# Analysis-I

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Course Name : Analysis-I

# Code(Credit) : CUTM-1513(3-0-3)

## Course Outcomes:

 COs Course outcomes Mapping Cos with POs (High-3, Medium-2, Low-1) CO1 Describe the real line as a complete, ordered field, Use the definitions of convergence as they apply to sequences, series, and functions, PO1(3), PO2(2) CO2 Determine the continuity, differentiability, and integrability of functions defined on subsets of the real line, produce rigorous proofs of results that arise in the context of real analysis. PO2(2), PO4(2)

## Course Syllabus:

Module-I   (5 hr+0 hr+4 hr)

Field Structure and Order Structure; Bounded and unbounded Sets, Supremum, Infimum;Completeness in the Set of Real numbers; Archimedean Property of Real numbers; Inequalities and Metric property of R

project 1: A study on completeness property of R and its application.

project 2: A Report on Some Inequalities of Real Numbers.

Module-II (5 hr+0 hr+6 hr)

Neighborhood of a point, Interior point of a set, Open Sets; Some Important Theorems on open Sets; Limit point of a Set, Closed Sets, Closure of a Set, Dense Set; Some Useful Theorems on Open Set and Closed Sets; Blozanoweirstrass theorem for Sets.

Project 3 : A study on some Useful Theorems on Limit point of a Sets.

Project 4:  A Study on  some Useful Theorems on Open Set and Closed Sets.

Project 5:  A study on the Blozanoweirstrass theorem for Sets.

Module-III   (6 hr+0 hr+6 hr)

Real Sequences and Theorems on Convergence of  Sequence;Limit point  a Sequence and Bolzano-Weierstrass theorem for Sequences; Convergent and Non-Convergent Sequences; Cauchy’s general Principle of Convergence and Cauchy Sequence;Some Important theorems on Real Sequences; Monotonic Sequences and Sub sequences.

Project 6: A Report on Cauchy’s general Principle of Convergence and Cauchy Sequence.

Project 7: A Report on Convergent and Non-Convergent Sequences.

Project 8: A study on Some Important Theorems of Real Sequences.

Module-IV   (5 hr+0 hr+6 hr)

Infinite Series and Some Preliminary Theorems; Positive term Series and Condition for Convergence ; Geometric Series and Comparison Series for Convergence of Infinite Series; Comparison Tests for Positive Term Series ; Cauchy’s Root test and D’ Alembert's Ratio Test.

Project 9:  A study on Some useful theorems of Infinite Series.

Project 10: A report on Comparison Test for Positive term Series.

Project 11: A Report on Cauchy’s Root test and D’ Alembert's Ratio Test.

Module-V   (3 hr+0 hr+4 hr)

Alternating Series and Leibnitz Test ; Absolute and Conditional Convergence; Solving problems on Absolute and Conditional Convergence.

Project 12: A Report on Alternating Series and Leibnitz Test.

Project 13: A Report on Absolute and Conditional Convergence.

Module-VI   (4 hr+0 hr+4 hr)

Limit and Continuity of Functions; Discontinuities and Types of Discontinuity;Uniform continuity and related Theorems; Differentiability of Real functions.

Project 14: A Report  on Finding Limit and continuity of Functions.

Project 15: A Report  on Differentiability of Real functions.

Module-VII (4 hr+0 hr+6 hr)

The Derivative and Higher Order Differentiation's; Darboux's Theorem and Roll's Theorem;Lagrange's Mean value theorems and Cauchy's Mean value Theorem ;Taylor’s Theorem with Reminder.

Project 16: A Report on Darboux's Theorem  and  Roll's Theorem .

Project 17: A Report on Lagrange's Mean value theorems and Cauchy's Mean value Theorem.

Project 18 : A study on Taylor's Theorem with Remainder and its Importance.

Text Book:

Mathematical Analysis (Wiley Eastern) : S.C. Malik and S.Arora (4 th Edition)

Chapters: 1 (except 4.3 and 4.4),  2, 3, 4 (upto Art.5 and Art 10.1, 10.2),  5, 6,

Reference Books:

Fundamentals of Real Analysis :S.L.Gupta& Nisha Rani

Mathematical Analysis-II : Sharma &Vasistha

Fundamental of Mathematical Analysis :G. Das&S.Pattanayak

## Session 1:

Field Structure and Order Structure

## Session 3:

Completeness in the Set of Real numbers

## Session 5:

Inequalities and Metric property of R.

## Project 1:

A study on completeness property of R and its application.

## Project 2:

A Report on Some Inequalities of Real Numbers.

## Session 10:

Real Sequence, Limit Points of a sequence Limit inferior and superior

## Session 11:

Some Important Theorems on open Sets.

## Session 12:

Limit point of a Set, Closed Sets, Closure of a set, Dense Set.

## Session 13:

Some Useful Theorems on Open Set and Closed Sets.

## Session 14:

Blozanoweirstrass theorem for Sets.

## Project 3:

A study on some Useful Theorems on Limit point of a Sets.

## Project 4:

A Study on  some Useful Theorems on Open Set and Closed Sets.

## Project 5:

A study on the Blozanoweirstrass theorem for Sets.

## Session 21:

Real Sequences and Theorems on Convergence of  Sequence

## Session 22:

Limit point  a Sequence and Bolzano-Weierstrass theorem for Sequences.

## Session 23:

Convergent and Non-Convergent Sequences.

## Session 24:

Cauchy’s general Principle of Convergence and Cauchy Sequence.

## Session 25:

Some Important theorems on Real Sequences.

## Project 6:

A Report on Cauchy’s general Principle of Convergence and Cauchy Sequence.

## Project 7:

A Report on Convergent and Non-Convergent Sequences.

## Project 8:

A study on Some Important Theorems of Real Sequences.

## Session 33:

Infinite Series and Some Preliminary Theorems

## Session 34:

Positive term Series and Condition for Convergence

## Session 35:

Geometric Series and Comparison Series for Convergence of Infinite Series

## Session 36:

Comparison Tests for Positive Term Series

## Session 37:

Cauchy’s Root test and D’ Alemberts Ratio Test.

## Project 9:

A study on Some useful theorems of Infinite Series.

## Project 10:

A report on Comparison Test for Positive term Series.

Solving Problems on Limit and continuity of functions

## Project 11:

A Report on Cauchy’s Root test and D’ Alembert's Ratio Test.

Discontinuities and Types of Discontinuity

## Session 45:

Absolute and Conditional Convergence;

## Session 46:

Solving problems on Absolute and Conditional Convergence.

## Project 12:

Solving problems on Uniform continuity

Solving problems on Uniform continuity

## Project 13:

Solving problems on Taylor’s Theorem

Solving problems on Taylor’s Theorem

## Session 51:

Limit and Continuity of Functions

## Session 52:

Discontinuities and Types of Discontinuity.

## Session 54:

Differentiability of Real functions.

## Project 14:

A Report  on Finding Limit and continuity of Functions.

## Project 15:

A Report  on Differentiability of Real functions.

## Session 60:

Darboux's Theorem and Roll's Theorem

## Session 61:

Lagrange's Mean value theorems and Cauchy's Mean value Theorem.

## Session 62:

Taylor’s Theorem with Reminder.

## Project 16 :

A Report on Darboux's Theorem  and  Roll's Theorem .

## Project 17:

A Report on Lagrange's Mean value theorems and Cauchy's Mean value Theorem.

## Project 18:

A study on Taylor's Theorem with Remainder and its Importance.

## Case Studies

### Dr. Bhairaba Kumar Majhi

##### Asst.Professor
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My Self Dr.Bhairaba Kumar Majhi,M,Sc,B.Ed,M.Tech(CSE),Ph.D(Mathematics ) ,Asst.Professor And HOD Department of Mathematics School Of Applied Sciences,Presently working at CUTM,Bolangir Campus.I have 16 Years of Experience in Teaching at Different Levels Of UG,PG & Engineering Mathematics.