Calculus I

Code:

M1001

Acronym:

M1001

Level:

100

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2018/2019 - 1S

Active?	Yes
Responsible unit:	Department of Mathematics
Course/CS Responsible:	Bachelor in Computer Science

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
L:B	0	Official Study Plan	3	-	6	56	162
L:CC	77	Plano de estudos a partir de 2014	1	-	6	56	162
L:F	0	Official Study Plan	2	-	6	56	162
			3
L:G	7	study plan from 2017/18	3	-	6	56	162
L:Q	0	study plan from 2016/17	3	-	6	56	162
MI:ERS	108	Plano Oficial desde ano letivo 2014	1	-	6	56	162

Teaching language

Suitable for English-speaking students

Objectives

To become acquainted with the basic concepts and techniques of calculus, at the level of real-valued functions of a single real variable, as well as sequences and series.

Learning outcomes and competences

Capacity of solving calculus problems. Autonomy on the solution of exercises.

Working method

Presencial

Program

0. Generalities on functions: 

Polynomial functions. Trigonometria functions. Exponential functions.

1. Limits and continuity:

Sequences of real numbers. Basic results on sequences. Real-valued functions of a real variable. Limits. Continuity. Intermediate Value Theorem and Weierstrass Extreme Value Theorem.

2. Derivatives and antiderivatives:

Derivatives. Geometric and physical interpretation of derivatives. Differentiation rules. Derivative of the inverse. Investes trigonometric functions and their derivatives.Theorems of Rolle, Lagrange and Cauchy, L ́Hôpital’s Rule. Applications to the study of the behaviour of a function and computation of minima and maxima. Antiderivatives and antiderivatives of elementary functions. Computing antiderivatives by substitution and by parts. Antiderivatives of rational functions.

3. Integration:

Riemann’s integral. Fundamental Theorem of Calculus. Integration by substitution and integration by parts. Computation of areas. Improper integrals.

4. Polynomial approximation and series:

Taylor polynomials. Numerical series. Basic properties. Convergence tests: Leibniz, ratio and integral.

Mandatory literature

Stewart James; Calculus. ISBN: 978-0-495-38273-7

Complementary Bibliography

Stewart James; Precalculus. ISBN: 978-0-495-55497-4
Spivak Michael; Calculus. ISBN: 0-914098-77-2
Chaves Gabriela; Cálculo Infinitesimal, Universidade do Porto

Teaching methods and learning activities

Exposition by the teacher, discussion of exercises.

Evaluation Type

Distributed evaluation without final exam

Assessment Components

designation	Weight (%)
Teste	100,00
Total:	100,00

Amount of time allocated to each course unit

designation	Time (hours)
Estudo autónomo	106,00
Frequência das aulas	52,00
Total:	158,00

Eligibility for exams

No requisites.

Calculation formula of final grade

The syllabus will be divided into two parts. each one evaluated by a test worth 10 points (2 hours long).

The first test will be held in the mid of the semester, the second test will be held during the first season exam period. The final mark of the first season is the sum of the marks obtained in each test (equal or above 9,5), except in the case of marks above 17, which will require an extra written proof.

Second season exam:

1. In the second season exam, students may repeat both tests or just one of them (except when they are just trying to improve their mark upon previous approval).

2. The mark of each part is the maximum of the marks obtained in the respective tests in both seasons (except when they are just trying to improve their mark upon previous approval).

3. The final mark of the second season is the sum of the marks obtained in both parts, rounded to integers, except in the case of marks above 17, which will require an extra written proof.

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Any other extraordinary evaluation may be preceded of an oral exam.