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In mathematics, more calculus, L'Hôpital's rule or L'Hospital's rule provides a technique to evaluate limits of indeterminate forms. Application of the rule converts an indeterminate form to an expression that can be evaluated by substitution; the rule is named after the 17th-century French mathematician Guillaume de l'Hôpital. Although the rule is attributed to L'Hôpital, the theorem was first introduced to him in 1694 by the Swiss mathematician Johann Bernoulli. L'Hôpital's rule states that for functions f and g which are differentiable on an open interval I except at a point c contained in I, if lim x → c f = lim x → c g = 0 or ± ∞, g ′ ≠ 0 for all x in I with x ≠ c, lim x → c f ′ g ′ exists lim x → c f g = lim x → c f ′ g ′; the differentiation of the numerator and denominator simplifies the quotient or converts it to a limit that can be evaluated directly. Guillaume de l'Hôpital published this rule in his 1696 book Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes, the first textbook on differential calculus.

However, it is believed. The general form of L'Hôpital's rule covers many cases. Let c and L be extended real numbers. Let I be an open interval containing c or an open interval with endpoint c; the real valued functions f and g are assumed to be differentiable on I except at c, additionally g ′ ≠ 0 on I except at c. It is assumed that lim x → c f ′ g ′ = L, thus the rule applies to situations in which the ratio of the derivatives has a finite or infinite limit, but not to situations in which that ratio fluctuates permanently as x gets closer and closer to c. If either lim x → c f = lim x → c g = 0 or lim x → c | f | = lim x → c | g | = ∞ lim x → c f g = L. Although we have written x → c throughout, the limits may be one-sided limits, when c is a finite endpoint of I. In the second case, the hypothesis that f diverges to infinity is not used in the proof; the hypothesis that g ′ ≠ 0 appears most in the literature, but some authors sidestep this hypothesis by adding other hypotheses elsewhere. One method is to define the limit of a function with the additional requirement that the limiting function is defined everywhere on the relevant interval I except at c.

Another method is to require that both f and g be differentiable everywhere on an interval containing c. The requirement that the limit lim x → c f ′ g ′ ( x

San Pascual the Municipality of San Pascual, is a 3rd class municipality in the province of Masbate, Philippines. According to the 2015 census, it has a population of 46,674 people. In addition to the northern part of Burias Island, the islands of Busing and another six small unpopulated islands belong to the municipality. San Pascual is politically subdivided into 22 barangays. In the 2015 census, the population of San Pascual, was 46,674 people, with a density of 190 inhabitants per square kilometre or 490 inhabitants per square mile. One of the popular tourist destinations in the municipality of San Pascual is the Sombrero Island with its pure white sand beach. Philippine Standard Geographic Code Philippine Census Information Local Governance Performance Management System

Oxycorynini are a tribe of American belids, primitive weevils of the family Belidae. Like in other belids, their antennae are not elbowed as in the true weevils. In contrast to other Belidae, Oxycorynini larvae feed on several types of fruit; the Oxycorynini contain the bulk of the subfamily Oxycoryninae. But while they are diverse, they are not speciose, with nearly 20 described and a few undescribed species compared to the more than 150 Hawaiian species of Proterhinus of the Aglycyderini; the fossil record shows that the three Oxycoryninae tribes were well distinct by the mid-Paleogene, about 50 million years ago. The belids as a whole are of Jurassic origin, the Oxycorynini must thus have evolved in the Late Cretaceous or Paleocene some 100-60 mya; the adult Oxycorynini have a number of characteristic traits, but they are not present in all members of this tribe. Many have a pronotum whose hind margin is at least as wide as the elytra bases wider. Except in the rather primitive Oxycraspedus, the first tarsomere is smaller than the second, microscopically the lack of spermatheca, with the spermathecal duct directly inserting on the common oviduct, is conspicuous.

Adults of Parallocorynus and Rhopalotria are peculiarly soft-bodied and possess several characteristic traits: their gular sutures are fused to a single suture, their elytra cannot be locked in open position and have round tips between which the pygidium can be seen. The sternite of their mesothorax slants forward, but the part protruding between the second leg pair is not raised, their spermathecal gland and spermathecal duct form a common tube. Alloxycorynus and Hydnorobius adults have two gular sutures which form a semicircle at the head base and end in distinct tips. Adults of Balanophorobius and Oxycorynus have a flattened hindside of the metathoracal sternite. More characteristic are the larvae, their labrum has two sensilla at the base and two more towards the center of the forward maring. They have a typical body shape with recurved hindparts. Except in Oxycraspedus, the dark forehead is bordered by a ridge towards the back, where the cuticle attaches, the body is widest in the middle of the abdomen.

And of course, Oxycorynini larvae differ from other belids in that their larvae do not feed on dead or decaying wood but on flowers and fruits. The Oxycorynini are tentatively divided into 3 subtribes:Oxycraspedina OxycraspedusAllocorynina Parallocorynus RhopalotriaOxycorynina Hydnorobius Alloxycorynus Oxycorynus BalanophorobiusHowever, most of this subdivision, in particular the monophyly of the Oxycorynina, is by no means reliably established. Oxycraspedus seems to be a basal genus. What seems probable is that Parallocorynus and Rhopalotria as well as Balanophorobius and Oxycorynus are sister genera, with the other genera are of rather uncertain position in regard to these two clades, but while the Allocorynina are quite a good clade, the biogeography of the Oxycorynina is puzzling, because Balanophorobius does not occur anywhere near its presumed sister taxon Oxycorynus, which instead ranges adjacent to the other two members of the proposed subtribe. Neither are all species of Oxycorynini discovered yet, nor is the knowledge about their distribution considered satisfactory by entomologists.

Oxycraspedus the most ancient living genus of Oxycorynini, has three described species limited to Chile between the Maule and Araucanía Regions and adjacent Argentina where they have been found in Neuquén Province. Parallocorynus with one and Rhopalotria with three described species - several undescribed species are known in each genus - occur across Central America and in some adjacent regions; the former only occurs in Mexico and Honduras, while the latter is widespread from Mexico to Panama occurring on Cuba and in Florida, in northern South America too. The two Alloxycorynus species are found from southern Peru to eastern Bolivia and northern Argentina. Hydnorobius with three species occurs to the southeast of Alloxycorynus south to central Argentina. Oxycorynus occurs with 5 known species northwards of Hydnorobius, from northern Argentina and southern Bolivia north into Brazil; the monotypic Balanophorobius, described only in 2005, has not yet been found outside the Osa Peninsula of Costa Rica.

Oxycraspedus utilizes the cones of the Monkey-puzzle tree. Adults seek shelter in the cones - female one but male ones too - and larvae feed on the parenchymatous tissue but not the seeds. Indeed, by the time the larvae have finished their development to pupation, the host cones are already discarded by the plant and have shed their seeds. Old cones that are rotting on the forest floor are taken, demonstrating the evolutionary link between Oxycorynini and the other Belidae which feed on decaying woody tissue, but of twigs and stems. Parallocorynus has to date only been collected from the cycad genus Dioon. Rhopalotria is associated with various species of Zamia cycads, though one of the as yet undescribed species was collected from Dioon spinulosum, it seems that the species of Rhopalotria are monophagous - like R. slossoni on Coontie or R. mollis on Cardboard Cycad - but more research in Zamia species in necessary to verify this. These two belid genera are important pollinators of their host cycads.

Alloxycorynus, Oxycorynus have been recorded from Balanophoraceae. Specific beetle-plant association are A. bruchi with Ombrophytum, O. nigripes and O. arm

It was bought by the New York Central Railroad system in 1915. From this point until around the 1930s, this railroad was among the causes of the expansion of Southern Michigan. However, starting in the 1930s the use of the train began to dwindle due to increasing automobile ridership; the railroad lost significant revenues from the lack of passenger service. The system still ran on exporting goods throughout the country. In 1963 and 1965 the tracks between Clinton and Jackson were removed, cutting the branch off from businesses. New York Central folded into Penn Central Railroad in 1968. In 1970 Penn Central filed for bankruptcy in 1970, in 1981 its successor, filed to abandon the track. In 1985 the Southern Michigan Railroad Society, led by three high school students, purchased the Clinton Branch and transformed it into an operating railroad museum; the society continues to preserve, to educate the public about the first railroad in Michigan. They offer various trips on the remaining tracks of what used to be an operating railroad, work on a volunteer basis.

Many of the events they offer are a tour during the Clinton Fall Festival, a tour during the Appleumpkin Festival, the Fall Colors tour. They feature a few winter excursions; these are the Tecumseh Holiday Open House, a tour through the Tecumseh Holiday Lighted Parade

Calvin Perry Stone was an American psychologist, known for his work in comparative and physiological psychology. He was a past president of the American Psychological Association and a member of the National Academy of Sciences. Stone was born on February 1892 on a farm in Jay County, Indiana, he was Emily Brinkerhoff Stone's eight children, the youngest boy. Stone's father died when he was 5 years old, while at the funeral, the family's house burned down; this resulted in years of struggle for his family, teaching Stone the potential that can result from great effort. While the only book in the family library was the Bible and his siblings received encouragement to pursue higher learning from family and the social environment, as well as practical support from their mother. Stone began school at the age of 6, by age 15 began studying at Valparaiso University, he obtained his bachelor of science in 3 years, began teaching at a local high school. While teaching he continued to take summer courses and obtained a bachelor of arts in classics from the same university.

Stone earned his master's, at Indiana University. Haggerty encouraged Stone to join him in his move to the University of Minnesota, where Stone began his doctoral work. During his early years as a doctoral student Stone accepted a position as director of research at the Indiana State Reformatory, which he held until his entrance into the Army. During this time Stone married his wife, Minnie Ruth Kemper, whom he had met at Valparaiso. Stone entered Officer's Candidate School in August 1917, where his knowledge of Psychology was recognized, he became a psychological examiner at Camp Taylor in Louisville, where he caught the attention of Robert Yerkes. Stone went on to become that adjunct and assistant director of training at Camp Greenleaf in Chattanooga Tennessee being discharged having attained the rank of captain. After leaving the Army Stone returned to the University of Minnesota to complete his doctoral work. However, he was under the advisement of Karl Lashley, while working with A. T. Rasmussen, a professor of Anatomy.

He received his Ph. D. in 1921, during that time he had 2 sons James and Robert. Shortly after obtaining his Ph. D. Stone began teaching at the University of Minnesota, but within a year received an offer from the chairman of the psychology department to begin teaching at Stanford University. Stone remained at Stanford for the remainder of his career. While he did receive other offers, Stone remained committed to teaching and research, his course on Freudian Psychology at Stanford was the first comprehensive class of its kind in an American University. Stone's research interests focused around the nervous system and glandular mechanisms, his main contributions relating to the relationship between physiological mechanisms and reproductive behavior, he was known to be thorough in his research, exploring all facets of his research questions. His interest in linking abnormal behavior to physiological psychology led to a number of animal and human studies, his experiments ranged from maze learning in rats to exploration of physical and mental development in humans during puberty.

One of his final studies was to study the effects of electroconvulsive shock, a technique he had seen during one of his sabbatical periods. In 1928 Stone took a sabbatical year and spent time at the Institute for Juvenile Research in Chicago, Illinois, he took a trip to Europe to attend the 1932 International Congress of Psychology in Oxford. In 1945, he spent the year in New York at the Psychiatric Institute, where he began his research on electroconvulsive shock. One of his main contributions to psychology on a national level was his participation on the seven-member Subcommittee on Survey and Planning of the Emergency Committee; this opportunity arose from his time spent working with the chair of the committee, Robert Yerkes, in WWI. This committee proposed the unification of psychology at the national level, suggesting an Intersociety Constitutional Convention, that resulted in a meeting in 1943; that convention, under the leadership of E. G. Boring, concluded that the American Psychological Association should remain the overall national organization, however suggested a divisional structure to introduce a certain level of independence for a number of specialized interests.

· 1931-1932: President of Western Psychological Association · 1938-1939: Vice President of the American Association for Advancement of Science · 1941-1942: President of APA · 1943: Elected to the National Academy of Science · 1947-1950: Journal of Comparative & Physiological Psychology · 1950: Annual Review of Psychology · 1951: Comparative Psychology Master's Thesis: “Notes on light discrimination in the dog” Doctoral Dissertation: “An experimental analysis of the congenital sexual behavior of the male Albino rat” Between 1922 and 1946 Stone and his research associates published 34 reports on sexual and maternal behavior in the albino rat. During this period Stone conducted studies on learning and incentive motivation using mazes and an apparatus to study discrimination. Stone published 8 papers between 1927 and 1939 addressing the physical and mental development in humans during puberty and adolescence. Fourteen papers were published between 1946 and 1956, including one posthumously, on Stone's work on the effects of electroconvulsive shock

Natasha Johns-Messenger is an Australian born conceptual artist and filmmaker, who lives and works in New York after winning the Green Card lottery in 2005. Johns-Messenger is best known for her large-scale site-determined installations that examine spatial perception and light, her work is a complex process of imitation and trickery activated by architectural interventions and optical physics. Johns-Messenger's practice includes digital painting and sculpture. Johns-Messenger is the Great-granddaughter of Australian Rugby legend Dally Messenger and sister of singer Julia Messenger. Johns-Messenger's mother, Catherine Marie Johns, is a poet and novelist in Australia and her works have been published in literary journals including Meanjin and Island Magazine, her father, Dally Messenger III is a published author noted for his books including Ceremonies and Celebrations, Murphy's Law And The Pursuit Of Happiness: A History Of The Civil Celebrant Movement and for his general contribution to the Australian civil celebrant movement.

Johns-Messenger employs the principles of architecture and the visual arts to produce a new experiential sculptural framework. She directly responds to the site as a pictorial object and creates representations or abstractions from the original architecture; the artist physically cuts into real space and uses material devices such as, periscopic mirrors, live video projections and architectural mimicry to produce large-scale installations. Her interest in quantum physics, mathematics and geometry is implemented within her practice. Johns-Messenger critically examines the role of the body in space and her primary concern with altering ordinary "ways of seeing" began during early childhood, where she spent days sketching and photographing shapes from ordinary objects. Johns-Messenger commenced her art practice as a painter, where she received a Bachelor of Fine Arts in 1994, with First Class Honors, in 2000 she completed an MFA, Masters of Fine Arts, from RMIT University, Australia, she has exhibited at major institutions including, Museum of Contemporary Art, Australian Centre for Contemporary Art, Gertrude Contemporary Art Spaces and the Contemporary Centre for Photography, Melbourne.

In 2012 Johns-Messenger completed an MFA in Film at Columbia University, New York, marking a shift into narrative film, where Johns-Messenger combines her interest with abstraction and the moving image into real time."Blackwood" her final graduate film at Columbia won numerous awards at the 2012 Columbia University Film Festival, including the Alumni Award for Best Film, National Board of Review Motion Pictures Award and the Adrienne Shelly Foundation Best Director Award. Blackwood went on to premiere at the Warsaw Film Festival and has been featured in 40 film festivals and won numerous awards, her first graduate film, "Off-Ramp" won Best Student Film and Best Actress at the LA International Underground Film Festival. Johns-Messenger has been curated into major international group exhibitions alongside artists such as Dan Graham, James Turrell and Lawrence Weiner. In 2009/10 Johns-Messenger was commissioned by the New York Public Art Fund, for her work ThisSideIn and in 2010 created Recollection for No Longer Empty, New York at Governor’s Island and was a studio resident at the ISCP in New York under the Australia Council for the Arts, studio residency program.

In 2007, she won the Den Haag Sculptuur Rabo Bank Prize presented to her by Queen Beatrix of the Netherlands and in 2006 Johns-Messenger won the Melbourne Prize for Urban Sculpture with her collaborative group OSW- Open Spatial Workshop. Johns-Messenger's exhibitions have taken place in Japan, China, The Netherlands, Taiwan and USA, her public works can be seen all over the world, with her most recent work Alterview, commissioned by Percent For Art and the New York Department of Cultural Affairs, permanently located at Hunters Point, New York. Her recent exhibitions include, Yellow, 2011, at ACCA, The Australian Centre for Contemporary Art, Melbourne Australia. 2011 Off Ramp Best Student Film at LA International Underground Film Festival Blackwood Williamsburg International Film Festival 2012 Blackwood Columbia University Alumni Award, Best Film, National Board of Review Prize and Adrienne Shelly Foundation Best Director 2008 AUSTRALIA COUNCIL FOR THE ARTS International Studio, ISCP New York.

2007 Winner, Den Haag Sculptuur, Rabobank Award, The Netherlands Backhouse, Megan. Making Gertrude’s Walls Talk, The Age, The Culture, November 27, 2002 Colless, Edward. Inside the fantasy world of edgy, The Australian, April 3, 2006 Coslovich, Gabriella; the Tyranny of distance for Australian Art, The Age, July 7, 2007 Crafti Stephen. If they can make it there, Sydney Morning Herald, 2 May 2010 Crafti, Stephen. Master of Illusion Indesign Magazine, Issue 43, 2010 Donald, Fiona. Strange Place, LIKE Art Magazine, No 11- Autumn 2000 Easton,Craig. Natasha Johns-Messenger:HERE, Eyeline Contemporary Visual Arts, 2001 Frost, Andrew. Super glossy, Australia Style Magazine, Issue 64 Mathews, Hannah. Power to the People, Australian Centre for Contemporary Art, Australia, 2011 Meade, John. Sense of, LIKE Art Magazine, No 12- Winter 2000 Miles, Melissa. Turning on Axes of Light: Natasha Johns-Messenge