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# Functional series. Uniform convergence
  
# Functional series. Uniform convergence
 
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== Intended Learning Outcomes (ILOs) ==
  
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=== What is the main purpose of this course? ===
  
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understand key principles involved in differentiation and integration of functions, solve problems that connect small-scale (differential) quantities to large-scale (integrated) quantities, become familiar with the fundamental theorems of Calculus, get hands-on experience with the integral and derivative applications and of the inverse relationship between integration and differentiation.
  
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=== ILOs defined at three levels ===
  
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==== Level 1: What concepts should a student know/remember/explain? ====
  
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By the end of the course, the students should be able to ...
  
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* Derivative. Differential. Applications
  
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* Indefinite integral. Definite integral. Applications
  
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* Sequences. Series. Convergence. Power Series
  
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==== Level 2: What basic practical skills should a student be able to perform? ====
  
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By the end of the course, the students should be able to ...
  
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* Derivative. Differential. Applications
  
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* Indefinite integral. Definite integral. Applications
  
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* Sequences. Series. Convergence. Power Series
  
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* Taylor Series
  
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==== Level 3: What complex comprehensive skills should a student be able to apply in real-life scenarios? ====
  
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By the end of the course, the students should be able to ...
  
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* Take derivatives of various type functions and of various orders
  
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* Integrate
  
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* Apply definite integral
  
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* Expand functions into Taylor series
  
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* Apply convergence tests
  
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== Grading ==
  
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=== Course grading range ===
  
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{| class="wikitable"
  
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! Grade !! Range !! Description of performance
  
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| A. Excellent || 90-100 || -
  
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| B. Good || 75-89 || -
  
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| C. Satisfactory || 60-74 || -
  
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| D. Poor || 0-59 || -
  
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=== Course activities and grading breakdown ===
  
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{| class="wikitable"
  
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! Activity Type !! Percentage of the overall course grade
  
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| Labs/seminar classes || 20
  
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| Interim performance assessment || 30
  
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| Exams || 50
  
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=== Recommendations for students on how to succeed in the course ===
  
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== Resources, literature and reference materials ==
  
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=== Open access resources ===
  
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* Zorich, V. A. “Mathematical Analysis I, Translator: Cooke R.” (2004)
  
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=== Closed access resources ===
  
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=== Software and tools used within the course ===
  
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= Teaching Methodology: Methods, techniques, & activities =
  
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== Activities and Teaching Methods ==
  
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{| class="wikitable"
  
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|+ Activities within each section
  
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! Learning Activities !! Section 1 !! Section 2 !! Section 3
  
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|-
  
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| Homework and group projects || 1 || 1 || 1
  
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| Midterm evaluation || 1 || 1 || 0
  
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| Testing (written or computer based) || 1 || 1 || 1
  
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| Discussions || 1 || 1 || 1
  
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|}
  
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== Formative Assessment and Course Activities ==
  
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=== Ongoing performance assessment ===
  
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==== Section 1 ====
  
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{| class="wikitable"
  
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! Activity Type !! Content !! Is Graded?
  
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| || A sequence, limiting value || 1
  
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| || Limit of a sequence, convergent and divergent sequences || 1
  
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| || Increasing and decreasing sequences, monotonic sequences || 1
  
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| || Bounded sequences. Properties of limits || 1
  
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| || Theorem about bounded and monotonic sequences. || 1
  
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| || Cauchy sequence. The Cauchy Theorem (criterion). || 1
  
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| || Limit of a function. Properties of limits. || 1
  
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| || The first remarkable limit. || 1
  
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| || The Cauchy criterion for the existence of a limit of a function. || 1
  
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| || Second remarkable limit. || 1
  
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| || Find a limit of a sequence || 2
  
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| || Find a limit of a function || 2
  
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==== Section 2 ====
  
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{| class="wikitable"
  
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! Activity Type !! Content !! Is Graded?
  
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| || A plane curve is given by <math>{\displaystyle x(t)=-{\frac {t^{2}+4t+8}{t+2}}}</math> , <math>{\textstyle y(t)={\frac {t^{2}+9t+22}{t+6}}}</math> . Find || 1
  
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| || the asymptotes of this curve; || 1
  
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| || the derivative <math>{\textstyle y'_{x}}</math> . || 1
  
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| || Derive the Maclaurin expansion for <math>{\textstyle f(x)={\sqrt[{3}]{1+e^{-2x}}}}</math> up to <math>{\textstyle o\left(x^{3}\right)}</math> . || 1
  
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| || Differentiation techniques: inverse, implicit, parametric etc. || 2
  
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| || Find a derivative of a function || 2
  
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| || Apply Leibniz formula || 2
  
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| || Draw graphs of functions || 2
  
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| || Find asymptotes of a parametric function || 2
  
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|}
  
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==== Section 3 ====
  
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{| class="wikitable"
  
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! Activity Type !! Content !! Is Graded?
  
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| || Find the indefinite integral <math>{\textstyle \displaystyle \int x\ln \left(x+{\sqrt {x^{2}-1}}\right)\,dx}</math> . || 1
  
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| || Find the length of a curve given by <math>{\textstyle y=\ln \sin x}</math> , <math>{\textstyle {\frac {\pi }{4}}\leqslant x\leqslant {\frac {\pi }{2}}}</math> . || 1
  
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| || Find all values of parameter <math>{\textstyle \alpha }</math> such that series <math>{\textstyle \displaystyle \sum \limits _{k=1}^{+\infty }\left({\frac {3k+2}{2k+1}}\right)^{k}\alpha ^{k}}</math> converges. || 1
  
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| || Integration techniques || 2
  
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| || Integration by parts || 2
  
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| || Calculation of areas, lengths, volumes || 2
  
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| || Application of convergence tests || 2
  
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| || Calculation of Radius of convergence || 2
  
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|}
  
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=== Final assessment ===
  
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'''Section 1'''
  
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# Find limits of the following sequences or prove that they do not exist:
  
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# <math>{\displaystyle a_{n}=n-{\sqrt {n^{2}-70n+1400}}}</math> ;
  
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# <math>{\textstyle d_{n}=\left({\frac {2n-4}{2n+1}}\right)^{n}}</math> ;
  
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# <math>{\textstyle 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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'''Section 2'''
  
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# Find a derivative of a (implicit/inverse) function
  
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# Apply Leibniz formula Find <math>{\textstyle y^{(n)}(x)}</math> if <math>{\textstyle y(x)=\left(x^{2}-2\right)\cos 2x\sin 3x}</math> .
  
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# Draw graphs of functions
  
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# Find asymptotes
  
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# Apply l’Hopital’s rule
  
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# Find the derivatives of the following functions:
  
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# <math>{\textstyle f(x)=\log _{|\sin x|}{\sqrt[{6}]{x^{2}+6}}}</math> ;
  
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# <math>{\textstyle y(x)}</math> that is given implicitly by <math>{\textstyle x^{3}+5xy+y^{3}=0}</math> .
  
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'''Section 3'''
  
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=== The retake exam ===
  
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'''Section 1'''
  
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'''Section 2'''
  
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'''Section 3'''
  

Revision as of 17:46, 19 April 2022

Mathematical Analysis I

  • Course name: Mathematical Analysis I
  • Code discipline:
  • Subject area: ['Differentiation', 'Integration', 'Series']

Short Description

Prerequisites

Prerequisite subjects

Prerequisite topics

Course Topics

Course Sections and Topics
Section Topics within the section
Sequences and Limits
  1. Sequences. Limits of sequences
  2. Limits of sequences. Limits of functions
  3. Limits of functions. Continuity. Hyperbolic functions
Differentiation
  1. Derivatives. Differentials
  2. Mean-Value Theorems
  3. l’Hopital’s rule
  4. Taylor Formula with Lagrange and Peano remainders
  5. Taylor formula and limits
  6. Increasing / decreasing functions. Concave / convex functions
Integration and Series
  1. Antiderivative. Indefinite integral
  2. Definite integral
  3. The Fundamental Theorem of Calculus
  4. Improper Integrals
  5. Convergence tests. Dirichlet’s test
  6. Series. Convergence tests
  7. Absolute / Conditional convergence
  8. Power Series. Radius of convergence
  9. Functional series. Uniform convergence
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