Difference between revisions of "IU:TestPage"

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| || Calculation of Radius of convergence || 2
  
| || Calculation of Radius of convergence || 2
 
|}
  
|}
  +
=== Final assessment ===
  +
'''Section 1'''
  +
# <math>{\displaystyle a_{n}=n-{\sqrt {n^{2}-70n+1400}}}</math>
  +
# <math>{\textstyle d_{n}=\left({\frac {2n-4}{2n+1}}\right)^{n}}</math>
  +
# x
  +
  +
n
  +
  +
  +
=
  +
  +
  +
  +
  +
  +
(
  +
  +
2
  +
  +
n
  +
  +
2
  +
  +
  +
+
  +
1
  +
  +
)
  +
  +
  +
6
  +
  +
  +
(
  +
n
  +
−
  +
1
  +
  +
)
  +
  +
2
  +
  +
  +
  +
  +
  +
(
  +
  +
  +
n
  +
  +
7
  +
  +
  +
+
  +
1000
  +
  +
n
  +
  +
6
  +
  +
  +
−
  +
3
  +
  +
)
  +
  +
  +
2
  +
  +
  +
  +
  +
  +
  +
{\textstyle x_{n}={\frac {\left(2n^{2}+1\right)^{6}(n-1)^{2}}{\left(n^{7}+1000n^{6}-3\right)^{2}}}}
  +
  +
.
  +
'''Section 2'''
  +
# <math>{\textstyle y^{(n)}(x)}</math>
  +
# Find the derivatives of the following functions:
  +
  +
  +
  +
  +
f
  +
(
  +
x
  +
)
  +
=
  +
  +
log
  +
  +
  +
|
  +
  +
sin
  +
⁡
  +
x
  +
  +
|
  +
  +
  +
  +
⁡
  +
  +
  +
  +
  +
x
  +
  +
2
  +
  +
  +
+
  +
6
  +
  +
  +
6
  +
  +
  +
  +
  +
  +
{\textstyle f(x)=\log _{|\sin x|}{\sqrt[{6}]{x^{2}+6}}}
  +
  +
;
  +
  +
  +
  +
  +
y
  +
(
  +
x
  +
)
  +
  +
  +
{\textstyle y(x)}
  +
  +
that is given implicitly by
  +
  +
  +
  +
  +
x
  +
  +
3
  +
  +
  +
+
  +
5
  +
x
  +
y
  +
+
  +
  +
y
  +
  +
3
  +
  +
  +
=
  +
0
  +
  +
  +
{\textstyle x^{3}+5xy+y^{3}=0}
  +
  +
.
  +
# <math>{\textstyle f(x)=\log _{|\sin x|}{\sqrt[{6}]{x^{2}+6}}}</math>
  +
# y
  +
(
  +
x
  +
)
  +
  +
  +
{\textstyle y(x)}
  +
  +
that is given implicitly by
  +
  +
  +
  +
  +
x
  +
  +
3
  +
  +
  +
+
  +
5
  +
x
  +
y
  +
+
  +
  +
y
  +
  +
3
  +
  +
  +
=
  +
0
  +
  +
  +
{\textstyle x^{3}+5xy+y^{3}=0}
  +
  +
.
  +
'''Section 3'''
  +
# <math>{\textstyle \int {\frac {{\sqrt {4+x^{2}}}+2{\sqrt {4-x^{2}}}}{\sqrt {16-x^{4}}}}\,dx}</math>
  +
# <math>{\textstyle \int 2^{2x}e^{x}\,dx}</math>
  +
# <math>{\textstyle \int {\frac {dx}{3x^{2}-x^{4}}}}</math>
  +
# <math>{\textstyle \sum \limits _{k=1}^{\infty }{\frac {3+(-1)^{k}}{k^{2}}}}</math>
  +
# <math>{\textstyle \sum \limits _{k=1}^{\infty }{\frac {k+1}{k^{2}+3}}}</math>
  +
# <math>{\textstyle \sum \limits _{k=1}^{\infty }{\frac {1}{16k^{2}-8k-3}}}</math>
  +
# ∑
  +
  +
k
  +
=
  +
1
  +
  +
  +
∞
  +
  +
  +
  +
  +
  +
k
  +
−
  +
  +
  +
  +
k
  +
  +
2
  +
  +
  +
−
  +
1
  +
  +
  +
  +
  +
  +
k
  +
  +
2
  +
  +
  +
+
  +
k
  +
  +
  +
  +
  +
  +
{\textstyle \sum \limits _{k=1}^{\infty }{\frac {k-{\sqrt {k^{2}-1}}}{\sqrt {k^{2}+k}}}}
  +
  +
.
  +
  +
=== The retake exam ===
  +
'''Section 1'''
  +
  +
'''Section 2'''
  +
  +
'''Section 3'''

Revision as of 16:54, 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

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

Intended Learning Outcomes (ILOs)

What is the main purpose of this course?

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.

ILOs defined at three levels

Level 1: What concepts should a student know/remember/explain?

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

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 ...

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

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. Poor 0-59 -

Course activities and grading breakdown

Activity Type Percentage of the overall course grade
Labs/seminar classes 20
Interim performance assessment 30
Exams 50

Recommendations for students on how to succeed in the course

Resources, literature and reference materials

Open access resources

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

Closed access resources

Software and tools used within the course

Teaching Methodology: Methods, techniques, & activities

Activities and Teaching Methods

Activities within each section
Learning Activities Section 1 Section 2 Section 3
Homework and group projects 1 1 1
Midterm evaluation 1 1 0
Testing (written or computer based) 1 1 1
Discussions 1 1 1

Formative Assessment and Course Activities

Ongoing performance assessment

Section 1

Activity Type Content Is Graded?
Second remarkable limit. 1
Find a limit of a function 2

Section 2

Activity Type Content Is Graded?
1
the derivative




y

x

′



{\textstyle y'_{x}}

.		 1
Derive the Maclaurin expansion for



f
(
x
)
=



1
+

e

−
2
x




3





{\textstyle f(x)={\sqrt[{3}]{1+e^{-2x}}}}

up to



o

(

x

3


)



{\textstyle o\left(x^{3}\right)}

.		 1
Find asymptotes of a parametric function 2

Section 3

Activity Type Content Is Graded?
1
1
Find all values of parameter



α


{\textstyle \alpha }

such that series





∑

k
=
1


+
∞




(



3
k
+
2


2
k
+
1



)


k



α

k





{\textstyle \displaystyle \sum \limits _{k=1}^{+\infty }\left({\frac {3k+2}{2k+1}}\right)^{k}\alpha ^{k}}

converges.		 1
Calculation of Radius of convergence 2

Final assessment

Section 1

  3. x

n


=



(

2

n

2


+ 1

)


6


( n − 1

)

2



(


n

7


+ 1000

n

6


− 3

)


2




{\textstyle x_{n}={\frac {\left(2n^{2}+1\right)^{6}(n-1)^{2}}{\left(n^{7}+1000n^{6}-3\right)^{2}}}}

. Section 2

  2. Find the derivatives of the following functions:



f ( x ) =

log


|

sin ⁡ x

|


⁡



x

2


+ 6


6



{\textstyle f(x)=\log _{|\sin x|}{\sqrt[{6}]{x^{2}+6}}}



y ( x )


{\textstyle y(x)} 

that is given implicitly by



x

3


+ 5 x y +

y

3


= 0


{\textstyle x^{3}+5xy+y^{3}=0}

.

  2. y

( x )


{\textstyle y(x)} 

that is given implicitly by



x

3


+ 5 x y +

y

3


= 0


{\textstyle x^{3}+5xy+y^{3}=0}

. Section 3

  7. ∑

k = 1


∞



k −


k

2


− 1



k

2


+ k



{\textstyle \sum \limits _{k=1}^{\infty }{\frac {k-{\sqrt {k^{2}-1}}}{\sqrt {k^{2}+k}}}}

.

The retake exam

Section 1

Section 2

Section 3

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