Calculus II

Code:

M1003

Acronym:

M1003

Level:

100

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2020/2021 - 2S

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	97	Plano de estudos a partir de 2014	1	-	6	56	162
L:F	0	Official Study Plan	2	-	6	56	162
L:G	1	study plan from 2017/18	2	-	6	56	162
			3
L:Q	0	study plan from 2016/17	3	-	6	56	162
MI:ERS	124	Plano Oficial desde ano letivo 2014	1	-	6	56	162

Teaching language

Portuguese

Objectives

Acquisition of the basic knowledge and skills of Differential and Integral Calculus in several real variables.

Learning outcomes and competences

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

Working method

Presencial

Program

1. Parametrized curves in R^n:
Velocity, acceleration, curvature and torsion.

2. 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. Maxima and minima of real-valued multivariate functions. Second derivative test to find the local extremes.

3. Multivariable integrals:
Definition of integral of a multivariate real-valued function over a rectangle and a bounded region. Fubini's theorem.  Change of coordinates.  

Mandatory literature

Stewart James; Cálculo. ISBN: 85-221-0479-4 (Vol. I)

Complementary Bibliography

Adams e Essex; Calculus - A Complete Course, 2010. ISBN: 978-0-321-54928-0 (7th edition)
Jerrold E. Marsden; Vector calculus. ISBN: 978-1-4292-2404-8

Teaching methods and learning activities

Lectures and classes: The contents of the syllabus are presented in the lectures, where examples are given to illustrate the concepts. There are also practical lessons, where exercises and related problems are solved. All resources are available for students at the unit’s web page.

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	103,00
Frequência das aulas	56,00
Trabalho escrito	3,00
Total:	162,00

Eligibility for exams

Attendance is not required.

Calculation formula of final grade

Assessment in "época normal" will be based on two face-to-face tests, worth 8 and 12 points and on 10 quizzes. 
The first test will take 50 minutes and the last 70 minutes. The first test will be on a day indicated at the begining of classes and the second on the day of "exame da época normal".
The 10 quizzes are together worth 2 points (so 0.2 each) and it will be one per week during 10 weeks on computer using Moodle.

If T1 is the score of the first test, T2 the score of the second, Q the sum of the scores obtained in the 10 quizzes, a student will pass the course if

                     T1+T2+Q ≥ 9.5

and the final mark is 

                           min{ T1+T2+Q , 20 }.

If for for some reason it won't be possible to do face to face writen tests during the semester, the two tests will be done on the day of the exam (also face to face).

Make-up exam:

Students who still need to pass: exam with two parts, each part corresponding to one of the tests:

• The student may choose to take any parts of the exam or use classifications of corresponding parts obtained in tests.
• If they choose to do a part of the exam, it will be the classification obtained there for the final grade.
• the final grade is computed as above. 

Students trying to improve their grade will have to write the whole exam (face to face)  and they will not be allowed to use testes or quizzes.

Special assessment (TE, DA, ...)

Written or oral exam.

Classification improvement

Exam. These students will not be allowed to take part of
the continuous assessment of the course