Difference between revisions of "BSc: Mathematical Analysis II.s23"
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==== Section 1 ==== |
==== Section 1 ==== |
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| + | # |
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| − | # A plane curve is given by <math>x(t)=-\frac{t^2+4t+8}{t+2}</math>, <math display="inline">y(t)=\frac{t^2+9t+22}{t+6}</math>. Find |
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| − | #: the asymptotes of this curve; |
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| − | #: the derivative <math display="inline">y'_x</math>. |
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| − | # Apply Leibniz formula Find <math display="inline">y^{(n)}(x)</math> if <math display="inline">y(x)=\left(x^2-2\right)\cos2x\sin3x</math>. |
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| − | #: Draw graphs of functions |
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| − | #: Find asymptotes |
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| − | # Find the derivatives of the following functions: |
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| − | #* <math display="inline">f(x)=\log_{|\sin x|}\sqrt[6]{x^2+6}</math>; |
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| − | #* <math display="inline">y(x)</math> that is given implicitly by <math display="inline">x^3+5xy+y^3=0</math>. |
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==== Section 2 ==== |
==== Section 2 ==== |
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| + | # |
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| − | # Find the following integrals: |
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| − | #*<math display="inline">\int\frac{\sqrt{4+x^2}+2\sqrt{4-x^2}}{\sqrt{16-x^4}}\,dx</math>; |
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| − | #*<math display="inline">\int2^{2x}e^x\,dx</math>; |
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| − | #*<math display="inline">\int\frac{dx}{3x^2-x^4}</math>. |
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| − | # Find the indefinite integral <math display="inline">\displaystyle\int x\ln\left(x+\sqrt{x^2-1}\right)\,dx</math>. |
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| − | # Find the length of a curve given by <math display="inline">y=\ln\sin x</math>, <math display="inline">\frac{\pi}4\leqslant x\leqslant\frac{\pi}2</math>. |
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==== Section 3 ==== |
==== Section 3 ==== |
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| + | # |
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| − | # Find limits of the following sequences or prove that they do not exist: |
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| − | #* <math>a_n=n-\sqrt{n^2-70n+1400}</math>; |
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| − | #* <math display="inline">d_n=\left(\frac{2n-4}{2n+1}\right)^{n}</math>; |
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| − | #* <math display="inline">x_n=\frac{\left(2n^2+1\right)^6(n-1)^2}{\left(n^7+1000n^6-3\right)^2}</math>. |
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=== Final assessment === |
=== Final assessment === |
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Revision as of 18:30, 23 June 2022
Mathematical Analysis II
- Course name: Mathematical Analysis II
- Code discipline: CSE203
- Subject area: Math
Short Description
- Series: convergence, approximation
- Multivariate calculus: derivatives, differentials, maxima and minima
- Multivariate integration
- Basics of vector analysis
Course Topics
| Section | Topics within the section |
|---|---|
| Infinite Series |
|
| Partial Differentiation |
|
| Multiple Integration |
|
| Vector Analysis |
|
Intended Learning Outcomes (ILOs)
What is the main purpose of this course?
The goal of the course is to study basic mathematical concepts that will be required in further studies. The course is based on Mathematical Analysis I, and the concepts studied there are widely used in this course. The course covers differentiation and integration of functions of several variables. Some more advanced concepts, as uniform convergence of series and integrals, are also considered, since they are important for understanding applicability of many theorems of mathematical analysis. In the end of the course some useful applications are covered, such as gamma-function, beta-function, and Fourier transform.
ILOs defined at three levels
We specify the intended learning outcomes at three levels: conceptual knowledge, practical skills, and comprehensive skills.
Level 1: What concepts should a student know/remember/explain?
By the end of the course, the students should be able to ...
- know how to find minima and maxima of a function subject to a constraint
- know how to represent double integrals as iterated integrals and vice versa
- know what the length of a curve and the area of a surface is
Level 2: What basic practical skills should a student be able to perform?
By the end of the course, the students should be able to ...
- find partial and directional derivatives of functions of several variables;
- find maxima and minima for a function of several variables
- use Fubini theorem for calculating multiple integrals
- calculate line and path integrals
Level 3: What complex comprehensive skills should a student be able to apply in real-life scenarios?
By the end of the course, the students should be able to ...
- find multiple, path, surface integrals
- find the range of a function in a given domain
- decompose a function into infinite series
Grading
Course grading range
| Grade | Range | Description of performance |
|---|---|---|
| A. Excellent | 90-100 | - |
| B. Good | 75-89 | - |
| C. Satisfactory | 60-74 | - |
| D. Fail | 0-59 | - |
Course activities and grading breakdown
| Activity Type | Percentage of the overall course grade |
|---|---|
| Midterm | 20 |
| Quizzes | 28 (2 for each) |
| Final exam | 50 |
| In-class participation | 7 (including 5 extras) |
Recommendations for students on how to succeed in the course
- Participation is important. Attending lectures is the key to success in this course.
- Review lecture materials before classes to do well.
- Reading the recommended literature is obligatory, and will give you a deeper understanding of the material.
Resources, literature and reference materials
Open access resources
- Jerrold E. Marsden and Alan Weinstein, Calculus I, II, and II. Springer-Verlag, Second Edition 1985
- Robert A. Adams, Christopher Essex (2017) Calculus. A Complete Course, Pearson
Software and tools used within the course
- No.
Activities and Teaching Methods
| Teaching Techniques | Section 1 | Section 2 | Section 3 | Section 4 |
|---|---|---|---|---|
| Problem-based learning (students learn by solving open-ended problems without a strictly-defined solution) | 0 | 0 | 0 | 0 |
| Project-based learning (students work on a project) | 0 | 0 | 0 | 0 |
| Modular learning (facilitated self-study) | 0 | 0 | 0 | 0 |
| Differentiated learning (provide tasks and activities at several levels of difficulty to fit students needs and level) | 1 | 1 | 1 | 1 |
| Contextual learning (activities and tasks are connected to the real world to make it easier for students to relate to them) | 0 | 0 | 0 | 0 |
| Business game (learn by playing a game that incorporates the principles of the material covered within the course) | 0 | 0 | 0 | 0 |
| Inquiry-based learning | 0 | 0 | 0 | 0 |
| Just-in-time teaching | 0 | 0 | 0 | 0 |
| Process oriented guided inquiry learning (POGIL) | 0 | 0 | 0 | 0 |
| Studio-based learning | 0 | 0 | 0 | 0 |
| Universal design for learning | 0 | 0 | 0 | 0 |
| Task-based learning | 0 | 0 | 0 | 0 |
| Learning Activities | Section 1 | Section 2 | Section 3 | Section 3 |
|---|---|---|---|---|
| Lectures | 1 | 1 | 1 | 1 |
| Interactive Lectures | 1 | 1 | 1 | 1 |
| Lab exercises | 1 | 1 | 1 | 1 |
| Experiments | 0 | 0 | 0 | 0 |
| Modeling | 0 | 0 | 0 | 0 |
| Cases studies | 0 | 0 | 0 | 0 |
| Development of individual parts of software product code | 0 | 0 | 0 | 0 |
| Individual Projects | 0 | 0 | 0 | 0 |
| Group projects | 0 | 0 | 0 | 0 |
| Flipped classroom | 0 | 0 | 0 | 0 |
| Quizzes (written or computer based) | 1 | 1 | 1 | 1 |
| Peer Review | 0 | 0 | 0 | 0 |
| Discussions | 1 | 1 | 1 | 1 |
| Presentations by students | 0 | 0 | 0 | 0 |
| Written reports | 0 | 0 | 0 | 0 |
| Simulations and role-plays | 0 | 0 | 0 | 0 |
| Essays | 0 | 0 | 0 | 0 |
| Oral Reports | 0 | 0 | 0 | 0 |
Formative Assessment and Course Activities
Ongoing performance assessment
Section 1
Section 2
Section 3
Final assessment
Section 1
- Apply the appropriate differentiation technique to a given problem.
- Find a derivative of a function
- Apply Leibniz formula
- Draw graphs of functions
- Find asymptotes of a parametric function
Section 2
- Apply the appropriate integration technique to the given problem
- Find the value of the devinite integral
- Calculate the area of the domain or the length of the curve
Section 3
- Find a limit of a sequence
- Find a limit of a function
The retake exam
Retakes will be run as a comprehensive exam, where the student will be assessed the acquired knowledge coming from the textbooks, the lectures, the labs, and the additional required reading material, as supplied by the instructor. During such comprehensive oral/written the student could be asked to solve exercises and to explain theoretical and practical aspects of the course.