Advanced Mathematics I (ING251) | GSÜ

Content

Objective

This course is the continuation of the Math I course.
In this context, the objectives of this course are:

- Demonstrate to the students the classical techniques [integration by parts and change of variables] to calculate a primitive,
- Teach students to handle the comparison relations "to be negligible in front of" and "to be equivalent to" on functions,
- Teach how to find a "" simple "" equivalent of a point function to find its limit,
- Demonstrate the different convergence criteria for the integrals of positive functions,
- Explain in which cases a limited expansion makes it possible to determine the nature of an integral,
- Demonstrate the different convergence criteria for series with positive terms,
- Explain in which cases a limited development makes it possible to determine the nature of a series

Course Content

1. Primitives: Definition, properties and first examples.
2. Primitives: Calculation rules [integration by parts and change of variable]
3. Comparison relations: function negligible in front of another, function equivalent to another
4. Comparison relations: calculation rules, comparative growth of logarithms, powers and exponential in 0 and infinity.
5. Comparison relations: Application to the search for limits.
6. Generalized integrals: definition, properties and first examples [Riemann integrals and Bertrand integrals].
7. Generalized integrals: comparison theorems for positive functions.
8. Generalized integrals: case of functions of any sign.
9. Partial Examination / Ara sinav
10. Generalized integrals: Integrals depending on a parameter
11. Numerical series: definition, properties and first examples [Riemann series and Bertrand series].
12. Numerical series: comparison theorems for series with positive terms.
13. Numerical series: Case of series of any sign. Convergence criterion of alternating series.
14. Digital Series: Series depending on a parameter

Course Learning Outcomes

The student who will take this course will develop the following skill elements and will be able to:

1. Make an integration by parts and / or a change of variable to calculate the integral of a function,
2. Compare two functions at a given point,
3. Determine a "simple" equivalent of a function to calculate its limit at a point,
4. Apply comparison theorems to determine if a positive function admits a generalized integral,
5. Know how to use a Taylor expansion to determine the nature of an integral [absolutely convergent, semi-convergent or divergent],
6. Apply comparison theorems to determine if a series with positive terms is convergent,
7. Know how to use a lTaylor expansion to determine the nature of a series [absolutely convergent, semi-convergent or divergent]

Teaching and Learning Methods

Lectures and supervised works/tutorials

References

1. Lectures notes ans worksheets
2. http://braise.univ-rennes1.fr/braise.cgi
3. http://www.unisciel.fr

Weekly Contents

Theory Topics
Week	Weekly Contents
1	Reminders: Derivation, usual functions and limited developments
2	Primitives: Definition, Interpretation and Properties
3	Primitives: Calculation methods (integration by part)
4	Primitives: Calculation methods (change of variable)
5	Primitives: Calculation methods (cases requiring several successive methods)
6	Comparison of functions: Definition and interpretation
7	Comparing functions: Practical search for the equivalent of a function
8	Comparing functions: Practical search for the equivalent of a function (continued)
9	Midterm exam
10	Generalized integrals: Definition, Interpretation and Properties
11	Generalized integrals: Case of positive functions
12	Generalized integrals: Case of functions of any sign
13	Generalized integrals: Semi-convergent integrals
14	Preparing for the final exam

Practice Topics
Week	Weekly Contents
1
2
3
4
5
6
7
8
9
10
11
12
13
14

Assessment System

Contribution to Overall Grade
Activities	Number	Contribution
Contribution of in-term studies to overall grade	0	0
Contribution of final exam to overall grade	0	0
Total	0	0

In-Term Studies
Activities	Number	Contribution
Assignments	0	0
Presentation	0	0
Midterm Examinations (including preparation)	0	0
Project	0	0
Laboratory	0	0
Other Applications	0	0
Quiz	0	0
Term Paper/ Project	0	0
Portfolio Study	0	0
Reports	0	0
Learning Diary	0	0
Thesis/ Project	0	0
Seminar	0	0
Other	0	0
Total	0	0

Relation of Proficiency

No	Program Learning Outcomes	Contribution
No	Program Learning Outcomes	1	2	3	4	5
1	Knowledge and understanding of a wide range of basic sciences (math, physics, ...) and the main concepts of engineering					X
2	Ability to combine the knowledge and skills to solve engineering problems and provide reliable solutions					X
3	Ability to select and apply methods of analysis and modeling to ask, reformulate and solve the complex problems of industrial engineering					X
4	Ability to conceptualize complex systems, processes or products under practical constraints to improve their performance, ability to use innovative methods of design			X
5	Ability to design, select and apply methods and tools needed to solve problems related to the practice of industrial engineering, ability to use computer technology					X
6	Ability to design experiments, collect and interpret data and analyze results				X
7	Ability to work independently, ability to participate in working groups and have a multidisciplinary team spirit				X
8	Ability to communicate effectively, ability to speak at least two foreign languages				X
9	Awareness of the need for continuous improvement of lifelong learning, ability to keep abreast of scientific and technological developments to use the tools of information management
10	Awareness of professional and ethical responsibility
11	Knowledge of the concepts of professional life as "project management", "risk management" and "management of change"
12	Knowledge on entrepreneurship, innovation and sustainability
13	Understanding of the effects of Industrial Engineering applications on global and social health, environment and safety.
14	Knowledge of the problems of contemporary society
15	Knowledge of the legal implications of the practice of industrial engineering