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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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| − | |+ |
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| − | |- |
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| − | ! Grade !! Range !! Description of performance |
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| − | |- |
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| − | | A. Excellent || 90-100 || - |
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| − | |- |
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| − | | B. Good || 75-89 || - |
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| − | |- |
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| − | | C. Satisfactory || 60-74 || - |
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| − | |- |
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| − | | D. Poor || 0-59 || - |
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| − | |} |
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| − | |||
| − | === Course activities and grading breakdown === |
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| − | {| class="wikitable" |
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| − | |+ |
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| − | |- |
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| − | ! Activity Type !! Percentage of the overall course grade |
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| − | |- |
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| − | | Labs/seminar classes || 20 |
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| − | |- |
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| − | | Interim performance assessment || 30 |
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| − | |- |
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| − | | Exams || 50 |
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| − | |} |
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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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| − | |- |
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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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| − | |- |
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| − | | Midterm evaluation || 1 || 1 || 0 |
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| − | |- |
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| − | | Testing (written or computer based) || 1 || 1 || 1 |
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| − | |- |
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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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| − | |+ |
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| − | |- |
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| − | ! Activity Type !! Content !! Is Graded? |
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| − | |- |
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| − | | || A sequence, limiting value || 1 |
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| − | |- |
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| − | | || Limit of a sequence, convergent and divergent sequences || 1 |
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| − | |- |
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| − | | || Increasing and decreasing sequences, monotonic sequences || 1 |
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| − | |- |
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| − | | || Bounded sequences. Properties of limits || 1 |
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| − | |- |
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| − | | || Theorem about bounded and monotonic sequences. || 1 |
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| − | |- |
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| − | | || Cauchy sequence. The Cauchy Theorem (criterion). || 1 |
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| − | |- |
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| − | | || Limit of a function. Properties of limits. || 1 |
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| − | |- |
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| − | | || The first remarkable limit. || 1 |
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| − | |- |
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| − | | || The Cauchy criterion for the existence of a limit of a function. || 1 |
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| − | |- |
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| − | | || Second remarkable limit. || 1 |
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| − | |- |
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| − | | || Find a limit of a sequence || 2 |
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| − | |- |
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| − | | || Find a limit of a function || 2 |
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| − | |} |
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| − | ==== Section 2 ==== |
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| − | {| class="wikitable" |
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| − | |+ |
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| − | |- |
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| − | ! Activity Type !! Content !! Is Graded? |
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| − | |- |
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| − | | || <math>{\displaystyle x(t)=-{\frac {t^{2}+4t+8}{t+2}}}</math> || 1 |
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| − | |- |
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| − | | || the asymptotes of this curve; || 1 |
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| − | |- |
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| − | | || <math>{\textstyle y'_{x}}</math> || 1 |
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| − | |- |
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| − | | || <math>{\textstyle f(x)={\sqrt[{3}]{1+e^{-2x}}}}</math> || 1 |
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| − | |- |
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| − | | || Differentiation techniques: inverse, implicit, parametric etc. || 2 |
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| − | |- |
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| − | | || Find a derivative of a function || 2 |
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| − | |- |
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| − | | || Apply Leibniz formula || 2 |
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| − | |- |
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| − | | || Draw graphs of functions || 2 |
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| − | |- |
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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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| − | |+ |
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| − | |- |
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| − | ! Activity Type !! Content !! Is Graded? |
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| − | |- |
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| − | | || <math>{\textstyle \displaystyle \int x\ln \left(x+{\sqrt {x^{2}-1}}\right)\,dx}</math> || 1 |
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| − | |- |
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| − | | || <math>{\textstyle y=\ln \sin x}</math> || 1 |
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| − | |- |
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| − | | || <math>{\textstyle \alpha }</math> || 1 |
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| − | |- |
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| − | | || Integration techniques || 2 |
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| − | |- |
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| − | | || Integration by parts || 2 |
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| − | |- |
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| − | | || Calculation of areas, lengths, volumes || 2 |
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| − | |- |
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| − | | || Application of convergence tests || 2 |
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| − | |- |
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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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| − | # <math>{\textstyle y^{(n)}(x)}</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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| − | # <math>{\textstyle f(x)=\log _{|\sin x|}{\sqrt[{6}]{x^{2}+6}}}</math> |
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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> |
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| − | '''Section 3''' |
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| − | # Find the following integrals: |
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| − | # <math>{\textstyle \int {\frac {{\sqrt {4+x^{2}}}+2{\sqrt {4-x^{2}}}}{\sqrt {16-x^{4}}}}\,dx}</math> |
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| − | # <math>{\textstyle \int 2^{2x}e^{x}\,dx}</math> |
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| − | # <math>{\textstyle \int {\frac {dx}{3x^{2}-x^{4}}}}</math> |
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| − | # <math>{\textstyle \sum \limits _{k=1}^{\infty }{\frac {3+(-1)^{k}}{k^{2}}}}</math> |
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| − | # <math>{\textstyle \sum \limits _{k=1}^{\infty }{\frac {k+1}{k^{2}+3}}}</math> |
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| − | # Find the sums of the following series: |
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| − | # <math>{\textstyle \sum \limits _{k=1}^{\infty }{\frac {1}{16k^{2}-8k-3}}}</math> |
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| − | # <math>{\textstyle \sum \limits _{k=1}^{\infty }{\frac {k-{\sqrt {k^{2}-1}}}{\sqrt {k^{2}+k}}}}</math> |
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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''' |
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Revision as of 16:53, 18 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
| Section | Topics within the section |
|---|---|
| Sequences and Limits |
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| Differentiation |
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| Integration and Series |
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