Calculus II

It is recommended some knowledge of subjects of year 1 first semester mathematics (Mathematical Analysis I and Linear Algebra).

By acquiring knowledge of Mathematical Analysis II (AM2), the student will develop among others the competencies of abstraction, deduction, demonstration, algorithmic and programming of methods associated with the subjects of the AM2 with a computational component.

It is intended that the student realizes the importance of Mathematics and its structuring role as a basic science and tool to support a logical and structured reasoning indispensable to the areas of Computer Engineering. Also, show that it is impossible to solve and properly code a poorly formulated problem, that has ambiguity in its terms or for which it is not possible to collect the necessary data.

In addition to classes (theoretical, theoretical-practice and laboratory practices), where students are constantly called to participation and thus are not at all and only exhibition classes, students are also encouraged and even evaluated regarding their activity in distance participation through e-learning platforms, thus AM2 includes in itself a component of b-learning.

Throughout the semester, learning and evaluation activities (programming and other work) are proposed. In the evaluation tests some questions will be programming on the subjects of the AM2 program.

It is intended that the student realizes the importance of Mathematics and its structuring role as a basic science and tool to support a logical and structured reasoning that is indispensable in the areas of Engineering. Also, show that it is impossible to solve and properly code a poorly formulated problem, that has ambiguity in its terms or for which it is not possible to collect the necessary data.

Differential equations (ED) are essential tools in the mathematical modeling of many engineering problems. Some will be studied analytical, numerically and computationally approached in parallel by teaching programming language of Matlab.

The student, will develop abstraction, demonstration, application, algorithmic and programming skills associated with the topics of the curricular unit and other related subjects: Differential Equations (ED), Systems of Differential Equations (SED); Mathematical modeling of engineering problems translated by ED and SED; Numerical methods for ED, SED, Polynomial Interpolation, Derivatives and Numerical Integration; Differential Calculation in IR^n; Multiple Integrals; 3D Representation and Visualization, Coordinate Systems: Cartesian, polar, cylindrical and spherical; Basic and fundamental programming in Matlab and Programming of Graphical Interfaces – App in Matlab.

1. Ordinary Differential Equations (EDO)
Definition, properties, directional fields and graphs of general and particular solutions. First order Differential Equations (ED): ED of Separable Variables, firt order Linear ED and others to be solved computationally through functions of type dsolve (differential solve). Order n Linear Differential Equations with constant coefficients: Homogeneous and Complete Equations; Wronskian, Fundamental Solution System, Characteristic Equation and Method of variation of arbitrary constants. Differential equations as mathematical models of description/modeling of systems or physical phenomena. Initial value issues. Computational treatment in CAS and Matlab programs.

2. Linear Differential Equation Systems
Linear ED reduction algorithm of order 2 for a 2-ED system of 1st order. Application problems (dynamic systems), mathematical modeling, analytical and numerical resolution.

3. Differential Calculus in IR^n
Brief topological notions in IRn. Functions of several variables: domains, boundaries, continuity, partial derivatives, straight lines and tangent planes, increments and differentials, composite function derivative, directional derivative, gradient, simple and conditioned extremes. Representation of level curves and domains in 2D, 3D visualization of simple and composite surfaces and their computational treatment in CAS programs and Matlab.

4. Multiple Integrals
Double and triple integral. Coordinate systems: cartesian, polar, cylindrical and spherical. Applications: calculation of areas, volumes of solids and center of mass. Representation and visualization of solids and their computational treatment using CAS programs and Matlab.

5. Numerical Methods
5.1 Polynomial Interpolation: Polynomials. Definition of polynomial interpolator. Newton interpolator of divided differences. Linear, quadratic and cubic interpolation (Spline). Extrapolation. Two-dimensional interpolation. Adjustment of curves: Linear approximation (Least Squares Method); Polynomial approach (polyfit).
5.2 Numerical methods for EDO and EDO Systems: Euler’s methods and Runge-Kutta’s methods.
5.3 Numerical derivation: Finite differences formulas in 2 and 3 points: progressive, regressive and centered.
5.4 Numerical integration: trapezoids’ rule and Simpson’s rule applied to double integrals.
6. Programming in Matlab

Throughout the semester essential and fundamental training will be given in Matlab programming. Exploration and use of Matlab’s Symbolic Math Toolbox. Programming numerical methods. Graphical interfaces in Matlab – Apps in Matlab. Creation of applications for automatic resolution of derivatives and integrals and resolution of applied problems mathematically modeled by ED or ED Systems.

In AM2 there are two evaluation options:

1st option: [continuous and periodic evaluation. The students enrolled for the first time in this course can only access this option if their attendance to classes is at least 75%]
» Tests/Exams with a weight of 60%  -12 points on a scale of 0 to 20
- Normal Exam
- Recurso Exam
- Special Exam
» Programming work, learning activities and evaluation distributed throughout the semester with a weight of 40% - 8 points on a scale of 0 to 20

2nd Option: [evaluation by exames]
» Tests/Exams weighing 100%  - 20 points
- Normal Exam
- Recurso Exam
- Special Exam

For the purpose of calculating the final grade, the student must have at least 5 (five) values in the exame (quoted for 20 values) about the entire curriculum program.

The learning and evaluation activities distributed throughout the semester (computational mathematics programming work) will be carried out in groups of 3 students, which must be submitted by one of the group members (upload on the e-learning platform) which should include the name of all the members of the group and their student number. There are other types of activities that are performed individually. Whenever necessary or randomly, students will have defend the work/activities they have submitted. Submissions may be penalized if submitted after the due date. The assessment of the assignments also requires the submission of self-evaluation and peer evaluation of the work and group members.

Exams are conducted in person (with the potential use of a computer if conditions allow), and the final grade will be the best of the grades with or without activities, i.e.,

Grade = Max(60%*Exam+40%Work&Act; 100%*Exam)

» Additional Evaluation
i. If the student has regular class attendance (aA), is punctual, and has a proactive, participative, and constructive attitude in class:

• if aA = 100%, then add 1 point to the final grade
• if 90% <= aA < 100%, then add 0.75 points to the final grade
• if 75% <= aA < 90%, then add 0.5 points to the final grade
• if 50% <= aA < 75%, then add 0.25 points to the final grade

ii. Regarding attendance (aGF) at office hours and/or active participation in the course forums on the MatApl1 page on the moodle@isec platform, according to the forum participation rules and during the lecture period:

• if aGF = 100% of classes, then add 1 point to the final grade
• if 75% <= aGF < 100%, then add 0.75 points to the final grade
• if 50% <= aGF < 75%, then add 0.5 points to the final grade

If the final grade exceeds 18 points, students must undergo a supplementary exam (oral and/or written), but if they choose not to take this exam, the final grade will be eighteen (18) points. Conducting additional orals will be another assessment tool to consider, whenever necessary, to resolve doubts and/or suspected cases of fraud or plagiarism in student assessments. Any student who achieves a final rounded grade of ten (10) or higher will be considered to have passed the course. As attendance is not mandatory in this course unit and there are two assessment options, students with working-student status are subject to the same assessment method as the remaining students.

NAO