SUMMARY / RELATED TOPICS

Tangent

In geometry, the tangent line to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More a straight line is said to be a tangent of a curve y = f at a point x = c on the curve if the line passes through the point on the curve and has slope f', where f' is the derivative of f. A similar definition applies to space curves in n-dimensional Euclidean space; as it passes through the point where the tangent line and the curve meet, called the point of tangency, the tangent line is "going in the same direction" as the curve, is thus the best straight-line approximation to the curve at that point. The tangent plane to a surface at a given point is the plane that "just touches" the surface at that point; the concept of a tangent is one of the most fundamental notions in differential geometry and has been extensively generalized. The word "tangent" comes from the Latin tangere, "to touch".

Euclid makes several references to the tangent to a circle in book III of the Elements. In Apollonius work Conics he defines a tangent as being a line such that no other straight line could fall between it and the curve. Archimedes found the tangent to an Archimedean spiral by considering the path of a point moving along the curve. In the 1630s Fermat developed the technique of adequality to calculate tangents and other problems in analysis and used this to calculate tangents to the parabola; the technique of adeqality is similar to taking the difference between f and f and dividing by a power of h. Independently Descartes used his method of normals based on the observation that the radius of a circle is always normal to the circle itself; these methods led to the development of differential calculus in the 17th century. Many people contributed. Roberval discovered a general method of drawing tangents, by considering a curve as described by a moving point whose motion is the resultant of several simpler motions.

René-François de Sluse and Johannes Hudde found algebraic algorithms for finding tangents. Further developments included those of John Wallis and Isaac Barrow, leading to the theory of Isaac Newton and Gottfried Leibniz. An 1828 definition of a tangent was "a right line which touches a curve, but which when produced, does not cut it"; this old definition prevents inflection points from having any tangent. It has been dismissed and the modern definitions are equivalent to those of Leibniz who defined the tangent line as the line through a pair of infinitely close points on the curve; the intuitive notion that a tangent line "touches" a curve can be made more explicit by considering the sequence of straight lines passing through two points, A and B, those that lie on the function curve. The tangent at A is the limit when point B approximates or tends to A; the existence and uniqueness of the tangent line depends on a certain type of mathematical smoothness, known as "differentiability." For example, if two circular arcs meet at a sharp point there is no uniquely defined tangent at the vertex because the limit of the progression of secant lines depends on the direction in which "point B" approaches the vertex.

At most points, the tangent touches the curve without crossing it. A point where the tangent crosses the curve is called an inflection point. Circles, parabolas and ellipses do not have any inflection point, but more complicated curves do have, like the graph of a cubic function, which has one inflection point, or a sinusoid, which has two inflection points per each period of the sine. Conversely, it may happen that the curve lies on one side of a straight line passing through a point on it, yet this straight line is not a tangent line; this is the case, for example, for a line passing through the vertex of a triangle and not intersecting it otherwise—where the tangent line does not exist for the reasons explained above. In convex geometry, such lines are called supporting lines; the geometrical idea of the tangent line as the limit of secant lines serves as the motivation for analytical methods that are used to find tangent lines explicitly. The question of finding the tangent line to a graph, or the tangent line problem, was one of the central questions leading to the development of calculus in the 17th century.

In the second book of his Geometry, René Descartes said of the problem of constructing the tangent to a curve, "And I dare say that this is not only the most useful and most general problem in geometry that I know, but that I have desired to know". Suppose that a curve is given as the graph of a function, y = f. To find the tangent line at the point p =, consider another nearby point q = on the curve; the slope of the secant line passing through p and q is equal to the difference quotient. As the point q approaches p, which corresponds to making h smaller and smaller, the difference quotient should approach a certain limiting value k, the slope of the tangent line at the point p. If k is known, the equation of the tangent line can be found in the point

Patrick Bruel

Patrick Maurice Benguigui, better known by his stage name Patrick Bruel, is a French singer-songwriter and professional poker player. Patrick is daughter of Elie and Céline ben Sidoun. In his youth, Bruel aspired to be a football player, but decided instead to pursue singing after seeing Michel Sardou in 1975, his first success came in 1979's Le Coup de sirocco. He continued acting in films, on television, in the theater while pursuing his singing career, his first single, "Vide", released in 1982, was not a success, but the follow-up, "Marre de cette nana-là", was a hit. In 2003, just before his partner, the writer and playwright Amanda Sthers, gave birth to his first child, Oscar, on 19 August, he changed his name to Bruel-Benguigui, combining his stage name with his birth name. On 21 September 2004, he wed the 26-year-old Sthers, his second child, Léon, was born on 28 September 2005. The couple separated in 2007; as of 2004, Bruel has acted in more than forty television and film productions, has made five studio albums and several live albums.

In 2002, Bruel released Entre Deux, a double CD of classic chanson that features duets with Charles Aznavour, Jean-Louis Aubert, Jean-Jacques Goldman, Alain Souchon and Renaud, among others. It made Bruel France's best paid singer of the year. At the beginning of 2005, in response to the South Asian tsunami of 26 December 2004, Bruel wrote the song "Et puis la terre" with Marie-Florence Gros, with whom he had begun a long-term collaboration in 1998, to benefit the Red Cross, his latest album, "Des souvenirs devant", was his fourth chart-topper in France. Bruel is a world-class professional poker player, he won. As of 2009, he has earned more than $900,000 in tournament play, of which his ten cash winnings at the WSOP account for $411,659. He comments on the World Poker Tour in France. Bruel is a member of the Les Enfoirés charity ensemble since 1993. Featured in and collective collaborations Official website Patrick Bruel on IMDb Biography of Patrick Bruel, from Radio France Internationale Photos

Ecclesbourne Valley Railway

The Ecclesbourne Valley Railway is a 9-mile long heritage railway in Derbyshire. The headquarters of the railway centre on Wirksworth station, services operate in both directions between Wirksworth and Duffield and from Wirksworth to Ravenstor. From April 2011 onward, passengers are now able to board and alight heritage services at Duffield where in recent years a station platform has been re-constructed. Heritage services are timed to connect with East Midlands Railway Nottingham - Derby - Matlock service at the adjacent Duffield. Network Rail platforms and therefore it is now possible for passengers to travel to and from Wirksworth by train from anywhere on the national network; the Ecclesbourne Valley Railway is named after the River Ecclesbourne and the track follows the river from its source to its confluence with the River Derwent at the Derbyshire village of Duffield. Despite being a branch in itself, there is a separate 1⁄2 mile branch operating from Platform 3 at Wirksworth Station up a 1 in 27 gradient incline to Ravenstor.

The line is operated by a large fleet of heritage Diesel Multiple Units, as well as diesel and visiting steam locomotives. Locomotive hauled trains only operated on Enthusiast and special event days alongside the DMU fleet, whereas now locomotive hauled services make up a larger part of the railway's timetable; the "Wirksworth Branch" was the product of early 19th century railway rivalry. Since 1835 Wirksworth's citizens had been promoting the idea, among others, for a branch line from the North Midland Railway the Midland Railway, at Duffield; the Midland was unenthusiastic, but realised that the branch could be extended to Rowsley, albeit with difficulty, avoiding the section from Ambergate, on its Manchester, Buxton and Midlands Junction Railway, shared with its rival the London and North Western Railway. It is for this reason that all of the bridges along the line, including the one which has a head shunt under it are built to double-tracked grand Midland Railway style; the 8 1⁄2-mile line was surveyed in 1862 and received Parliamentary assent the following year.

It would follow the valley of the River Ecclesbourne with no major obstacles apart from the final climb into Wirksworth. A cutting was required, some buildings were demolished, while there was considerable upheaval in Duffield; the final inspection of the line was carried out by Colonel J. A. Rich of the Royal Engineers on 26 September 1867, who approved the line for opening; the line was opened to Wirksworth on 1 October 1867 and was worked by the Staff System. Under the original scheme, it would have descended from Wirksworth to Cromford using a 1,503-yard tunnel and a 280-yard long viaduct, proceed parallel to the existing line, but on the west side of the river through Matlock to Rowsley. However, when the lease expired on the original Ambergate line, the LNWR withdrew, the Midland acquired complete control, thus the section beyond Wirksworth was never built. The Midland was left with one of its few branch lines, one which, it felt, was of questionable viability; the presence of the line allowed Wirksworth's limestone business to develop, the carriage of, its mainstay until the middle of the 20th century.

There was farm produce milk, some 800,000 imp gal daily, a number of textile mills. It saw a regular passenger service, with stations at Hazlewood and Idridgehay. There were three, rising to six, passenger trains from Derby each way, with one on Sunday, two goods trains. By 1939, however milk was carried instead by road, during World War II passenger travel was curtailed. There was the hourly "number 37" bus, which led to a decline in passenger numbers. Passenger trains were temporarily suspended in 1947 and were ceased in 1949. An hourly direct bus still operates between Derby with a journey time of 50 minutes; however this runs via Belper rather than directly along the main road. In the early 1950s people near the line were treated to the eerie sight of a railway carriage ghosting along by itself, it must be said that there would be some who remembered the use of steam motor carriages from the Morecambe and Heysham Railway at the beginning of the century, steam railmotors from the Yarmouth and North Norfolk Railway.

However this was the test vehicle for the new diesel railcars being designed in Derby - nothing more than a standard coach with the mechanism fitted and a windscreen cut in each end for the driver - that presaged a major change in British rail travel. When the so-called Derby Lightweights were produced they were each tested on the line after leaving the workshop. One of the only three surviving of those built, M79900, was converted from being the IRIS test car back to passenger carrying standard and has been joined by the other two, residing on the line on which they were tested some 60 years ago. On 25 August 1981, a rail accident occurred when a laden freight train derailed 300 yards south of Wirksworth. Although most of the goods had transferred to the roads, limestone traffic continued, including that hauled by the Cromford and High Peak Railway, when it closed in 1967. Though the amount of traffic justified the installation of some continuous wel