Calculus I

Code:

EQ0059

Acronym:

AM I

Keywords
Classification	Keyword
OFICIAL	Physical Sciences (Mathematics)

Instance: 2015/2016 - 1S

Active?	Yes
Responsible unit:	Department of Chemical and Biological Engineering
Course/CS Responsible:	Master in Chemical Engineering

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
MIEQ	86	Syllabus	1	-	6	63	162

Teaching language

Portuguese

Objectives

This course aims to endow students with fundamental knowledge on mathematics. It also aims to complete students’ education on IR defined functions, and develop the problem of primitives and function integration and study its representation

Objectives

- Understand, manipulate and apply the concepts of integration of functions of one variable and series.

- Provide a base set of mathematical fundamental to the proper functioning of other courses of integrated Masters.

- Develop scientific and mathematical reasoning and ability to apply mathematical concepts acquired.

Learning outcomes and competences

Ability to describe the main results in the field of differential and integral calculus, the series of real numbers and the polynomial approximation of functions of one variable by use of Taylor polynomials.

Ability to identify and properly apply the techniques to use in solving problems.

Obtaining a set of fundamental mathematical tools with direct application in other courses of the integrated MSc.

Working method

Presencial

Program

1. Functions, limits and derivatives
Revision of concepts studied in Secondary School about functions, limits, continuity and differentiation.
2. Complements on functions and differential calculus
New functions and their derivatives. Inverse functions and their derivatives. Local linear approximation and differentials. Relative extrema. Abolute extrema. Indeterminate forms and L’Hôpital’s rule.
3. Antiderivative of a function
Antiderivative of an elementary function. Integration by parts. Integation by substitution. Integration of rational functions. Rational substitution. Trigonometric antiderivatives. Trigonometric substitutions.
4. Integral Calculus
Approximate calculation of areas. Riemann sum. Definite Integral. Basic properties of definite integrals; Assessment of integrals of continuous functions. Mean value of f in (a,b). Mean value theorem. Integration of discontinuous functions. Geometric applications of integrals. Deduction of integral formulas. Functions defined by integrals. Improper integrals.
5. Series of real numbers
Series of real numbers. Study of series by its definition. Series of non-negative terms. Series of non-positive terms. Alternating series. Series of positive and negative terms.
6. Taylor and Maclaurian series
Function approximation by polynomials. Estimative of the nth remainder for a Taylor polynomial. Taylor’s theorem and Taylor's formula with remainder. Taylor series (or Maclaurian series) for a function. Power series.

Mandatory literature

João Mendonça;Matemática I (Módulo de Análise Matemática) - Apontamentos das aulas, 2007/08

Complementary Bibliography

Edwards, C. Henry; Calculus. ISBN: 0-13-095006-8
Anton, Howard; Calculus. ISBN: 0-471-48237-4
Ana Alves de Sá e Bento Louro; Sucessões e Séries, Escolar Editora, 2009. ISBN: 978-972-592-222-4

Teaching methods and learning activities

General theoretical-practical classes are based on the presentation of the themes of the course, where examples are given. The practical classes are based on problem solving and application of the themes that have been taught during theoretical-practical classes

Evaluation Type

Distributed evaluation with final exam

Assessment Components

Designation	Weight (%)
Exame	75,00
Teste	25,00
Total:	100,00

Eligibility for exams

Obtaining frequency for regular students depends on:

• not exceeding a maximum of 3 laboratory class misses and
• obtain a minimum grade of 6 in distributed evaluation (AD).

Students with frequency from previous years do not need to attend classes. Students who wish to attend must obey the above rules.

Calculation formula of final grade

Final mark (FM) will be based on one of the following formulas:

• FM = max(0.25 x MT + 0.35 x T1 + 0.4 x T2, 0.5 x T1 + 0.5 x T2)

or

• FM = max(0.25 x MT + 0.75 x EF, EF)

wherein:

MT = arithmetic average of the three best marks obtained in mini-tests

T1 = mark obtained in Test 1

T2 = mark obtained in Test 2

EF = mark obtained in final exam

Conditions for course approval:

• minimum grade of 6 values on MT
• minimum grade of 6 values on both T1 and T2 or EF

The realization of the mini-tests is mandatory for all students without previous frequency. The other may choose as venue and must communicate its decision to the teacher in the first week of classes. Not performing a mini-test on the date set corresponds to zero mark.

Special assessment (TE, DA, ...)

An exam at the corresponding seasons.

Classification improvement

Improvement of classification can be attempted in the Recurso season. The computation formula is identical to the one used in final classification described above.

Observations

Students cannot use calculators on mini-tests, tests and exams.