BSc: Mathematical Analysis II.s23

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

Course Sections and Topics

Section	Topics within the section
Infinite Series	The Sum of an Infinite Series The Comparison Test The Integral and Ratio Tests Alternating Series Power Series Taylor's Formula
Partial Differentiation	Limits of functions of several variables Introduction to Partial Derivatives The Chain Rule Gradients Level Surfaces and Implicit Differentiation Maximas and Minimas Constrained Extrema and Lagrange Multipliers
Multiple Integration	The Double Integral and Iterated Integral The Double Integral over General Region Integrals in Polar coordinates, Substitutions in the double integrals Integrals in Cylindrical and Spherical Coordinates Applications of the Double and Triple Integrals
Vector Analysis	Line Integrals, Path Independence Exact Differentials Green’s Theorem Circulation and Stoke’s Theorem Flux and Divergence Theorem

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 ...

• remember the differentiation techniques
• remember the integration techniques
• remember how to work with sequences and series

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 ...

• Take derivatives of various type functions and of various orders
• Integrate
• Apply definite integral
• Expand functions into Taylor series
• Apply convergence tests

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
• Zorich, V. A. Mathematical Analysis I, Translator: Cooke R. (2004)

Software and tools used within the course

• No.

Activities and Teaching Methods

Teaching and Learning Methods within each section

Teaching Techniques	Section 1	Section 2	Section 3
Problem-based learning (students learn by solving open-ended problems without a strictly-defined solution)	0	0	0
Project-based learning (students work on a project)	1	1	1
Modular learning (facilitated self-study)	0	0	0
Differentiated learning (provide tasks and activities at several levels of difficulty to fit students needs and level)	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
Business game (learn by playing a game that incorporates the principles of the material covered within the course)	0	0	0
Inquiry-based learning	0	0	0
Just-in-time teaching	0	0	0
Process oriented guided inquiry learning (POGIL)	0	0	0
Studio-based learning	0	0	0
Universal design for learning	0	0	0
Task-based learning	0	0	0

Activities within each section

Learning Activities	Section 1	Section 2	Section 3
Lectures	1	1	1
Interactive Lectures	1	1	1
Lab exercises	1	1	1
Experiments	0	0	0
Modeling	0	0	0
Cases studies	0	0	0
Development of individual parts of software product code	0	0	0
Individual Projects	0	0	0
Group projects	0	0	0
Flipped classroom	0	0	0
Quizzes (written or computer based)	0	0	0
Peer Review	0	0	0
Discussions	1	1	1
Presentations by students	0	0	0
Written reports	0	0	0
Simulations and role-plays	0	0	0
Essays	0	0	0
Oral Reports	0	0	0

Formative Assessment and Course Activities

Ongoing performance assessment

Section 1

1. A plane curve is given by $x(t)=-{\frac {t^{2}+4t+8}{t+2}}$, ${\textstyle y(t)={\frac {t^{2}+9t+22}{t+6}}}$. Find

   the asymptotes of this curve;

   the derivative ${\textstyle y'_{x}}$.

2. Apply Leibniz formula Find ${\textstyle y^{(n)}(x)}$ if ${\textstyle y(x)=\left(x^{2}-2\right)\cos 2x\sin 3x}$.

   Draw graphs of functions

   Find asymptotes

3. Find the derivatives of the following functions:
   • ${\textstyle f(x)=\log _{|\sin x|}{\sqrt[{6}]{x^{2}+6}}}$;
   • ${\textstyle y(x)}$ that is given implicitly by ${\textstyle x^{3}+5xy+y^{3}=0}$.

Section 2

1. Find the following integrals:
   • ${\textstyle \int {\frac {{\sqrt {4+x^{2}}}+2{\sqrt {4-x^{2}}}}{\sqrt {16-x^{4}}}}\,dx}$;
   • ${\textstyle \int 2^{2x}e^{x}\,dx}$;
   • ${\textstyle \int {\frac {dx}{3x^{2}-x^{4}}}}$.
2. Find the indefinite integral ${\textstyle \displaystyle \int x\ln \left(x+{\sqrt {x^{2}-1}}\right)\,dx}$.
3. Find the length of a curve given by ${\textstyle y=\ln \sin x}$, ${\textstyle {\frac {\pi }{4}}\leqslant x\leqslant {\frac {\pi }{2}}}$.

Section 3

1. Find limits of the following sequences or prove that they do not exist:
   • $a_{n}=n-{\sqrt {n^{2}-70n+1400}}$;
   • ${\textstyle d_{n}=\left({\frac {2n-4}{2n+1}}\right)^{n}}$;
   • ${\textstyle x_{n}={\frac {\left(2n^{2}+1\right)^{6}(n-1)^{2}}{\left(n^{7}+1000n^{6}-3\right)^{2}}}}$.

Final assessment

Section 1

1. Apply the appropriate differentiation technique to a given problem.
2. Find a derivative of a function
3. Apply Leibniz formula
4. Draw graphs of functions
5. Find asymptotes of a parametric function

Section 2

1. Apply the appropriate integration technique to the given problem
2. Find the value of the devinite integral
3. Calculate the area of the domain or the length of the curve

Section 3

1. Find a limit of a sequence
2. 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.