I.konyukhov: /* What should a student be able to apply at the end of the course? */

2022-06-23T15:26:09Z

What should a student be able to apply at the end of the course?

← Older revision		Revision as of 15:26, 23 June 2022
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	By the end of the course, the students should be able to ...		By the end of the course, the students should be able to ...

		+	* find multiple, path, surface integrals
−	* Take derivatives of various type functions and of various orders
		+	* find the range of a function in a given domain
−	* Integrate
		+	* decompose a function into infinite series
−	* Apply definite integral
−	* Expand functions into Taylor series
−	* Apply convergence tests

	=== Course evaluation ===		=== Course evaluation ===

M.petrishchev: M.petrishchev moved page BSc: Mathematical Analysis I.F21 to BSc: Mathematical Analysis I.F21.test

2022-06-22T13:19:59Z

M.petrishchev moved page BSc: Mathematical Analysis I.F21 to BSc: Mathematical Analysis I.F21.test

← Older revision	Revision as of 13:19, 22 June 2022
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N.askarbekuly: Created page with "Category:TRD = Mathematical Analysis I = == Course Characteristics == === Key concepts of the class === * Differentiation * Integration * Series === What is the purpo..."

2022-06-09T14:52:17Z

Created page with "Category:TRD = Mathematical Analysis I = == Course Characteristics == === Key concepts of the class === * Differentiation * Integration * Series === What is the purpo..."

New page

[[Category:TRD]]

= Mathematical Analysis I =

== Course Characteristics ==

=== Key concepts of the class ===

* Differentiation
* Integration
* Series

=== What is the purpose of this course? ===

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.

This calculus course will provide an opportunity for participants to:

* 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.

== Course Objectives Based on Bloom’s Taxonomy ==

=== What should a student remember at the end of the course? ===

By the end of the course, the students should be able to ...

* Derivative. Differential. Applications
* Indefinite integral. Definite integral. Applications
* Sequences. Series. Convergence. Power Series

=== What should a student be able to understand at the end of the course? ===

By the end of the course, the students should be able to ...

* Derivative. Differential. Applications
* Indefinite integral. Definite integral. Applications
* Sequences. Series. Convergence. Power Series
* Taylor Series

=== What should a student be able to apply at the end of the course? ===

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

=== Course evaluation ===

{|
|+ Course grade breakdown
!
!
!align="center"| '''Proposed points'''
|-
| Labs/seminar classes
| 20
|align="center"|
|-
| Interim performance assessment
| 30
|align="center"|
|-
| Exams
| 50
|align="center"|
|}

If necessary, please indicate freely your course’s features in terms of students’ performance assessment.

=== Grades range ===

{|
|+ Course grading range
!
!
!align="center"| '''Proposed range'''
|-
| A. Excellent
| 90-100
|align="center"|
|-
| B. Good
| 75-89
|align="center"|
|-
| C. Satisfactory
| 60-74
|align="center"|
|-
| D. Poor
| 0-59
|align="center"|
|}

If necessary, please indicate freely your course’s grading features.

=== Resources and reference material ===

* Zorich, V. A. “Mathematical Analysis I, Translator: Cooke R.” (2004)
*
*

== Course Sections ==

The main sections of the course and approximate hour distribution between them is as follows:

{|
|+ Course Sections
!align="center"| '''Section'''
! '''Section Title'''
!align="center"| '''Teaching Hours'''
|-
|align="center"| 1
| Sequences and Limits
|align="center"| 28
|-
|align="center"| 2
| Differentiation
|align="center"| 24
|-
|align="center"| 3
| Integration and Series
|align="center"| 28
|}

=== Section 1 ===

==== Section title: ====

Sequences and Limits

=== Topics covered in this section: ===

* Sequences. Limits of sequences
* Limits of sequences. Limits of functions
* Limits of functions. Continuity. Hyperbolic functions

=== What forms of evaluation were used to test students’ performance in this section? ===

{|
!
!align="center"| '''Yes/No'''
|-
| Development of individual parts of software product code
|align="center"| 0
|-
| Homework and group projects
|align="center"| 1
|-
| Midterm evaluation
|align="center"| 1
|-
| Testing (written or computer based)
|align="center"| 1
|-
| Reports
|align="center"| 0
|-
| Essays
|align="center"| 0
|-
| Oral polls
|align="center"| 0
|-
| Discussions
|align="center"| 1
|}

=== Typical questions for ongoing performance evaluation within this section ===

# A sequence, limiting value
# Limit of a sequence, convergent and divergent sequences
# Increasing and decreasing sequences, monotonic sequences
# Bounded sequences. Properties of limits
# Theorem about bounded and monotonic sequences.
# Cauchy sequence. The Cauchy Theorem (criterion).
# Limit of a function. Properties of limits.
# The first remarkable limit.
# The Cauchy criterion for the existence of a limit of a function.
# Second remarkable limit.

=== Typical questions for seminar classes (labs) within this section ===

# Find a limit of a sequence
# Find a limit of a function

=== Test questions for final assessment in this section ===

# Find limits of the following sequences or prove that they do not exist:
# <math>a_n=n-\sqrt{n^2-70n+1400}</math>;
# <math display="inline">d_n=\left(\frac{2n-4}{2n+1}\right)^{n}</math>;
# <math display="inline">x_n=\frac{\left(2n^2+1\right)^6(n-1)^2}{\left(n^7+1000n^6-3\right)^2}</math>.

=== Section 2 ===

==== Section title: ====

Differentiation

=== Topics covered in this section: ===

* Derivatives. Differentials
* Mean-Value Theorems
* l’Hopital’s rule
* Taylor Formula with Lagrange and Peano remainders
* Taylor formula and limits
* Increasing / decreasing functions. Concave / convex functions

=== What forms of evaluation were used to test students’ performance in this section? ===

{|
!
!align="center"| '''Yes/No'''
|-
| Development of individual parts of software product code
|align="center"| 0
|-
| Homework and group projects
|align="center"| 1
|-
| Midterm evaluation
|align="center"| 1
|-
| Testing (written or computer based)
|align="center"| 1
|-
| Reports
|align="center"| 0
|-
| Essays
|align="center"| 0
|-
| Oral polls
|align="center"| 0
|-
| Discussions
|align="center"| 1
|}

=== Typical questions for ongoing performance evaluation within this section ===

# 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
## the asymptotes of this curve;
## the derivative <math display="inline">y'_x</math>.
# 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>.

=== Typical questions for seminar classes (labs) within this section ===

# Differentiation techniques: inverse, implicit, parametric etc.
# Find a derivative of a function
# Apply Leibniz formula
# Draw graphs of functions
# Find asymptotes of a parametric function

=== Test questions for final assessment in this section ===

# Find a derivative of a (implicit/inverse) function
# 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>.
# Draw graphs of functions
# Find asymptotes
# Apply l’Hopital’s rule
# Find the derivatives of the following functions:
## <math display="inline">f(x)=\log_{|\sin x|}\sqrt[6]{x^2+6}</math>;
## <math display="inline">y(x)</math> that is given implicitly by <math display="inline">x^3+5xy+y^3=0</math>.

=== Section 3 ===

==== Section title: ====

Integration and Series

==== Topics covered in this section: ====

* Antiderivative. Indefinite integral
* Definite integral
* The Fundamental Theorem of Calculus
* Improper Integrals
* Convergence tests. Dirichlet’s test
* Series. Convergence tests
* Absolute / Conditional convergence
* Power Series. Radius of convergence
* Functional series. Uniform convergence

=== What forms of evaluation were used to test students’ performance in this section? ===

{|
!
!align="center"| '''Yes/No'''
|-
| Development of individual parts of software product code
|align="center"| 0
|-
| Homework and group projects
|align="center"| 1
|-
| Midterm evaluation
|align="center"| 0
|-
| Testing (written or computer based)
|align="center"| 1
|-
| Reports
|align="center"| 0
|-
| Essays
|align="center"| 0
|-
| Oral polls
|align="center"| 0
|-
| Discussions
|align="center"| 1
|}

=== Typical questions for ongoing performance evaluation within this section ===

# Find the indefinite integral <math display="inline">\displaystyle\int x\ln\left(x+\sqrt{x^2-1}\right)\,dx</math>.
# 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>.
# 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.

==== Typical questions for seminar classes (labs) within this section ====

# Integration techniques
# Integration by parts
# Calculation of areas, lengths, volumes
# Application of convergence tests
# Calculation of Radius of convergence

==== Test questions for final assessment in this section ====

<ol>
<li>Find the following integrals:</li>
<li><math display="inline">\int\frac{\sqrt{4+x^2}+2\sqrt{4-x^2}}{\sqrt{16-x^4}}\,dx</math>;</li>
<li><math display="inline">\int2^{2x}e^x\,dx</math>;</li>
<li><math display="inline">\int\frac{dx}{3x^2-x^4}</math>.</li>
<li>Use comparison test to determine if the following series converge.
<math display="inline">\sum\limits_{k=1}^{\infty}\frac{3+(-1)^k}{k^2}</math>;</li>
<li>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.</li>
<li>Find the sums of the following series:</li>
<li><math display="inline">\sum\limits_{k=1}^{\infty}\frac1{16k^2-8k-3}</math>;</li>
<li><math display="inline">\sum\limits_{k=1}^{\infty}\frac{k-\sqrt{k^2-1}}{\sqrt{k^2+k}}</math>.</li></ol>

BSc: Mathematical Analysis I.F21.test - Revision history

I.konyukhov: /* What should a student be able to apply at the end of the course? */

M.petrishchev: M.petrishchev moved page BSc: Mathematical Analysis I.F21 to BSc: Mathematical Analysis I.F21.test

N.askarbekuly: Created page with "Category:TRD = Mathematical Analysis I = == Course Characteristics == === Key concepts of the class === * Differentiation * Integration * Series === What is the purpo..."