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Fundamentals of Calculus

Code:

QAA18

Sigla:

FC

Áreas Científicas
Classificação	Área Científica
OFICIAL	Matemática

Ocorrência: 2023/2024 - 1S

Ativa?	Yes
Página Web:	https://moodle.ips.pt/2324/course/view.php?id=2263
Unidade Responsável:	Departamento de Matemática
Curso/CE Responsável:	Professional Technical Higher Education Courses in Environment and Safety Management

Ciclos de Estudo/Cursos

Sigla	Nº de Estudantes	Plano de Estudos	Anos Curriculares	Créditos UCN	Créditos ECTS	Horas de Contacto	Horas Totais
QAA	33	Plano_estudos_2018_19	1	-	6	60	162

Língua de trabalho

Portuguese

Objetivos

Providing students with the basic mathematical knowledge on real number calculus and functions of a single real variable required for task fulfillment and as basis of continuous  professional updating of a top professional technician.

Resultados de aprendizagem e competências

1. Recalling of logical operators, quantificators, and interpreting them in mathematical statements.
2. Recognizing the different ordered number sets, its arithmetic operations and properties.
3. Solving equations, inequations and systems of linear equations.
4.  Identifying a real function of a single real variable and recognizing the elementary mathematical functions.
5. Analyzing the properties of a real function of a single real variable and performing operations with elementary mathematical functions.
6. Computing the derivative function of a real function of a
single real variable.
7. Applying calculus of derivatives to solve problems of geometrical or physical nature, or from real life situations.
8. Interpreting and describing characteristics of objects present in calculus.

Modo de trabalho

Presencial

Pré-requisitos (conhecimentos prévios) e co-requisitos (conhecimentos simultâneos)

Mathematical knowledge corresponding to introductory  secondary education, namely number arithmetical operations and ordering notions, rational numbers, and  equation and inequation solving skills.

Programa

1. Logic and set theory

1.1 Algebra of statements: Operators, priorities and properties. 1.2 Predicates and sets; universal and existential quantificators.
1.3 Operators and relations in set theory.
1.4 Mappings between sets. Injective, surjective and bijective mappings.

2. Real numbers and operations.

2.1 Natural numbers. Integer, rational, irrational numbers. 
2.2 Arithmetic operations and order relation for real numbers.
2.3 Roots and its properties.
2.4 Power with rational exponent and its properties..

3. Equations and Inequations.

3.1 Pollynomial division and Ruffini algorithm; roots and multiplicity; polynomial factorization.
3.2 Equations and inequations of 1st and 2nd degree.
3.3 Systems of equations.

4. Real functions of a single real variable.

4.1 Real function of a single real variable: its graphic and analytical expression.
4.2 Composition of functions and inverse function.
4.3 Monotony, extremal values, concavity, parity and periodicity.
4.4 The trigonometrical circle and functions.
4.5 Functions described by branches and modulus function.
4.6 Exponential and logarithmic functions. Power and logarithms in arbitrary base.

5. Derivatives of real functions of a single real variable.

5.1 Mean variation rate of a function on an interval. Instantaneous variation rate at a point. Geometrical and physical interpretation.
5.2 Derivative function. Derivation rules and calculus of derivatives.
5.3 Tangent and normal lines to a function's graphic.

Bibliografia Obrigatória

Michael Sullivan; Precalculus, Pearson, 2020. ISBN: 9780136872733

Bibliografia Complementar

João Paulo Santos; Cálculo numa variável real, IST Press, 2013. ISBN: 9789898481184
Edward T. Dowling; Cálculo para economia, gestão e ciências sociais, McGraw-Hill, 1994. ISBN: 972-9241-29-5
Jaime Campos Ferreira; Introdução à Análise Matemática, Fundação Calouste Gulbenkian,, 2008. ISBN: 9789723101799
Ron Larson, Robert P. Hostetler, Bruce H. Edwards; Cálculo Vol.1, McGraw-Hill, 2006. ISBN: 85-86804-56-8

Observações Bibliográficas

Lecture notes, exercise book, slides and other supporting electronic materials are avaliable at the moodle page of the curricular unit.

Métodos de ensino e atividades de aprendizagem

Classes are theoretical-practical where the fundamental concepts of the different subjects of the program are presented and some exercises that illustrate the topics are solved. Then students will carry out, under the guidance of the teacher, a set of exercises that will allow them to obtain a deeper understanding of the topics under consideration.

Central notions are introduced with a rigorous mathematical language, and its properties are analysed, identifying the most relevant results regarding them, with the help of examples and counter-examples.

Solving of exercises, first by direct application of the studied properties, and later by extension of these ideas to broader areas, eanbles the identification and resolution of doubts the students may have about the meaning of these objects and its properties.

Proposal of practical exercises for the students to solve outside of the classroom and the presentation of these resolutions within the classroom allows to secure the adquired knowledge.

Tools for the lesson: Whiteboard, Computer with projector, virtual teaching environments (Moodle/Teams), Lesson Notes and Exercise collection.

Palavras Chave

Physical sciences > Mathematics > Mathematical analysis > Functions
Physical sciences > Mathematics > Algebra > Set theory
Physical sciences > Mathematics > Applied mathematics > Engineering mathematics

Tipo de avaliação

Distributed evaluation without final exam

Componentes de Avaliação

Designation	Peso (%)
Participação presencial	0,00
Teste	100,00
Total:	100,00

Componentes de Ocupação

Designation	Tempo (Horas)
Estudo autónomo	102,00
Frequência das aulas	60,00
Total:	162,00

Obtenção de frequência

Students have two systems for evaluation: continuous evaluation or evaluation by exam.  

 

CONTINUOUS EVALUATION  

Continuous evaluation assumes a compulsory presence in at least 75% ot presential lessons, for students of 1st inscription in the 1st year, and consists on the execution of 2 tests and 1 minitest (open answer questions)

Taking as T1, T2 the test results (evaluated from 0 to 8 points, rounded to the decimal unit), and MT3 the mini-test result (evaluated from 0 to 4 points, rounded to the decimal unit), the final score FS (rounded to units) is computed as follows:  

 

FS = T1 + T2 + MT3.  

 

Conditions to obtain positive evaluation are:  

1. If FS is greater than or equal to 10 and lower than 18, the student has positive evaluation with FS as final grade, provided that T1 and T2 are both greater than or equal to 2.5 points. 
2. In the case that the student doesn't have positive evaluation due to conditions indicated on (1.), he/she may choose to take an alternative exam to improve the lowest of the results T1 or T2, provided that one of these  is greater than or equal to 2.5 points.
3. If FS is greater than or equal to 18, the student must do an oral test. The final grade will be the average of these two grades. If the student does not attend the oral test, the final grade will be 17 values.

 

EVALUATION BY EXAM  

 

Students that don't reach a positive result in the continuous evaluation may take an exam, with results in the integer scale between 0 and 20 points, and where the positive evaluation is for those students that get 10 or more points.  

 

In any of these evaluation systems, if the final score is greater or equal 18, the student will be subject to an additional oral exam. The final grade in this case will be the mean of the written and oral evaluations. If the student does not attend the oral exam, its final score will be 17 points.  

Fórmula de cálculo da classificação final

FS = T1 + T2 + MT3.  

FS - Final score (integer values range 0-20)

T1 - test 1 score (values range 0-8 rounded to decimal unit)

T2 - test 2 score (values range 0-8 rounded to decimal unit)

MT3 - Mini-test 3 score (values range 0-4 rounded to decimal unit)

Avaliação especial (TE, DA, ...)

Workers, top-level athletes, association leaders and students under the Law of Religious Freedom must present to the Head of the curricular unit their relevant specificities until the second week of the semester, in accordance with the respective diplomas, under penalty of being unable to be executed for lack of objective conditions.

Melhoria de classificação

As indicated in 11th article of "Regulamento das Atividades Académicas e Linhas Orientadoras de Avaliação de Desempenho Escolar dos Estudantes do IPS", the option to improve the avaliaton results may occur in the same year of inscription through the "Época de Recurso" Exam, or in the next year following  the positive avaliation of the curricular unit, using any of the regular avaliation processes, except for the "Época Especial" exams.

Observações

• Each test has a duration of 90 minutes, the mini-test has a duration of 45 minutes; The alternative exam has 90 minutes, and the exam is 2 hours and 30 minutes.  


• In all evaluation processes the presentation of an ID with photo is compulsory.


• In all written exams student may consult an specific formula sheet provided by the teachers.


• The use or merely reaching for any electronic devices during the evaluation proofs is forbidden.


• In case of fraud suspicion or other circumstance that leads to the need to assess the knowledge evidenced in the resolution of a test or exam, the student may be called to a session, in person, which will focus only on that knowledge. If the student does not attend this session, without due justification, the test or exam will be considered invalid.