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# Calculus I

 Code: M1001 Acronym: M1001 Level: 100

Keywords
Classification Keyword
OFICIAL Mathematics

## Instance: 2019/2020 - 1S

 Active? Yes Responsible unit: Department of Mathematics Course/CS Responsible: First Degree 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 study plan from 2016/17 3 - 6 56 162
L:CC 81 Plano de estudos a partir de 2014 1 - 6 56 162
L:F 0 study plan from 2017/18 2 - 6 56 162
3
L:G 0 study plan from 2017/18 2 - 6 56 162
3
L:Q 0 study plan from 2016/17 3 - 6 56 162
MI:ERS 110 Plano Oficial desde ano letivo 2014 1 - 6 56 162

### Teaching Staff - Responsibilities

Teacher Responsibility
Jorge Manuel Meneses Guimarães de Almeida

### Teaching - Hours

 Theoretical classes: 2,00 Theoretical and practical : 2,00
Type Teacher Classes Hour
Theoretical classes Totals 2 4,00
Jorge Manuel Meneses Guimarães de Almeida 2,00
Theoretical and practical Totals 5 10,00
Jorge Manuel Meneses Guimarães de Almeida 2,00
António José de Oliveira Machiavelo 2,00
Mário Alexandre Duarte Magalhães 2,00

Portuguese

### 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.

Presencial

### Program

0. Generalities on functions:

Polynomial functions. Trigonometric 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. Inverse 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
Joseph W. Kitchen Jr.; Calculus
Chaves Gabriela; Cálculo Infinitesimal, Universidade do Porto

### Teaching methods and learning activities

Presentation of the course material by the teacher, discussion of exercises.

sage

### 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

No requisites.

### Calculation formula of final grade

The student evaluation wil be done in two parts. During the semester, it will consist of two tests, equally vallued. The students that obtain a grade of at least 10 on the average of the two tests are exempt from taking the final exam but may, nevertheless, choose to take it.

All students are admitted to the final exam. Both the final exam and the makeup exam will also be divided in two parts, roughly corresponding to the material covered by each test, with the exception of material covered after the second test which will. also be covered by the second parts of the exams.

In each part of the exams, the grade will be the best grade obtained for that part of the course material in any test or exam taken up to that point during the academic year. In particular, a student who is already happy with the grade obtained for a part of the course material may choose to take an exam only on the other part, a rule that applies also to students exempted from the final exam.

All grade values are obtained by truncation to centesimals and successive rounding to decimals and units.

### Special assessment (TE, DA, ...)

Any extraordinary evaluation may be complemented or preceded by an oral exam.