Difference between revisions of "BSc:Mathematical Analysis (Computer Engineering)"
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(Created page with "= Mathematical Analysis I = * <span>'''Course name:'''</span> Course Title * <span>'''Course number:'''</span> BS-MA1 == Course Characteristics == === Key concepts of the c...") |
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| − | = Mathematical Analysis I = |
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| − | * <span>'''Course name:'''</span> Course Title |
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| − | * <span>'''Course number:'''</span> BS-MA1 |
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| − | |||
| − | == Course Characteristics == |
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| − | |||
| − | === Key concepts of the class === |
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| − | |||
| − | * Differentiation |
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| − | * Integration |
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| − | * Series |
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| − | |||
| − | === What is the purpose of this course? === |
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| − | This calculus course covers differentiation and integration of functions of one variable, with applications. The basic objective of Calculus is to relate small-scale (differential) quantities to large-scale (integrated) quantities. This is accomplished by means of the Fundamental Theorem of Calculus. Should be understanding of the integral as a cumulative sum, of the derivative as a rate of change, and of the inverse relationship between integration and differentiation. |
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| − | This calculus course will provide an opportunity for participants to: |
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| − | * understand key principles involved in differentiation and integration of functions |
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| − | * solve problems that connect small-scale (differential) quantities to large-scale (integrated) quantities |
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| − | * become familiar with the fundamental theorems of Calculus |
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| − | * get hands-on experience with the integral and derivative applications and of the inverse relationship between integration and differentiation. |
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| − | |||
| − | === Course objectives based on Bloom’s taxonomy === |
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| − | |||
| − | === - What should a student remember at the end of the course? === |
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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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| − | === - What should a student be able to understand at the end of the course? === |
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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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| − | |||
| − | === - What should a student be able to apply at the end of the course? === |
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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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| − | === Course evaluation === |
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| − | {| |
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| − | |+ Course grade breakdown |
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| − | ! |
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| − | ! |
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| − | !align="center"| '''Proposed points''' |
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| − | |- |
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| − | | Labs/seminar classes |
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| − | | 20 |
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| − | |align="center"| |
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| − | |- |
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| − | | Interim performance assessment |
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| − | | 30 |
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| − | |align="center"| |
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| − | |- |
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| − | | Exams |
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| − | | 50 |
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| − | |align="center"| |
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| − | |} |
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| − | If necessary, please indicate freely your course’s features in terms of students’ performance assessment. |
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| − | |||
| − | === Grades range === |
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| − | |||
| − | {| |
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| − | |+ Course grading range |
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| − | ! |
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| − | ! |
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| − | !align="center"| '''Proposed range''' |
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| − | |- |
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| − | | A. Excellent |
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| − | | 90-100 |
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| − | |align="center"| |
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| − | |- |
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| − | | B. Good |
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| − | | 75-89 |
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| − | |align="center"| |
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| − | |- |
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| − | | C. Satisfactory |
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| − | | 60-74 |
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| − | |align="center"| |
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| − | |- |
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| − | | D. Poor |
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| − | | 0-59 |
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| − | |align="center"| |
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| − | |} |
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| − | If necessary, please indicate freely your course’s grading features. |
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| − | === Resources and reference material === |
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| − | * Zorich, V. A. “Mathematical Analysis I, Translator: Cooke R.” (2004) |
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| − | * |
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| − | * |
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| − | == Course Sections == |
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| − | The main sections of the course and approximate hour distribution between them is as follows: |
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| − | |||
| − | {| |
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| − | |+ Course Sections |
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| − | !align="center"| '''Section''' |
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| − | ! '''Section Title''' |
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| − | !align="center"| '''Teaching Hours''' |
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| − | |- |
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| − | |align="center"| 1 |
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| − | | Sequences and Limits |
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| − | |align="center"| 28 |
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| − | |- |
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| − | |align="center"| 2 |
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| − | | Differentiation |
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| − | |align="center"| 24 |
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| − | |- |
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| − | |align="center"| 3 |
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| − | | Integration and Series |
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| − | |align="center"| 28 |
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| − | |} |
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| − | |||
| − | === Section 1 === |
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| − | |||
| − | ==== Section title: ==== |
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| − | Sequences and Limits |
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| − | === Topics covered in this section: === |
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| − | * Sequences. Limits of sequences |
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| − | * Limits of sequences. Limits of functions |
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| − | * Limits of functions. Continuity. Hyperbolic functions |
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| − | === What forms of evaluation were used to test students’ performance in this section? === |
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| − | <div class="tabular"> |
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| − | <span>|a|c|</span> & '''Yes/No'''<br /> |
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| − | Development of individual parts of software product code & 1<br /> |
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| − | Homework and group projects & 1<br /> |
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| − | Midterm evaluation & 1<br /> |
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| − | Testing (written or computer based) & 1<br /> |
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| − | Reports & 0<br /> |
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| − | Essays & 0<br /> |
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| − | Oral polls & 0<br /> |
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| − | Discussions & 1<br /> |
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| − | |||
| − | |||
| − | |||
| − | </div> |
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| − | === Typical questions for ongoing performance evaluation within this section === |
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| − | # A sequence, limiting value |
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| − | # Limit of a sequence, convergent and divergent sequences |
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| − | # Increasing and decreasing sequences, monotonic sequences |
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| − | # Bounded sequences. Properties of limits |
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| − | # Theorem about bounded and monotonic sequences. |
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| − | # Cauchy sequence. The Cauchy Theorem (criterion). |
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| − | # Limit of a function. Properties of limits. |
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| − | # The first remarkable limit. |
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| − | # The Cauchy criterion for the existence of a limit of a function. |
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| − | # Second remarkable limit. |
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| − | === Typical questions for seminar classes (labs) within this section === |
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| − | # Find a limit of a sequence |
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| − | # Find a limit of a function |
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| − | === Test questions for final assessment in this section === |
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| − | # Find limits of the following sequences or prove that they do not exist: |
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| − | # <math display="inline">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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| − | === Section 2 === |
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| − | |||
| − | ==== Section title: ==== |
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| − | |||
| − | Differentiation |
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| − | === Topics covered in this section: === |
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| − | * Derivatives. Differentials |
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| − | * Mean-Value Theorems |
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| − | * l’Hopital’s rule |
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| − | * Taylor Formula with Lagrange and Peano remainders |
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| − | * Taylor formula and limits |
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| − | * Increasing / decreasing functions. Concave / convex functions |
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| − | === What forms of evaluation were used to test students’ performance in this section? === |
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| − | |||
| − | <div class="tabular"> |
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| − | |||
| − | <span>|a|c|</span> & '''Yes/No'''<br /> |
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| − | Development of individual parts of software product code & 1<br /> |
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| − | Homework and group projects & 1<br /> |
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| − | Midterm evaluation & 1<br /> |
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| − | Testing (written or computer based) & 1<br /> |
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| − | Reports & 0<br /> |
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| − | Essays & 0<br /> |
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| − | Oral polls & 0<br /> |
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| − | Discussions & 1<br /> |
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| − | |||
| − | |||
| − | |||
| − | </div> |
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| − | === Typical questions for ongoing performance evaluation within this section === |
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| − | # A plane curve is given by <math display="inline">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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| − | # Derive the Maclaurin expansion for <math display="inline">f(x)=\sqrt[3]{1+e^{-2x}}</math> up to <math display="inline">o\left(x^3\right)</math>. |
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| − | |||
| − | === Typical questions for seminar classes (labs) within this section === |
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| − | # Differentiation techniques: inverse, implicit, parametric etc. |
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| − | # Find a derivative of a function |
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| − | # Apply Leibniz formula |
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| − | # Draw graphs of functions |
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| − | # Find asymptotes of a parametric function |
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| − | === Test questions for final assessment in this section === |
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| − | # Find a derivative of a (implicit/inverse) function |
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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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| − | # Apply l’Hopital’s rule |
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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 3 === |
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| − | |||
| − | ==== Section title: ==== |
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| − | Integration and Series |
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| − | |||
| − | ==== Topics covered in this section: ==== |
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| − | |||
| − | * Antiderivative. Indefinite integral |
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| − | * Definite integral |
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| − | * The Fundamental Theorem of Calculus |
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| − | * Improper Integrals |
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| − | * Convergence tests. Dirichlet’s test |
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| − | * Series. Convergence tests |
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| − | * Absolute / Conditional convergence |
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| − | * Power Series. Radius of convergence |
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| − | * Functional series. Uniform convergence |
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| − | |||
| − | === What forms of evaluation were used to test students’ performance in this section? === |
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| − | |||
| − | <div class="tabular"> |
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| − | |||
| − | <span>|a|c|</span> & '''Yes/No'''<br /> |
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| − | Development of individual parts of software product code & 1<br /> |
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| − | Homework and group projects & 1<br /> |
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| − | Midterm evaluation & 1<br /> |
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| − | Testing (written or computer based) & 1<br /> |
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| − | Reports & 0<br /> |
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| − | Essays & 0<br /> |
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| − | Oral polls & 0<br /> |
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| − | Discussions & 1<br /> |
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| − | |||
| − | |||
| − | |||
| − | </div> |
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| − | === Typical questions for ongoing performance evaluation within this section === |
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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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| − | # Find all values of parameter <math display="inline">\alpha</math> such that series <math display="inline">\displaystyle\sum\limits_{k=1}^{+\infty}\left(\frac{3k+2}{2k+1}\right)^k\alpha^k</math> converges. |
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| − | |||
| − | ==== Typical questions for seminar classes (labs) within this section ==== |
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| − | # Integration techniques |
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| − | # Integration by parts |
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| − | # Calculation of areas, lengths, volumes |
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| − | # Application of convergence tests |
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| − | # Calculation of Radius of convergence |
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| − | |||
| − | ==== Test questions for final assessment in this section ==== |
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| − | |||
| − | <ol> |
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| − | <li><p>Find the following integrals:</p></li> |
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| − | <li><p><math display="inline">\int\frac{\sqrt{4+x^2}+2\sqrt{4-x^2}}{\sqrt{16-x^4}}\,dx</math>;</p></li> |
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| − | <li><p><math display="inline">\int2^{2x}e^x\,dx</math>;</p></li> |
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| − | <li><p><math display="inline">\int\frac{dx}{3x^2-x^4}</math>.</p></li> |
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| − | <li><p>Use comparison test to determine if the following series converge.</p> |
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| − | <p><math display="inline">\sum\limits_{k=1}^{\infty}\frac{3+(-1)^k}{k^2}</math>;</p></li> |
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| − | <li><p>Use Cauchy criterion to prove that the series <math display="inline">\sum\limits_{k=1}^{\infty}\frac{k+1}{k^2+3}</math> is divergent.</p></li> |
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| − | <li><p>Find the sums of the following series:</p></li> |
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| − | <li><p><math display="inline">\sum\limits_{k=1}^{\infty}\frac1{16k^2-8k-3}</math>;</p></li> |
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| − | <li><p><math display="inline">\sum\limits_{k=1}^{\infty}\frac{k-\sqrt{k^2-1}}{\sqrt{k^2+k}}</math>.</p></li></ol> |
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