martins dos Reis Responsável (2º Semestre)

Weekly workload Hours/week T TP P PL L TC THE EL OT OT/PL TPL S Type of classes 0 0 Lectures Type Teacher Classes Hours Theorethical and Practical classes Totals 1 0,00 Laboratories Totals 1 0,00

Teaching language

Portuguese

Intended learning outcomes (Knowledges, skills and competencies to be developed by the students)

The aim of this course is to familiarize students with the mathematical method, providing them with skills to deal glibly with the mechanism of differential and integral calculus of functions of one real variable, in order to provide them the conditions to apply their knowledge in real life situations and in engineering.

Syllabus

1. Real functions of one real variable
Basic notions. Limits; properties. Continuous functions, properties, continuous extension. Intermediate-Value, Extreme-Value and inverse function theorem. Inverse trigonometric functions.
2. Differential calculus
Derivative and its interpretations; tangent and normal lines to a graph. Differentiable f., properties; differentiation rules; chain rule and inverse function theorem; derivatives of inverse trigonometric functions; differential. Rolle’s, Mean-Value and Cauchy’s Mean-Value theorem; Hospital’s rule. Higher-order derivatives; Taylor’s and Maclaurin’s formulas (Lagrange’s remainder). Monotony, local extrema, concavities.
3. Integral calculus
Antiderivatives; properties. Immediate antiderivatives. Methods: parts, substitution, decomposition. Riemann integral; properties. Indefinite integral; properties. Fundamental Theorem of Calculus, Barrow’s formula. Integration by parts and by substitution. Areas and volumes of solids of revolution. Improper integrals.

Demonstration of the syllabus coherence with the UC intended learning outcomes

In the theoretical-practical classes are presented the basic concepts of the different subjects of the syllabus and the proofs of the main results, followed by problems solving. In this type of classes students will acquire an overview of the themes and their interconnections.
In practical/laboratorial classes students will solve a set of exercises, under the guidance of a teacher and occasionally using the software MATLAB, allowing them to gain a deeper understanding of the subjects
discussed.

Teaching methodologies

In the theoretical-practical classes are presented the basic concepts of the different subjects of the syllabus and the proofs of the main results, followed by problems solving. In this type of classes students will acquire an overview of the themes and their interconnections.
In practical classes students will solve under the guidance of a teacher, a set of exercises, allowing them to gain a deeper understanding of the subjects discussed.

Demonstration of the teaching methodologies coherence with the curricular unit's intended learning outcomes

The student should be able to:
Understand basic notions on rf1rv;
Understand the concept and evaluate limits;
Identify continuous f., find continuous extensions;
Apply Intermediate-Value and Extreme-Value theor.;
Use inverse trigonometric f.;
Understand the meaning of the derivative;
Identify differentiable f. and evaluate derivatives;
Find and apply the differential;
Understand and apply Rolle’s, Mean-Value and Cauchy’s Mean-Value theor.;
Evaluate limits with Hospital’s rule;
Find and apply Taylor’s and Maclaurin’s formulas (Lagrange’s remainder);
Study f. monotony, local extrema and concavities;
Identify, formulate solve optimization problems;
Understand the notion of antiderivative and its properties;
Identify immediate antiderivatives and use methods for finding antiderivatives;
Understand Riemann integral;
Understand and apply the indefinite integral and its properties;
Evaluate integrals;
Evaluate areas and volumes of solids of revolution;
Identify and evaluate improper integrals

Assessment methodologies and evidences

The use of the Curricular Unit (UC) can be obtained through two evaluation processes: Continuous Assessment and Examination Evaluation.

Continuous evaluation
The continuous evaluation is based on the performance of two tests (with grades rounded to tenths). The approval conditions for continuous assessment are as follows:
1. If the average (rounded to the units) of the test scores is greater than or equal to 10 and less than 17, the student is approved with a final grade equal to said average, provided that in any of the tests the grade was higher or equal to 6.5;
2. If the average (rounded to the units) of the test scores is greater than or equal to 17, the student will have to take an oral test, the final grade being the average of these two marks. If you do not attend the oral test the final classification will be 16 values.

Retrieving one of the tests
In order to meet the approval conditions (final grade greater than or equal to 10 and a mark in both tests greater than or equal to 6.5 marks), a student who has a mark of 8.0 or higher in one of the tests has the option to perform the recovery of one and only one of the tests, on the same day and time as the normal period examination. In these conditions, if the student had a grade lower than 8.0 in one of the tests, could not perform or has given up, can only recover the test. You will not be able to recover a test for note improvement.

Examination Evaluation
The examination evaluation is based on the performance of an examination, the conditions of approval being as follows:
1. If the exam grade (rounded to the units) is greater than or equal to 10 and less than 17, the student is approved with a final grade equal to the grade of the exam (rounded to the units);
2. If the exam grade (rounded to the units) is greater than or equal to 17, the student will have to take an oral test, obtaining as a final mark the average of the classifications of said oral test and the exam. If you do not attend the oral test, the final grade will be 16 points.
Comments:
1. The tests have a duration of 2 hours and the tests 2 hours and 30 minutes, both of development and are rated on the scale of 0 to 20.
Working students, high-level athletes, association leaders and students under the Religious Freedom Act must, by the second week of the semester, contact the person responsible for the discipline, either personally or by email anabela.pereira@estsetubal.ips.pt , in order to present their specific characteristics, in accordance with the terms of the respective diplomas, failing which they can not be implemented due to lack of objective conditions.

Dates and times of evaluation:
Test 1: November 24, 2018 (at 10 o'clock)
Test 2: January 19, 2019 (at 10 o'clock)
Exam of Normal Season: February 4, 2019 at 9:30 am
Time Period Exam: February 18, 2019 at 9:30 am
Special Time Exam: date to be defined by the Pedagogical Council

Bibliography

Notes edited by the Department of Mathematics (available on the discipline page and Moodle).
Elementos de Cálculo Diferencial e Integral em R e R^n, Azenha, A., Jerónimo, M. A., McGraw Hill, 2000.
Larson, R. E., Hostetler, R. P., Edwards, B. H., Cálculo, Vol. I – 8ª edição, McGraw Hill, 2006.
Swokowski, E. W., Cálculo com Geometria Analítica, Vol. 1– 2ª edição, Makron Books, 1995.
Apostol, T, Cálculo – Vol. I – 2ª edição, Reverté, 1994.
Campos Ferreira, J., Introdução à Análise Matemática – 11ª edição, F. Calouste Gulbenkian, 2014.
Demidovitch, B., Problemas e Exercícios de Análise Matemática, Escolar Editora, 2010.
Stewart, J., Cálculo – 7ª edição, Cengage Learning, 2014.
Thomas, G. B., Finney, R. L., Cálculo, Vols. I e II – 11ª edição, Pearson-Addison Wesley, 2009.
Anton H., Bivens I., Stephen, D., Cálculo Vol. I – 10ª edição, Bookman, 2014.

Observations

Doubtful time in class:

Anabela Pereira: monday from 2 p.m. to 2.30 p.m. and tuesday from 1 p.m. to 2.30 p.m.
Ana Barros: monday from 11 am to 00:30 p.m. and tuesday from 4 to 5:30 p.m.
Artur Cruz: Friday from 9 a.m. to 11 a.m. and from 2 p.m. to 3 p.m.
Carla Rodrigues: thursday from 11 a.m. to 1 p.m. and friday form 10 a.m. to 11 a.m.
Cristina Almeida: tuesday from 5:30 p.m. to 7 p.m. and thursday from 3 p.m. to 4:30 p.m.
Teresa Ribeiro: tuesday from 2:30 p.m. to 4 p.m. and thursday from 11 a.m. to 00:30 p.m. Vanda Rosado: tuesday from 3:30 p.m. to 5.30 p.m.