Is all differentiable functions are integrable?

While attending teleconference as a resource person in bangalore. I and panel members was asked to answer the following questions by my fellow lecturers.
Is integrable functions are continous?, Is every continous functions are integrable?, Is all functions are integrable etc.
Really I find these questions are good and we should have to clarify this questions. Here I tried to answer this : students are competent can give your suggestion
<!– /* Style Definitions */ p.MsoNormal, li.MsoNormal, div.MsoNormal {mso-style-parent:""; margin:0in; margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:12.0pt; font-family:"Times New Roman"; mso-fareast-font-family:"Times New Roman";} @page Section1 {size:8.5in 11.0in; margin:1.0in 1.25in 1.0in 1.25in; mso-header-margin:.5in; mso-footer-margin:.5in; mso-paper-source:0;} div.Section1 {page:Section1;}
–>1. <!– /* Style Definitions */ p.MsoNormal, li.MsoNormal, div.MsoNormal {mso-style-parent:""; margin:0in; margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:12.0pt; font-family:"Times New Roman"; mso-fareast-font-family:"Times New Roman";} @page Section1 {size:8.5in 11.0in; margin:1.0in 1.25in 1.0in 1.25in; mso-header-margin:.5in; mso-footer-margin:.5in; mso-paper-source:0;} div.Section1 {page:Section1;} –>Are all functions that can be differentiated, integratable?
All differentiable functions are integrable.
True because all differentiable functions are continuous(Because Differentiability implies continuity but continuity need not implies differentiability) and by FTC, fundamental theorem of integral calculus all continuous functions are integrable.

2.Is every continuous functions are integrable?
True by fundamental theorem of integral calculus

3. Is all integrable functions are continuous?
<!– /* Style Definitions */ p.MsoNormal, li.MsoNormal, div.MsoNormal {mso-style-parent:""; margin:0in; margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:12.0pt; font-family:"Times New Roman"; mso-fareast-font-family:"Times New Roman";} @page Section1 {size:8.5in 11.0in; margin:1.0in 1.25in 1.0in 1.25in; mso-header-margin:.5in; mso-footer-margin:.5in; mso-paper-source:0;} div.Section1 {page:Section1;} –>This doesn’t follow from the FTC, but I’m having trouble thinking of a counter-example. I looked around on the web and saw a couple people say that this is false, but never explain why. Can you integrate piecewise functions? If so then I can think of an easy counter-example. We’ve never talked about doing so in class. but think!

Some of the following integrals are not integrable:
LIST OF SUCH INTEGRALS <!– /* Style Definitions */ p.MsoNormal, li.MsoNormal, div.MsoNormal {mso-style-parent:""; margin:0in; margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:12.0pt; font-family:"Times New Roman"; mso-fareast-font-family:"Times New Roman";} @page Section1 {size:8.5in 11.0in; margin:1.0in 1.25in 1.0in 1.25in; mso-header-margin:.5in; mso-footer-margin:.5in; mso-paper-source:0;} div.Section1 {page:Section1;} –> <!– /* Style Definitions */ p.MsoNormal, li.MsoNormal, div.MsoNormal {mso-style-parent:""; margin:0in; margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:12.0pt; font-family:"Times New Roman"; mso-fareast-font-family:"Times New Roman";} @page Section1 {size:8.5in 11.0in; margin:1.0in 1.25in 1.0in 1.25in; mso-header-margin:.5in; mso-footer-margin:.5in; mso-paper-source:0;} div.Section1 {page:Section1;} –> <!– /* Style Definitions */ p.MsoNormal, li.MsoNormal, div.MsoNormal {mso-style-parent:""; margin:0in; margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:12.0pt; font-family:"Times New Roman"; mso-fareast-font-family:"Times New Roman";} p {mso-margin-top-alt:auto; margin-right:0in; mso-margin-bottom-alt:auto; margin-left:0in; mso-pagination:widow-orphan; font-size:12.0pt; font-family:"Times New Roman"; mso-fareast-font-family:"Times New Roman";} span.postbody {mso-style-name:postbody;} @page Section1 {size:8.5in 11.0in; margin:1.0in 1.25in 1.0in 1.25in; mso-header-margin:.5in; mso-footer-margin:.5in; mso-paper-source:0;} div.Section1 {page:Section1;} –> 1. int sinx / x dx

2. int cosx/ x dx

3. int root (sinx) dx

4. int root (cosx) dx

5.int sin (x ² ) dx

6. int cos (x ² ) dx

7. int e ^{-x 2} dx

8. int e ^{x 2} dx

10. int root(1+x ³ ) dx

11. int x tan x dx

12. int 1/ log x dx . and many more ok dear. try to remember it.

WHAT IS INTEGRALS?

Calculus is the study of how things change.

In differentiation we are finding a derivative i.e. slope of tangent to the curve i.e how the curve changes at different points for a given function

In integration we are finding a original function whose derivative or slope of the tangent or change at different points is known

Slopes of curves is also can be considered as rate of change at different cases. This is called instantaneous change. Derivative of a curves tell us the  instantaneous rate of change of a curve.

A curvy function changes at different rates throughout its domain—sometimes it’s increasing quickly and the tangent line is steep (causing a high-valued derivative). At other places the curve may be increasing shallowly or even decreasing, causing the derivative to be small or negative, respectively.

Look at the graph of f(x) in Figure 

If instantaneous change is given to find the original state we need to integrate.

To find the change at different points we need to differentiate.

Calculus analyses things that change, and physics is much concerned with changes. For physics, you’ll need at least some of the simplest and most important concepts from calculus.

A typical course in calculus covers the following topics:
1. How to find the instantaneous change (called the “derivative”) of various functions. (The process of doing so is called “differentiation”.)
2. How to use derivatives to solve various kinds of problems.
3. How to go back from the derivative of a function to the function itself. (This process is called “integration”.)
4. Study of detailed methods for integrating functions of certain kinds.
5. How to use integration to solve various geometric problems, such as computations of areas and volumes of certain regions.

We know how to find the derivative of a function by different methods, Now we are going back in integration.

If we know that

 dy/dx=2x

and we need to know the function this derivative came from, then we “undo” the differentiation process. (Think: “What would I have to differentiate to get this result?”)

y=x²  is ONE antiderivative of   dy/dx =2x

There are infinitely many other antiderivatives which would also work, for example:

y=x²+4

y=x²+π

y=x²+27.3

In general, we say y=x²+Kis the indefinite integral of 2x. The number K is called the constant of integration.

Note: Most math text books use C for the constant of integration, but for questions involving electrical engineering, we prefer to write “+K“, since C is normally used for capacitance and it can get confusing.

Notation for the Indefinite Integral

We write: ∫2x  dx=x²+K and say in words:

“The integral of 2x with respect to xequals x² + K.”

The Integral Sign

The ∫sign is an elongated “S”, standing for “sum”. (In old German, and English, “s” was often written using this elongated shape.) Later we will see that the integral is the sum of the areas of infinitesimally thin rectangles.
∑is the symbol for “sum”. It can be used for finite or infinite sums.
∫is the symbol for the sum of an infinite number of infinitely small areas (or other variables).
This “long s” notation was introduced by Leibniz when he developed the concepts of integration.
GEOMETRICAL INTERPRETATION OF INDEFINITE INTEGRAL:
Let f (x) = 2x. Then ∫ f (x) dx = x² + C. For different values of C, we get different integrals. But these integrals are very similar geometrically.
Thus, y = x² + C, where C is arbitrary constant, represents a family of integrals. By assigning different values to C, we get different members of the family. These together constitute the indefinite integral. In this case, each integral represents a parabola with its axis along y-axis.

Clearly, for C = 0, we obtain y = x², a parabola with its vertex on the origin.
The curve y = x² + 1 for C = 1 is obtained by shifting the parabola y = x² one unit along y-axis in positive direction.
For C = – 1, y = x² – 1 is obtained by shifting the parabola y = x² one unit along y-axis in the negative direction.
Thus, for each positive value of C, each parabola of the family has its vertex on the positive side of the y-axis and for negative values of C, each has its vertex along the negative side of the y-axis. Some of these have been shown in the Fig
 

Let us consider the intersection of all these parabolas by a line x = a. In the Fig 7.1, we have taken a > 0. The same is true when a < 0. If the line x = a intersects the parabolas y = x²,
y = x² + 1, y = x² + 2, y = x² – 1, y = x²– 2 at P₀, P₁, P₂, P_-1, P_-2etc., respectively, then dy / dx at these points equals 2a.
This indicates that the tangents to the curves at these points are parallel. Thus, ∫2x² dx = x + C = F_C (x) (say), implies that the tangents to all the curves y = F_C (x), C ∈ R, at the points of intersection of the curves by the line x = a, (a ∈ R), are parallel.

Further, the following equation (statement) ∫ f (x) dx = F (x) + C = y (say) , represents a family of curves. The different values of C will correspond to different members of this family and these members can be obtained by shifting any one of the curves parallel to itself. This is the geometrical interpretation of indefinite integral.
∫ f (x) dx = F (x) + C = Family of all curves which are geometrically similar and the tangents drawn to all curves by the line x=a (at any particular point) are parallel.

NOW THE QUESTIONS?

1.      Is All differentiable functions are integrable.

Ans: True because all differentiable functions are continuous and by FTC all continuous functions are integrable.

2.      Is All integrable functions are continuous.?

Ans:  This doesn’t follow from the FTC, Now for this it is difficult to answer at this level. We have to learn advance mathematics

Can you integrate piecewise functions? If so then I can think of an easy counter-example. We’ve never talked about doing so in class.

3.      Is All integrable functions are differentiable.

Ans: Even though 1 is true this doesn’t follow from it. Same difficulty as 2.

{Differentiable functions} ⊂ {Continuous functions} ⊂ {Integrable functions}

In order for some function f(x) to be continuous at x = c, then the following two conditions must be true:

i)                    f(c) is defined and the limit of f(x) as x approaches c is equal to f(c).

In order for some function f(x) to be differentiable at x = c, then it must be continuous at x = c and it must not be a corner point (i.e., it’s right-side and left-side derivatives must be equal).

Continuity implies integrability; if some function f(x) is continuous on some interval [a,b], then the definite integral from a to b exists.

While all continuous functions are integrable, not all integrable functions are continuous. To understand this idea we need to study advanced mathematics i.e Reimann integral.

Hope this will satisfy your needs    : Happy learning   : by KHV, SAGAR

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CHRNOLOGY OF LIFE OF SRINIVASA RAMANUJAN

1887	Dec. 22 Born of Komalathamal and K.Srinivasa Iyangar at Erode in his maternal grandmother¡¯s house. He was the first of the three surviving sons. They are Lakshmi Narashiman and Tirunarayanan.
1882	Oct.1, on a Vijayathasami day, he was enrolled in the Thinnai Palli Koodam (Pial school)
1884	Enrolled in Kangeyam Primary School in Kumbokonam. He was quiet and contemplative. He was fond of questions, and liked to be by himself. Discouraged by his mother from going out to play, he talked to his friends only from the window overlooking the street. He lacked interest in sports, so he was fat and plump.
1897	Stood first in the Tanjore District Primary examination and earned a half-fee concession for a high school study.
1898	Joined Town High School, Kumbokonam in I Form.
1900	During his school days, he impressed everyone by his extraordinary intuition and astounding proficiency in several branches of mathematics. In III form, studied by himself Algebra, Geometric series, Loney¡¯s Part II Trigonometry, and learnt sine and cosine series before knowing them as ratios.
1903	In VI form, obtained G.S.Carr¡¯s: A Synopsis of Elementary Results, a book on Pure Mathematics, which contained propositions, formulae and methods of analysis with abridged demonstrations published in 1886. It was this book which awakened the genius in him.
1904	Passed Madras Matriculation examination and joined the F.A., class in the Government Arts College in Kumbakonam. Obtained Junior Subramaniam scholarship.
1905	He did not pass in English and hence was not promoted to senior F.A. class, so lost his scholarship.
1906	Joined Pachaiappa¡¯s College, after about three months, he fell ill and discontinued his studies.
1907	Appeared privately for F.A., examination, secured centum in mathematics, but failed to secure pass marks in other subjects. Never tried again.
1909	July 14 Married to 9 year old Janaki in Karur.
1911	His first contribution to the Journal of the Indian Mathematical Society appeared.
1912	Jan. 12 Joined as an officiating clerk in A.G¡¯s Office on Rs.20 per month.
	Feb. 09 Applied to Madras Port Trust for a job.
	Mar. 01 Joined Madras Port Trust in the Account Section in Class III, Grade IV on Rs.30 per month.
	Wrote to E.W.Hobson, a Senior Wrangler in Cambridge, but received no reply. Wrote to Henry Frederick Backer, an elected Fellow of the Royal Society and Cayley Lecturer in Mathermatics at Cambridge. He also did not acknowledge.
1913	Jan. 14 Ramanujan tumbled upon G.H.Hardy¡¯s Tract on Orders of Infinity through Prof. Seshu Aiyar. He was excited, since he had already discovered the result which gave the order of p(x). This made Prof. Seshu Aiyar suggest communication of this and other results to G.H.Hardy.
	Jan. 16 Wrote his first letter to Prof. G.H.Hardy, a Fellow of the Royal Society and Cayley Lecturer in Mathermatics at Cambridge.
	Feb. 08 Hardy acknowledged and asked for rigorous proofs.
	Feb. 27 Ramanujan wrote to Hardy, stating that he has found in him a friend. Hardy wrote to the Secretary of the Indian students, in the India Office, London, suggesting that some means be found to get Ramanujan to Cambridge.
	Ap. 07 The Government sanctioned the proposal of the Syndicate of the Madras University to grant Ramnaujan a special scholarship of Rs. 75 per month for two years. This sanction gave Ramanujan free access to the Mathematical books in the University Library.
	May 01 Ramanujan was granted two years leave on loss of pay and started working as the first research scholar of the Madras University.
	Aug. 05 Sent his first Quarterly report to the University.
	Nov. 07 Sent his second Quarterly report to the University.1915
1914	Jan. 14 Ramanujan expressed his willingness to leave for Cambridge.
	Feb. 12 Government sanctioned the appropriation of a sum not exceeding Rs.10,000 from the University Vacation Lecturer¡¯s Fund for the grant to Ramanujan of a scholarship of £250 a year tenable in England for a period of two years, a free passage and a reasonable outfit.
	Mar. 17 Sailed to England by s.s.Nevasa. The cost of the II Class ticket was Rs.440.
	Ap. 14 Arrived in London, admitted in Trinity College and stays in Prof.Nevilli¡¯s house.
	Ap. 17 Went to Cambridge.
1915	Jan. 07 Worked on Arithmetical Functions.
	Dec. 15 On recommendation of Francis Spring to Dewsbury, Ramanujan¡¯s scholarship was extended up to the end of March 1918.
1916	Mar. 15 Ramanujan was conferred upon, B.A., degree by research.
1917	Mar. 12 Fell ill and hospitalized in Sanatoria at Wells, Matlock and London.
	Oct. 12 Showed improvement and resumed active work.
	Oct. 18 Conferred upon, Trinity Fellowship consisting of £ 250 a year with no conditions about duties or residence.
1918	Feb. 28 Elected Fellow of the Royal Soceity.
	Nov.26 Hardy wrote to Dewsbury, about Ramnaujan¡¯s continued illness and need for his short return to India.
	Dec. 09 On the recommendation made by Dewsbury, the Syndicate granted Ramanujan a sum of £ 250 a year for 5 years from April 1, 1919 in recognition of his service to the Science of Mathematics.
1919	Mar. 13 Left England by s.s.Nagoya.
	Mar. 27 Arrived in India and given a warm reception at Bombay.
	Ap. 02 Reached Madras by train.
	Aug.12 Based on the report of Dr.Pires, C.F.Fearnside, the D.M.O., shifted Ramanujan to Coimbatore.
1920	Jan. 20 Returned to Madras.
	Ap. 26 Passed away leaving behind his wife Janaki Ammal aged jut 20 years. He had no issue

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66	Indian Institute of Management Raipur, Govt. Engg. College Campus, Old Dhantari Road, Sejbahar, Raipur – 492 015. Chhattisgarh	http://www.iimraipur.ac.in
67	Indian Institute of Management, Ranchi, Suchna Bhawan, Audrey House Campus, Meur’s Road, Ranchi Jharkhand	http://www.iimranchi.ac.in
68	Indian Institute of Management Tiruchirappalli, National Institute of Technology Campus, Tiruchirappalli – 620 015. Tamil Nadu	http://www.iimtrichy.ac.in
69	Indian Institute of Management Kashipur, Uttarakhand	http://www.iimkashipur.ac.in
70	Indian Institute of Management Udaipur, Rajasthan	http://www.iimu.ac.in/
71	ABV -Indian Institute of Information Technology and Management Morena Link Road Gwalior – 474 003.	http://www.iiitm.ac.in/
72	Indian Institute of Information Tehnology (IIIT) Deoghat, Jhalwa, Allahabad – 211 002	http://www.iiita.ac.in/
73	Pandit Dwarka Prasad Mishra Indian Institute of Information Technology, Design & Manufacturing (IIITDM) IT Bhavan, Jabalpur Engg. Campus, Ranjhi, Jabalpur – 482 011, Madhya Pradesh	http://www.iiitdm.in/
74	Indian Institute of Information Tehnology, Design & Manufacturing (IIITDM), Kancheepuram, Melakottaiyur, Chennai – 600048	http://www.iiitdm.ac.in/
75	Indian Institute of Science Bengaluru 560 012.	http://www.iisc.ernet.in/
76	Indian Institute of Science Education & Research (IISER) Pune, Temporarily at : National Chemical Laboratory, Dr. Homi Bhabha Road, Pune – 411008.	http://www.iiserpune.ac.in/
77	Indian Institute of Science Education & Research (IISER) Kolkata, Temporarily at : IIT Kharagpur Kolkata Campus, HC Block, Sector-III, Kolkata – 700106.	http://www.iiserkol.ac.in/
78	Indian Institute of Science Education & Research (IISER) Mohali, Temporarily at : MGSIPA Complex, Sector-26, Chandigarh -160019.	http://www.iisermohali.ac.in
79	Indian Institute of Science Education & Research (IISER) Bhopal Temporarily at: ITI (Gas Rahat) Building Govindpura, Bhopal 462 023	http://www.iiserbhopal.ac.in/
80	Indian Institute of Science Education & Research (IISER) Thiruvananthapuram Temporarily at: CET Campus, Thiruvananthapuram – 695016	http://iisertvm.ac.in/
81	National Institute of Technical Teachers’ Training & Research Block FC, Sector – III, Salt Lake, Bidhan Nagar, Kolkata – 700 091.	http://www.nitttrkol.ac.in/
82	National Institute of Technical Teachers’ Training & Research Southern Region, Taramani PO, Chennai- 600 113.	http://www.nitttrc.ac.in/
83	National Institute of Technical Teachers’ Training & Research Shamla Hills, Bhopal – 462 002.	http://www.nitttrbhopal.org/.
84	National Institute of Technical Teachers’ Training & Research Sector 26, Chandigarh- 160 019.	http://www.nitttrchd.ac.in/index.php
85	Board of Apprenticeship Training Western Region, New Admn. Building,2nd Floor, ATI Campus,Sion-Trombay Road, Sion, MUMBAI – 400 022.	http://www.apprentice-engineer.com/
86	Board of Practical Training (BOPT), Eastern Region Block EA, Sector I (OPP. Labony Estate) PO Salt Lake City, Kolkata – 700 064.	http://bopter.gov.in/html/ImpAuth.htm
87	Board of Apprenticeship Training (BOAT), Plot No.16, Block-1-A, Lakhanpur, GT Road, Kanpur – 208024.	http://boatnr.org/
88	Board of Apprenticeship Training, (BOAT) CIT Campus, Taramani, Chennai – 600 113.	https://www.boat-srp.com/boat/
89	Indian School of Mines University Dhanbad – 826004, JHARKHAND	http://www.ismdhanbad.ac.in/
90	National Institute of Foundry and Forge Technology (NIFFT) P.O. Hatia, Ranchi – 834003, Jharkhand.	http://www.nifft.ernet.in
91	National Institute of Industrial Engineering Vihar Lake, PO- NITIE, MUMBAI – 400 087.	http://www.nitie.edu/
92	School of Planning & Architecture I.P. Estate, New Delhi – 110 002.	http://spa.ac.in/
93	School of Planning & Architecture Bhopal. Temporary at NIT Campus, Bhopal.	http://www.spabhopal.ac.in/
94	National Institute of Technology CALICUT – 673601.	http://www.nitc.ac.in
95	School of Planning & Architecture Vijayawada. (Mentored by SPA, New Delhi)	http://www.spav.ac.in
96	Sant Longowal Institute of Engineering & Technology (SLIET) Village Longowal, Distt: Sangrur Punjab 148106	http://www.sliet.ac.in/
97	North Eastern Regional Institute of Science & Technology (NERIST) Nirjuli – 79110 (Itanagar), Arunachal Pradesh	http://www.nerist.ac.in/
98	National Institute of Technology Hazaratbal, SRINAGAR – 190006, J&K.	http://www.nitsri.net
99	Motilal Nehru National Institute of Technology ALLAHABAD – 211004, (UP).	http://www.mnnit.ac.in
100	Rashtriya Sanskrit Sansthan 56-57, Institutional Area, Pankha Road, Janak Puri, NEW DELHI.	http://www.sanskrit.nic.in/
01	National Institute of Technology DURGAPUR – 713209, (WEST BENGAL).	http://www.nitdgp.ac.in
102	Shri Lal Bahadur Shastri Rashtriya Sanskrit Vidyapeeth Katwaria Sarai, Near Qutub Hotel, New Mehrauli Road, NEW DELHI-110067.	http://www.slbsrsv.ac.in/home.asp
103	Rashtriya Sanskrit Vidyapeetha TIRUPATI, (A.P.).	http://rsvidyapeetha.ac.in/
104	National Institute of Technology JAMSHEDPUR-831014, JHARKHAND	http://www.nitjsr.ac.in
105	Maharshi Sandeepani Rashtriya Veda Vidya Pratishthan Ujjayini Development Authority, Administrative Building, Bharatpur, Ujjain – 456010.	http://msrvvp.nic.in/default.htm
106	Visvesvaraya National Institute of Technology NAGPUR – 440001.	http://www.vnit.ac.in/
107	National Institute of Technology Srinivasanagar SURTHAKAL – 575 025.	http://www.nitk.ac.in
108	Kendriya Hindi Sansthan Hindi Sansthan Marg, AGRA – 282005.	http://khsindia.org/index.php?lang=hi
109	National Council for Promotion of Urdu Language West Block No.I, R.K. Puram, New Delhi – 110 066.	http://www.urducouncil.nic.in/
110	National Institute of Technology WARANGAL – 506004, (AP)	http://www.nitw.ac.in/
111	Malaviya National Institute of Technology JAIPUR – 302017. Rajasthan	http://www.mnit.ac.in
112	National Council for Promotion of Sindhi Language 5th Floor, Darpan Building, R.C.Dutt Road, Alkapuri, Vadodra – 390005.	http://www.ncpsl.gov.in/
113	Central Institute of Classical Tamil Plot.No.40, IRT Campus,100 Ft Road,Taramani,Chennai 600 005	http://www.cict.in/
114	National University of Educational Planning and Administration (NUEPA 17-B, Sri Aurobindo Marg, NIE Camp, NEW DELHI – 110016.	http://www.nuepa.org/
115	Auroville Foundation Bharat Nivas, P.O. Auroville, Distt. Villupuram, AUROVILLE – 605101, Tamil Nadu.	http://www.auroville.org/
116	Educational Consultants of India Limited (EdCIL) Plot No. 18A,Sector – 16A, NOIDA – 201301, (UP).	http://www.edcilindia.co.in/
117	Central Institute of Indian Languages Manasagangotri, Mysore – 570 006	http://www.ciil.org/
118	Central Hindi Directorate R.K.Puram, New Delhi.	http://hindinideshalaya.nic.in/
119	Commission for Scientific and Technical Terminology R.K.Puram, New Delhi.	http://www.cstt.nic.in/
120	National Book Trust of India A-15, Green Park, NEW DELHI – 110016.	http://www.nbtindia.org.in/
	National Institute of Technology ROURKELA – 769008, ORISSA	http://www.nitrkl.ac.in/
122	Maulana Azad National Institute of Technology BHOPAL – 462007.	http://www.manit.ac.in
123	All India Council of Technical Education (AICTE) 7th Floor, Chanderlok Building, Janpath, New Delhi – 110 001.	http://www.aicte-india.org/
124	Indian Institute of Advanced Studies (IIAS) Rashtrapati Nivas, Shimla – 171 005.	http://www.iias.org/
125	National Institute of Technology Tiruchirapalli- 620 015, TAMIL NADU	http://www.nitt.edu
126	National Commission for Minority Educational Institutions 1st Floor, Jeevan Tara Building, 5, Sansad Marg, Patel Chowk, New Delhi – 110 001	http://ncmei.gov.in/
127	National Institute of Technology Kurukshetra – 132119, HARYANA	http://www.nitkkr.ac.in/
128	Council of Architecture India Habitat Centre, Core-6-A, Ist Flooor, Lodhi Road, New Delhi – 110 003.	http://www.coa.gov.in/
129	National Institute of Technology Silchar – 788010, ASSAM	http://www.nits.ac.in
130	National Institute of Technology Hamirpur – 177001 , HIMACHAL PRADESH	http://nith.ac.in/
131	National Institute of Technology Patna – 800 005, BIHAR	http://www.nitp.ac.in/
132	Dr. B.R. Ambedkar National Institute of Technology G.T. Road, Bye Pass, Jallandhar – 144 011, PUNJAB.	http://www.nitj.ac.in
133	National Institute of Technology Raipur, Chhattisgarh	http://www.nitrr.ac.in
134	National Institute of Technology Agartala, Tripura	http://www.tec.nic.in
135	National Institute of Technology Sikkim, Barfung Block, Ravangla Sub-Division, South Sikkim – 737 139.	http://www.nitsikkim.ac.in/
136	National Institute of Technology Yupia, District Papum Pare, Arunachal Pradesh – 791112.	http://www.nitap.in/
137	National Institute of Technology Meghalaya	http://www.nitm.ac.in/
138	National Institute of Technology Nagaland	http://nitnagaland.ac.in/home/
139	National Institute of Technology Manipur	http://www.nitmanipur.in/
140	National Institute of Technology Mizoram, Chaltlang, Aizawl-796012.	http://www.nitmz.ac.in/
	National Institute of Technology Govt. Polytechnic, Srinagar (Garhwal) – 246174, Uttarkhand	http://nituk.com/
142	National Institute of Technology Goa	http://www.nitgoa.ac.in/
143	National Institute of Technology Delhi	http://nitdelhi.ac.in/
144	National Institute of Technology Puducherry	http://www.nitt.edu/home/nitp/
145	Central Board of Secondary Education 2 Community Centre, Preet Vihar New Delhi – 110092	http://www.cbse.nic.in/
146	National Council for Educational Research and Training (NCERT) Sri Aurbindo Marg New Delhi – 1100162	http://ncert.nic.in/
147	National Institute of Open Schooling (NIOS) A-24/25, Institutional Area Sector – 62 NOIDA-201309 Uttar Pradesh	http://www.nos.org/
148	Central Tibetan School Administration (CTSA) Plot No. 1, Community Centre Sector-3, Rohini Delhi – 110085	http://ctsa.nic.in/
149	Navodaya Vidyalaya Samiti (NVS) A-28, Kailash Colony New Delhi – 110048 Sector-3, Rohini Delhi – 110085	http://navodaya.nic.in/
150	Kendriya Vidyalaya Sangathan (KVS 18 – Shaheed Jeet Singh Marg New Delhi – 110016	http://kvsangathan.nic.in/
151	Central Institute of Education Technology (CIET) NCERT Campus Sri Aurbindo Marg New Delhi – 110016	http://ciet.nic.in/
152	National Bal Bhawan Kotla Road, New Delhi-110002	http://nationalbalbhavan.nic.in/
153	Sardar Vallabhbhai National Institute of Technology Ichchhanath, SURAT-395 007 (Gujarat)	http://www.svnit.ac.in/

LINKS TO USEFUL WEBSITES FOR EDUCTION

Links to useful websites

.	Ministry Of Education, Government Of India
.	National Center for Technology in Education, Dublin City University, Dublin
.	Center for Research & Studies on Educational Technology, Open & Distance Education and Teacher Training
.	National Council of Teachers of English, USA
.	Support for New Teachers Ideas that work
.	Education sources for theory, research, policy and practice
.	Serving educators with ideas, tools, and resources for integrating technology into classroom and curriculum
.	On Line Learning Resources for Teachers and students
.	Project Based Learning
.	Global School House
.	Good Education Research Material
.	Better Teachers, Better Schools
.	REFORM : VOICE OF A SCHOOL
.	US DEPARTMENT OF EDUCATION
.	Society for Information Technology and Teacher Education international association of teacher educators who are interested in the use of IT in teacher education
.	ASOCIATION OF TEACHER EDUCATORS – TEACHER EDUCATORS SPECIAL INTEREST GROUP
.	Society for ADVANCEMENT OF COMPUTING IN EDUCATION
.	ISATT – THE INTERNATIONAL STUDY ASSOCIATION ON TEACHERS AND TEACHING
.	Teacher Learning Home Page
.	GATEWAY TO SEARCH EDUCATION MATERIALS OF THE WEB
.	Education Initiatives
.	High-Performing, High-Poverty Schools
.	DISNEY : Learning Partnership ALL INDIA ASSOCIATION FOR EDUCATIONAL RESEARCH
.	A VISIT TO VIGYAN PRASAR (For Daily Science News From India) Read ComCom February Issue : INTERVIEW with Prof. E.V. Chinis (Let Us Reach 50,000 Science Teachers) MASS Higher EDUCATION (but with a difference) By Dr. D.K. Passi
.	Indian News Papers
.	World News papers
	LINKS TO EDUCATION DEPARTMENTS OF INDIA:

FACTORIAL N: FACTS

Definition

In mathematics, the factorial of a natural number n is the product of the positive integers less than or equal to n. This is written as n! and pronounced ‘n factorial’.
The factorial function is formally defined by

 

       =1 × 2 × 3 × . . . × (n – 1) × n

For example,

5! = 1 * 2 * 3 *4 * 5 = 120

This definition implies in particular that

0! = 1

because the product of no numbers at all is 1. Proper attention to the value of the empty product is important in this case, because

• it makes the above recursive relation work for n = 1;
• many identities in combinatorics would not work for zero sizes without this definition

The factorial of an integer n is denoted by n!. This n! notation was first used by a French mathematician. Who was he?

Christian Kramp. Kramp was born in Strasbourg, France in 1760. He used the notation n! in 1808 in one of his books, “Elements d’arithmétique Universelle”. 

 Multiple scientists worked on this subject, but the principal inventors are J. Stirling in 1730 who    gives the asymptotic formula after some work in collaboration with De Moivre, then Euler in 1751 and finally C. Kramp and Arbogast who introduces between 1808 and 1816 the actual notation: n!. Of course other scientists such as Taylor also worked a lot with this notation.

Given that n! = n(n-1)(n-2)…(2)(1). Instead of multiplication, there exists an analog series that is defined by addition, that is, f(n) = n + (n-1) + (n-2) + … + 2 + 1. This type of number is known as a/an?

Triangular number. The first triangular number is 1. The second one is 2 + (2-1) = 2 + 1 = 3. The third triangular number is given by 3 + (3-1) + (3-2) = 3 + 2 + 1 = 6. This is followed by 10, 15, 21. Another simpler way to calculate the nth triangular number would be by using the formula [(n)(n+1)]/2. For instance, the 5th triangular number is [(5)(6)]/2 = 15. 

A prime number that is 1 more or 1 less than the value of a factorial is called a factorial prime. Which of the following numbers is not a factorial prime?

5! – 1 = 119. 119 is the product of 2 smaller primes, which are 7 and 17. Some other factorial primes are such as 1! + 1 = 2, 2! + 1 = 3, 3! + 1 = 7.

 

Factorial operations are used widely in combinations and permutations problems. To find the number of different ways one can select r items from a total of n items, we use the combination formula, which is n C r = (n!)/{[(n-r)!](r!)}. On the other hand, the formula for permutation, given by n P r, is?

n!/(n-r)!. For permutation, we wish to find the number of different ways one can arrange r items in different orders, selected from a total of n items.

The factorial of 0, namely 0! is 0.

F. Actually, 0! = 1. We notice the followings: 1! = 1 

2!= 2 x 1 = 2 

3! = 3 x 2 x 1 = 6 

4! = 4 x 3 x 2 x 1 = 24 

5! = 5 x 4 x 3 x 2 x 1 = 120  Working backwards, we will get:

 4! = 5!/5 = 24 

 3! = 4!/4 = 6 

 2! = 3!/3 = 2

 1! = 2!/2 = 1 

 0! = 1!/1 = 1 

 Therefore, the operation of  C(5, 0)  is valid and possible, where the answer is 1. This means that we have only 1 way of choosing 0 items from a total of 5 items. 

Factorial operations are used widely in combinations and permutations problems. To find the number of different ways one can select r items from a total of n items, we use the combination formula, which is n C r = (n!)/{[(n-r)!](r!)}. On the other hand, the formula for permutation, given by n P r, is?

n!/(n-r)!. For permutation, we wish to find the number of different ways one can arrange r items in different orders, selected from a total of n items.

 

A positive integer that is in the form of n^{(n-1)^[(n-2)^…^{2^(1)}]} is called a/an ?

Exponential factorial. Let f(n) be the exponential factorial function for an integer n. When n = 1, f(n) = 1 When n = 2, f(n) = 2^1 = 2 When n = 3, f(n) = 3^(2^1) = 9 When n = 4, f(n) = 4^[3^(2^1)] = 262144 It should be noticed that when calculating the exponential factorial for n = 4, the operation involved is 4^[3^(2^1)], not [(4^3)^2]^1

Factorials also come in useful in calculus. Some exponential and trigonometry functions can be expressed as a power series of x, as what is stated and defined in which of the following theorems?

Taylor’s theorem. Some of the famous power series that is derived from this Taylor’s theorem is cos x = 1 – (x^2)/2! + (x^4)/4! – (x^6)/6! + (x^8)/8! -… Apart from this, the functions sin x, tan x, and e^x can also be expressed as power series. 

Calculating factorials

The numeric value of n! can be calculated by repeated multiplication if n is not too large. That is basically what pocket calculators do. The largest factorial that most calculators can handle is 69!, because 70! > 10¹⁰⁰.
When n is large, n! can be estimated quite accurately using Stirling’s approximation:

The gamma function

The related gamma function Γ(z) is defined for all complex numbers z except for the nonpositive integers (z = 0, −1, −2, −3, …). It is related to factorials in that it satisfies a recursive relationship similar to that of the factiorial function:

n! = n(n – 1)!

Γ(n + 1) = nΓ(n)

Together with the definition Γ(1) = 1 this yields the equation

Because of this relationship, the gamma function is often thought of as a generalization of the factorial function to the domain of complex numbers. This is justified for the following reasons.

• Shared meaning—The canonical definition of the factorial function is the mentioned recursive relationship, shared by both.
• Uniqueness—The gamma function is the only function which satisfies the mentioned recursive relationship for the domain of complex numbers and is holomorphic and whose restriction to the positive real axis is log-convex. That is, it is the only function that could possibly be a generalization of the factorial function.
• Context—The gamma function is generally used in a context similar to that of the factorials (but, of course, where a more general domain is of interest).

Multifactorials

A common related notation is to use multiple exclamation points to denote a multifactorial, the product of integers in steps of two (n!!), three (n!!!), or more.
n!! denotes the double factorial of n and is defined recursively by

For example, 8!! = 2 · 4 · 6 · 8 = 384 and 9!! = 1 · 3 · 5 · 7 · 9 = 945. The sequence of double factorials for n = 0, 1, 2,… starts

1, 1, 2, 3, 8, 15, 48, 105, 384, 945, 3840, …

Some identities involving double factorials are:

n! = n!!(n – 1)!!

(2n)!! = 2ⁿn!

One should be careful not to interpret n!! as the factorial of n!, which would be written (n!)! and is a much larger number.
The double factorial is the most commonly used variant, but one can similarly define the triple factorial (n!!!) and so on. In general, the k-th factorial, denoted by n!^(k), is defined recursively as

<img alt=" n!^{(k)}= \left\{ \begin{matrix} 1,\qquad\qquad\ &&\mbox{if }0\le n

Hyperfactorials

Occasionally the hyperfactorial of n is considered. It is written as H(n) and defined by

For n = 1, 2, 3, 4,… the values of H(n) are 1, 4, 108, 27648,…
The hyperfactorial function is similar to the factorial, but produces larger numbers. The rate of growth of this function, however, is not much larger than a regular factorial.

Superfactorials

The superfactorial of n, written as n$ (the $ should really be a factorial sign ! with an S superimposed) has been defined as

n$ = n⁽⁴⁾n

where the ⁽⁴⁾ notation denotes the hyper4 operator, or using Knuth’s up-arrow notation.

The sequence of superfactorials starts:

1$ = 1

2$ = 2² = 4

  source: collections from books, internet etc

 

MATH FUN

A Mathematicians Watch