Calculus I

Code:

M1001

Acronym:

M1001

Level:

100

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2014/2015 - 1S

Active?	Yes
Web Page:	http://moodle.up.pt/course/view.php?id=3221
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:CC	84	Plano de estudos a partir de 2014	1	-	6	56	162
MI:ERS	130	Plano Oficial desde ano letivo 2014	1	-	6	56	162

Teaching language

Portuguese

Objectives

With this course, it is intended that students will know and understand
some of the main classical results of Calculus. These results,  for their historical
importance in the explanation of scientific phenomena and on the resolution of scientific problems, should be known by every student of the area of science.

Learning outcomes and competences

The student knows, understands and is able to use concepts that explain scientific phenomena, as well as techniques therefrom derived for solving scientific problems. In the process, he develops his knowledge of the physical world and his ability and taste for analysis, in addition to his ability to understand, use and develop abstract structures.

Working method

Presencial

Pre-requirements (prior knowledge) and co-requirements (common knowledge)

Basic techniques of calculus and basic knowledge of mathematics, which will be revised at the beginning of the course.

Program

0. Preliminaries
1. Calculus - Preliminaries
1.1. Functions, domain, codomain and image ; injective, surjective and bijective functions; graphs of functions; composite functions; the inverse of a bijective function.
1.2. Polynomial functions and their graphs. Dividing polynomials. Zeros of polynomials.
1.3. Trigonometric functions; properties. Inverse trigonometric functions. Exponential and Logarithmic functions.
1.4. Sequences and their limits.
1.5. Limits; definition and properties.
1.6. Continuous functions; definitions and properties, the Intermediate-Value Theorem, the Max-Min Theorem.
2. Differentiation
2.1. Definition of derivative, differentiation rules (sums and constant multiples, the product rule, the quotient rule, inverse functions). Derivatives of Trigonometric functions and of their inverse functions. Derivatives of exponential and logarithmic functions.
2.2. Rolle, Lagrange and Cauchy's theorem. Increasing and decreasing functions, maximum and minimum values.
2.3. Extreme-value problems.
2.4. Concavity and inflexictions.
2.5. Sketching the graph of a function.
3. Taylor polynomials.
4. Integration
4.1. The definite integral: definition and properties.
4.2. The fundamental Theorem of Calculus. The method of substitution, integration by parts. Integral of rational functions.
4.3. Improper integrals.
4.4. Areas and volumes of solids of revolution, length of  graphs of functions.
5. Series.
5.1. Definitions and properties.
5.2. Criteria for convergence of series: ratio, integral and Leibnitz criterion.

Mandatory literature

Spivak Michael; Calculus. ISBN: 0-914098-89-6
Stewart James; Calculus. ISBN: 978-0-495-38273-7
Adams Robert A.; Calculus. ISBN: 0-201-82823-5
Stewart James; Precalculus. ISBN: 978-0-495-55497-4

Teaching methods and learning activities

Lectures and classes: the contents of the syllabus will be presented in lectures, where numerous examples will be given to illustrate the concepts. There will also be practical classes, where exercises and related problems are solved. On the course site in the internet other supporting materials (texts, resolutions of the quizzes, etc..) will be available.

The first part of the course, reviewing the basic concepts and techniques of mathematics, will be supported by a program of e-learning specially created for this purpose, which will allow students to practice the basic techniques of calculus and simultaneously to self-assess their progress.

Evaluation Type

Distributed evaluation with final exam

Assessment Components

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

Eligibility for exams

For being accepted at the final exam, must be attend at least half of the practical classes (at least 7).

Students who were considered acceptable for exam in 2012/13 may be excused from attending TP classes. For this purpose, they must apply until October 17 following the instructions in Moodle.

Calculation formula of final grade

Final exam( for the 1st and 2nd exam time and for students that are not trying to improve the score), where two groups of questions can be replaced, at student's will, independently of one another, by the scores obtained in two querys:

. The first one, worth 3/20,  will be held on computer and assesses the basic knowledge and mastery of basic techniques that are reviewed in the beginning of the course.  It will correspond to chapters 1 of the syllabus.

. The second one, worth 5/20, will be held in November and will focus on the part of the syllabus after the initial revision hitherto taught.

In no other case will the student be allowed to replace part of the exam by a test. 

Any special exam can be either an oral or a written exam.

Special assessment (TE, DA, ...)

Any special exam can be either an oral and/or a written exam. It will not be allowed to replace any part of the exame by a test done during the semester.

Classification improvement

Exam. For these students it will not be allowed to replace any part of the exam by any test done during the semester.