Calculus II

Code:

M1003

Acronym:

M1003

Level:

100

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2015/2016 - 2S

Active?	Yes
Web Page:	http://moodle.up.pt/course/view.php?id=2968
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	89	Plano de estudos a partir de 2014	1	-	6	56	162
MI:ERS	155	Plano Oficial desde ano letivo 2014	1	-	6	56	162

Teaching language

Portuguese

Objectives

Understanding and ability to make use of the concepts and results covered in the syllabus, namely through the resolution of exercises of practical nature.

Learning outcomes and competences

Ability to make use of the concepts and results covered in the syllabus.

Working method

Presencial

Program

I Parametrized curves. 
Velocity, acceleration, curvature, Frenet frame. 

II. Differential calculus of vector-valued multivariate functions. 
Graphs of real-valued functions of two variables, contour lines of functions of two variables and level surfaces of functions of three variables. Open and closed subsets of R^n. Accumulation point and isolated point. Limits and continuity of functions. Directional derivatives and partial derivatives. Derivative function at a point of a real-valued multivariate function. Gradient vector and derivability. Tangent plane to the graph of a function of two variables. Interpretation of the gradient vector. Normal line and tangent hiperplane at a point on the level surface of a function. Higher order derivatives. Derivative function at a point of a vector-valued multivariate function. Jacobian matrix. Derivation of composition of functions. Examples. Inverse function theorem.

Maxima and minima of real-valued multivariate functions. Second derivative test to find the local extremes. The method of Lagrange multipliers for finding extreme values of constrained functions. 

III. Multiple integrals. 
Definition of integral of a multivariate real-valued function over a rectangle and a bounded region. Fubini's theorem. Calculation of double and triple integrals via iterated integrals. Integration and the change of coordinates. Applications: double integrals in polar coordinates, and triple integrals in cylindrical and spherical coordinates.

Mandatory literature

000094380. ISBN: 0-321-27000-2
000102128. ISBN: 7167-4992-0
000098594. ISBN: 85-221-0479-4 (Vol. I)
www.stewartcalculus.com
Chaves, G.; Cálculo Infinitesimal
Jorge Picado; Apontamentos de geometria diferencial, Departamento de Matemática, Universidade de Coimbra
Maria do Rosário Pinho, Maria Margarida Ferreira; Análise Matemática II, Nots das aulas teóricas, FEUP, Fev 2004, 2004

Teaching methods and learning activities

Exposition by the lecturers. Exercise sheets will also be available to students, as well as a list of recommended exercises which will tentatively be covered in the practical sessions. The latter will be available in advance, thus encouraging students to prepare for the class. Other materials will be accessible to students, such as quizzes and exams from previous years, solutions and resolutions.

The teachers involved in the class will have regular office hours to help the students and will be available to discuss the students’ performance in quizzes and in class. This should help the students to assess their progress and to promote a timely intervention in case of poor performance.

Evaluation Type

Distributed evaluation with final exam

Assessment Components

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

Eligibility for exams

Students are required to attend at least 8 practical classes/discussion sections. 

Students having met attendence requirements in the previous academic year will be exempt from the above rule.

Calculation formula of final grade

Final exam. In the first exam(for students that are not trying to improve the final grade), two groups of questions (out of 3) can be replaced independently of one another, by the scores obtained in two short test:

- the first one, worth 4/20, about the first chapter;

- the second one, worth 8/20, will focus on the second topic of the syllabus.

In the second exam (also for student that are not trying to improve their final grade) the first two groups of questions can be replacede either by the correspondent test or by the correspondent ones of the first exam. The third and last group of this exam can be replaced by the score obtained in third group of questions of  the first exam. The decision of replacing a group of questions by the score obtained previously is taken only by the student after he sees the whole exame.

In no other case will the student be allowed to replace part of the exam. Students trying to improve the score have to write the whole exam.

Any special exam can be either an oral or a written exam. No part of these exams can be replaced by the score obtained in a test.

Examinations or Special Assignments

Any special exam can be either an oral or a written exam. No part of these exams can be replaced by the score obtained in a test.

Special assessment (TE, DA, ...)

Special exams will consist of a written test, which might be preceded by an eliminatory oral test to assess whether the student satisfies minimum requirements to tentatively pass the written test.
No part of these exams can be replaced by the score obtained in a test.

Classification improvement

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

Observations

Any student can be asked to do an oral examination in case there are some dougts about the written examination. If such a exam happens it will be worth 50% of the final score.