The rate of change of the function

We are now ready to interpret the derivative in terms of the average rate of change of a function.

Rf(A, Bi) = (^x)/(^y) gives the slope of the straight line passing through A and Bi (i= 1,2,3,.....,n). These lines are called secant lines. By definition of the derivative, lim ^x->0 (^y)/(^x)= f'(a).

Also as ^x approaches zero, the points Bi aproach the point A.

These two statments taken together mean that f'(a) gives the limit of the average rates of change of y=f(x) between A and Bi, as Bi approaches A along the graph of the function. this limit is called simply Rf(A).




Bibliotheca Publica

Milton had strong ideas about the place of a "Public library" (Bibliotheca Publica) in the cultivation of virtue, with the librarian as a "faithful guardian of works eternal." Even though Milton had good words for the Bodleian Library in his 'Ode to Rouse', he lamented that the students were making poor use of it. In a cautionary letter to his former pupil who had come up to Oxford to study, Milton wrote "The library there is rich in books, but unless the minds of the students be improved by a more rational mode of education, it may better deserve the name of a book-repository than of a library."

Oxford Univ. News, Opening of Bodleain Milton Exhibition, 17 Dec.




Wisemen's advice

Philip Pullman advice to teachers to put the class in order: "If you work out quickly in the first couple of days who the king and queen are, and you direct all your attention to them in the first week or so, get them on your side, you won't have any discipline problems because everyone follows them. They don't follow you. They follow them."

He noticed that if a girl fell out of one group and joined the other, she instantly took on the attributes of the new group.

Source: An interview with Pullman, INTELLIGENT LIFE magazine, December 2007
http://moreintelligentlife.com/node/697






Milton charges Cromwell with the duty of maintaining in peacetime all that has been won in war.
Tanner 831, Bodleian Library, John Milton, 'To Oliver Cromwell,' Letters of State (1694) , Oxford Univ.








A look at statistics

The average UK person will this year have a greater income than their US counterpart for the first time since the 19th Century, figures suggest (BBC,07 Jan 2008).

UK’s Gross national income and net national income is higher than other OECD countries after US and Norway and Switzerland (2005).

Percentage of elderly population is highest in Japan and Germany 19%, in UK is 16% and US 12%. (2005)

In 2002, OECD universities awarded some 5.9 million degrees at university level, of which 156 000 doctorates. In other words, fewer than one person in three at the typical age of graduation completed a university degree, while one out of 100 received a doctoral degree.

In 2002, the United States hosted 79 000 foreign doctoral students, the largest number in the OECD. The United Kingdom is the second major host with over 22 000 students.

International mobility of doctoral students has increased over the past five years, particularly so in Norway and in Spain. The share of foreign students enrolled in advanced research programmes grew in most countries between 1998 and 2002, with the notable exception of two of the main European host countries (Belgium and the United Kingdom). In the United Kingdom, this was due to a sharp change from 2001 to 2002.

On average, 28% of persons employed in the OECD area had a tertiary-level degree in 2003. Canada and Japan (over 40%) and the United States (38%) ranked far ahead of the European Union, where less than one worker out of four holds a tertiary-level degree.

Between 1998 and 2003, employment of tertiary-level graduates grew at an annual pace of about 4% in the OECD area. The population of tertiary-level workers is aging.

In 2004, almost 54 million persons were employed in an Science &Technology occupation, across the EU25, almost 42 million in the United States and about 10 million in Japan. In Europe, two-thirds were concentrated in the four largest economies, i.e. Germany (23%), France (14%), the United Kingdom (13%) and Italy (12%). The Czech Republic, Hungary, Poland and the Slovak Republic together employed 12%.

European natives are more likely to go abroad, especially if they are highly educated. Two-thirds of OECD-area highly skilled expatriates are European. Emigration is particularly frequent from the United Kingdom (19% as percentage of native highly skilled workers) and Austria, and is also common from eastern Europe.

In the vast majority of OECD countries, the number of researchers rises at a faster rate than the number of total R&D personnel.

In 2000, approximately 3.4 million researchers were engaged in research and development (R&D) in the OECD area. Around 38% of all OECD area researchers reside in the United States, 29% in the EU15 and 19% in Japan.

In the United States, four out of five researchers work in the business sector but only one out of two in the European Union.

Share of countries in Biotech patents filed in 2001, Germany 12.8, UK 5.9, Netherland, 2.2, France 5.3 - (US 41.5% and EU 34%).

The United States is first direction for migration, with over 7.8 million highly skilled expatriates. The European Union follows with 4.7 million, before Canada and Australia, with 2 and 1.4 million highly skilled foreign residents, respectively. Over half of these migrants come from outside the OECD area. In addition to the 6.7 million highly educated persons involved in intra-OECD skill flows, the region has attracted 10.1 million from non-OECD countries.

Source: OECD Statisitcs, 2007

Interpreting the derivative geometrically, we see that it gives the slope of the straight line that touches the curve at A = (a, f(a)) and occupies the limiting position of the secnt lines. The line just described is called the tangent line to the curve at A.

Finally, the two interpretaitons discussed above are summarized y the following statement. The point rate of change of y = f(x) at the point A equals the slope of the tangent line to the graph of y = f(x) at the point A.