MM304 : Real Analysis 2

Department

Department of Mathematics

Academic Program

Bachelor in mathematics

Type

Compulsory

Credits

03

Prerequisite

MM303

Overview

This course introduces the student to the basic concepts of the derivative of real functions - and the derivative of a function in space Rp, also it deals with the middle value theorem - the derivative connection - the chain rule - and the L'Hubetal rule . This course aims to develop the student's ability to determine Derivation of Higher Orders - Taylor's Theorem - Maximum and Minimum Limits, Integration, and Sequences and Series of Functions.

Intended learning outcomes

By the end of the course, the student should be able to

1. Recognize the derivatives of functions and the directional derivatives of functions on Partial derivatives, derivability of functions, matrix representation of the total derivative, function gradient, some theorems on derivation, and the chain rule.
2. Explain the concept of integration, the Riemann integral and its properties, improper integrals of the first and second kind and their properties, and convergence tests for improper integrals.
3. Explain sequences of functions, regular convergence, integration, uniform convergence, and derivation.
4. Define series of functions, regular series convergence, power series and Taylor series.
5. Explain derivatives of functions, directional derivatives of functions over R , partial derivatives, derivability of functions, matrix representation of the total derivative, function gradient, some theorems on derivation and the chain rule.
6. prove some theories and theorems related to integration, the Riemann integral and its properties, improper integrals of the first and second kind and their properties, and convergence tests for improper integrals.
7. Analyze sequences of functions, regular convergence, integration, regular convergence and derivation.
8. Distinguish between series of functions, regular convergence of series, power series, Taylor series and Maclaurin series.
9. Solve a number of exercises and problems with more than one idea about derivatives of functions, vector derivatives of functions on R , and partial derivatives.
10. give solutions to new, different and multiple problems related to integration, Riemann integral and its properties, improper integrals of the first and second kind and their properties, and convergence tests for improper integrals.
11. Use some ideas to prove and solve problems related to sequences of functions, uniform convergence, integration, uniform convergence, and derivation.
12. Apply theories related to series of functions, regular series convergence, power series, Taylor series and Maclaurin series

Teaching and learning methods

1. Theoretical lectures
2. Discussion and dialogue
3. Brain storming
4. Using mathematical proof methods
5. Exercises, training and multi-idea problem-solving

Methods of assessments

1. A written exam (essay + objective) = 25 marks, or its assessment is left to the course instructor.
2. Short tests (written or oral), demonstration tasks, applications, exercises, and presentation = 15 marks, or the assessment is left to the course instructor.
3. Written final exam (essay + objective) = 60 marks

Course content:

the week	Scientific subject	The number of hours	a lecture	exercises
1	Derivative functions on R	3	2	1
2	Vector derivatives of functions on R	3	2	1
3	Partial derivatives and the differentiability of functions	3	2	1
4	Matrix representation of the total derivative	3	2	1
5	First midterm exam (2 hours)
5-6	The function and some theorems on derivation and the chain rule are included.	4	3	1
7	The concept of integration and Riemann integration and its properties	3	2	1
8	Improper integrals of the first and second kind and their properties	3	2	1
9	Convergence tests for improper integrals	3	2	1
10	Second midterm exam (2 hours)
10-11	Function sequences and series	4	3	1
12	Function sequences and regular convergence	3	2	1
13-14	Integration, regular convergence, and derivation.	6	4	2
15-16	final exam
	the total	42

References

Title of References	publisher	version	Author
Real analysis	International House for Publishing and Distribution	The second	Dr . Ramadan Mohamed Juhayma
Mathematical analysis	University of Tripoli Publications	The first	Translated by Ali Mohamed Ibrahim

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